mm-4628 === Subject: Re: ::Cyclic subgroups -- finite vs. infinite:: days. My association with the Department is that of an alumnus. >Given a group G, and an element x of G, >consider the subset consisting of all the integral >powers of x. It is standard that is indeed a >subgroup of G, using the subgroup criterion. What is it that you call the subgroup criterion? >But there's something about this proof >that bothers me a bit (appearing in >every book that I've seen) especially in >the case when is finite; >i.e. when not all powers of x are >distinct. Every author doesn't make this >distinction, however, and I'm trying to understand why. That is, when start with a given a subset H >of G, and wish to >apply the subgroup criterion, a priori, >aren't all the elements of H supposed be distinct? Distinct elements of H are of course distinct. There is nothing, however, about them having distinct ->names<-; there is absolutely no problem with the same element being called several different things. For example, one standard way to check that a subset H of G is a subgroup of G is to check that: (i) It is nonempty; (ii) If x and y are in H, then xy is in H; and (iii) If x is in H, then x^{-1} is in H. Even though you use 'x' and 'y', there is no problem with x and y being different names for the same thing. >For only then does it become >meaningful to compute products and >inverses accordingly, in my opinion. Why is it not meaningful to multiply the same thing by itself, even if you call it different things each time? >Suppose you have more than one representation >of the same element -- as in , >when it's finite, for instance --- how >can one tell whether it's really >closed under products and inverses, if we >know already that not all the elements >are distinct? What makes you think that giving the same thing several different names somehow makes it not be in H? >Wouldn't it make sense first to list all >the *distinct* elements of that >subset, *before* we proceed to show that >it is a bonafide subgroup? No. = { x^i | i is an integer } So: every element that can be expressed as an integer power of x is in ; everything in can be expressed, in at least one way, as an integer power of x. That's all. So what if the same element gets called by several different names? is closed under products because if you take two elements a and b in , then you know that a=x^i for some i, and b = x^j for some j, and then ab = x^ix^j = x^{i+j}. This being expressible as an integer power of x, it must, per force, be in . This shows that if a and b are in , then ab will necessarily be in as well. It doesn't matter if it has a hundred other names besides x^{i+j}. is closed under inverses, because if a is in , then a can be written as an integer power of x, a=x^i. Then a^{-1} can also be written as an integral power of x, namely a^{-1} = x^{-i}, so a^{-1} must be in (since contains EVERYTHING that can be expressed, in at least one way, as an integer power of x). So is closed under inverses. Doesn't matter if a^{-1} can be written a million different ways as a power of x, just one is enough. -- magidin-at-member-ams-org === Subject: Re: ::Cyclic subgroups -- finite vs. infinite:: If G is a group, then the elements of G are distinct; if H is a subset of G, then the elements of H, too, are distinct, yes? The definition of , for any x in G (where G could be any group), is merely that it consists of all the powers of x; and there's no stipulation on its order, either. A priori, we don't know whether the powers of x themselves are distinct or not. In case x is finite, however, the powers of x are not all distinct; so how is it, when we list all the powers of x, and declare to be the set consisting of all those powers, that is really a subset of G? In other words, how do we account for the redundancy -- in light the definition of the cardinality of a finite set -- when thinking of as a *subset* of G, and not just a *list* of elements of G? === Subject: Re: ::Cyclic subgroups -- finite vs. infinite:: days. My association with the Department is that of an alumnus. Please quote the message you are replying to to provide context. In mathforum, use the quote original button. >If G is a group, then the elements of G are distinct; >if H is a subset of G, then the elements of H, too, >are distinct, yes? This is a red herring. Distinct elements of a set are, of course, distinct. But that is not really an issue. Have you read Alice Through the Looking Glass? While the has a name, and is called something else, it is still the same song. Each element of the set may have many different names. ->It does not matter<-. The issue of multiple names for a single thing arises only if your definitions somehow assume that names are unique. For instance, you have both a first and a last name. If I were to try to define a function with you in the domain and said The value of the function is the first letter of your name, then this could be a problem if your first name and last name do not begin with the same letter; in that situation, you will want to specify a single name. But the fact that you are a distinct individual is not in any way, shape, or form affected by the fact that you may be called by many different names. >The definition of , for any x in G (where G could be any group), >is merely that it consists of all the powers of x; Indeed. It consists exactly of each and every element of G that, among its many possible names, has a name of the form x^i for some integer i. Anything in has at least one name of the form x^i with an integer; anything that has at least one name of the form x^i with i an integer is in . That's all. Doesn't matter if they are also called Clyde. >and there's no stipulation on its order, either. A priori, we don't >know whether the powers of x themselves are distinct or not. No, we don't. And for some things, that may be important. What it is not important for, however, is to check whether the set is closed under multiplication, inverses, and is nonempty, because the definition of the multiplication and the definition of the inverses does not care what any other possible names the element of H may have. All that matters is that if a has a name of the form x^i, and b has a name of the form x^j, then a*b will have a name of the form x^{i+j}, and a^{-1} will have a name of the form x^{-i}. These are things that you already know, and that ->also<- do not depend on whether the distinct powers of x are distinct elements or not. All that matters is that x^i * x^j is ALWAYS the same as x^{i+j} (and it can also be equal to other things, yes, but it will ALWAYS be equal to x^{i+j}). >In case >x is finite, however, the powers of x are not all distinct; so how is >it, when we list all the powers of x, Who said you are to list them? > and declare to be the set >consisting of all those powers, that is really a subset of G? Because each element of the set is in the set G. That's all that it takes to be a subset. Doesn't matter if each element has a myriad of names, so long as it is in G. As it happens, no matter what you call the element, you can tell that it is in G (and can be called that same thing in G as well). If x^7 happens to be the same as x^1958943, does that somehow mean that x^7 (and that x^1958943) is no longer an element of the set G? Of course not! It means that it happens to have (at least) two names, but it is still enrolled in the set G under (both those) name(s). > In >other words, how do we account for the redundancy There is no need to account for redundancy. There is no requirement in a set that there be a single name for each thing. Remember, the set {a,a,a,a} is exactly the same as the set {a}, because the condition for two sets A and B to be equal is For all x( x is in A if and only if x is in B). It doesn't matter if you emphatically write down x 1650 times in A, and only once in B; the only thing that matters is whether it is in A. > -- in light the >definition of the cardinality of a finite set Why would the definition of cardinality be relevant? Does the definition of group, set, operation, subgroup, power, etc, ever rely on the cardinality of the set in question? No. So why would the definition of cardinality matter? It matters exactly as much as the definition of an economic recession: not at all. > -- when thinking of every element of A is also an element of B<-. Likewise, it doesn't matter what you call : doesn't matter if you list it, if you write it out using roman numerals instead of arabic numerals, if you repeat every element 15834820321 times. Each element of is in G, so is a subset of G, period. -- magidin-at-member-ams-org === Subject: Re: Analysis with integral before.. > OR) > int{0 to 1} log(x) / (1-x) dx > = lim{a->0+, b->1-} int{a to b} log(x) / (1-x) dx > = lim{a->0+, b->1-} int{a to b} sum{k=0 to oo} (x^k).log(x) dx > by log(x) / (1-x) = sum{k=0 to oo} (x^k).log(x), Sum{k=0 to oo} (x^k).log(x) is uniformly convergence on [a, b] > by Weierstrass M-test. so, int <==> sum (exchangeable) Calculate the int{a to b} (x^k).log(x) dx. > Integrating by parts, > = [{x^(k+1)/(k+1)}*log(x)]_{a to b} - int{a to b} (x^k)/(k+1) dx > = -1/(k+1)^2 as a -> 0 , b -> 1. Consequently, > answer is -sum{k=1 to oo} 1/k^2 > No, you still have to show sum and limit can be interchanged. > I already used the Weierstrass M-test by |(x^k).log(x)| <= (a^k).log(a) > for 0 and Sum{k=1 to oo} (a^k).log(a) converges. > so, converges uniformly. You're missing the point; you have lim int (sum log(x)*x^k) dx. > Uniform convergence implies this = lim sum int log(x)*x^k dx. But you > still have to worry about the lim <-> sum interchange. Hm... This is good point. Maybe, for this, I need the facts about uniform convergence of integral in advanced calculus book. > A nice way to see int{0 to 1} log(x) /(1-x) dx = sum{k=0 to oo} int{0 > to 1} x^k log(x) dx is to use the dominated convergence theorem. > Yes, > By dominated convergence theorem, > int{0 to 1} lim{n->oo} sum{k=0 to n} x^k.log(x) dx > = lim{n->oo} int{0 to 1} sum{k=0 to n} x^k.log(x) dx > Because, > Sum{k=0 to n} |x^k.log(x)| <= Sum{k=0 to n} |log(x)| on [0, 1] > and Sum{k=0 to n} |log(x)| = n.|log(x)| > and int{0 to 1} n.|log(x)| is integrable. No, that's a bad mistake. Sum{k=0 to n} |x^k.log(x)| <= > |log(x)|/(1-x), which is integrable. Yes, you're right. I need the integrable g(x) with |f_n(x)| <= g(x) for all n. and I already showed that int{0 to 1} log(x) / (1-x) dx converges === Subject: LHC: absurd hype One expects nonsense about black-holes from the lower tabloids, but absurd claims for the LHC seem to have invaded serious media. What is the basis for the widely-repeated statement - obviously the product of some PR department - that It will smash atoms together to create energies and temperatures not seen since moments after the Big Bang that created the universe and space-time about 14 billion years ago. I am not a physicist, but this seems patent nonsense to me. Also, the statements that it will help in some way in cancer treatment, and in the disposal of nuclear material, would do credit to a snake-oil salesman. As far as I can see, the only outcome at all likely Since that seems to have been incorporated into the accepted model already this would hardly change the face of science. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: LHC: absurd hype > One expects nonsense about black-holes from the lower tabloids, > but absurd claims for the LHC seem to have invaded serious media. What is the basis for the widely-repeated statement - > obviously the product of some PR department - that > It will smash atoms together to create energies and temperatures > not seen since moments after the Big Bang > that created the universe and space-time about 14 billion years ago. For suitable values of the word moments this is not terribly bad for a main-stream-media outlet. I think we are probably into first three minutes type territory here. Maybe the second three minutes. I have not looked at the specific energy levels and where that fits compared to current cosmology models. > Also, the statements that it will help in some way in cancer treatment, > and in the disposal of nuclear material, > would do credit to a snake-oil salesman. This accelerator is unlikely to do anything, at least not directly, for those issues. At best, it will train up a bunch of people who know about accelerators, and they will do stuff that will contribute to them. It is very special purpose, and would be very wasteful to try to do those things with it. Cancer treatment using accelerators was going on very early in their history. Lawrence used one to treat his own mother's cancer, by all accounts with good results. She had been given only a few months, and wound up living many years more, and her death was not related to cancer. The disposal of nuclear material is, at present, a suggestion made by some accelerator physicists. It has not been tried, even in prototype, so far as I know. I'm not sure it would be reasonable to do it, nor of any net benefit since it would likely cost a huge amount to do it, maybe using energy comparable to the energy generated in the use of the Uranium in a reactor. > As far as I can see, the only outcome at all likely Since that seems to have been incorporated into the accepted model already > this would hardly change the face of science. They might see supersymmetry. Or they might get a big old null result on both. Or they might see something entirely new. If it's any of those, then we learn something. Or it might be confusing and ambiguous, meaning that the new stuff is still outside the energy range. That would be disappointing, but would still tell us stuff. Just not quite as much. The facility has value. It is possible to argue that this value does or does not justify the cost. But it's hard to do this usefully without a lot of context. And I will does not give me any special insight into this. I've been in the nuclear industry for 18 years, and much of what my limited understanding. The politics of the siutation is largely off topic here. If you want to argue about govt funding of science, you might like to find a politics forum. Socks === Subject: Re: LHC: absurd hype > One expects nonsense about black-holes from the lower tabloids, > but absurd claims for the LHC seem to have invaded serious media. > What is the basis for the widely-repeated statement - > obviously the product of some PR department - that > It will smash atoms together to create energies and temperatures > not seen since moments after the Big Bang > that created the universe and space-time about 14 billion years ago. For suitable values of the word moments this is > not terribly bad for a main-stream-media outlet. > I think we are probably into first three minutes > type territory here. Maybe the second three minutes. > I have not looked at the specific energy levels and > where that fits compared to current cosmology models. What exactly are you saying? with greater energy than these protons since just after the big bang? Seems very unlikely to me. And are you claiming that the temperature in the LHC is not only higher than anywhere else in the universe, but is even higher than it has been anywhere since just after the big bang? Seems even more unlikely. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: LHC: absurd hype > One expects nonsense about black-holes from the lower tabloids, > but absurd claims for the LHC seem to have invaded serious media. > What is the basis for the widely-repeated statement - > obviously the product of some PR department - that > It will smash atoms together to create energies and temperatures > not seen since moments after the Big Bang > that created the universe and space-time about 14 billion years ago. For suitable values of the word moments this is > not terribly bad for a main-stream-media outlet. > I think we are probably into first three minutes > type territory here. Maybe the second three minutes. > I have not looked at the specific energy levels and > where that fits compared to current cosmology models. What exactly are you saying? > with greater energy than these protons since just after the big bang? Seems very unlikely to me. it is false even indeed one of the biggest points against the creation of black holes is that cosmic rays cause collision events of these energies all the time > And are you claiming that the temperature in the LHC > is not only higher than anywhere else in the universe, > but is even higher than it has been anywhere > since just after the big bang? Seems even more unlikely. temperature in colliders is a weird thing the beam itself has temperatures that must be separated into transverse and collinear parts because it is not in the same equilibrium in both directions (it's not in equilibrium at all but physicists are content enough with a pseudoequilibrium based on time-scale differences) these temperatures are usually relatively cool so temperature comparison really only applies to the collision event but the answer is no with the suitable clauses attached not just for cosmic ray reasons (which it's actually hard to compare to here) but it's they are all transitory events or involve things difficult to see (like accretion disks of black holes) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: path of maximum length > Your phrase length of the path along these k points seems unclear. > Suppose that by path you mean an ordered list of k distinct points, > and path length is the sum of the k-1 point-to-point successive > distances. This is indeed how I meant it to be. Sorry if i was not clear in > my definition. > If k is less than N, one can make a longer path by adding > another point to the path; mentioning subsets of points is superfluous. > The value of the integer k is given and is less than or equal to N. So for > a given k, one should find k distinct points with a path length > (sum of the k-1 point-to-point successive distances) that is as > large as possible. Adding another point is thus not an option, > since k is fixed. Next to the above description, I think I can also describe my problem more in terms of graph-theory: Given a complete weighted graph with N vertices, find the path of given length k <= N-1 which has the largest weight. How is this problem known in the literature? What's the best approach to solve this? Bart -- === Subject: Error surface question I am doing some optimization to fit my experimental data to a model. Someone asks me what is the error surface of this? I am not sure what does this mean. I.e., which forms of error surface is good, and which is not? and I can i decide the form of my error surface? I searched from the Internet and got a few information, but none has a overall description on the error surface on my questions above, may be just because I didn't get the right one. Can anyone here kindly answer my qustions or recommend some good links or tutorials on this. Hongyu === Subject: Re: Error surface question > .... > I am doing some optimization to fit my experimental data to a model. > Someone asks me what is the error surface of this? I am not sure what > does this mean.... The news group might be a good place for your question. Ken Pledger. === Subject: Re: Error surface question > I am doing some optimization to fit my experimental data to a model. > Someone asks me what is the error surface of this? I am not sure what > does this mean.... The news group might be a good place for your question. === Subject: Re: Error surface question > I am doing some optimization to fit my experimental data to a model. > Someone asks me what is the error surface of this? I am not sure what > does this mean. I.e., which forms of error surface is good, and which > is not? and I can i decide the form of my error surface? I searched from the Internet and got a few information, but none has a > overall description on the error surface on my questions above, may be > just because I didn't get the right one. Can anyone here kindly answer > my qustions or recommend some good links or tutorials on this. > Hongyu Sorry, this and I can i decide the form of my error surface? should be and How can I decide the form of my error surface? === Subject: Re: More with Quadratic Diophantine Theorem > i think you would be surprised at gauss' elementariness > i figured that if you spent some time reading him > you too could be that paranoid asshole > inflating yourself with me me me > and finally be a fair mathematician to boot >I'm contemplating this awesome result linking all quadratic contemplating means you have nothing. >Diophantine equations in 2 variables and wondering how far it goes. Google for it, lazy. >You're mumbling. You're trolling. >History will remember me. But what about you? History will remember JSH will never f*cking do it -UA How long will History remember JSH? About 3 days after his last post. >Oh, you don't care, right? >Hundreds of years from now, some people may be wondering about who I >am and what kind of person I was, where one may hero worship, and >another may mouth off about my many failings. ...may be... but in fact, not at all. No one will be wondering about JSH, long gone troll on the internet. >But who will even know about you? She is famous, JSH is known crackpot/troll and self-admitted non-mathematical person. >James Harris === Subject: Solutions manual to Fundamentals of Applied Electromagnetics 5th edition by Fawwaz T. Ulaby solution manual solutions manual (To search click in keyboard Ctrl+F) Solutions Manuals in Electronic (PDF)Format! Just contact with , solutionpayfee (at) hotmail.com (my email address), these are parts of our solutions, if the solution you want is on the list, please email to me. NOTICE: if the solutions manual that in my list ,please note it in your email . 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Within this environment, pure mathematics arose up to an anti-practical attitude. Cantor was a talented actor. I have a simple explanation the obvious fact that most of the nationalist-feeling German intellectuals were jews: Several of them were not just more intelligent than average, well educated and rich. As a not fully tolerated minority, they were also forced to arrange with Christian belief in order to have a chance of a good job. This selected the most flexible ones. If I recall correctly, G. Cantor's father converted to protestantism and married a catholic woman. Why was G. Cantor accepted at all? In contrast to the popular Weierstrass who and whose pupils protected him, his rich opponent Kronecker did not have pupils who understood and admired Kronecker. Nobody disproved Cantor's claims. G. Cantor's arguments were seemingly flawless and overly astonishing. In the emphatic time after the second German Reich was founded, students felt a need to create something new and admired mostly the most bizarre ideas. Cantor's periods of insanity even increased his nimbus of a genius. He managed to become an international president. Finally, the majority avoided to admit that they did not understand Cantor. Just single outstanding mathematicians like Poincar.8e and Brouwer continued to reject transfinite numbers. Hilbert, Zermelo, Goedel, Fraenkel, Robinso(h)n, the Bourbakis and many others preferred to defend set theory with no convincing success. Salviati: ... in ultima conclusione, gli attributi di eguale maggiore e minore non aver luogo ne gl'infiniti, ma solo nelle quantit.88 terminate. IR>|>IR+>|>IR === Subject: Re: An unwise decision æit's all a big conspiracy, innit? > So who's behind it? attitude. Cantor was a talented actor. I have a simple explanation the > obvious fact that most of the nationalist-feeling German intellectuals were > jews: Aha! > Nobody disproved Cantor's claims. Exactly. > Cantor's periods of insanity even increased his nimbus of a genius. > He managed to become an international president. Wow! An international government, headed by a mathematician, in the 19th century. I must have missed that in my history books! Victor Meldrew I don't believe it! === Subject: Re: A wise decision schrieb im Newsbeitrag > Nobody disproved Cantor's claims. >Exactly. This does not mean that they were correct. He managed to show that there is no complete list of numbers with indefinitely many decimals. Just his interpretation was circular. He did not take into account what Fraenkel called the 4th l.9agical possibkility after >, =, or <: incomparable. The mathematicians were perhaps too much confined within their formal rules of thinking as to get aware of the blunder. Cantor got support from his friend Hurwitz and also from Hadamard who appreciated the freedom to calculate with reals as if they were rationals. > Cantor's periods of insanity even increased his nimbus of a genius. > He managed to become an international president. >Wow! An international government, headed by a mathematician, in the >19th century. I must have missed that in my history books! Cantor achieved the foundation of Deutsche Mathematiker-Vereinigung in 1890. In Halle 1891, he became their elected president and held this post until 1893. In 1897 Cantor attended the first International Congress of Mathematicians in Zurich. He lectured on the paradoxes of set theory to a meeting of the Deutsche Mathematiker-Vereinigung in 1903, and he attended the International Congress of Mathematicians at Heidelberg in August 1904. Salviati: ... in ultima conclusione, gli attributi di eguale maggiore e minore non aver luogo ne gl'infiniti, ma solo nelle quantit.88 terminate. IR>|>IR+>|>IR === Subject: Re: An unwise decision > æNobody disproved Cantor's claims. >Exactly. This does not mean that they were correct. He managed to show that > there is no complete list of numbers with indefinitely many decimals. Yes, he showed there was no bijection between N and R. > Cantor got support from his friend Hurwitz and also from Hadamard > who appreciated the freedom to calculate with reals as if they were > rationals. All mathematicians appreciate freedom, but not Herr von Bumschiesse. > Cantor's periods of insanity even increased his nimbus of a genius. > He managed to become an international president. > Cantor achieved the foundation of Deutsche Mathematiker-Vereinigung in 1890. > In Halle 1891, he became their elected president and held this post until > 1893. > In 1897 Cantor attended the first International Congress of Mathematicians > in Zurich. > He lectured on the paradoxes of set theory to a meeting of the Deutsche > Mathematiker-Vereinigung in 1903, and he attended the International Congress > of Mathematicians at Heidelberg in August 1904. And that makes him an international president ?? Victor Meldrew I don't believe it! === Subject: Re: A wise decision Nobody disproved Cantor's claims. >Exactly. This does not mean that they were correct. He managed to show that > there is no complete list of numbers with indefinitely many decimals. Yes, he showed there was no bijection between N and R. < Of course. However while Salviati correctly concluded that one must not < compare infinite quantities, Cantor concluded that there must be more < than indefinitely many real numbers. Cantors largest aleph is still not yet < really infinite. The same is true for *IR. All mathematicians appreciate freedom, but not Herr von Bumschiesse. < Genuine freedom is based on correct insight rather than intention. < In the end we have to look whether or not an idea is fertile as < what Heaviside introduced or obviously futile as Cantor's alephs. < EOD. === Subject: Breadline schmuck ad nauseam > < Of course. However while Salviati correctly concluded that one must not ONE MUST NOT! ON PAIN OF DEATH!! > < compare infinite quantities, Cantor concluded that there must be more > < than indefinitely many real numbers. more than indefinitely many? Vague waffle even by your own lamentable standards, Herr Bulmchein. >Cantors largest aleph is still not > yet > < really infinite. Cantor didn't have a largest aleph. > The same is true for *IR. And what is *IR, save yet another undefined term from the egregious Bumschein. > < In the end we have to look whether or not an idea is fertile as > < what Heaviside introduced or obviously futile as Cantor's alephs. obviously futile: More argument by insult. the fact is that Cantor's work has been far more central and essential to the deveoplment of modern mathematics as that of Heaviside. Victor Meldrew I don't believe it! === Subject: number of perfect powers <=n , lookuping sloane or older post i could get the closed form as n - sigma (k = 2 to log2(n)) mu(k)(n^1/k -1) Example : 9 -> 4 ( 1, 4,8,9) 16 ->5 (1,4,8,9,16) 36 -> 9 (1,4,8,9,16,25,27,32,36) A rough estimate with duplicates is 1 + Sigma { k from 2 to log2 N) (N^1/k -1) , now how do you remove duplicates. It would be good if you can elaborate onthis derivation. I didn't know where the mobius mu came from . Can someone explain and give examples. What i could figure out is that all powers of 4 are powers of 2 , so that could be excluded , since mu(4)=0 , those terms are removed for all perfect kth power numbers.but i'm not clear how to get that closed form === Subject: Re: number of perfect powers <=n , > lookuping sloane or older post i could get the closed form as n - > sigma (k = 2 to log2(n)) mu(k)(n^1/k -1) > Example : > 9 -> 4 ( 1, 4,8,9) > 16 ->5 (1,4,8,9,16) > 36 -> 9 (1,4,8,9,16,25,27,32,36) > A rough estimate with duplicates is 1 + Sigma { k from 2 to log2 N) (N^1/k -1) , now how do you remove > duplicates. > It would be good if you can elaborate onthis derivation. I didn't > know where the mobius > mu came from . The mu function is exactly how you remove duplicates. Keyphrase: Principle of Inclusion-Exclusion. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: number of perfect powers <=n , > lookuping sloane or older post i could get the closed form as n - > sigma (k = 2 to log2(n)) mu(k)(n^1/k -1) Example : > 9 æ -> 4 æ ( 1, 4,8,9) > 16 ->5 æ (1,4,8,9,16) > 36 -> 9 æ(1,4,8,9,16,25,27,32,36) A rough estimate with duplicates is 1 + Sigma { k from 2 to log2 N) (N^1/k -1) æ, now how do you remove > duplicates. > æIt would be good if you can elaborate onthis derivation. I didn't > know where the mobius > mu came from . æ The mu function is exactly how you remove duplicates. > Keyphrase: Principle of Inclusion-Exclusion. > I don't understand how to form inclusion-exclusion here . All i could figure out is that for non squarefree numbers like 4^2,12^2 etc, they appear as power of some other term like 2^4 , 6^4 etc but how do we get the complete closed form. ok, now i understand that signs of inclusion exclusion are controlled by mu but i don't get it. Can you explain it. === Subject: Re: number of perfect powers <=n , > lookuping sloane or older post i could get the closed form as n - > sigma (k = 2 to log2(n)) mu(k)(n^1/k -1) > Example : > 9 æ -> 4 æ ( 1, 4,8,9) > 16 ->5 æ (1,4,8,9,16) > 36 -> 9 æ(1,4,8,9,16,25,27,32,36) > A rough estimate with duplicates is > 1 + Sigma { k from 2 to log2 N) (N^1/k -1) æ, now how do you remove > duplicates. > æIt would be good if you can elaborate onthis derivation. I didn't > know where the mobius > mu came from . æ The mu function is exactly how you remove duplicates. > Keyphrase: Principle of Inclusion-Exclusion. I don't understand how to form inclusion-exclusion here . All i could > figure out is that for non squarefree numbers like 4^2,12^2 etc, they > appear as power of some other term like 2^4 , 6^4 etc but how do we > get the complete closed form. ok, now i understand that signs of inclusion exclusion are controlled > by mu but i don't get it. Can you explain it. Let's count the numbers up to n that aren't perfect powers. Start with all n numbers up to n. Subtract n^(1/2) for the perfect squares. Subtract n^(1/3) for the perfect cubes. But now you've removed the 6th powers twice, so add in n^(1/6). Subtract n^(1/5) for the 5th powers. Add n^(1/10) for the 10th powers, and n^(1/15) for the 15th powers. Think about the 30th powers: they got subtracted three times (as squares, cubes, and 5th powers) and then subtracted three times (as 6th, 10th, and 15th powers), so we have to subtract n^(1/30). Now look at the pattern so far: n - n^(1/2) - n^(1/3) - n^(1/5) + n^(1/6) + n^(1/10) + n^(1/15) - n^(1/30) The coefficient of n^(1/k) is precisely mu(k). PS: 12^2 is not 6^4. -- GM === Subject: Re: number of perfect powers <=n , > lookuping sloane or older post i could get the closed form as n - > sigma (k = 2 to log2(n)) mu(k)(n^1/k -1) Example : > 9 ? -> 4 ? ( 1, 4,8,9) > 16 ->5 ? (1,4,8,9,16) > 36 -> 9 ?(1,4,8,9,16,25,27,32,36) A rough estimate with duplicates is 1 + Sigma { k from 2 to log2 N) (N^1/k -1) ?, now how do you remove > duplicates. > ?It would be good if you can elaborate onthis derivation. I didn't > know where the mobius > mu came from . ? The mu function is exactly how you remove duplicates. > Keyphrase: Principle of Inclusion-Exclusion. Sure. But the number of powers is strongly dominated by the squares. #(cubes + 5thpowers + ...) = o(#squares). The number of powers less than N is O(N^(1/2 + epsilon)) for any epsilon > 0. === === Subject: Schrodinger's Universe, the new book written by Dr. Milo Wolff, may Schrodinger's Universe, the new book written by Dr. Milo Wolff, may change science. http://www.amperefitz.com/schrodi.htm (Click link above to read.) === Subject: Re: Short Mars travel times at high speed. > Unable to concentrate on my studies, I began to explore Eastern > religions, specifically the religious synthesism of RamaKrishna. In > the spring of 1969 I dropped out of the University of the South and > hitchhiked to California, a stop on my way to India. In May I arrived > in Berkeley, with a suitcase and a navy duffel bag and a copy of the > Gospel According to Rama Krishna. I didn't know it, but it was the end > of the line. Okay, when the effects of the LSD wore off, what happened next? I'd take Vishnu's Krishna incarnation any day of the week over some > Koran whacko who marries people a thousand at a time. There is a saddle in the Grand Canyon where Vishnu Temple and Krishna Shrine meet. I have never been there as it takes at least two days to get there by foot and I haven't heard of any pack animals going to that remote place either. > Of course, if we are going for Vishnu avatars, then my favorite has to > be Hanuman - the Cute, Personality-Filled Monkey God, who could kick ass > to a degree that King Kong could only dream of. > Hanuman Rules! I do not believe that Clarence Dutton named a butte or temple in the Grand Canyon after Hanuman. > Meanwhile, time for installment #3 of Monkey News; a occasional > special adjunct to my postings describing interesting events in the > world of monkeys - written by, for, and about monkeys. > Today we have a very scary monkey photo from France; ranking up with the > scary photo of the blue-white eyed Afghani girl that caused such a stir > on the cover of National Geographic Magazine a couple of decades back:http://www.nature.com/news/2008/080910/full/455145a.html > In my search for Clarence Dutton I first came across Clarence Darrow, speaking on the subject of moneys. :) === Subject: Re: Short Mars travel times at high speed. I heard, there was a flatware Flat Earther in hear, I mean THERE; peeling aorund the edges ... the map? what are the odds of throwing snake-eyes mod two? > on the cover of National Geographic Magazine a couple of decades back: http://www.nature.com/news/2008/080910/full/455145a.html In my search for Clarence Dutton I first came across Clarence Darrow, what kind of index was that !?! how old were you? where do you come from & who do you work for -- the ten foot Easter Bunny? taller? don't make me count the syllables, Kimosabe! === Subject: Re: Short Mars travel times at high speed. > I heard, there was a flatware Flat Earther in hear, > I mean THERE; peeling aorund the edges ... the map? what are the odds of throwing snake-eyes mod two? > on the cover of National Geographic Magazine a couple of decades back: http://www.nature.com/news/2008/080910/full/455145a.html In my search for Clarence Dutton I first came across Clarence Darrow, what kind of index was that !?! Google > how old were you? Old enough to know just how full of poop you are... > where do you come from & who do you work for -- > the ten foot Easter Bunny? > taller? don't make me count the syllables, Kimosabe! I wouldn't think of it! === Subject: Re: Short Mars travel times at high speed. > wow, maybe you could advise me. æso, > which is better, LSD or mild ergot poisoning -- > will the latter flashforward to cure flashbacks > from the former? Sorry, it just seemed like the right comment to make based upon where your adventure began and where it ended. most of which I did not read & that I just noticed, is, > that little statement about synthesizm; > no synesthezia required? also, Moog just came-out with a guitar (saw the ad; > at 6500US, it's time to roll one's own .-) >http://www.allentwood.com/essays/lordofflies.html > religions, specifically the religious synthesism of RamaKrishna. In > Okay, when the effects of the LSD wore off, what happened next? thus quoth: > Fannie Mae was created by President Franklin D. Roosevelt in 1938, as > a government agency to buy mortgages from lenders, as a way of funding > the purchase of homes in the Great Depression. In more recent years, > Fannie Mae and its sibling Freddie Mac, were taken over by what FDR > attacked as the economic royalists, and turned into vehicles for > derivatives speculation. Under the great Greenspan bubble, Fannie and > Freddie were turned into money machines to feed the run-up in real > estate values to provide assets.84in the form of mortgage debt.84as fuel > to the derivatives markets. This scheme was bound to fail, as it > spectacularly has, leaving Fannie and Freddie, and the U.S. banking > system, utterly bankrupt. However, Fannie and Freddie are at the heart of Treasury Secretary > Henry Paulson's and Federal Reserve chairman Ben Bernanke's insane > scheme to bail out the banks by dumping all their bad mortgage paper > into the two government-sponsored enterprises, effectively > transferring the banks' losses to the government, and ultimately to > the taxpayer. The government is not really bailing out Fannie and > Freddie, but merely funding their conversion into the largest toxic > waste dumps in history. Far from being saved, Fannie Mae and Freddie > Mac are being destroyed.http://larouchepub.com/other/editorials/2008/3536tantamount treason.html It goes back to this administration and things being big in Texas, especially the government. This administration has caused more problems that they themselves feel they must rescue us from to the point where we'd all have been better off had they done nothing in the first place. They are beyond being self-serving to the point of being total bungling. Resorting to this sort of socialism would have been better served by saving Anheiser-Busch from the Belguins rather than trying to bail out Fannie Mae and Freddie Mac. === Subject: Re: Extending a group's operation beyond the group Distribution: world Keywords: Algebra >I'd like to define the extension of a binary operation defined on a group, >G, to a superset of G. >If the superset is a subset of a group, H, that has G as a sub-group, the >extension is simple enough, since the operation's already defined on H. >I would assume that such a concept already exists. What's it called? Your H will be a monoid, Ah, I've heard of those. Never knew what they were good for. Now, I do. > with G a subgroup (a submonoid which is >also a group). Apparently, according to both you and Dr. Israel, I was looking at things inside out. Instead of partially extending a group's operation, the simpler way to look at it is as the group being a specialized part of a bigger entity -- which comes first. And really, this is the case with the particular example that I'm looking at, since multiplication of non-singular nxn matrices *is* just a special case of multiplication of nxn matrices. >There is an entire hierarchy of >sets-with-a-single-binary-operation. I was vaguely aware of this. > If all you have is a binary >operation, and you do not even have a warrant for associativity, you >get a magma. Quasigroups are magmas in which division is >possible. Assuming that the group operation is called multiplication rather than addition? -- Michael F. Stemper #include 91.2% of all statistics are made up by the person quoting them. === Subject: Re: Extending a group's operation beyond the group Keywords: Algebra days. My association with the Department is that of an alumnus. [...] > If all you have is a binary >operation, and you do not even have a warrant for associativity, you >get a magma. Quasigroups are magmas in which division is >possible. Assuming that the group operation is called multiplication rather >than addition? Yes. Addition usually includes a strong implication of commutativity, so the 'standard' is to denote arbitrary binary operations by juxtaposition/multiplication. -- magidin-at-member-ams-org === Subject: Re: Statistics > What is the difference between Mutually Exclusive and Independant > Events ? Events that are mutually exclusive cannot be independent (except when the > probabilities are both zero). You mean, except when the probability of at least one is zero. === Subject: Re: Statistics > What is the difference between Mutually Exclusive and Independant > Events ? Events that are mutually exclusive cannot be independent (except when the > probabilities are both zero). You mean, except when the probability of at least one is zero. > -- > Robert Israel æ æ æ æ æ æ æisr...@math.MyUniversitysInitials.ca > Department of Mathematics æ æ æ æhttp://www.math.ubc.ca/~israel > University of British Columbia æ æ æ æ æ æVancouver, BC, Canada Ok, it makes sense now. Need to pick a simple topic for a short project. Must use true data from true sources. Topics may include but no limited to Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression. === Subject: Re: Statistics >What is the difference between Mutually Exclusive and Independant >Events ? This confusion comes from not reading the definitions. Mutually exclusive means that it is impossible for both to happen. Independent means that there is no probability information for one from the other, but it is usually given as P(A & B) = P(A)*P(B). Now if A&B is the empty event, what does this tell you about independence in this case? -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Statistics What is the difference between Mutually Exclusive and Independant >Events ? This confusion comes from not reading the definitions. Mutually exclusive means that it is impossible for both > to happen. æIndependent means that there is no probability > information for one from the other, but it is usually > given as æ æ æ æ P(A & B) = P(A)*P(B). Now if A&B is the empty event, what does this tell you > about independence in this case? > -- > This address is for information only. æI do not claim that these views > are those of the Statistics Department or of Purdue University. > Herman Rubin, Department of Statistics, Purdue University > hru...@stat.purdue.edu æ æ æ æ Phone: (765)494-6054 æ FAX: (765)494-0558 Correct me if I am wrong. I am new to this topic. So there is no independence in this case, event A dependence on event B So if A & B is the empty event than the probabilty is 0. === Subject: Distances and directions and constraint satisfaction I am thinking about a problem wherein we would like to use measurements of distinguishable reference points of known location in order to determine where we are and how we are oriented in three-dimensional Euclidean space. For now I'll brush under the rug awkward minority cases like when points are coplanar or collinear. It seems to me that we have a continuous six-dimensional phase space: three dimensions for our position, three for our orientation. So, if we can take six independent measurements, we should have enough equations to be able to determine our position and orientation. This seems to fit okay when I think about how just knowing our distance from three different reference points lets us figure out our position (at least down to finitely many solutions); each distance gives us an independent measurement. (I don't know if it takes three more to give us our orientation, my imagination is failing.) Also, when I consider how when we know the direction (both azimuth and inclination, as a unit vector or something) of three distinguishable reference points, we can determine our position and orientation; each three-space direction gives us two parameters, providing six in total. However, when I think about measuring both direction and distance of two reference points of known location, I can only get that to fix *five* dimensions of phase space. My position ends up on a circular locus where all points on it have a satisfactory orientation available if any do. Am I wrong in that? If so, how? Or, if I am correct, is there a simple way to calculate, given x distance measurements to some points, y direction measurements to other points, and z direction+distance measurements to still other points, how many dimensions of phase space those fix for us? Or, some easy path to an intuition that makes it all obvious? I am surprised to seem to end up with, x y z fixed 3 0 0 3 0 3 0 6 0 0 2 5 which isn't what I'd expected. I suppose to expand the position-and-orientation problem beyond three-space into n-space, directions end up being in terms of n-1 parameters. But perhaps I shouldn't think about that yet. Mark === Subject: :: Conjugacy classes of a finite group:: Suppose G is a finite group and S is the set of representatives of the distinct conjugacy classes of G. How do we show that S, in fact, generates G, i.e. G = ? Or, is this even true ? === Subject: Algebra textbook where Rings/Fields comes before Groups Can anyone recommend a decent algebra textbook where Rings/Fields comes before Groups? I tried to go to Part 2 (Rings Theory) of Dummit/Foote directly but it defines many things in there in terms of groups (e.g., first axiom for a ring says the set and the + make an abelian group). A lot of the discussion in the ring section also calls upon knowledge of groups. === Subject: Re: Algebra textbook where Rings/Fields comes before Groups > Can anyone recommend a decent algebra textbook where Rings/Fields > comes before Groups? I tried to go to Part 2 (Rings Theory) of Dummit/Foote directly but it > defines many things in there in terms of groups (e.g., first axiom for > a ring says the set and the + make an abelian group). A lot of the > discussion in the ring section also calls upon knowledge of groups. > Dummit/Foote is a good book. It would be good if you understood most of the results on groups before seriously studying rings; given that rings contain a group and semigroup operation. Any treatment of rings which precedes without an introduction to group theory would therefore be necessarily basic. === Subject: Re: Algebra textbook where Rings/Fields comes before Groups > Can anyone recommend a decent algebra textbook where Rings/Fields > comes before Groups? Peter J. Cameron, Introduction to Algebra is good IMO. === Subject: Re: Algebra textbook where Rings/Fields comes before Groups > Can anyone recommend a decent algebra textbook where Rings/Fields > comes before Groups? I tried to go to Part 2 (Rings Theory) of Dummit/Foote directly but it > defines many things in there in terms of groups (e.g., first axiom for > a ring says the set and the + make an abelian group). A lot of the > discussion in the ring section also calls upon knowledge of groups. M. Artin, Algebra. That's the textbook at MIT, and he still teaches there. The book starts with a brief overview of groups, then goes - also fairly briefly - through rings and fields, and vector spaces. Once you have a 'bird eye's view' on abstract algebra, it does some 'fun staff' like symmetry. Then it goes back and start moving slowly and systematically through the same staff. === Subject: Re: Algebra textbook where Rings/Fields comes before Groups > I must have a different book. (Got a copy recently, but haven't > read it.) Correct me if I'm wrong, but it looks as if the word > ring is only defined on page 345 - by which time he's covered > linear algebra and tons of group theory, up to representation > theory! (His dad's book on Galois Theory might be worth a > recommendation, though. I've only read a bit of it so far, > but it's nice, it's a classic, it doesn't mess about - and > Dover have reprinted it.) -- > Angus Rodgers > Contains mild peril I'll check it out. The book is at home, but I'm not. :-) My recollection: Artin goes very briefly through basics, groups, rings, and fields, to get to vector spaces, so at least you feel you are on familiar ground (if you had a course of Linear Algebra, you should). But since finite-dimensional vector spaces are in a sense trivial, he covers the whole thing fairly rapidly as well. Then he says: now let's have fun, and start discussing some applications like symmetry. And few other things. Then he start all over, in much greater depth. === Subject: quick favor: review of a real analysis HW answer(involves limit of a sequence) quick favor: review of a real analysis HW answer(involves limit of a sequence) The question: Using the .83Ì - N definition of a limit show that the following limit is true (A) lim (5n+7)/(8n+4) = 5/8 Solution: Need to show that |((5n+7)/(8n+4)) - 5/8| < .83Ì To get a value for .83Ì... |((5n+7)/(8n+4)) - 5/8| = (36/(64n+32))<.83Ì n > ((36/.83Ì)-32)/64 So, if we choose n> N > ((36/.83Ì)-32)/64 we have |((5n+7)/(8n+4)) - 5/8| < 36/(64n+32) = .83Ì QED Does this look correct? Any errors, comments? Any input would be so very greatly appreciated! === Subject: Re: quick favor: review of a real analysis HW answer(involves limit of a sequence) > quick favor: review of a real analysis HW answer(involves limit of a > sequence) > The question: > Using the .8c - N definition of a limit show that the following limit is > true (A) lim (5n+7)/(8n+4) = 5/8 Solution: > Need to show that |((5n+7)/(8n+4)) - 5/8| < .8c To get a value for .8c... You mean ''to get a value for N ...''. .8c is given. > |((5n+7)/(8n+4)) - 5/8| = (36/(64n+32))<.8c > n > ((36/.8c)-32)/64 > So, if we choose n> N > ((36/.8c)-32)/64 > we have > |((5n+7)/(8n+4)) - 5/8| < 36/(64n+32) = .8c To be a bit more formal, say Let N be a natural number such that N > ((36/.8c)-32)/64. Then for all n > N we have |((5n+7)/(8n+4)) - 5/8| < 36/(64n+32) = .8c. > QED Does this look correct? Yes. > Any errors, comments? Except for those minor glitches, no. > Any input would be so very greatly appreciated! === Subject: Re: quick favor: review of a real analysis HW answer(involves limit of a sequence) > quick favor: review of a real analysis HW answer(involves limit of a > sequence) > The question: > Using the .8c - N definition of a limit show that the following limit is > true > (A) lim (5n+7)/(8n+4) = 5/8 > Solution: > Need to show that |((5n+7)/(8n+4)) - 5/8| < .8c > To get a value for .8c... You mean ''to get a value for N ...''. .8c is given. > |((5n+7)/(8n+4)) - 5/8| = (36/(64n+32))<.8c > n > ((36/.8c)-32)/64 > So, if we choose n> N > ((36/.8c)-32)/64 > we have > |((5n+7)/(8n+4)) - 5/8| < 36/(64n+32) = .8c To be a bit more formal, say > Let N be a natural number such that N > ((36/.8c)-32)/64. > Then for all n > N we have > |((5n+7)/(8n+4)) - 5/8| < 36/(64n+32) = .8c. > QED > Does this look correct? Yes. > Any errors, comments? Except for those minor glitches, no. porky_pig_jr@my-deja.com, material. === Subject: Estimation of parameters using method of maximum likelihood imagine that we have random value X defined by its density function which (beside the value 'x') depends on some parameter theta. Let's say that estimation of theta using method of maximum likelihood gives me this: theta_MLE = sqrt(sample_mean) Now I would like to find asymptotic distribution of this estimation of theta (theta_MLE) and express confidence interval for real value of theta. I have to use Fisher information to do this. Well, according to some theorem, we know that the asymptotic distribution of theta_MLE is N(theta, 1/n*F(theta)) where 'n' is size of the sample and F(theta) is Fisher information of theta. I can compute Fisher information of theta, it gives me that F(theta) = 4n/(theta^2) But I have problem now since I don't know how the confidence interval for theta should look like. Would somebody help me please? === Subject: Re: Estimation of parameters using method of maximum likelihood > .... random value X defined by its density function > which (beside the value 'x') depends on some parameter theta.... The news group might be a good place for your question. Ken Pledger. === === Subject: Intermediate Accounting 12th Ed. The teacher has stressed to us that they would like us to use this website, but they are not personally allowed to give us the site and password. Usually, we can get it from another student who knows, but this book is fairly new and I was wondering if anyone knew the website and password to the accounting book Intermediate Accounting 12th edition Authors are Donald E. Kieso, Jerry J. Weygandt and Terry D. === Subject: Re: ultrafinitism and ultraformalism > Well, I still don't understand how you would avoid either formalism, > or using sentences with the word set in them. Avoid what formalism? As to your not being able to see how to avoid > the word 'set', perhaps you would provide an example of a sentence in > context of ZFC that uses the word 'set' but that you don't know how to > express the thought behind that sentence but without the word. Well, I don't know how to express the thought behind the sentence > anyway, regardless. That's the point. You're not making sense when you say That's the point. I said that for any use of the word 'set' in context of ZFC, I bet I can come up with a way not to use the word. You said you don't understand how that instance in which you see the word 'set' used in context of ZFC but don't understand how the word could be avoided. Surely, if you're objecting to the word 'set' and done trust my claim that the word can where you don't see how it could be avoided. > No, it's quantification full stop that doesn't make sense to me. E.g. > for any number x in N, x+2 is in N. That I understand. The statement > for any x, x=x makes no sense to me. That is handled in the interpretation for a language. Given an interpretation for the language, the universal quantifier ranges over all and only those objects in a certain set. And if the interepretation is specified, then the set is defined by an explicilty articulated property, so you can refer to any (natural) number rather than the set of (natural) numbers, if that better suits you. > Also, the statement 2 is in > the set of all sets of numbers makes no sense to me. That is my > consideration. The word 'set' there is dispensible. We could instead say: 2 is a member of the object whose members are all and only those objects whose members are only natural numbers. > Every there is sentence is indeed equivalent to a sentence that uses > for every. In classical logic, yes. Not in intuitionistic logic. > For me the sentence not for every x in X not phi(x) > does not have literally the same meaning as there is an x in X such > that phi(x), so *semantically* should not be a definition, but > *formally* it works fine to replace one with the other, negated. I Semantically, they are extensionally equivalent, though I suppose (I don't know enough about it) that one could say that they are not intensionally equivalant. > but you never know. Certainly using both for all and there is is > more elegant. But considering only one is convenient for some Godel- > style proofs, I believe. I don't know what Godel proofs you have in mind. But, as I said, classically, one quantifier is sufficient, but intutionistically we need both quantifiers. > Indeed, just a quantifier without a matrix following it is not even > syntactical in the language for ZFC. Though, contrary to your > preference, it is not required that every quantifier be relativized; > however a quantifier does only range over æa given set per an > interpretation for the language. > Okay. Then perhaps now quantification makes sense to you. I answered that above. Then why did you write 'okay'? And above you did NOT respond to the point that the range of a quantifier is limited in the INTERPRETATION for the language. > You also can't compare numbers to sets of numbers, for example, just > like you can't say or denounce 2>{the set of primes}. > Predicate symbols such as '>' can be defined in a way that eliminates > such problems as how to regard such formulas as you mention above. > My point was that 2={the set of primes} is as coherent as 2>{the > set of primes}. The former is not allowed, just like the latter. Not allowed in what context? What is allowed as a well formed > formula is determined by the syntax rules of whatever system. What > particular system do you have in mind? I mean it doesn't make sense. It's not allowed if you want to make > sense. Goto next response. We can allow such expressions and make sense. You've done little more than to declare, without much reason, that certain things don't make sense to you. Okay, they don't make sense to you, but meanwhile they make quite fine sense to me. You've not shown any breakdown in my sense, which is given by the method of interpretation for languages. > Oh, sorry. Put in a schema the axioms for nth order arithmetic, for > each, n, and take into account the type-like restrictions on the > allowable sentences. That is a sketch. By this time, I don't even recall the question I asked was. See the allowed question. So that is supposed to be a sketch of some system in which certain things don't make sense. So (1), please let me know when you have more than the above utterly vague and circular (you're using the word 'allowed' to explicate your use of the word 'allowed') sketch, and (2) note that the fact that something doesn't make sense in some system of your making doesn't entail that said thing doesn't make sense in all other systems. > There's no help in or need to read a book on the subject. How do you know? You have strong opinions on a subject of which it seems you haven't studied and of which you have misconceptions. > I do > actually own computability and logic by Boolos et al, 2nd edition. Which is not a textbook in set theory. > Most of my reading has been in algebraic number theory, Galois theory, > and analytic number theory, and complex function theory. But anyway, > my consideration is more fundamental, I believe, than those dealt with > in any book. At least you could inform yourself on the subject and also start ridding yourself of your misconceptions about it. As to fundamental considerations dealt with in books, you are simply unaware of the extent of the literature. > Maybe you, and maybe even every student of logic think > differently, fine. Well, in your pretty much thoroughgoing ignorance of the subject you don't bring much of interest to the conversation, not even as cogent CRITICISM of set theory, fine. MoeBlee === Subject: Re: ultrafinitism and ultraformalism > Well, I still don't understand how you would avoid either formalism, > or using sentences with the word set in them. > Avoid what formalism? As to your not being able to see how to avoid > the word 'set', perhaps you would provide an example of a sentence in > context of ZFC that uses the word 'set' but that you don't know how to > express the thought behind that sentence but without the word. Well, I don't know how to express the thought behind the sentence > anyway, regardless. That's the point. You're not making sense when you say That's the point. I said that > for any use of the word 'set' in context of ZFC, I bet I can come up > with a way not to use the word. You said you don't understand how that > instance in which you see the word 'set' used in context of ZFC but > don't understand how the word could be avoided. Surely, if you're > objecting to the word 'set' and done trust my claim that the word can > where you don't see how it could be avoided. Well, if it can be avoided, that may not be an improvement. > No, it's quantification full stop that doesn't make sense to me. E.g. > for any number x in N, x+2 is in N. That I understand. The statement > for any x, x=x makes no sense to me. That is handled in the interpretation for a language. Given an > interpretation for the language, the universal quantifier ranges over > all and only those objects in a certain set. And if the > interepretation is specified, then the set is defined by an explicilty > articulated property, so you can refer to any (natural) number > rather than the set of (natural) numbers, if that better suits you. Also, the statement 2 is in > the set of all sets of numbers makes no sense to me. That is my > consideration. The word 'set' there is dispensible. We could instead say: 2 is a member of the object whose members are all and only those > objects whose members are only natural numbers. By the use of language, it's implicit that object has to mean something like set. But even if that avoids the word set, but I still have a problem with the words object and members. > Every there is sentence is indeed equivalent to a sentence that uses > for every. In classical logic, yes. Not in intuitionistic logic. For me the sentence not for every x in X not phi(x) > does not have literally the same meaning as there is an x in X such > that phi(x), so *semantically* should not be a definition, but > *formally* it works fine to replace one with the other, negated. I Semantically, they are extensionally equivalent, though I suppose (I > don't know enough about it) that one could say that they are not > intensionally equivalant. but you never know. Certainly using both for all and there is is > more elegant. But considering only one is convenient for some Godel- > style proofs, I believe. I don't know what Godel proofs you have in mind. But, as I said, > classically, one quantifier is sufficient, but intutionistically we > need both quantifiers. > Indeed, just a quantifier without a matrix following it is not even > syntactical in the language for ZFC. Though, contrary to your > preference, it is not required that every quantifier be relativized; > however a quantifier does only range over a given set per an > interpretation for the language. > Okay. > Then perhaps now quantification makes sense to you. I answered that above. Then why did you write 'okay'? And above you did NOT respond to the > point that the range of a quantifier is limited in the INTERPRETATION > for the language. > You also can't compare numbers to sets of numbers, for example, just > like you can't say or denounce 2>{the set of primes}. > Predicate symbols such as '>' can be defined in a way that eliminates > such problems as how to regard such formulas as you mention above. > My point was that 2={the set of primes} is as coherent as 2>{the > set of primes}. The former is not allowed, just like the latter. > Not allowed in what context? What is allowed as a well formed > formula is determined by the syntax rules of whatever system. What > particular system do you have in mind? I mean it doesn't make sense. It's not allowed if you want to make > sense. Goto next response. We can allow such expressions and make sense. You've done little more > than to declare, without much reason, that certain things don't make > sense to you. Okay, they don't make sense to you, but meanwhile they > make quite fine sense to me. You've not shown any breakdown in my > sense, which is given by the method of interpretation for languages. I didn't claim there could be much reasoned argument behind saying something was nonsensical. If someone says something that is gibberish to you, you can't really say anything other than that's gibberish. > Oh, sorry. Put in a schema the axioms for nth order arithmetic, for > each, n, and take into account the type-like restrictions on the > allowable sentences. That is a sketch. > By this time, I don't even recall the question I asked was. See the allowed question. So that is supposed to be a sketch of some system in which certain > things don't make sense. So (1), please let me know when you have more > than the above utterly vague and circular (you're using the word > 'allowed' to explicate your use of the word 'allowed') sketch, and > (2) note that the fact that something doesn't make sense in some > system of your making doesn't entail that said thing doesn't make > sense in all other systems. There's no help in or need to read a book on the subject. How do you know? You have strong opinions on a subject of which it > seems you haven't studied and of which you have misconceptions. Because I have a hunch, they wouldn't even mention what I have been saying. > I do > actually own computability and logic by Boolos et al, 2nd edition. Which is not a textbook in set theory. Most of my reading has been in algebraic number theory, Galois theory, > and analytic number theory, and complex function theory. But anyway, > my consideration is more fundamental, I believe, than those dealt with > in any book. At least you could inform yourself on the subject and also start > ridding yourself of your misconceptions about it. As to fundamental > considerations dealt with in books, you are simply unaware of the > extent of the literature. If you knew this was dealt with in the literature, you could have posted something dealing with it. But whatever you have posted has not dealt with it, not for me. > Maybe you, and maybe even every student of logic think > differently, fine. Well, in your pretty much thoroughgoing ignorance of the subject you > don't bring much of interest to the conversation, not even as cogent > CRITICISM of set theory, fine. I didn't claim it was interesting, and I'm not criticising anything. I find the phrases for any set X, X=X, and for any object x, x=x incoherent. === Subject: Re: ultrafinitism and ultraformalism > Well, I still don't understand how you would avoid either formalism, > or using sentences with the word set in them. > Avoid what formalism? As to your not being able to see how to avoid > the word 'set', perhaps you would provide an example of a sentence in > context of ZFC that uses the word 'set' but that you don't know how to > express the thought behind that sentence but without the word. > Well, I don't know how to express the thought behind the sentence > anyway, regardless. That's the point. You're not making sense when you say That's the point. I said that > for any use of the word 'set' in context of ZFC, I bet I can come up > with a way not to use the word. You said you don't understand how that > instance in which you see the word 'set' used in context of ZFC but > don't understand how the word could be avoided. Surely, if you're > objecting to the word 'set' and done trust my claim that the word can > where you don't see how it could be avoided. Well, if it can be avoided, that may not be an improvement. I really don't know what you're on about. First you object to couching notions in terms of sets. Then I say that we don't have to mention sets. Then you say that's not an improvement. Perhaps get back to me when you've decided just what it is you want to say. > 2 is a member of the object whose members are all and only those > objects whose members are only natural numbers. By the use of language, it's implicit that object has to mean > something like set. implicit, something like, blah blah. Perhaps get back to me when you have a SPECIFIC objection. > But even if that avoids the word set, but I > still have a problem with the words object and members. Don't need the word 'members' either. Just need the primitive 'is a member of'. As to 'objects', it's as neutral as I can imagine anything. The only thing more neutral is using variables, which, of course we could do also. Instead of mention 'objects' we could say 'for any x...' etc. I really don't see that what you object to is not so pervasive as to be an objection even to mathematical use of variables. I mean, are you sure you're interested in mathematics at all? > Indeed, just a quantifier without a matrix following it is not even > syntactical in the language for ZFC. Though, contrary to your > preference, it is not required that every quantifier be relativized; > however a quantifier does only range over æa given set per an > interpretation for the language. > Okay. > Then perhaps now quantification makes sense to you. > I answered that above. Then why did you write 'okay'? And above you did NOT respond to the > point that the range of a quantifier is limited in the INTERPRETATION > for the language. No response from you here. > We can allow such expressions and make sense. You've done little more > than to declare, without much reason, that certain things don't make > sense to you. Okay, they don't make sense to you, but meanwhile they > make quite fine sense to me. You've not shown any breakdown in my > sense, which is given by the method of interpretation for languages. I didn't claim there could be much reasoned argument behind saying > something was nonsensical. If someone says something that is gibberish > to you, you can't really say anything other than that's gibberish. But you've never even studied the syntax and semantics of first order language. So you're just saying that's gibberish without having even looked into the matter. I've never studied Russian. By your method, I might as well say to people speaking Russian, that's gibberish. > Because I have a hunch, they wouldn't even mention what I have been > saying. You have such a wonderfully intellectual approach. You have hunches about this and that about logic and mathematics, so you don't even bother to look into it while still broadcasting your strong opinions about it. You should be talk radio host. > At least you could inform yourself on the subject and also start > ridding yourself of your misconceptions about it. As to fundamental > considerations dealt with in books, you are simply unaware of the > extent of the literature. If you knew this was dealt with in the literature, you could have > posted something dealing with it. But whatever you have posted has > not dealt with it, not for me. The comment you made is as to MORE fundamental (emphasis added). The literature is vast and gets deep into fundamentals. I can't promise that you'll find precisely your own notions addressed, but a familiarity with the thoughts of other human beings who have thought and worked hard on such areas may contribute even to a reformation of what you even take to be fundamental. On the other hand, if you feel that being an strongly opionated ignoramus more suits your style, then, of course, you will have at it. > I > find the phrases for any set X, X=X, and for any object x, x=x > incoherent. So I should say I find it incoherent when people from Russia utter those weird combinations of sounds and syllables. I find it inchorent when people use a bunch of terminology about marine microbiology. I find it incoherent when computer programmers write statements such as x=x+1. Etc. By the way, you skipped again my remark about quantification being limited by interpretation. MoeBlee === Subject: Re: ultrafinitism and ultraformalism > Well, I still don't understand how you would avoid either formalism, > or using sentences with the word set in them. > Avoid what formalism? As to your not being able to see how to avoid > the word 'set', perhaps you would provide an example of a sentence in > context of ZFC that uses the word 'set' but that you don't know how to > express the thought behind that sentence but without the word. > Well, I don't know how to express the thought behind the sentence > anyway, regardless. That's the point. > You're not making sense when you say That's the point. I said that > for any use of the word 'set' in context of ZFC, I bet I can come up > with a way not to use the word. You said you don't understand how that > instance in which you see the word 'set' used in context of ZFC but > don't understand how the word could be avoided. Surely, if you're > objecting to the word 'set' and done trust my claim that the word can > where you don't see how it could be avoided. Well, if it can be avoided, that may not be an improvement. I really don't know what you're on about. First you object to couching > notions in terms of sets. Then I say that we don't have to mention > sets. Then you say that's not an improvement. Perhaps get back to me > when you've decided just what it is you want to say. > 2 is a member of the object whose members are all and only those > objects whose members are only natural numbers. By the use of language, it's implicit that object has to mean > something like set. implicit, something like, blah blah. Perhaps get back to me when > you have a SPECIFIC objection. My objection is as specific as anyone can be when objecting to gibberish. > But even if that avoids the word set, but I > still have a problem with the words object and members. Don't need the word 'members' either. Just need the primitive 'is a > member of'. As to 'objects', it's as neutral as I can imagine > anything. The only thing more neutral is using variables, which, of > course we could do also. Instead of mention 'objects' we could say > 'for any x...' etc. Yes, I object to the phrase objects. I don't find meaningful phrases like for any object, or for any set, or for any x,. Is that specific enough for you? > I really don't see that what you object to is not so pervasive as to > be an objection even to mathematical use of variables. I mean, are you > sure you're interested in mathematics at all? Yeah, I am. Mathematics can be carried out without such talk that I object to. > Indeed, just a quantifier without a matrix following it is not even > syntactical in the language for ZFC. Though, contrary to your > preference, it is not required that every quantifier be relativized; > however a quantifier does only range over a given set per an > interpretation for the language. > Okay. > Then perhaps now quantification makes sense to you. > I answered that above. > Then why did you write 'okay'? And above you did NOT respond to the > point that the range of a quantifier is limited in the INTERPRETATION > for the language. No response from you here. It still depends on phrases such as a set or (presumably) ZF-talk of models. Such talk is what I am saying is incoherent. > We can allow such expressions and make sense. You've done little more > than to declare, without much reason, that certain things don't make > sense to you. Okay, they don't make sense to you, but meanwhile they > make quite fine sense to me. You've not shown any breakdown in my > sense, which is given by the method of interpretation for languages. I didn't claim there could be much reasoned argument behind saying > something was nonsensical. If someone says something that is gibberish > to you, you can't really say anything other than that's gibberish. But you've never even studied the syntax and semantics of first order > language. So you're just saying that's gibberish without having even > looked into the matter. I've never studied Russian. By your method, I > might as well say to people speaking Russian, that's gibberish. I've never studied it very much because I read the first few sentences (which are supposed to be taken as given or self explanatory) and am led to object in the way I have done earlier in this thread. > Because I have a hunch, they wouldn't even mention what I have been > saying. You have such a wonderfully intellectual approach. You have hunches > about this and that about logic and mathematics, so you don't even > bother to look into it while still broadcasting your strong opinions > about it. You should be talk radio host. I am not going to go looking through every book on the subject to find a discussion relevant to what I am saying, when I have *never* seen anyone raise it before, and have no confidence that it will be raised anywhere. > At least you could inform yourself on the subject and also start > ridding yourself of your misconceptions about it. As to fundamental > considerations dealt with in books, you are simply unaware of the > extent of the literature. If you knew this was dealt with in the literature, you could have > posted something dealing with it. But whatever you have posted has > not dealt with it, not for me. The comment you made is as to MORE fundamental (emphasis added). The > literature is vast and gets deep into fundamentals. I can't promise > that you'll find precisely your own notions addressed, but a > familiarity with the thoughts of other human beings who have thought > and worked hard on such areas may contribute even to a reformation of > what you even take to be fundamental. On the other hand, if you feel > that being an strongly opionated ignoramus more suits your style, > then, of course, you will have at it. You can't promise me, and I know why. What I object to is taken as self-explanatory in the literature. It's not as if I haven't thought about this. It first occurred to me two and a bit years ago. Staring at the phrase for any x, x=x is not going to start making sense to me just because I stare and stare at it. I can modify it to make a sensible statement for numbers such as for any number x in N, x=x or for any function f:N->N in {functions f:N->N}, f=f}. There is no reason for me to suspect that being insulted is going to change this state of affairs. > I > find the phrases for any set X, X=X, and for any object x, x=x > incoherent. So I should say I find it incoherent when people from Russia utter > those weird combinations of sounds and syllables. I find it inchorent > when people use a bunch of terminology about marine microbiology. I > find it incoherent when computer programmers write statements such as > x=x+1. Etc. What people from Russia is not supposed to be self-explanatory to an English-speaking person. === Subject: Re: ultrafinitism and ultraformalism > My objection is as specific as anyone can be when objecting to > gibberish. You intentionally DON'T investigate the subject, so you continually ensure that it remains, to you, gibberish. I intentionally don't investigate the strange sounds made by people from Russia, so that those sounds remain, to me, as gibberish. > Yes, I object to the phrase objects. I don't find meaningful phrases > like for any object, or for any set, or for any x,. Is that > specific enough for you? You've not given a specific REASON for such objection. You just declare that they are gibberish and further that, since they are gibberish, you can't give any more specific reason. However, you intentionally avoid investigating explanations of quantification, so that you ensure that it remains gibberish to you. > I really don't see that what you object to is not so pervasive as to > be an objection even to mathematical use of variables. I mean, are you > sure you're interested in mathematics at all? Yeah, I am. Mathematics can be carried out without such talk that I > object to. Please point me to a specific example of such mathematical writing. Further, mathematics can be carried out leads to the question of what you mean by carrying out mathematics. Moreover, what EXTENT of mathematics? For that matter, what do you take mathematical proof itself to be? > It still depends on phrases such as a set or (presumably) ZF-talk of > models. Such talk is what I am saying is incoherent. We've gone in circles now. I showed that 'set' is not needed. Now you're just avoiding the point I made: The limitation on quantification is achieved at the stage of interpretation. And, you could even take such interpretations in a sense of relying not on a notion of sets or even on a notion of membership, but rather on referring to properties, just as you have said you don't object to quantification limited by such properties as 'is a natural number'. Moreover, YOU can be taken as using a notion of quantification in your own informal discussion about mathematics. Your pronouncements use such phrases as something, that, etc. One might as well say to you, I find such use of words 'something', 'that', etc. to be gibberish. > I've never studied it very much because I read the first few sentences > (which are supposed to be taken as given or self explanatory) and am > led to object in the way I have done earlier in this thread. Yes, poor attention span. Unwillingness to take some time and effort to look further into a matter, to eventually digest thoughts and concepts that seem at first to be alien. Typical crank mentality. > I am not going to go looking through every book on the subject to find > a discussion relevant to what I am saying, when I have *never* seen > anyone raise it before, and have no confidence that it will be raised > anywhere. Indeed, a thoroughgoing lack of intellectual curiosity about a subject you nevertheless wish to broadcast your uninformed opinons. The very point is not to look through every book to find something that immediately addresses your own precious concerns, but rather to famliarize yourself at least with a sense of the range and depth of thinking that is available in the literature. On the other hand, one should at least applaud your steadfastness in your dedication to remaining ignorant. > You can't promise me, and I know why. What I object to is taken as > self-explanatory in the literature. It's not as if I haven't thought > about this. It first occurred to me two and a bit years ago. Staring > at the phrase for any x, x=x is not going to start making sense to > me just because I stare and stare at it. It's good that you describe your approach so frankly. Indeed, just staring and staring won't work. > I can modify it to make a > sensible statement for numbers such as for any number x in N, x=x (1) That is EXACTLY how it works at the stage of interpretation. (2) How is your in different from 'is a member of', and what does your 'N' stand for if not a set? > or > for any function f:N->N in {functions f:N->N}, f=f}. Same (1) and (2) as above. > There is no > reason for me to suspect that being insulted is going to change this > state of affairs. Being insulted won't change the mathematics. But learning about the mathematics will obviate the FAIR insult that you are an opionated blowhard on a subject of which you are a practiced ignoramus. Meanwhile, you insult intelligence by posting a bunch of silliness premised on your misconceptions about a subject you haven't even honestly looked into. > What people from Russia is not supposed to be self-explanatory to an > English-speaking person. Where is it claimed that mathematics, set theory, and mathematical logic are self-explanatory and do not require study, thought, intellectual labor, and time to digest, as with just about any other field of study? MoeBlee === Subject: Re: ultrafinitism and ultraformalism My objection is as specific as anyone can be when objecting to > gibberish. You intentionally DON'T investigate the subject, so you continually > ensure that it remains, to you, gibberish. I intentionally don't > investigate the strange sounds made by people from Russia, so that > those sounds remain, to me, as gibberish. But Russian people can and do go around explaining those things (which are not supposed to be self explanatory) to English-speaking people. > Yes, I object to the phrase objects. I don't find meaningful phrases > like for any object, or for any set, or for any x,. Is that > specific enough for you? You've not given a specific REASON for such objection. You just > declare that they are gibberish and further that, since they are > gibberish, you can't give any more specific reason. Yeah, something like that. > I really don't see that what you object to is not so pervasive as to > be an objection even to mathematical use of variables. I mean, are you > sure you're interested in mathematics at all? Yeah, I am. Mathematics can be carried out without such talk that I > object to. Please point me to a specific example of such mathematical writing. > Further, mathematics can be carried out leads to the question of > what you mean by carrying out mathematics. Moreover, what EXTENT of > mathematics? Real and complex analysis, ring theory, algebraic number theory, analytic number theory, group theory, combinatorics, non ZF-dependent topology. > It still depends on phrases such as a set or (presumably) ZF-talk of > models. Such talk is what I am saying is incoherent. We've gone in circles now. I showed that 'set' is not needed. Now > you're just avoiding the point I made: The limitation on > quantification is achieved at the stage of interpretation. And, you > could even take such interpretations in a sense of relying not on a > notion of sets or even on a notion of membership, but rather on > referring to properties, just as you have said you don't object to > quantification limited by such properties as 'is a natural number'. Moreover, YOU can be taken as using a notion of quantification in your > own informal discussion about mathematics. Your pronouncements use > such phrases as something, that, etc. One might as well say to > you, I find such use of words 'something', 'that', etc. to be > gibberish. When I discuss mathematics I quantify over sets of numbers, functions from N to N, etc. You could call *that* gibberish, if you wanted to, although you might not really believe it. > I've never studied it very much because I read the first few sentences > (which are supposed to be taken as given or self explanatory) and am > led to object in the way I have done earlier in this thread. Yes, poor attention span. Unwillingness to take some time and effort > to look further into a matter, to eventually digest thoughts and > concepts that seem at first to be alien. Typical crank mentality. Umm, well either you do understand exactly what the sentence for any x, x=x means or you don't. And I don't. > I am not going to go looking through every book on the subject to find > a discussion relevant to what I am saying, when I have *never* seen > anyone raise it before, and have no confidence that it will be raised > anywhere. Indeed, a thoroughgoing lack of intellectual curiosity about a subject > you nevertheless wish to broadcast your uninformed opinons. The very > point is not to look through every book to find something that > immediately addresses your own precious concerns, but rather to > famliarize yourself at least with a sense of the range and depth of > thinking that is available in the literature. On the other hand, one > should at least applaud your steadfastness in your dedication to > remaining ignorant. It's not as if I haven't read anything. But there is only so much spin you can put upon for any x, x=x and other such sentences. I get the distinct impression that these things are meant to be, after perhaps a short discussion, self-explanatory. > You can't promise me, and I know why. What I object to is taken as > self-explanatory in the literature. It's not as if I haven't thought > about this. It first occurred to me two and a bit years ago. Staring > at the phrase for any x, x=x is not going to start making sense to > me just because I stare and stare at it. It's good that you describe your approach so frankly. Indeed, just > staring and staring won't work. I can modify it to make a > sensible statement for numbers such as for any number x in N, x=x (1) That is EXACTLY how it works at the stage of interpretation. (2) > How is your in different from 'is a member of', and what does your > 'N' stand for if not a set? or > for any function f:N->N in {functions f:N->N}, f=f}. Same (1) and (2) as above. These sentences are coherent, but for any x, x=x is not. That is my point. The phrase for any makes perfect sense as part of the phrase I mentioned: for any number x in N, x=x. N is the universal set of numbers, or N={0,1,2,3,...}. > There is no > reason for me to suspect that being insulted is going to change this > state of affairs. Being insulted won't change the mathematics. But learning about the > mathematics will obviate the FAIR insult that you are an opionated > blowhard on a subject of which you are a practiced ignoramus. My objections are not about mathematics. That's the point. I do know about mathematics, for instance, I understand in detail the proofs of Dirichlet's theorem on primes in arithmetic progression and the Erdos- Selberg proof of the prime number theorem. Do you? > Meanwhile, you insult intelligence by posting a bunch of silliness > premised on your misconceptions about a subject you haven't even > honestly looked into. Uh, well I have thought about it. I really don't know what else to say. > What people from Russia is not supposed to be self-explanatory to an > English-speaking person. Where is it claimed that mathematics, set theory, and mathematical > logic are self-explanatory and do not require study, thought, > intellectual labor, and time to digest, as with just about any other > field of study? I said that the specific sentences I objected to were supposed to be self-explanatory. Not mathematics and mathematical logic. === Subject: Re: ultrafinitism and ultraformalism But Russian people can and do go around explaining those things (which > are not supposed to be self explanatory) to English-speaking people. Explaining what things? My point is that if one doesn't learn the language, then it's gibberish to one. And if one does attempt to learn the concepts of marine microbiology, then it's gibberish to one. And if one does not attempt to learn the concepts of mathematical foundations, then they may be gibberish to one. > Yes, I object to the phrase objects. I don't find meaningful phrases > like for any object, or for any set, or for any x,. Is that > specific enough for you? You've not given a specific REASON for such objection. You just > declare that they are gibberish and further that, since they are > gibberish, you can't give any more specific reason. Yeah, something like that. How intellectually enervating. > I really don't see that what you object to is not so pervasive as to > be an objection even to mathematical use of variables. I mean, are you > sure you're interested in mathematics at all? > Yeah, I am. Mathematics can be carried out without such talk that I > object to. Please point me to a specific example of such mathematical writing. > Further, mathematics can be carried out leads to the question of > what you mean by carrying out mathematics. Moreover, what EXTENT of > mathematics? Real and complex analysis, ring theory, algebraic number theory, > analytic number theory, group theory, combinatorics, non ZF-dependent > topology. No, I didn't ask about whole fields of study. I asked for an example of specific piece of mathematical writing. Ordinary writing in such fields as you mentioned use ordinary mathematical quantifiers all the time. Also, you didn't answer as to what you mean by carrying out mathematics and the extent of mathematics. > It still depends on phrases such as a set or (presumably) ZF-talk of > models. Such talk is what I am saying is incoherent. We've gone in circles now. I showed that 'set' is not needed. Now > you're just avoiding the point I made: The limitation on > quantification is achieved at the stage of interpretation. And, you > could even take such interpretations in a sense of relying not on a > notion of sets or even on a notion of membership, but rather on > referring to properties, just as you have said you don't object to > quantification limited by such properties as 'is a natural number'. Moreover, YOU can be taken as using a notion of quantification in your > own informal discussion about mathematics. Your pronouncements use > such phrases as something, that, etc. One might as well say to > you, I find such use of words 'something', 'that', etc. to be > gibberish. When I discuss mathematics I quantify over sets of numbers, functions > from N to N, etc. You could call *that* gibberish, if you wanted to, > although you might not really believe it. > I've never studied it very much because I read the first few sentences > (which are supposed to be taken as given or self explanatory) and am > led to object in the way I have done earlier in this thread. Yes, poor attention span. Unwillingness to take some time and effort > to look further into a matter, to eventually digest thoughts and > concepts that seem at first to be alien. Typical crank mentality. Umm, well either you do understand exactly what the sentence for any > x, x=x means or you don't. And I don't. specific question of whether one understands a particular formulation. > I am not going to go looking through every book on the subject to find > a discussion relevant to what I am saying, when I have *never* seen > anyone raise it before, and have no confidence that it will be raised > anywhere. Indeed, a thoroughgoing lack of intellectual curiosity about a subject > you nevertheless wish to broadcast your uninformed opinons. The very > point is not to look through every book to find something that > immediately addresses your own precious concerns, but rather to > famliarize yourself at least with a sense of the range and depth of > thinking that is available in the literature. On the other hand, one > should at least applaud your steadfastness in your dedication to > remaining ignorant. It's not as if I haven't read anything. But there is only so much spin > you can put upon for any x, x=x and other such sentences. I get the > distinct impression that these things are meant to be, after perhaps a > short discussion, self-explanatory. Where did you get this distinct impression? And again you skipped > I can modify it to make a > sensible statement for numbers such as for any number x in N, x=x (1) That is EXACTLY how it works at the stage of interpretation. (2) > How is your in different from 'is a member of', and what does your > 'N' stand for if not a set? > or > for any function f:N->N in {functions f:N->N}, f=f}. Same (1) and (2) as above. These sentences are coherent, but for any x, x=x is not. > That is my > point. The phrase for any makes perfect sense as part of the phrase > I mentioned: for any number x in N, x=x. N is the universal set of > numbers, or N={0,1,2,3,...}. Now you're not even trying to make it look like you're responding to my points. Your'e just in repeater mode now. > There is no > reason for me to suspect that being insulted is going to change this > state of affairs. Being insulted won't change the mathematics. But learning about the > mathematics will obviate the FAIR insult that you are an opionated > blowhard on a subject of which you are a practiced ignoramus. My objections are not about mathematics. That's the point. I do know > about mathematics, for instance, I understand in detail the proofs of > Dirichlet's theorem on primes in arithmetic progression and the Erdos- > Selberg proof of the prime number theorem. Do you? No, but I'm not shooting my mouth off about them. That, in contrast to you, is MY point. And that reminds me, I mentioned that I don't know what your notion of mathematical proof IS. Which also reminds me, as you might explain proofs of such theorems as you mention, if I keep asking you to prove the results upon which these proofs themselves depend, at what point are you going to declare that we've hit axioms and primitives? In the various fields of mathematics that interest you, what are the most basic claims that you take as givens, and what principles of logic do you take to be determinative? > Meanwhile, you insult intelligence by posting a bunch of silliness > premised on your misconceptions about a subject you haven't even > honestly looked into. Uh, well I have thought about it. I really don't know what else to > say. Oh, okay, as long as you've thought about it, then surely that should be enough. I just thought about prime numbers the other day; now I'm sure that all the talk about them is gibberish. > Where is it claimed that mathematics, set theory, and mathematical > logic are self-explanatory and do not require study, thought, > intellectual labor, and time to digest, as with just about any other > field of study? I said that the specific sentences I objected to were supposed to be > self-explanatory. Not mathematics and mathematical logic. But you don't object to just specific sentences. Where do you find it expressed that the methods of set theory and mathematical logic that you object to are supposed to be self-explanatory? MoeBlee === Subject: Re: ultrafinitism and ultraformalism > But Russian people can and do go around explaining those things (which > are not supposed to be self explanatory) to English-speaking people. Explaining what things? My point is that if one doesn't learn the > language, then it's gibberish to one. And if one does attempt to learn > the concepts of marine microbiology, then it's gibberish to one. And > if one does not attempt to learn the concepts of mathematical > foundations, then they may be gibberish to one. The point is, the phrases (for all x, x=x) I was talking about are supposed to be self-explanatory. There's not *supposed* to be any teaching to be done, and I don't see anyone trying to teach it. > Yes, I object to the phrase objects. I don't find meaningful phrases > like for any object, or for any set, or for any x,. Is that > specific enough for you? > You've not given a specific REASON for such objection. You just > declare that they are gibberish and further that, since they are > gibberish, you can't give any more specific reason. Yeah, something like that. How intellectually enervating. > I really don't see that what you object to is not so pervasive as to > be an objection even to mathematical use of variables. I mean, are you > sure you're interested in mathematics at all? > Yeah, I am. Mathematics can be carried out without such talk that I > object to. > Please point me to a specific example of such mathematical writing. > Further, mathematics can be carried out leads to the question of > what you mean by carrying out mathematics. Moreover, what EXTENT of > mathematics? Real and complex analysis, ring theory, algebraic number theory, > analytic number theory, group theory, combinatorics, non ZF-dependent > topology. No, I didn't ask about whole fields of study. I asked for an example > of specific piece of mathematical writing. Ordinary writing in such > fields as you mentioned use ordinary mathematical quantifiers all the > time. Also, you didn't answer as to what you mean by carrying out > mathematics and the extent of mathematics. Carrying out might as well mean formal verifiability. Formally the sentences involved can be written in the form for any number x in N: x=x. I already said earlier that the formalization I have in mind is more or less a schema including nth order arithmetic for each n. Like the system for second order arithmetic on wikipedia, but for each n in N. > It still depends on phrases such as a set or (presumably) ZF-talk of > models. Such talk is what I am saying is incoherent. > We've gone in circles now. I showed that 'set' is not needed. Now > you're just avoiding the point I made: The limitation on > quantification is achieved at the stage of interpretation. And, you > could even take such interpretations in a sense of relying not on a > notion of sets or even on a notion of membership, but rather on > referring to properties, just as you have said you don't object to > quantification limited by such properties as 'is a natural number'. > Moreover, YOU can be taken as using a notion of quantification in your > own informal discussion about mathematics. Your pronouncements use > such phrases as something, that, etc. One might as well say to > you, I find such use of words 'something', 'that', etc. to be > gibberish. When I discuss mathematics I quantify over sets of numbers, functions > from N to N, etc. You could call *that* gibberish, if you wanted to, > although you might not really believe it. > I said in response to a later point: These sentences are coherent, but for any x, x=x is not. That is my point. The phrase for any makes perfect sense as part of the phrase I mentioned: for any number x in N, x=x. N is the universal set of numbers, or N={0,1,2,3,...}. > I've never studied it very much because I read the first few sentences > (which are supposed to be taken as given or self explanatory) and am > led to object in the way I have done earlier in this thread. > Yes, poor attention span. Unwillingness to take some time and effort > to look further into a matter, to eventually digest thoughts and > concepts that seem at first to be alien. Typical crank mentality. Umm, well either you do understand exactly what the sentence for any > x, x=x means or you don't. And I don't. specific question of whether one understands a particular formulation. am I supposed to say to that? > I am not going to go looking through every book on the subject to find > a discussion relevant to what I am saying, when I have *never* seen > anyone raise it before, and have no confidence that it will be raised > anywhere. > Indeed, a thoroughgoing lack of intellectual curiosity about a subject > you nevertheless wish to broadcast your uninformed opinons. The very > point is not to look through every book to find something that > immediately addresses your own precious concerns, but rather to > famliarize yourself at least with a sense of the range and depth of > thinking that is available in the literature. On the other hand, one > should at least applaud your steadfastness in your dedication to > remaining ignorant. It's not as if I haven't read anything. But there is only so much spin > you can put upon for any x, x=x and other such sentences. I get the > distinct impression that these things are meant to be, after perhaps a > short discussion, self-explanatory. Where did you get this distinct impression? And again you skipped I get that impression because no further explanation seems to be (in the text) forthcoming! What you said was that I was ignorant and should familiarize myself with the sort of literature which I had already articulated my problems with. I don't have a problem with the idea of self-explanatory-ness *in general*, by the way. > I can modify it to make a > sensible statement for numbers such as for any number x in N, x=x > (1) That is EXACTLY how it works at the stage of interpretation. (2) > How is your in different from 'is a member of', and what does your > 'N' stand for if not a set? > or > for any function f:N->N in {functions f:N->N}, f=f}. > Same (1) and (2) as above. These sentences are coherent, but for any x, x=x is not. > That is my > point. The phrase for any makes perfect sense as part of the phrase > I mentioned: for any number x in N, x=x. N is the universal set of > numbers, or N={0,1,2,3,...}. Now you're not even trying to make it look like you're responding to > my points. Your'e just in repeater mode now. Well, that's my response. > There is no > reason for me to suspect that being insulted is going to change this > state of affairs. > Being insulted won't change the mathematics. But learning about the > mathematics will obviate the FAIR insult that you are an opionated > blowhard on a subject of which you are a practiced ignoramus. My objections are not about mathematics. That's the point. I do know > about mathematics, for instance, I understand in detail the proofs of > Dirichlet's theorem on primes in arithmetic progression and the Erdos- > Selberg proof of the prime number theorem. Do you? No, but I'm not shooting my mouth off about them. That, in contrast to > you, is MY point. Actually, I don't usually talk about this. > And that reminds me, I mentioned that I don't know what your notion of > mathematical proof IS. See earlier in this reply. > Which also reminds me, as you might explain proofs of such theorems as > you mention, if I keep asking you to prove the results upon which > these proofs themselves depend, at what point are you going to declare > that we've hit axioms and primitives? In the various fields of > mathematics that interest you, what are the most basic claims that you > take as givens, and what principles of logic do you take to be > determinative? There indeed is a formal system I adhere to (i.e. in principle it could be formalised as this or that). I do have axioms and primitives. For example, 0, s0, ss0, sss0, etc. Also N, PN, PPN, etc. Also ordered n-tuples of various sorts. I take as given, for all n in N: not sn=0. When the +1 function is defined this just means for all n in N: not n+1=0. > Meanwhile, you insult intelligence by posting a bunch of silliness > premised on your misconceptions about a subject you haven't even > honestly looked into. Uh, well I have thought about it. I really don't know what else to > say. Oh, okay, as long as you've thought about it, then surely that > should be enough. I just thought about prime numbers the other day; > now I'm sure that all the talk about them is gibberish. > Where is it claimed that mathematics, set theory, and mathematical > logic are self-explanatory and do not require study, thought, > intellectual labor, and time to digest, as with just about any other > field of study? I said that the specific sentences I objected to were supposed to be > self-explanatory. Not mathematics and mathematical logic. But you don't object to just specific sentences. Where do you find it > expressed that the methods of set theory and mathematical logic that > you object to are supposed to be self-explanatory? I do object to sentences of a particular kind. It (the intention that what is being said is meant to be self-explanatory) isn't expressed explicitly, but I take it as implicit! === Subject: Re: ultrafinitism and ultraformalism > The point is, the phrases (for all x, x=x) I was talking about are > supposed to be self-explanatory. There's not *supposed* to be any > teaching to be done, and I don't see anyone trying to teach it. I asked before, where do you get the idea that quantification is supposed to be self-explanatory and there is not supposed to be, nor is there, any explanation of it? Sheesh, this discussion with you is making me feel like I'm Charlie Gibson interviewing Sarah Palin! > Please point me to a specific example of such mathematical writing. > Further, mathematics can be carried out leads to the question of > what you mean by carrying out mathematics. Moreover, what EXTENT of > mathematics? > Real and complex analysis, ring theory, algebraic number theory, > analytic number theory, group theory, combinatorics, non ZF-dependent > topology. No, I didn't ask about whole fields of study. I asked for an example > of specific piece of mathematical writing. Ordinary writing in such > fields as you mentioned use ordinary mathematical quantifiers all the > time. Also, you didn't answer as to what you mean by carrying out > mathematics and the extent of mathematics. Carrying out might as well mean formal verifiability. Formally the > sentences involved can be written in the form for any number x in N: > x=x. I already said earlier that the formalization I have in mind is > more or less a schema including nth order arithmetic for each n. Like > the system for second order arithmetic on wikipedia, but for each n in > N. Then I can only await your formalization of real and complex analysis, ring theory, algebraic number theory, analytic number theory, group theory, combinatorics, and topology in such a way that is not equivalent to or does not subsume set theory and ordinary quantification. Meanwhile, you've still not pointed out a single specific piece of mathematical writing in which mathematics is carried out without the talk you object to. I said in response to a later point: These sentences are coherent, but for any x, x=x is not. That is my > point. The phrase for any makes perfect sense as part of the phrase > I mentioned: for any number x in N, x=x. N is the universal set of > numbers, or N={0,1,2,3,...}. We've gone in circles now. I showed that 'set' is not needed. Now you're just avoiding the point I made: The limitation on quantification is achieved at the stage of interpretation. And, you could even take such interpretations in a sense of relying not on a notion of sets or even on a notion of membership, but rather on referring to properties, just as you have said you don't object to quantification limited by such properties as 'is a natural number'. Moreover, YOU can be taken as using a notion of quantification in your own informal discussion about mathematics. Your pronouncements use such phrases as something, that, etc. One might as well say to you, I find such use of words 'something', 'that', etc. to be gibberish. Also now, you object to the notion of 'set' but use it yourself. > specific question of whether one understands a particular formulation. am I supposed to say to that? I refer to this: We've gone in circles now. I showed that 'set' is not needed. Now you're just avoiding the point I made: The limitation on quantification is achieved at the stage of interpretation. And, you could even take such interpretations in a sense of relying not on a notion of sets or even on a notion of membership, but rather on referring to properties, just as you have said you don't object to quantification limited by such properties as 'is a natural number'. Moreover, YOU can be taken as using a notion of quantification in your own informal discussion about mathematics. Your pronouncements use such phrases as something, that, etc. One might as well say to you, I find such use of words 'something', 'that', etc. to be gibberish. And added now, that you object to use of 'set' but use it yourself. > It's not as if I haven't read anything. But there is only so much spin > you can put upon for any x, x=x and other such sentences. I get the > distinct impression that these things are meant to be, after perhaps a > short discussion, self-explanatory. Where did you get this distinct impression? And again you skipped I get that impression because no further explanation seems to be (in > the text) forthcoming! There's PLENTY of explanation of quantification in books on logic. I've even given you one point of explanation, tailored specifically for your particular concern, that you keep ignoring here. > Actually, I don't usually talk about this. You have been here. > There indeed is a formal system I adhere to (i.e. in principle it > could be formalised as this or that). I do have axioms and > primitives. For example, 0, s0, ss0, sss0, etc. Also N, PN, PPN, etc. > Also ordered n-tuples of various sorts. I take as given, for all n in N: not sn=0. When the +1 function is > defined this just means for all n in N: not n+1=0. Perhaps one day you'll tell us what your axioms are. Perhaps start us off with a specification of the syntax of your language. And of course, you'll do that without recourse to the notion of 'set' or of plain quantification. > Where is it claimed that mathematics, set theory, and mathematical > logic are self-explanatory and do not require study, thought, > intellectual labor, and time to digest, as with just about any other > field of study? > I said that the specific sentences I objected to were supposed to be > self-explanatory. Not mathematics and mathematical logic. But you don't object to just specific sentences. Where do you find it > expressed that the methods of set theory and mathematical logic that > you object to are supposed to be self-explanatory? I do object to sentences of a particular kind. It (the intention that > what is being said is meant to be self-explanatory) isn't expressed > explicitly, but I take it as implicit! Oh come on! Of course you object to sentences of a particular kind, but you ALSO have general objections. So if self-explanatoriness is not expressed explicitly, then why do you take self-explanatoriness as implicit, especially when there IS plenty of explanation and discussion about the notion of quantification? MoeBlee === Subject: Re: ultrafinitism and ultraformalism The point is, the phrases (for all x, x=x) I was talking about are > supposed to be self-explanatory. There's not *supposed* to be any > teaching to be done, and I don't see anyone trying to teach it. I asked before, where do you get the idea that quantification is > supposed to be self-explanatory and there is not supposed to be, nor > is there, any explanation of it? Sheesh, this discussion with you is making me feel like I'm Charlie > Gibson interviewing Sarah Palin! > Please point me to a specific example of such mathematical writing. > Further, mathematics can be carried out leads to the question of > what you mean by carrying out mathematics. Moreover, what EXTENT of > mathematics? > Real and complex analysis, ring theory, algebraic number theory, > analytic number theory, group theory, combinatorics, non ZF-dependent > topology. > No, I didn't ask about whole fields of study. I asked for an example > of specific piece of mathematical writing. Ordinary writing in such > fields as you mentioned use ordinary mathematical quantifiers all the > time. > Also, you didn't answer as to what you mean by carrying out > mathematics and the extent of mathematics. Carrying out might as well mean formal verifiability. Formally the > sentences involved can be written in the form for any number x in N: > x=x. I already said earlier that the formalization I have in mind is > more or less a schema including nth order arithmetic for each n. Like > the system for second order arithmetic on wikipedia, but for each n in > N. Then I can only await your formalization of real and complex analysis, > ring theory, algebraic number theory, analytic number theory, group > theory, combinatorics, and topology in such a way that is not > equivalent to or does not subsume set theory and ordinary > quantification. Meanwhile, you've still not pointed out a single specific piece of > mathematical writing in which mathematics is carried out without the > talk you object to. Second order arithmetic, for example, can carry out much of mathematics. I said in response to a later point: These sentences are coherent, but for any x, x=x is not. That is my > point. The phrase for any makes perfect sense as part of the phrase > I mentioned: for any number x in N, x=x. N is the universal set of > numbers, or N={0,1,2,3,...}. > We've gone in circles now. I showed that 'set' is not needed. Now > you're just avoiding the point I made: The limitation on > quantification is achieved at the stage of interpretation. And, you > could even take such interpretations in a sense of relying not on a > notion of sets or even on a notion of membership, but rather on > referring to properties, just as you have said you don't object to > quantification limited by such properties as 'is a natural number'. > Moreover, YOU can be taken as using a notion of quantification in your > own informal discussion about mathematics. Your pronouncements use > such phrases as something, that, etc. One might as well say to > you, I find such use of words 'something', 'that', etc. to be > gibberish. Also now, you object to the notion of 'set' but use it yourself. I don't object to the word set at all. What I do object to is certain phrases which include the word set. For example, for any set x, x=x. I don't say for any something x, x=x. I also don't find the property is a natural number coherent. I don't say x is a thing, that is also a natural number. I do use the word set *when* I say things like ...for any set of numbers in the set of sets of numbers PN... or ...for any set of functions f:N->N in the set of sets of functions f:N->N.... > specific question of whether one understands a particular formulation. am I supposed to say to that? I refer to this: We've gone in circles now. I showed that 'set' is not needed. Now > you're just avoiding the point I made: The limitation on > quantification is achieved at the stage of interpretation. And, you > could even take such interpretations in a sense of relying not on a > notion of sets or even on a notion of membership, but rather on > referring to properties, just as you have said you don't object to > quantification limited by such properties as 'is a natural number'. > Moreover, YOU can be taken as using a notion of quantification in your > own informal discussion about mathematics. Your pronouncements use > such phrases as something, that, etc. One might as well say to > you, I find such use of words 'something', 'that', etc. to be > gibberish. And added now, that you object to use of 'set' but use it yourself. The words set and something and that are not incoherent, it is the phrases in which they are used which are coherent or incoherent. > It's not as if I haven't read anything. But there is only so much spin > you can put upon for any x, x=x and other such sentences. I get the > distinct impression that these things are meant to be, after perhaps a > short discussion, self-explanatory. > Where did you get this distinct impression? And again you skipped I get that impression because no further explanation seems to be (in > the text) forthcoming! There's PLENTY of explanation of quantification in books on logic. > I've even given you one point of explanation, tailored specifically > for your particular concern, that you keep ignoring here. Actually, I don't usually talk about this. You have been here. Yes, I have. > There indeed is a formal system I adhere to (i.e. in principle it > could be formalised as this or that). I do have axioms and > primitives. For example, 0, s0, ss0, sss0, etc. Also N, PN, PPN, etc. > Also ordered n-tuples of various sorts. I take as given, for all n in N: not sn=0. When the +1 function is > defined this just means for all n in N: not n+1=0. Perhaps one day you'll tell us what your axioms are. Perhaps start us > off with a specification of the syntax of your language. And of > course, you'll do that without recourse to the notion of 'set' or of > plain quantification. I said before, I don't have a problem with using the word set. I already said earlier that the formalization I have in mind is more or less a schema including nth order arithmetic for each n. Like the system for second order arithmetic on wikipedia, but for each n in N. > Where is it claimed that mathematics, set theory, and mathematical > logic are self-explanatory and do not require study, thought, > intellectual labor, and time to digest, as with just about any other > field of study? > I said that the specific sentences I objected to were supposed to be > self-explanatory. Not mathematics and mathematical logic. > But you don't object to just specific sentences. Where do you find it > expressed that the methods of set theory and mathematical logic that > you object to are supposed to be self-explanatory? I do object to sentences of a particular kind. It (the intention that > what is being said is meant to be self-explanatory) isn't expressed > explicitly, but I take it as implicit! Oh come on! Of course you object to sentences of a particular kind, > but you ALSO have general objections. So if self-explanatoriness is > not expressed explicitly, then why do you take self-explanatoriness as > implicit, especially when there IS plenty of explanation and > discussion about the notion of quantification? I can't find any explanation anywhere of the kind of language I am objecting to (and I wouldn't expect to find any, because it is clearly meant to be self-explanatory). In particular, whatever explanation that you say there is, hasn't satisfied me in this thread. === Subject: Re: ultrafinitism and ultraformalism > Then I can only await your formalization of real and complex analysis, > ring theory, algebraic number theory, analytic number theory, group > theory, combinatorics, and topology in such a way that is not > equivalent to or does not subsume set theory and ordinary > quantification. Meanwhile, you've still not pointed out a single specific piece of > mathematical writing in which mathematics is carried out without the > talk you object to. Second order arithmetic, for example, can carry out much of > mathematics. Second order arithmetic is essentially first order set theory. Second order arithmetic has full quantification and not just on individual varialbles but on predicate variables too. And the ordinary semantics for second order arithmetic are set theoretical. If you say to a bunch of mathematicians who work in foundations. You've given no rational reason why you deem unrelativized quantification as nonsensical in first order but deem it sensical (PLUS OVER PREDICATES) in second order. By the way, you STILL didn't answer the question. I haven't given an actual piece of mathematical writing you have in mind that is an example of mathematics done without what you object to. > We've gone in circles now. I showed that 'set' is not needed. Now > you're just avoiding the point I made: The limitation on > quantification is achieved at the stage of interpretation. And, you > could even take such interpretations in a sense of relying not on a > notion of sets or even on a notion of membership, but rather on > referring to properties, just as you have said you don't object to > quantification limited by such properties as 'is a natural number'. > Moreover, YOU can be taken as using a notion of quantification in your > own informal discussion about mathematics. Your pronouncements use > such phrases as something, that, etc. One might as well say to > you, I find such use of words 'something', 'that', etc. to be > gibberish. Also now, you object to the notion of 'set' but use it yourself. I don't object to the word set at all. What I do object to is > certain phrases which include the word set. For example, for any > set x, x=x. But that makes no sense, since in ordinary set theory, for any set x can be replaced by for any x. So 'set' has nothing to do with it. So what you object to, as we've been discussing, is for any x. But then, when I mention that we interpret that as relative to some domain of discourse, you object that the domain of discourse is a set. So we're back full circle again, as indeed you do object to 'set'. > I don't say for any something x, x=x. So what? No one said you do. But you do use unrestricted quantification even in your own informal remarks. For example, you write, One can only talk about obviously true w.r.t something that makes sense. You object to 'set', and 'object', but there you quantify over things. > I also don't find the property > is a natural number coherent. Whether you call it a description of a property, you take 'is a natural number' and the 'the set of natural numbers' as meaningful, but for any given interpretation for the language, evaluation of each formula is relativized to a set of objects or even to an actual formula with one free variable (such as 'x is a natural number' is a formula with one free variable). > I don't say x is a thing, that is > also a natural number. I do use the word set *when* I say things like > ...for any set of numbers in the set of sets of numbers PN... or > ...for any set of functions f:N->N in the set of sets of functions > f:N->N.... (1) You use unrelativized quantification in your own discussion about such things. (2) You've not shown how we could even define and/or give axioms about such things as 'function', 'natural number', 'N', 'PN', etc. without unrelativized quantification. > I said before, I don't have a problem with using the word set. I believe that phrases such as let X be a set or let X be a class make no sense. So 'set' is okay in certain uses but not in let x be a set' and other uses. Would you please then give a rule to determine in which uses 'set' is okay? Meanwhile, I take it that you recognize, 'Let x be a natural number' as meaningful. So why do you hold that 'Let x be a set' is not meaningful? > I already said earlier that the > formalization I have in mind is more or less a schema including nth > order arithmetic for each n. Like the system for second order > arithmetic on wikipedia, but for each n in N. Something that is more or less something else is not a specification of an actual formal system and axioms. We have no way to evaluate your claims about your system until you tell us your system. Anyway, previously you mentioned second order arithmetic as being an example of mathematics carried out without your objection. But that makes no sense, since second order arithmetic has even MORE unrelativized quantification than any first order theory. > I can't find any explanation anywhere of the kind of language I am > objecting to (and I wouldn't expect to find any, because it is clearly > meant to be self-explanatory). You just keep repeating because it is clearly meant to be self- explanatory as DOGMA. > In particular, whatever explanation > that you say there is, hasn't satisfied me in this thread. I don't know any explanation about ANYTHING that would satisfy someone who says yes to second order but no to first order. MoeBlee === Subject: Re: ultrafinitism and ultraformalism > My objection is as specific as anyone can be when objecting to > gibberish. You intentionally DON'T investigate the subject, so you continually > ensure that it remains, to you, gibberish. I intentionally don't > investigate the strange sounds made by people from Russia, so that > those sounds remain, to me, as gibberish. But Russian people can and do go around explaining those things (which > are not supposed to be self explanatory) to English-speaking people. Only to those who agree to listen to those explanations. > Yes, I object to the phrase objects. I don't find meaningful phrases > like for any object, or for any set, or for any x,. Is that > specific enough for you? You've not given a specific REASON for such objection. You just > declare that they are gibberish and further that, since they are > gibberish, you can't give any more specific reason. Yeah, something like that. > I really don't see that what you object to is not so pervasive as to > be an objection even to mathematical use of variables. I mean, are you > sure you're interested in mathematics at all? > Yeah, I am. Mathematics can be carried out without such talk that I > object to. Please point me to a specific example of such mathematical writing. > Further, mathematics can be carried out leads to the question of > what you mean by carrying out mathematics. Moreover, what EXTENT of > mathematics? Real and complex analysis, ring theory, algebraic number theory, > analytic number theory, group theory, combinatorics, non ZF-dependent > topology. I find very little of any of them that is not made easier to carry out by allowing both quantifiers and set terminology. Perhaps, as an example, you can give a definition of, say, the Reimann integral with no mention of there exists or for all or is a member of in English, or the equivalent in Russian or any other language. Until we see some examples of how you would carry out your restricted form of mathematics, we remain justifiably dubious of your ability to do so. Your move! === Subject: Re: ultrafinitism and ultraformalism > Perhaps, as an example, you can give a definition of, say, the Reimann > integral with no mention of there exists or for all or is a member > of in English, or the equivalent in Russian or any other language. Until we see some examples of how you would carry out your restricted > form of mathematics, we remain justifiably dubious of your ability to > do so. Your move! I didn't say the phrases you mention made no sense. I said, for example, for any number x in the set of numbers N, x=x makes sense, and for any x, x=x does not make sense. The quantifiers there exists, for every are very much allowed. === Subject: Re: ultrafinitism and ultraformalism > Perhaps, as an example, you can give a definition of, say, the Reimann > integral with no mention of there exists or for all or is a member > of in English, or the equivalent in Russian or any other language. Until we see some examples of how you would carry out your restricted > form of mathematics, we remain justifiably dubious of your ability to > do so. Your move! I didn't say the phrases you mention made no sense. I said, for > example, for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. The quantifiers there > exists, for every are very much allowed. Does it help you to think of for any x, x=x as saying for any entity x, x=x? Or for every object x under consideration, x=x? Or even everything is what it is? -- Alan Smaill === Subject: Re: ultrafinitism and ultraformalism > Perhaps, as an example, you can give a definition of, say, the Reimann > integral with no mention of there exists or for all or is a member > of in English, or the equivalent in Russian or any other language. > Until we see some examples of how you would carry out your restricted > form of mathematics, we remain justifiably dubious of your ability to > do so. > Your move! I didn't say the phrases you mention made no sense. I said, for > example, for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. The quantifiers there > exists, for every are very much allowed. Does it help you to think of for any x, x=x as saying > for any entity x, x=x? Or for every object x under consideration, x=x? Or even everything is what it is? -- > Alan Smaill === Subject: Re: ultrafinitism and ultraformalism > Perhaps, as an example, you can give a definition of, say, the Reimann > integral with no mention of there exists or for all or is a member > of in English, or the equivalent in Russian or any other language. > Until we see some examples of how you would carry out your restricted > form of mathematics, we remain justifiably dubious of your ability to > do so. > Your move! > I didn't say the phrases you mention made no sense. I said, for > example, for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. The quantifiers there > exists, for every are very much allowed. Does it help you to think of for any x, x=x as saying > for any entity x, x=x? Or for every object x under consideration, x=x? Or even everything is what it is? -- > Alan Smaill > Do you find these English statements nonsensical in themselves? Or do you not think they relate to the logic statement? You are you, I am I, my toothbrush is my toothbrush -- in fact everything is what it is. -- Alan Smaill === Subject: Re: ultrafinitism and ultraformalism > for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. > Does it help you to think of for any x, x=x as saying > for any entity x, x=x? > Or even everything is what it is? > in fact everything is what it is. Mathematicians who speak this way (i.e., in total disregard of some specified universe of discourse), must also say things like the following, in order to avoid contradictions: Theorem 50. 0 = {x: x = x} That's from Suppes, Axiomatic Set Theory. 0 is the empty set. Does the above theorem seem intuitive to you? === Subject: Re: ultrafinitism and ultraformalism > for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. > Does it help you to think of for any x, x=x as saying > for any entity x, x=x? > Or even everything is what it is? > in fact everything is what it is. Mathematicians who speak this way (i.e., in total disregard of > some specified universe of discourse), must also say things like > the following, in order to avoid contradictions: Theorem 50. 0 = {x: x = x} The source of such theorems is not unrelativized quantification but rather Fregean method for handling improperly referring temrs, specifically those that result from the set abstraction operator. > That's from Suppes, Axiomatic Set Theory. 0 is the empty set. > Does the above theorem seem intuitive to you? It is baffling UNLESS you include the context, which is the Fregean method for handling improperly referring terms. MoeBlee === Subject: Re: ultrafinitism and ultraformalism > for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. > Does it help you to think of for any x, x=x as saying > for any entity x, x=x? > Or even everything is what it is? > in fact everything is what it is. Mathematicians who speak this way (i.e., in total disregard of > some specified universe of discourse), must also say things like > the following, in order to avoid contradictions: Theorem 50. 0 = {x: x = x} That's from Suppes, Axiomatic Set Theory. 0 is the empty set. > Does the above theorem seem intuitive to you? If x is drawn from the natural numbers, the assertion is false. If x is drawn from {}, the assertion is true. -- Michael Press === Subject: Re: ultrafinitism and ultraformalism > Does it help you [kleptomaniac666] to > think of for any x, x=x as saying > for any entity x, x=x? > Or even everything is what it is? > in fact everything is what it is. > Mathematicians who speak this way (i.e., in total disregard of > some specified universe of discourse), must also say things like > the following, in order to avoid contradictions: > Theorem 50. 0 = {x: x = x} > That's from Suppes, Axiomatic Set Theory. 0 is the empty set. > Does the above theorem seem intuitive to you? If x is drawn from the natural numbers, > the assertion is false. > If x is drawn from {}, the assertion is true. (Actually, Virgil's reply was quite correct, in that the above theorem can hardly be intuitive without knowing some particulars of Suppes' development.) For Suppes, the above theorem holds with the variable x ranging over all objects, or all entitities (whatever that means), and is a consequence of (1) there being no nonempty *set* whose members are just those entities with the stated property, and of (2) Suppes' definition schema for the abstraction operation {- : -}, by which the result is *always* a set. I seem to have misjudged when I thought that the OP's concern was just about the question of ambiguity in references to all entities without mention of any domain of discourse. === Subject: Re: ultrafinitism and ultraformalism > for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. > Does it help you to think of for any x, x=x as saying > for any entity x, x=x? > Or even everything is what it is? > in fact everything is what it is. > Mathematicians who speak this way (i.e., in total disregard of > some specified universe of discourse), must also say things like > the following, in order to avoid contradictions: > Theorem 50. 0 = {x: x = x} > That's from Suppes, Axiomatic Set Theory. 0 is the empty set. > Does the above theorem seem intuitive to you? If x is drawn from the natural numbers, > the assertion is false. > If x is drawn from {}, the assertion is true. I think that what his trying to claim is that unless some universe for x is defined or declared a priori, that 'x = x' cannot have any meaning at all. While that sort of restriction may be required by ,e.g., computers, he has not made his case that it need hold otherwise. === Subject: Re: ultrafinitism and ultraformalism > for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. > Does it help you to think of for any x, x=x as saying > for any entity x, x=x? > Or even everything is what it is? > in fact everything is what it is. > Mathematicians who speak this way (i.e., in total disregard of > some specified universe of discourse), must also say things like > the following, in order to avoid contradictions: > Theorem 50. 0 = {x: x = x} > That's from Suppes, Axiomatic Set Theory. 0 is the empty set. > Does the above theorem seem intuitive to you? > If x is drawn from the natural numbers, > the assertion is false. > If x is drawn from {}, the assertion is true. I think that what his trying to claim is that unless some universe for x > is defined or declared a priori, that 'x = x' cannot have any meaning at > all. While that sort of restriction may be required by ,e.g., computers, he > has not made his case that it need hold otherwise. -- Michael Press === Subject: Re: ultrafinitism and ultraformalism > for any number x in the set of numbers N, x=x makes sense, > and for any x, x=x does not make sense. > Does it help you to think of for any x, x=x as saying > for any entity x, x=x? > Or even everything is what it is? > in fact everything is what it is. Mathematicians who speak this way (i.e., in total disregard of > some specified universe of discourse), must also say things like > the following, in order to avoid contradictions: Theorem 50. 0 = {x: x = x} That's from Suppes, Axiomatic Set Theory. 0 is the empty set. > Does the above theorem seem intuitive to you? If it is 'theorem 50', it presumably comes from a set of axioms, definitions, and 49 prior theorems without which it it does not seem at all intuitive. === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) > 1. Brouwer's fixed-point theorem. > 2. Robertson and Seymour's graph minor theorem. > 3. Martin's Borel-determinacy theorem. > 4. Every commutative ring with unit has a prime ideal. > 5. The product of compact topological spaces is compact. > 6. ZFC is consistent. > 7. Projective determinacy. [...] >Assuming what you say is true, (1) and (2) are proven mathematical >theorems. (3) is incoherent. (4), and (5) can be given schematically >and are dependent on choice principles, and whether they are true or >not depends on refinement of the phrase set in the phrases in which >it is used. (6) is unknown. (7) I do not know the statement of so I >don't know. What's incoherent about (3)? The notion of a Borel set? Surely not, because the Borel sigma-algebra is definable with only a handful of iterations of the powerset operation on the naturals. The notion of an infinite game? Surely that's even more concrete than the notion of a Borel set. >Axioms are (or are not) so basic there's no helping you, not >theorems. E.g. any set of numbers has a least element. My line is >drawn at the axioms for numbers, sets of numbers, sets of sets of >numbers, etc, including ordered n-tuples of. And you are so blind to your own prejudices that you can't see that someone who thinks that an arbitrary set of integers is incoherent is being no less silly than you are when you claim that the notion of an arbitrary set is incoherent? And you believe that your line in the sand is God-given and that people who draw their lines elsewhere are either beyond help or incoherent? -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: ultrafinitism and ultraformalism > 1. Brouwer's fixed-point theorem. > 2. Robertson and Seymour's graph minor theorem. > 3. Martin's Borel-determinacy theorem. > 4. Every commutative ring with unit has a prime ideal. > 5. The product of compact topological spaces is compact. > 6. ZFC is consistent. > 7. Projective determinacy. > [...] >Assuming what you say is true, (1) and (2) are proven mathematical >theorems. (3) is incoherent. (4), and (5) can be given schematically >and are dependent on choice principles, and whether they are true or >not depends on refinement of the phrase set in the phrases in which >it is used. (6) is unknown. (7) I do not know the statement of so I >don't know. What's incoherent about (3)? The notion of a Borel set? Surely not, because > the Borel sigma-algebra is definable with only a handful of iterations of the > powerset operation on the naturals. The notion of an infinite game? Surely > that's even more concrete than the notion of a Borel set. I thought it required Z or ZF to even be stated. If it can be stated in nth order arithmetic for some n, then I guess it's either like (4) or (5) or unknown as yet. >Axioms are (or are not) so basic there's no helping you, not >theorems. E.g. any set of numbers has a least element. My line is >drawn at the axioms for numbers, sets of numbers, sets of sets of >numbers, etc, including ordered n-tuples of. And you are so blind to your own prejudices that you can't see that someone > who thinks that an arbitrary set of integers is incoherent is being no less > silly than you are when you claim that the notion of an arbitrary set is > incoherent? And you believe that your line in the sand is God-given and that > people who draw their lines elsewhere are either beyond help or incoherent? It's simple. If they say it's incoherent then they are wrong. There is finding incoherent, and a truth/falsity objection. Those two are different. People tend to try and give reasons for the latter. In the case of Nelson, his reasons are nonsensical. There are two types of things I say: (1) If someone says abitrary set of integers is incoherent then I say no, you are wrong, and (2) If someone says such and such a principle is invalid because of this reason I say your reason is incoherent. === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) [Re: Borel determinacy] >I thought it required Z or ZF to even be stated. If it can be stated >in nth order arithmetic for some n, then I guess it's either like (4) >or (5) or unknown as yet. The *statement* doesn't require a lot of heavy set theory. The *proof* does, and is therefore incoherent by your standards. In any normal conversation, though, Borel determinacy is a proven mathematical theorem, and to say that the proof is incoherent would be a serious insult to Martin. Martin's proof is certainly not incoherent in the green ideas sleep furiously sense, because people are able to understand it, formalize it, generalize it, etc. It can be incoherent only in an idiosyncratic kleptomaniac666 sense. This is why it is so laughable that you toss around terms like proven mathematical theorem as if it were a totally objective concept, when secretly you assign the phrase a private meaning that differs from what the rest of the mathematical community means by it. But, you've admitted that you're beyond help, and blind to your own prejudices, so I'll stop trying to help you. I am curious about one thing, though. Is is true that the continuum hypothesis is either true or false, and we just don't know which? Extrapolating from what you've said, I would guess yes, because we're only at the level of bijections between sets of reals (or sets of sets of integers), but I'm wondering if you can figure out a way to make the continuum hypothesis incoherent. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: ultrafinitism and ultraformalism > [Re: Borel determinacy] > Martin's proof (not the D. A. Martin you mention), which might be relevant to the present discussion ... Existential quantification and the 'regimentation' of ordinary language ( first page viewable online at http://www.jstor.org/pss/2251892 ) === Subject: Re: ultrafinitism and ultraformalism [Re: Borel determinacy] I thought it required Z or ZF to even be stated. If it can be stated >in nth order arithmetic for some n, then I guess it's either like (4) >or (5) or unknown as yet. The *statement* doesn't require a lot of heavy set theory. The *proof* does, > and is therefore incoherent by your standards. Yeah. > In any normal conversation, > though, Borel determinacy is a proven mathematical theorem, and to say that > the proof is incoherent would be a serious insult to Martin. If it hurts his feelings, it hurts his feelings. > Martin's proof is certainly not incoherent in the green ideas sleep > furiously sense, because people are able to understand it, formalize > it, generalize it, etc. It can be incoherent only in an idiosyncratic > kleptomaniac666 sense. This is why it is so laughable that you toss > around terms like proven mathematical theorem as if it were a totally > objective concept, when secretly you assign the phrase a private meaning > that differs from what the rest of the mathematical community means by it. Admittedly, the phrase does call for a little further explanation. There are certain axioms which I see as true (and are sufficient to derive any mathematical truth of which I am capable of deriving, whilst either proving (nontrivially, or as axioms) every mathematical truth which is obviously true to me). If someone says that axiom is meaningless I say no, you're wrong. If someone says that axiom is false I say no, you're wrong. If they give a reason for the latter that is nonsensical, I will say that the reason they have given is nonsensical (e.g. Nelson). If someone states a nonsensical axiom, then I will say (possible without any further ado) what you are postulating is nonsensical. > But, you've admitted that you're beyond help, and blind to your own > prejudices, so I'll stop trying to help you. I am curious about one > thing, though. Is is true that the continuum hypothesis is either > true or false, and we just don't know which? Extrapolating from what > you've said, I would guess yes, because we're only at the level of > bijections between sets of reals (or sets of sets of integers), but > I'm wondering if you can figure out a way to make the continuum > hypothesis incoherent. The statement for any subset X of R that is infinite, either X is equipollent to N or X is equipollent to R is certainly sensible. It is allowed in my system (I already said earlier that the formalization I have in mind is more or less a schema including nth order arithmetic for each n. Like the system for second order arithmetic on wikipedia, but for each n in N). My view is that there is nothing particular about the meaning of the word set in the phrases in which it is used in the language of the system I refer to, which says anything on CH. Thus you can do mathematics with the assumption that CH is true, or the assumption that CH is false. === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >The statement for any subset X of R that is infinite, either X is >equipollent to N or X is equipollent to R is certainly sensible. It >is allowed in my system (I already said earlier that the >formalization I have in mind is more or less a schema including nth >order arithmetic for each n. Like the system for second order >arithmetic on wikipedia, but for each n in N). So let me ask a followup question. Let S_0 be the set of integers. For i > 0, let S_i be the set of all subsets of S_{i-1}. I take it that you don't have any problems so far? Now let S be the union of the S_i. Did I just say something nonsensical? -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: ultrafinitism and ultraformalism The statement for any subset X of R that is infinite, either X is >equipollent to N or X is equipollent to R is certainly sensible. It >is allowed in my system (I already said earlier that the >formalization I have in mind is more or less a schema including nth >order arithmetic for each n. Like the system for second order >arithmetic on wikipedia, but for each n in N). So let me ask a followup question. Let S_0 be the set of integers. > For i > 0, let S_i be the set of all subsets of S_{i-1}. I take it > that you don't have any problems so far? Now let S be the union of the S_i. Did I just say something nonsensical? Yes. === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) > So let me ask a followup question. Let S_0 be the set of integers. > For i > 0, let S_i be the set of all subsets of S_{i-1}. I take it > that you don't have any problems so far? > Now let S be the union of the S_i. > Did I just say something nonsensical? Yes. But what is it about what I said that is nonsensical? You don't object to *all* talk about sets. Here I'm referring to a specific set S, and I'm telling you exactly how to determine whether x is in S. Namely, x is in S if and only if x is in S_i for some i. It is one thing to say that you *don't know* if such a set S exists, or to insist that there is no *proof* that such a set S exists. But why is it *nonsense* to say, Let S be the union of the S_i? If such a line were to appear in a mathematics paper you were reading, you wouldn't even blink. You would still be able to follow the intended line of reasoning, even if you felt that the existence of S needed proof. In contrast, if instead you encountered the sentence Twas brillig, and the slithy toves did gyre and gimble in the wabe in the middle of a technical proof in a math paper, you would assume that there was a typographical error, because that really would be nonsense. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: ultrafinitism and ultraformalism > So let me ask a followup question. Let S_0 be the set of integers. > For i > 0, let S_i be the set of all subsets of S_{i-1}. I take it > that you don't have any problems so far? > Now let S be the union of the S_i. > Did I just say something nonsensical? Yes. But what is it about what I said that is nonsensical? You don't object to > *all* talk about sets. Here I'm referring to a specific set S, and I'm > telling you exactly how to determine whether x is in S. Namely, x is in S > if and only if x is in S_i for some i. It is one thing to say that you *don't know* if such a set S exists, or > to insist that there is no *proof* that such a set S exists. But why is > it *nonsense* to say, Let S be the union of the S_i? It just is. 2={2} falls into the same bracket of phrases as 2=Julius Caesar or twas brillig - it is just incoherent. === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >It just is. 2={2} falls into the same bracket of phrases as >2=Julius Caesar or twas brillig - it is just incoherent. Wait, what is incoherent about 2={2}? And what does this have to do with the incoherence of a set S such that x is in S if and only if x is in S_i for some i? My guess would be something like this. 2 is an integer, while {2} is a set of integers. Implicitly you have in mind the notion of some formal language (call it K666) in which all coherent mathematical statements can be expressed. In K666, any sentence that equates an integer with a set of integers is not syntactically well-formed. Also, nothing in K666 allows you to express a set that has elements of the type that I suggested. Assuming this conjectural reconstruction of your point of view is correct, does K666 allow objects such as {2, {2}}? It would seem that part of the reason to allow yourself to ascend to third-order objects is so that you can encode concepts such as a set of reals in a natural way---i.e., directly as a third-order object---instead of having to figure out how to reword or encode what you want to say in second-order arithmetic. More generally, when you get to something like a Lie group, which is typically defined as an ordered triple (G, T, x) where G is a set, T is the topology, and x is the group operation, you can encode this directly without having to worry about exactly how many levels up everything is, as long as the number of levels is finite. But you would lose most of this flexibility if you insisted that nth-order objects had to have members that were *exactly* (n-1)st-order, as to being *at most* (n-1)st-order. If that is the case, then the objection to the set S I proposed can't be that it contains elements of different order. So its alleged incoherence is of a different nature from that of 2 = {2}, and citing 2 = {2} doesn't shed any light on why you think talking about S is incoherent. It must instead have to do with the illegality of quantifying over infinitely many sets of different types in K666. Even this doesn't make talk about S incoherent. It just means that one cannot talk directly about S in one's formal language. By analyzing the context, one may be able to see how to reconstruct whatever argument that involves S without ever directly speaking of a set of infinite type. My sense from the kinds of things you've been saying is that your approach to foundational questions is to pick some arbitrary sharp cutoff, and declare everything below that cutoff to be legitimate and everything above that cutoff to be incoherent. If someone presses you for reasons for picking that cutoff and not some other cutoff, you simply refuse. This is a more-or-less tenable approach (though by sticking your head in the sand in this way, you block yourself from learning some things that might turn out to be useful to you one day), as long as you have a clear picture of what exactly your own cutoff is and what falls below and what falls above. It seems to me that you haven't thought through your position very far; in particular, you probably couldn't specify the exact syntax and axioms of K666, let alone figure out how one would translate ordinary mathematical language into it. The result is that you're poorly equipped to handle searching questions about boundary cases. Given your chosen strategy, you probably can't admit in public that you would, in fact, have no trouble following an argument in which something like S was invoked (even though this is obviously true), because S looks like something above your boundary line, and you're committed to shooting down such things. If you went to the trouble of fleshing out the details of K666 more clearly, then I think it's highly likely you would respond to my question in the way I suggested. That is, S, invoked informally in a mathematical argument, would not be *incoherent*, because one could make sense of it and follow the logic. However, if there appeared to be no way to reword the argument so that it could be formalized in K666, then it would be rejected. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: anharmonic quantum oscillator I was trying to solve the solution of the Schrodinger equation for an anharmonic oscillator, with an asymmetrical potential, such as Morse or Lennard-Jones or something like x^2 - A x^3, not the classical x^4, i.e. : -y'' + V(x) y = E y I was considering e.g. the shooting method, but I did'n know which boundary conditions I have to consider (typically y(-Inf)=y(+Inf)=0, but since the potential is not symmetrical, y'(0)=0 cannot be considered). Luca P. === Subject: Re: anharmonic quantum oscillator <20289232.1221157609494.JavaMail.jakarta@nitrogen.mathforum.org>, Luca I was trying to solve the solution of the Schrodinger equation for an > anharmonic oscillator, with an asymmetrical potential, such as Morse or > Lennard-Jones or something like x^2 - A x^3, not the classical x^4, i.e. : -y'' + V(x) y = E y I was considering e.g. the shooting method, but I did'n know which boundary > conditions I have to consider (typically y(-Inf)=y(+Inf)=0, but since the > potential is not symmetrical, y'(0)=0 cannot be considered). Luca P. EnergyRB allows you to easily find and display energy eigenvalues and eigenfunctions for the one-dimensional time-independent Schroedinger equation. Incorporating REALbasic's RBScript, almost any potential energy function can be entered by the user. This function does not need to be even. Freeware. Download page: http://homepage.mac.com/delaneyrm/EnergyRB.html But note that x^2 - A x^3 cannot have bound states, since it goes to +infinity in one direction, and -infinity in the other. Bob === Subject: Re: -- Wrong limits do not commute > I've made quite clear what I really want to dispute: the significance of > those nested limits with respect to other issues than pure formalistics. In other words, the significance of mathematics outside of mathematics. Within mathematics, definitions rule. Without mathematics, what rules? === Subject: Re: -- Wrong limits do not commute > So, what is > lim f(x,y) ? > x -> oo > As you've said, it's the function g(y) = 0. Now, surely, you have no > doubt what the limit of that function is as y -> oo. Surely. And I have no doubt that IT MAKES NO SENSE (literally) as well. What you probably mean is that it does not have any physical interpretation that you are comfortable with, but that there be any physical interpretation is a hang up almost exclusively limited to physicists like you. In mathematics itself, there is no need for any such physical interpretation. And to suggest that there ought to be is a common sin of physicists. === Subject: Re: -- Wrong limits do not commute > I'm a bit dishonest in suggesting that I don't really know things, but I > am never dishonest in trying to fulfill my destination here. May I suggest that your destination, which is ultimately the grave, may be unrelated to what you see as your destiny. === Subject: Re: -- Wrong limits do not commute > Any definition other than with sinc(0) = 1 will never have applications > outside mathematics. That's why default mathematics should only contain > _this_ definition and no others. You are quite free to define your own default mathematics, so long as you do not insist that anyone else be bound by it. But the you must equally allow anyone else to define his or her own default mathematics without let or hindrance as long as they do not bind you to it. But there is a non-default standard for mathematics in general, and in that standard the definition of a double limit (two variables simultaneously) is different from the corresponding definitions for iterated limits (in which the limiting for one variable precedes the limiting of the other). === Subject: Re: -- Wrong limits do not commute >And then we do y -> oo , so I must go up through a line with a slope of >90 degrees, while having arrived at an infinite horizontal distance from >the origin. How can we do that ? > it would make some (pedagogical) fun to explain these issues to a > first year student. > The two-level operation > lim lim f(x, y) > y->oo x->oo > simply behaves this way: First you keep y fixed and let x go to > infinity, that might give you some value g(y). In your example you > simply obtain g(y) = 0 for all values of y. > Next you observe the behaviour of this function g(y), as y approaches > infinity. > Since you obtain different results when you approach (oo, oo) > differently, e.g. along a ray with a fixed slope, you know, that the > function doesn't have an unique limit for (oo, oo). > There is nothing wrong about that, you simple have to be aware that > taking limits needs proper care. Yes I can understand that, formally. But now we try the other way around > and substitute (VERY LARGE, VERY LARGE) for (oo, oo). Because that's the > _physical_ (or, if you prefer: the geometrical) _meaning_ of the limits. Since the value of x*y/(x^2-y^2) varies between 0 (acheived) and oo (only approachable) depending on the path towards (oo,oo) one follows, there can be no single limit., and the limit cannot exist. However each iterated limit exists and is zero. At least as long as one goes by the formal definitions and does not try to impose one's own geometric intuition on top of the mathematics. > And then we see that the answer = 0 is kind of nonsense. I've decided to > simply call it wrong. (Why show mercy: _they_ have so often said it to > this author). And here is another occasion on which that author is spouting nonsense, at least regarding the mathematics involved. === Subject: Re: -- Wrong limits do not commute > At least as long as one goes by the formal definitions and does not try > to impose one's own geometric intuition on top of the mathematics. I've always thought that geometric intuition is _part of_ mathematics. Han de Bruijn === Subject: Re: -- Wrong limits do not commute At least as long as one goes by the formal definitions and does not try > to impose one's own geometric intuition on top of the mathematics. I've always thought that geometric intuition is part of mathematics. Han de Bruijn ************************************************************* Big...huge...humongous mistake, Han: intuition, ANY intuition, is no part of ANY part of mathematics, and hasn't been ever, and it looks like it never will. Of course, you can say YOU use your intuition, geometric or whatever, to work out this or that problem in maths in this or that fashion, but it still isn't part of mathematics. This sheds a little more light on your rather thinking that much of mathematics HAS to be wrong simply because it doesn't fit some intuitive idea you have. I'm beginning to arrive to the enlighting conclusion that you treat mathematics as you'd treat one of your kids: don't do that, behave like this and that, don't open your mouth while eating, etc., but it only sounds more like don't say this and that parts in limits theory correct since they MUST be wrong because I can't see how this can do this or that thing, don't define stuff like that because one half of brain doesn't understand that, behave properly and don't talk about infinite stuff since I can't see how I will use it in physcis or while buying stuff in the market, etc. You are pissed off at maths because it doesn't behave as you have decided, the hell knows why, it should behave...amazing! Of course, it could also be that you're pissed off at mathematics because it is stuff way above your head, and thus you feel it must be wrong, otherwise such an ashtonishing developed intelligence like yours should grasp it...but it doesn't, so it is wrong or else it all is nonsense! Oh, well.... Tonio === Subject: Re: -- Wrong limits do not commute > At least as long as one goes by the formal definitions and does not try > to impose one's own geometric intuition on top of the mathematics. I've always thought that geometric intuition is _part of_ mathematics. Han de Bruijn ************************************************************* Big...huge...humongous mistake, Han: intuition, ANY intuition, is no > part of ANY part of mathematics, and hasn't been ever, and it looks > like it never will. Of course, you can say YOU use your intuition, geometric or whatever, > to work out this or that problem in maths in this or that fashion, but > it still isn't part of mathematics. This sheds a little more light on your rather thinking that much of > mathematics HAS to be wrong simply because it doesn't fit some > intuitive idea you have. > I'm beginning to arrive to the enlighting conclusion that you treat > mathematics as you'd treat one of your kids: don't do that, behave > like this and that, don't open your mouth while eating, etc., but it > only sounds more like don't say this and that parts in limits theory > correct since they MUST be wrong because I can't see how this can do > this or that thing, don't define stuff like that because one half of > brain doesn't understand that, behave properly and don't talk about > infinite stuff since I can't see how I will use it in physcis or while > buying stuff in the market, etc. You are pissed off at maths because it doesn't behave as you have > decided, the hell knows why, it should behave...amazing! Of course, it could also be that you're pissed off at mathematics > because it is stuff way above your head, and thus you feel it must > be wrong, otherwise such an ashtonishing developed intelligence like > yours should grasp it...but it doesn't, so it is wrong or else it all > is nonsense! Oh, well.... how do you negotiate repeatable symbol manipulation? do you think it is definitions all the way down? why do mathematicians always fall back to some naive theory like a metalogic for formation or application? do you think language can form without some faculty (no matter how faulty) for establishing bisimulation? -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: -- Wrong limits do not commute But then we have to shift that parallel line towards infinity and _that_ >stumps me, because now I can't see anymore what's happening. (This is >merely a consequence of _both_ my brain halves working. I can understand >it with the formal half but not with the visual half). Consequently, >I still find that the limit A is indeed wrong. Flame shield ... well, >think I can take it. Do we have to live with counter intuition or can >we refuse and let the other brain half speak ? I don't know why you have problems with it. Take a function f(x,y). A graph of that function is a surface. If we >consider the limit as x -> oo of f(x,y), we get a curve: a function >g(y). Now, you can visualize what it means to take lim_{y->oo} g(y), >so there you go. >(Seems that you have two brain halves working as well here.) Okay, I can >see that for _some_ functions. But how about f(x,y) = x.y/(x^2 + y^2) , >being equal to sin(2.phi)/2 in polar coordinates. Quite a messy thing >for x -> oo . Here are a few isolines (a few due to ASCII limitations): > -1 0 +1 > | / > | / > | / > | / > | / > |/ > 0 ------------------- 0 > /| > / | > / | > / | > / | > / | > +1 0 -1 >Now you can think that for x -> oo the function becomes zero. Allright. >But then for y -> oo you go upwards, and my geometrical brain half says >that you will meet the isoline y = x somewhere. Resulting in a value > + 1 (and others)eventually. But no, the official (i.e. formal) result >is 0 . Confused .. > Your geometric brain should have told you that once you have gone to the > limit indicated by x -> oo while y remains fixed and finite, that you > are on a line through origin with slope equal to 0, and must remain > there. And then we do y -> oo , so I must go up through a line with a slope of > 90 degrees, while having arrived at an infinite horizontal distance from > the origin. How can we do that ? For one thing, 90 degrees is not a slope at all. Do you mean a slope of tan(90 degrees)? For another, how does one get off of a line with slope zero at a point for which y is still finite but x has already gone to infinity? === Subject: Re: -- Wrong limits do not commute I do. Please look up my Internet profile, if you want. You will see then >that my objectives are always evolving around the same theme, which has >become kind of an obsession: the _meaning_ of mathematics in a _science_ >environment. The whole thread fits precisely in this perspective, and it >is not a surprise for the older 'sci.math' debaters that it does. Maybe >I'm a bit dishonest in suggesting that I don't really know things, but I >am never dishonest in trying to fulfill my destination here. > I think that your feigned ignorance of taking nested limits is indeed > dishonest. In this respect, you're crossing the line from crank to > troll. Why not stick to what you really dispute and stop pretending > that this evil brain half keeps muddling your calculus? I've made quite clear what I really want to dispute: the > significance of those nested limits with respect to other issues > than pure formalistics. Perhaps you want to claim: every nested limit in physics (and other sciences) is a limit that commutes. Perhaps this is so, perhaps not. I'll leave that discussion to others. But that claim has nothing to do with your feigned confusion over the limit lim_y->oo lim_x->oo xy/(x^2 + y^2). > When talking about honesty, I didn't expect _you_ would sink into this: http://phiwumbda.org/~jesse/teaching/BusEthics-F08/syll.html And hey ! To be honest, mind the supermarket paradigm in your > lessons ! Teaching business ethics is shameful? News to me. -- It's my belief that when religion and pseudoscience achieve an official status within a culture [...], then genocide, war, oppression, injustice, and economic stagnation are sure to follow. -- David Petry, on why |X| < |P(X)| is bad, bad, bad. === Subject: Re: -- Wrong limits do not commute Teaching business ethics is shameful? æNews to me. Capitalism is compatible with ethics? News to me. Han de Bruijn === Subject: Re: -- Wrong limits do not commute > Teaching business ethics is shameful? æNews to me. Capitalism is compatible with ethics? News to me. Rather off-topic, but I see nothing inherently immoral about trading goods freely. In any case, I don't really give a what you think about (a) capitalism or (b) the courses I teach. -- I liked the world a lot better over ten years ago. I believed in a lot more things. Hell, most people believed in a lot more things. Back then the United States was still, well, known as most people used to know the United States. -- James S. Harris in a nostalgic mood === Subject: Re: -- Wrong limits do not commute > Teaching business ethics is shameful? æNews to me. Capitalism is compatible with ethics? æNews to me. Rather off-topic, but I see nothing inherently immoral about trading > goods freely. Neither do I. Define: freely. > In any case, I don't really give a what you think about > (a) capitalism or (b) the courses I teach. Han de Bruijn === Subject: Re: -- Wrong limits do not commute > Teaching business ethics is shameful? æNews to me. > Capitalism is compatible with ethics? æNews to me. > Rather off-topic, but I see nothing inherently immoral about trading > goods freely. Neither do I. Define: freely. As I said, I don't see any reason to debate this point with you, especially not on sci.math. I don't really care about your opinion. > In any case, I don't really give a what you think about > (a) capitalism or (b) the courses I teach. -- Jesse F. Hughes It doesn't even pay to be brilliant any more. Every goddamn moron with an attitude has a computer and can chatter you to the point of wishing suicide. --- James S. Harris, one or the other. === Subject: Re: -- Wrong limits do not commute >But then we have to shift that parallel line towards infinity and _that_ >stumps me, because now I can't see anymore what's happening. (This is >merely a consequence of _both_ my brain halves working. I can understand >it with the formal half but not with the visual half). Consequently, >I still find that the limit A is indeed wrong. Flame shield ... well, >think I can take it. Do we have to live with counter intuition or can >we refuse and let the other brain half speak ? >I don't know why you have problems with it. >Take a function f(x,y). A graph of that function is a surface. If we >consider the limit as x -> oo of f(x,y), we get a curve: a function >g(y). Now, you can visualize what it means to take lim_{y->oo} g(y), >so there you go. (Seems that you have two brain halves working as well here.) > Nonsense. Imagination and visualization is a pretty common tool in > understanding mathematics, but I don't regard it as two brain > halves. *Most* folks trying to understand calculus rely on visual > imagery to some extent, I'd wager. Which doesn't mean that both abilities are located in the same brain > center. Okay, I can see that for _some_ functions. But how about f(x,y) = >x.y/(x^2 + y^2) , being equal to sin(2.phi)/2 in polar >coordinates. Quite a messy thing for x -> oo . Here are a few >isolines (a few due to ASCII limitations): -1 0 +1 > | / > | / > | / > | / > | / > |/ > 0 ------------------- 0 > /| > / | > / | > / | > / | > / | > +1 0 -1 Now you can think that for x -> oo the function becomes zero. Allright. >But then for y -> oo you go upwards, and my geometrical brain half says >that you will meet the isoline y = x somewhere. Resulting in a value > + 1 (and others)eventually. But no, the official (i.e. formal) result >is 0 . Confused .. > Again, I don't see *why* you're confused, really. > lim lim f(x,y) > y -> oo x -> oo > means: take the limit with respect to x first. The result is a > function of y. Take the limit with respect to y now. > Your geometrical brain half seems determined to forget what it's > doing here. Or perhaps you think that the function > lim f(x,y) > x -> oo > really is a cross-section of f(x,y) found at some point x = a. It > isn't. > So, what is > lim f(x,y) ? > x -> oo > As you've said, it's the function g(y) = 0. Now, surely, you have no > doubt what the limit of that function is as y -> oo. Surely. And I have no doubt that IT MAKES NO SENSE (literally) as well. Oh, you know what _senses_ are ? Sure. And yet, I do not come to the same conclusion as you. Take the limit in terms of x. You get a function g(y). Take its limit. Neither my senses (which report nothing at all about this) nor my imagination leads me to any conclusion but that the latter limit is 0. -- But you people are scum of the earth who pretend to be something that is clearly beyond you--real mathematicians. I wouldn't be having these problems with Gauss or Euler. I wouldn't be having these problems with Fermat or Archimedes. -- James S. Harris on pretending === Subject: Re: -- Wrong limits do not commute > Neither my senses (which report nothing at all about this) nor my > imagination leads me to any conclusion but that the latter limit is 0. It's typical that your senses report nothing at all about mathematical truth. Be not surprised then that the mathematics you come up with may turn out to be be non-sense. Han de Bruijn === Subject: Re: -- Wrong limits do not commute > Neither my senses (which report nothing at all about this) nor my > imagination leads me to any conclusion but that the latter limit is 0. It's typical that your senses report nothing at all about mathematical > truth. Be not surprised then that the mathematics you come up with may > turn out to be be non-sense. It's true! It's true! I've never seen the curve x y f(x,y) = --------- x^2 + y^2 I've seen two-dimensional projections of part of that curve, and that's pretty close. But I only knew that the projection represented the curve by reasoning about the behavior of the curve. And such projections were bounded. They didn't show me what happened arbitrarily far out. So, I reiterate: My senses did not lead me to any conclusion about the limit at all. -- No sane person actually believes that religion mumbo-jumbo[...] Of course, very few people [...] would ever admit that they don't actually believe any of it. Of course I can't prove this, so don't ask. But you know it's true as well as I do. -- Mensanator === Subject: Re: -- Wrong limits do not commute > Neither my senses (which report nothing at all about this) nor my > imagination leads me to any conclusion but that the latter limit is 0. It's typical that your senses report nothing at all about mathematical > truth. Be not surprised then that the mathematics you come up with may > turn out to be be non-sense. It's true! It's true! I've never seen the curve x y > f(x,y) = --------- > x^2 + y^2 How is it a curve at all? A surface, maybe, if interpreted as {(x,y,z) in R^3: z = f(x,y)}. === Subject: Re: -- Wrong limits do not commute Neither my senses (which report nothing at all about this) nor my > imagination leads me to any conclusion but that the latter limit is 0. > It's typical that your senses report nothing at all about mathematical > truth. Be not surprised then that the mathematics you come up with may > turn out to be be non-sense. > It's true! It's true! I've never seen the curve > x y > f(x,y) = --------- > x^2 + y^2 > How is it a curve at all? A surface, maybe, if interpreted as {(x,y,z) in R^3: z = f(x,y)}. You're right, of course. I meant surface or function or something similar. -- Jesse F. Hughes The American people would have been incredibly proud of watching our military folks dispense with basic health care needs to people who needed help. --George W. Bush, March 13, 2007 === Subject: Re: -- Wrong limits do not commute > Neither my senses (which report nothing at all about this) nor my > imagination leads me to any conclusion but that the latter limit is 0. > It's typical that your senses report nothing at all about mathematical > truth. Be not surprised then that the mathematics you come up with may > turn out to be be non-sense. > It's true! It's true! I've never seen the curve > x y > f(x,y) = --------- > x^2 + y^2 > How is it a curve at all? A surface, maybe, if interpreted as {(x,y,z) in R^3: z = f(x,y)}. You're right, of course. I meant surface or function or something > similar. I've made worse errors. === Subject: Re: -- Wrong limits do not commute >Wrong limits do not commute. Almost dare to say now that non commuting >limits are simply .. wrong. Any counter examples are quite welcome. Given A transistor dissipates power propotional to its linear dimension. Let T = linear dimension of transistor, C = linear dimension of chip, > P_T = power dissipated by transistor, P_T = power dissipated by chip, > k = certain constant, N = number of transistors on chip. Then we have: > P_T = k.T^3 , nope P_T = (power per transistor) * (number of transistors) = (k * T) * (area of chip / area of transistor) = (k * T) * ( C^2 / T^2) = kC^2/T > N = C/T^2 (approximately), P_C = N.P_T = k.C.T . Right? > I hope so. My knowledge of electronics is a bit rusty. a) for a fixed chip size what is the power dissipated by the chip > as the transistor size gets very small? lim P_C = lim k.C.T = 0 > T->0 T->0 > lim P_T = lim k.C^2/T = oo T->0 T->0 > b) for a fixed transistor size what is the power dissipated by the chip > as the chip size gets very small? lim P_C = lim k.C.T = 0 > C->0 C->0 lim P_C = lim k.C^2/T = 0 T->0 C->0 > The limits in a) and b) are hardly wrong. > However, c) what is the power dissipated by the chip as the transistor size > and the chip size get very small? > lim P_T = lim kC^2/T C->0,T->0 C->0,T->0 Has no answer. So the limits are meaninful (for a fixed chip size as the transistor size gets small the power dissipated by the chip gets very large; for a fixed transistor size as the chip size gets small the power dissipated by the chip gets very small) but they do not commute. - William Hughes === Subject: Re: -- Wrong limits do not commute >Wrong limits do not commute. Almost dare to say now that non commuting >limits are simply .. wrong. Any counter examples are quite welcome. >Given >A transistor dissipates power propotional to its linear dimension. >Let T = linear dimension of transistor, C = linear dimension of chip, >P_T = power dissipated by transistor, P_T = power dissipated by chip, >k = certain constant, N = number of transistors on chip. Then we have: >P_T = k.T^3 , Typo, must be: P_C = power dissipated by chip. I'm too hasty .. > nope Whew ! Sorry ! Ooops ! How could I misread the question so much: > P_T = (power per transistor) * (number of transistors) = (k * T) * (area of chip / area of transistor) = (k * T) * ( C^2 / T^2) = kC^2/T Yeah, you're quite right, of course, if what you mean is: P_T = k.T and P_C = k.C^2/T . >N = C/T^2 (approximately), P_C = N.P_T = k.C.T . Right? >I hope so. My knowledge of electronics is a bit rusty. >a) for a fixed chip size what is the power dissipated by the chip > as the transistor size gets very small? >lim P_C = lim k.C.T = 0 >T->0 T->0 lim P_T = lim k.C^2/T = oo > T->0 T->0 lim P_C = lim k.C^2/T = oo T->0 T->0 So that limit _does not exist_. Right ? >b) for a fixed transistor size what is the power dissipated by the chip > as the chip size gets very small? >lim P_C = lim k.C.T = 0 >C->0 C->0 lim P_C = lim k.C^2/T = 0 > T->0 C->0 lim P_C = lim k.C^2/T = 0 C->0 C->0 >The limits in a) and b) are hardly wrong. Apart from the fact that the first one (a) does _not_ exist. >However, >c) what is the power dissipated by the chip as the transistor size > and the chip size get very small? > lim P_T = lim kC^2/T > C->0,T->0 C->0,T->0 Has no answer. Correct: 0/0 is anything. But now we divide the power by the chip area, being C^2 , and get zero for the limit of the power _density_, right ? > So the limits are meaninful (for a fixed chip size as the transistor > size gets small the power dissipated by the chip gets very large; > for a fixed transistor size as the chip size gets small the power > dissipated by the chip gets very small) but they do not commute. Not at all. Your meaningful limit (a) does not exist, namely. Han de Bruijn === Subject: Re: -- Wrong limits do not commute >Wrong limits do not commute. Almost dare to say now that non commuting >limits are simply .. wrong. Any counter examples are quite welcome. Given A transistor dissipates power propotional to its linear dimension. >Let T = linear dimension of transistor, C = linear dimension of chip, >P_T = power dissipated by transistor, P_T = power dissipated by chip, >k = certain constant, N = number of transistors on chip. Then we have: >P_T = k.T^3 , Typo, must be: P_C = power dissipated by chip. I'm too hasty .. nope Whew ! Sorry ! Ooops ! How could I misread the question so much: P_T = (power per transistor) * (number of transistors) = (k * T) * (area of chip / area of transistor) = (k * T) * ( C^2 / T^2) = kC^2/T Yeah, you're quite right, of course, if what you mean is: P_T = k.T and P_C = k.C^2/T . >N = C/T^2 (approximately), P_C = N.P_T = k.C.T . Right? >I hope so. My knowledge of electronics is a bit rusty. a) for a fixed chip size what is the power dissipated by the chip > as the transistor size gets very small? >lim P_C = lim k.C.T = 0 >T->0 T->0 lim P_T = lim k.C^2/T = oo > T->0 T->0 lim P_C = lim k.C^2/T = oo > T->0 T->0 So that limit _does not exist_. Right ? Nope. The limit is oo (e.g. as the size of the transisor gets small the power dissipated by the chip gets large, and you can make this as large as you want). If you insist on finite limits take P_C = min(M, k.C^2/T) where M is some maximum power. - William Hughes === Subject: Re: -- Wrong limits do not commute [ whatever ] with linear dimension C containing transistors with linear dimension T. The formula is (apart from a constant): P_C = C^2/T . Here the quotient N = C^2/T^2 is the number of transistors on the chip. One thing should be noted: N is a natural. Meaning that William Hughes' formula is only valid for C = sqrt(N).T : straight lines through the origin in the (C,T) plane. And since the minimal value of N is 1, only the following are relevant: C = T , C = sqrt(2).T , C = sqrt(3).T , .. Consequently also T > 0 , T > 0 , C > T (last one greater or equal). C>0 C>T | / | / | / | / | / forbidden |/ --------------------- T>0 | | | The search for a MEANINGFUL limit could end in the following: consider the number of transistors on chip as fixed (therefore travel along one of the lines C = sqrt(N).T , for fixed N). What is the limit of P_C if the transistor size approaches zero then ? Answer: lim (N.T) = 0 . T->0 Han de Bruijn === Subject: Re: -- Wrong limits do not commute > [ whatever ] Ok, we will go back to accepting oo as a valid limit. with linear dimension C containing transistors with linear dimension > T. The formula is (apart from a constant): P_C = C^2/T . Here the quotient N = C^2/T^2 is the number of transistors on the > chip. > One thing should be noted: N is a natural. Meaning that William > Hughes' > formula is only valid for C = sqrt(N).T Nope. If you insist that N be a whole number the correct number of transistors is N = floor(C^2/T^2) Of course the minumum number of transistors on a chip is not 1 but 0. So the limits i: lim P_C = lim k* floor(C^2/T) = oo T->0 T->0 and ii: lim P_C = lim k* floor(C^2/T) = 0 C->0 C->0 are meaningful and do not commute. Whether another limit is meaningful or not has nothing to do with whether i and ii are meaningful. - William Hughes === Subject: Re: -- Wrong limits do not commute > So the limits i: æ lim æP C = lim æk* floor(C^2/T) = oo > æ æ æT->0 æ æ æ T->0 > and ii: lim æP C = ælim æk* floor(C^2/T) = 0 > æ æ æC->0 æ æ æ C->0 are meaningful and do not commute. I rest my case. Han de Bruijn === Subject: Re: -- Wrong limits do not commute > So the limits i: æ lim æP_C = lim æk* floor(C^2/T) = oo > æ æ æT->0 æ æ æ T->0 > and ii: lim æP_C = ælim æk* floor(C^2/T) = 0 > æ æ æC->0 æ æ æ C->0 are meaningful and do not commute. I rest my case. Han de Bruijn You just lost your case. === Subject: Re: -- Wrong limits do not commute So the limits i: lim P_C = lim k* floor(C^2/T) = oo > T->0 T->0 > and ii: lim P_C = lim k* floor(C^2/T) = 0 > C->0 C->0 are meaningful and do not commute. I rest my case. Good. Note that you now agree that your statement properly posed limits _do_ commute is nonsense. - William Hughes === Subject: Re: -- Wrong limits do not commute > So the limits > i: æ lim æP C = lim æk* floor(C^2/T) = oo > æ æ æT->0 æ æ æ T->0 > and > ii: lim æP C = ælim æk* floor(C^2/T) = 0 > æ æ æC->0 æ æ æ C->0 > are meaningful and do not commute. I rest my case. Good. æNote that you now agree that your > statement properly posed limits do commute > is nonsense. æ æ æ æ æ æ æ æ æ æ æ- William Hughes Might be wrong about the meaning of I rest my case. Anyway, these are your words, not mine. Han de Bruijn === Subject: Re: -- Wrong limits do not commute > So the limits > i: lim P_C = lim k* floor(C^2/T) = oo > T->0 T->0 > and > ii: lim P_C = lim k* floor(C^2/T) = 0 > C->0 C->0 > are meaningful and do not commute. > I rest my case. Good. Note that you now agree that your > statement properly posed limits _do_ commute > is nonsense. - William Hughes Might be wrong about the meaning of I rest my case. I assume it means that you do not indend to present any more arguments. As you have not presented any argument that the limits are not meaningful, I assume you accept the example. - William Hughes === Subject: Re: -- Wrong limits do not commute So the limits >i: lim P_C = lim k* floor(C^2/T) = oo > T->0 T->0 >and >ii: lim P_C = lim k* floor(C^2/T) = 0 > C->0 C->0 >are meaningful and do not commute. >I rest my case. >Good. Note that you now agree that your >statement properly posed limits _do_ commute >is nonsense. > - William Hughes >Might be wrong about the meaning of I rest my case. I assume it means that you do not indend to present > any more arguments. As you have not presented any argument > that the limits are not meaningful, I assume you accept the example. I accept the example as a confirmation of the fact that wrong limits do not commute. And the reverse is true as well. Han de Bruijn === Subject: Re: -- Wrong limits do not commute >So the limits >i: lim P_C = lim k* floor(C^2/T) = oo > T->0 T->0 >and >ii: lim P_C = lim k* floor(C^2/T) = 0 > C->0 C->0 >are meaningful and do not commute. >I rest my case. >Good. Note that you now agree that your >statement properly posed limits _do_ commute >is nonsense. > - William Hughes >Might be wrong about the meaning of I rest my case. > I assume it means that you do not indend to present > any more arguments. As you have not presented any argument > that the limits are not meaningful, I assume you accept the example. I accept the example as a confirmation of the fact that wrong limits do > not commute. And the reverse is true as well. HdB has not yet proven anything is wrong with them except that he doesn't like them. But having HdB's approval is not even close to being a requirement for mathematical rightness. And physics is irrelevant here. === Subject: Re: -- Wrong limits do not commute > I accept the example as a confirmation of the fact that wrong limits do > not commute. Since the limits in the example are not wrong this is nonsensical. - William Hughes === Subject: Re: -- Wrong limits do not commute >I accept the example as a confirmation of the fact that wrong limits do >not commute. Since the limits in the example are not > wrong this is nonsensical. - William Hughes According to _common_ calculus, a limit that is infinite does not exist. So _your_ argument is nonsensical, always has been. Han de Bruijn === Subject: Re: -- Wrong limits do not commute >I accept the example as a confirmation of the fact that wrong limits do >not commute. > Since the limits in the example are not > wrong this is nonsensical. > - William Hughes According to _common_ calculus, a limit that is infinite does not exist. > So _your_ argument is nonsensical, always has been. Han de Bruijn The right (iterated) limits lim__{x->oo} lim+{y->oo} x*y/(x^2 - y^2) and lim__{y->oo} lim+{x->oo} x*y/(x^2 - y^2) both exist and both equal zero, so it is HdB's argument that is nonsensical, and always has been. === Subject: Re: -- Wrong limits do not commute >I accept the example as a confirmation of the fact that wrong limits do >not commute. Since the limits in the example are not > wrong this is nonsensical. - William Hughes According to _common_ calculus, a limit that is infinite does not exist. > So _your_ argument is nonsensical, always has been. Absolute piffle. Infinite limits are meaningful and anyway I pointed out a simple way to avoid the infinite limit limit and you said whatever. - William Hughes === Subject: Re: -- Wrong limits do not commute >I accept the example as a confirmation of the fact that wrong limits do >not commute. Since the limits in the example are not > wrong this is nonsensical. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ- William Hughes According to common calculus, a limit that is infinite does not exist. > So your argument is nonsensical, always has been. Han de Bruijn ********************************************************** Honestly and seriously, Han: stopping making such a an ass of yourself...it already is embarrasing! (1) Lim n (n --> oo) = 00 exists perfectly well in common calculus, just as lim (in) (n-->oo) = 0 perfectly well exists. (2) Lim sin(n) (n-->oo) DOES NOT exist. And both things are so because of MATHEMATICAL definitions. Period. Nobody cares whether it seems nice, neat, intuitive or whatever to you so or not, or whether one of your galore of brain halves tells you this or that. (1) and (2): two different things...and it'd be nice if you could FINALLY learn what's the difference between these two. Tonio === Subject: Re: -- Wrong limits do not commute >I accept the example as a confirmation of the fact that wrong limits do >not commute. >Since the limits in the example are not >wrong this is nonsensical. > - William Hughes >According to _common_ calculus, a limit that is infinite does not exist. >So _your_ argument is nonsensical, always has been. >Han de Bruijn ********************************************************** Honestly and seriously, Han: stopping making such a an ass of > yourself...it already is embarrasing! I'm not ashamed in whatever respect. > (1) Lim n (n --> oo) = 00 exists perfectly well in common calculus, False. n can approach infinity, but it's not a limit. > just as lim (in) (n-->oo) = 0 perfectly well exists. ?? ; don't even know what _that_ means. > (2) Lim sin(n) (n-->oo) DOES NOT exist. Agreed. > And both things are so because of MATHEMATICAL definitions. Period. > Nobody cares whether it seems nice, neat, intuitive or whatever to you > so or not, or whether one of your galore of brain halves tells you > this or that. It's just a figure of speech. What I really mean to say is: in whatever way you're looking at it, you will always find the same result. There's not just ONE (formalist) way of doing mathematics. I vehemently protest against that view. > (1) and (2): two different things...and it'd be nice if you could > FINALLY learn what's the difference between these two. The only one who has to learn here is _you_. Han de Bruijn === Subject: Re: -- Wrong limits do not commute >I accept the example as a confirmation of the fact that wrong limits do >not commute. Since the limits in the example are not >wrong this is nonsensical. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - William Hughes >According to common calculus, a limit that is infinite does not exist. >So your argument is nonsensical, always has been. >Han de Bruijn ********************************************************** Honestly and seriously, Han: stopping making such a an ass of > yourself...it already is embarrasing! I'm not ashamed in whatever respect. (1) Lim n (n --> oo) = 00 exists perfectly well in common calculus, False. n can approach infinity, but it's not a limit. just as lim (in) (n-->oo) = 0 perfectly well exists. ?? ; don't even know what that means. (2) Lim sin(n) (n-->oo) DOES NOT exist. Agreed. And both things are so because of MATHEMATICAL definitions. Period. > Nobody cares whether it seems nice, neat, intuitive or whatever to you > so or not, or whether one of your galore of brain halves tells you > this or that. It's just a figure of speech. What I really mean to say is: in whatever > way you're looking at it, you will always find the same result. There's > not just ONE (formalist) way of doing mathematics. I vehemently protest > against that view. (1) and (2): two different things...and it'd be nice if you could > FINALLY learn what's the difference between these two. The only one who has to learn here is you . Han de Bruijn- ************************************************* ***Sigh***...That last line in your post is, sadly, true, Han: the only one here that has to learn something is me. So let us begin: I give up. As someone once said: There's no blind man as the one who doesn't want to see. Tonio === Subject: Re: - conjugacy classes representatives generate a finite group ? crossedproduct asked at http://mathforum.org/kb/thread.jspa?messageID=6393717 and at http://mathforum.org/kb/thread.jspa?messageID=6394766 > Suppose G is a finite group and S is the set of > representatives of the distinct conjugacy classes of > G. How do we show that S, in fact, generates G, i.e. > G = ? G is the union of the G-conjugates of the subgroup , however a simple counting argument shows that no group is the union of the conjugates of a proper finite index subgroup. Indeed, your problem is equivalent to this covering by conjugate subgroups formulation. Covers of groups by subgroups is an interesting area of group theory. There are (somewhat easy) examples of infinite groups G that are the union of the G-conjugates of an infinite index subgroup. === Subject: Re: - conjugacy classes representatives generate a finite group ? Suppose G is a finite group and S is the set of > representatives of the distinct conjugacy classes > of > G. How do we show that S, in fact, generates G, > i.e. > G = ? G is the union of the G-conjugates of the subgroup > , > however a simple counting argument shows that no > group > is the union of the conjugates of a proper finite > index > subgroup. I know that in a _finite_ group G that G cannot be expressed as a union of all the conjugates of a proper subgroup H (it's easy to count the number of distinct elements in these conjugates). But how does this result extend to all groups G, and proper finite-index subgroups? That is, if G is any group, then why can't G be expressed as the union of G-conjugates of a proper finite-index subgroup? === Subject: Re: - conjugacy classes representatives generate a finite group ? > I know that in a _finite_ group G that G cannot be > expressed as a union of all the conjugates of a > proper subgroup H (it's easy to count the number of > distinct elements in these conjugates). But how does this result extend to all groups G, and > proper finite-index subgroups? That is, if G is any > y group, then why can't G be expressed as the union > of G-conjugates of a proper finite-index subgroup? This is simply the lattice isomorphism theorem. Alternatively, if N is a normal subgroup of G contained in H, then N is contained in every conjugate of H. Instead of counting elements of G, count cosets of N, where N is the G-core of H. === Subject: Constructing a matrix with a prescribed eigenspace Consider the 1-dimensional subspace S of R^3 with basis (5, -1, 1); that is, S consists of all elements of the form (5t, -t, t) for t in R. Suppose S is the eigenspace E_3 associated to the eigenvalue 3 for some matrix A. How would I go about finding A itself (or something similar to A)? I know that A must satisfy A(5, -1, 1) = (15, -3, 3) but this wouldn't be enough to determine A completely. Any suggestions? === Subject: Re: Constructing a matrix with a prescribed eigenspace <20244338.1221159658655.JavaMail.jakarta@nitrogen.mathforum.org>, > Consider the 1-dimensional subspace S of R^3 with basis (5, -1, 1); that is, > S consists of all elements of the form (5t, -t, t) for t in R. Suppose S is the eigenspace E_3 associated to the eigenvalue 3 for some > matrix A. How would I go about finding A itself (or something similar to A)? I know that A must satisfy A(5, -1, 1) = (15, -3, 3) but this wouldn't be > enough to determine A completely. What do you mean by determine A completely? You're not under the impression that A is unique, are you? Dave has given a good way to find lots of answers. Here's a pretty good way to find one answer: 0 0 15 0 3 0 3/5 0 0 Can you see why it works? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Constructing a matrix with a prescribed eigenspace > Consider the 1-dimensional subspace S of R^3 with > basis (5, -1, 1); that is, > S consists of all elements of the form (5t, -t, t) > for t in R. > Suppose S is the eigenspace E_3 associated to the > eigenvalue 3 for some > matrix A. > How would I go about finding A itself (or something > similar to A)? > Dave has given a good way to find lots of answers. > Here's a pretty > good way to find one answer: 0 0 15 > 0 3 0 > 3/5 0 0 Can you see why it works? > Hm, does it? The eigenspace associated to the eigenvalue 3 is 2-dimensional, not 1-dimensional. === Subject: Re: Constructing a matrix with a prescribed eigenspace <18975428.1221175050471.JavaMail.jakarta@nitrogen.mathforum.org>, > Consider the 1-dimensional subspace S of R^3 with > basis (5, -1, 1); that is, > S consists of all elements of the form (5t, -t, t) > for t in R. > Suppose S is the eigenspace E_3 associated to the > eigenvalue 3 for some > matrix A. > How would I go about finding A itself (or something > similar to A)? > > Dave has given a good way to find lots of answers. > Here's a pretty > good way to find one answer: > 0 0 15 > 0 3 0 > 3/5 0 0 > Can you see why it works? > Hm, does it? The eigenspace associated to the eigenvalue 3 is 2-dimensional, not > 1-dimensional. Right you are. OK, try this: The matrix 1 5 0 1 0 -5 0 0 0 obviously has a 1-dimensional nullspace generated by (5, -1, 1). Now just add 3 down the diagonal: 4 5 0 1 3 -5 0 0 3 -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Constructing a matrix with a prescribed eigenspace === Subject: Re: Constructing a matrix with a prescribed eigenspace > Consider the 1-dimensional subspace S of R^3 with > basis (5, -1, 1); that is, > S consists of all elements of the form (5t, -t, t) > for t in R. > Suppose S is the eigenspace E_3 associated to the > eigenvalue 3 for some > matrix A. > How would I go about finding A itself (or something > similar to A)? > I know that A must satisfy A(5, -1, 1) = (15, -3, > 3) but this wouldn't be > enough to determine A completely. What do you mean by determine A completely? You're > not under > the impression that A is unique, are you? I thought to myself at first that this A cannot be unique, so no. === Subject: Re: Constructing a matrix with a prescribed eigenspace > Consider the 1-dimensional subspace S of R^3 with basis (5, -1, 1); that is, S consists of all elements of the form (5t, -t, t) for t in R. Suppose S is the eigenspace E 3 associated to the eigenvalue æ3 for some matrix A. How would I go about finding A itself (or something similar to A)? I know that A must satisfy A(5, -1, 1) = (15, -3, 3) but this wouldn't be enough to determine A completely. Any suggestions? Choose two additional eigenvalue-eigenvector pairs, making sure that the three eigenvectors are linearly independent. Form a 3-by-3 diagonal matrix, Lambda, of the eigenvalues and a 3-by-3 matrix X whose columns are the eigenvectors in the same order as the eigenvalues are in Lambda. Then A = X * Lambda * X^(-1) Dave === Subject: Re: The Computable Reals (beta) > Now, and between me and you, I'd indeed claim that even irrational > sequences, and actually all computable sequences, are their in the > list, or rather a little step away: the transfinite step. > So are they there or not? Well, they're a little step away. What > does that mean, exactly? > The transfinite step. What is your mother tongue, ancient Greek? My mother tongue is English - how is that relevant? I don't understand > (at all!) what you think you mean by transfinite step, so I'm asking > for a _mathematical_ explanation. http://en.wikipedia.org/wiki/Transfinite_induction > I don't see any evidence that anyone > else understands what you mean either, so I can't see that the help > you are looking for is likely to come. Me neither. -LV Since julio is trying to prove something about a list, ordinary induction would be quite sufficient as he need not run into any non-successor ordinals in his quest. But neither ordinary or transfinite induction can prove false what has already been proved true beyond julio's powers to refute. === Subject: Re: The Computable Reals (beta) > Now, and between me and you, I'd indeed claim that even irrational > sequences, and actually all computable sequences, are their in the > list, or rather a little step away: the transfinite step. So are they there or not? Well, they're a little step away. æWhat > does that mean, exactly? The transfinite step. What is your mother tongue, ancient Greek? To call a transfinite step a little step seems a bit oxymoronic. Listing the rationals is the little part, and has been done many times and in many ways. Listing irrationals is the hard part that julio makes light or but cannot do. I was just hoping for some help in ***extending a finite-inductive > definition to the transfinite***. That's all, the rest is the usual > noise, as usual. Why should anyone who accepts that it has been proved impossible spend any time or energy trying to achieve it? === Subject: Re: The Computable Reals (beta) > Now, and between me and you, I'd indeed claim that even irrational > sequences, and actually all computable sequences, are their in the > list, or rather a little step away: the transfinite step. > So are they there or not? Well, they're a little step away. What > does that mean, exactly? > The transfinite step. What is your mother tongue, ancient Greek? My mother tongue is English - how is that relevant? I don't understand > (at all!) what you think you mean by transfinite step, so I'm asking > for a mathematical explanation. http://en.wikipedia.org/wiki/Transfinite induction Quote: Let P(.87) be a property defined for all ordinals .87. Suppose that whenever P(.89) is true for all .89 < .87, then P(.87) is also true. Then transfinite induction tells us that P is true for all ordinals. Yes, I understand that. (I don't think you do, of course.) I don't see what transfinite induction has to do with your attempts to generate the computable reals (Where we still don't really know what you mean by this) -- in particular, if you are going to use transfinite induction, you need to specify a property defined for all ordinals. Can you? I think that rather than using transfinite induction, it would be simpler for you to rely on the principle of manondulation, since you don't seem too bothered about convincing anyone else of anything. Whatever. Brian Chandler === Subject: Re: The Computable Reals (beta) > Now, and between me and you, I'd indeed claim that even irrational > sequences, and actually all computable sequences, are their in the > list, or rather a little step away: the transfinite step. > So are they there or not? Well, they're a little step away. What > does that mean, exactly? > The transfinite step. What is your mother tongue, ancient Greek? > My mother tongue is English - how is that relevant? I don't understand > (at all!) what you think you mean by transfinite step, so I'm asking > for a mathematical explanation. http://en.wikipedia.org/wiki/Transfinite induction Quote: > Let P(.87) be a property defined for all ordinals .87. Suppose that > whenever P(.89) is true for all .89 < .87, then P(.87) is also true. Then > transfinite induction tells us that P is true for all ordinals. Yes, I understand that. (I don't think you do, of course.) Wanderful. And, of course. > I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? For all n < w (w the limit ordinal), list(n) is the list of all the binary strings of length n (provable by finite induction). Now we want to *extend* to a list(w) that is the infinite list of all the binary strings of infinite length (length w). The reason why we are after this extension is that we are after an exception-free system, so we want our set of computables to be closed under all operations. Now, how to go to write down such extended case? Intuitively, it's clear how it looks like, at least in that the first entry must be (0), the last entry must be (1), and all in between is lexicographically ordered. Still: Maybe a limit case can work here? Or rather a definition rethought in terms of transfinite recursion? What would you think? > I think that rather than using transfinite induction, it would be > simpler for you to rely on the principle of manondulation, since you > don't seem too bothered about convincing anyone else of anything. I am not trying to convince anyone of anything, I was rather looking for some support if not collaboration. Eresia, now I shall suppose. > Whatever. -LV === Subject: Re: The Computable Reals (beta) > I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). Okay then. Since every list(n) for n < w is a countable list, can you show that list(w) is also countable? If not, then you can't claim that list(w) is an exmple that contradicts Cantor's theorem. > The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. If list(w) is a denumerable list, you're going to have to show what element immediately follows the first element (0) in the lexographical ordering, or at least how to compute it from its predecessor (0). Otherwise you won't be able to show that the list is denumerable. === Subject: Re: The Computable Reals (beta) > I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). Okay then. æSince every list(n) for n < w is a countable list, > can you show that list(w) is also countable? If not, then you can't claim that list(w) is an exmple that > contradicts Cantor's theorem. The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. If list(w) is a denumerable list, you're going to have to show what > element immediately follows the first element (0) in the lexographical > ordering, or at least how to compute it from its predecessor (0). > Otherwise you won't be able to show that the list is denumerable. We are talking about infinite strings in the binary alphabet. We can write the first string in the list as: 0...0, which is an infinite succession of zeros. We will rewrite, in compact form: (0). A more complicated example: the string 0110...00110 becomes, in compact form, 011(0)110. Now, the successor of (0) is (0)1. Actually, given the lexicographical order, the first few entries are (indexing is conventional throughout): 0+0: (0) 0+1: (0)1 0+2: (0)10 0+3: (0)11 0+4: (0)100 ... and so on. Similarly, we can enumerate backwards from the last entry: oo-0: (1) oo-1: (1)0 oo-2: (1)01 oo-3: (1)00 oo-4: (1)011 ... Actually, a much stronger property holds: we can enumerate backwards/ forwards from any rational starting point. For instance, let 01(10) (corresponding to the rational 5/12) be the starting point: An intermediate passage to explicit the mechanics of the enumeration (again, please note that index labels are conventional): ... (5/12)-4: 01(10)0110 (5/12)-3: 01(10)0111 (5/12)-2: 01(10)1000 (5/12)-1: 01(10)1001 (5/12)~0: 01(10)1010 (5/12)+1: 01(10)1011 (5/12)+2: 01(10)1100 (5/12)+3: 01(10)1101 (5/12)+4: 01(10)1110 ... The enumeration rewritten in most compact form: ... (5/12)-4: 01(10)0110 (5/12)-3: 01(10)0111 (5/12)-2: 01(10)00 (5/12)-1: 01(10)01 (5/12)+0: 01(10) (5/12)+1: 01(10)11 (5/12)+2: 01(10)1100 (5/12)+3: 01(10)1101 (5/12)+4: 01(10)1110 ... Incidentally: with reference to the thread on infinite binary strings, maybe it is worth noting that the above procedure is perfectly defined as far as infinite strings are taken as valid objects. We are still not enumerating the digits in this case, we straight enumerate the strings. The problem remains that irrational strings, that is the non- periodic strings, are not reachable by this procedure of enumeration from the rational points either (unless, again, in an infinite amout of steps), and that's -I guess- where sequences of digits have to come into the scene, and the existence of the corresponding infinite strings (infinite in the string sense of their representation being infinite) is still in predicatus. -LV === Subject: Re: The Computable Reals (beta) > We are talking about infinite strings in the binary alphabet. We can > write the first string in the list as: 0...0, which is an infinite > succession of zeros. Not in my book. Such a list, being sequentially ordered with both a first and a last, is necessarily finite. By standard mathematical definition, a string is a mapping from either the naturals or some finite initial segment of naturals to some set of characters, and any such string with a last character is the image of a finite initial segment of the naturals, and is, therefore , finite. Much garbage deleted. === Subject: Re: The Computable Reals (beta) > I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? > For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). Okay then. ?Since every list(n) for n < w is a countable list, > can you show that list(w) is also countable? If not, then you can't claim that list(w) is an exmple that > contradicts Cantor's theorem. > The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. If list(w) is a denumerable list, you're going to have to show what > element immediately follows the first element (0) in the lexographical > ordering, or at least how to compute it from its predecessor (0). > Otherwise you won't be able to show that the list is denumerable. We are talking about infinite strings in the binary alphabet. We can > write the first string in the list as: 0...0, which is an infinite > succession of zeros. Ah, well, there's a big problem here. In normal mathematics, the problem is that infinite means without end, you know, infinito, endless, and this means that while an infinite string has one end, where it starts, it doesn't have a second end where it stops, because it doesn't stop. So it is not possible to represent anything that normal mathematics calls an infinite succession of zeros by writing 0....0. > We will rewrite, in compact form: (0). A more > complicated example: the string 0110...00110 becomes, in compact > form, 011(0)110. Now, the successor of (0) is (0)1. So, again in normal mathematics, since there isn't a right end, while we could write (0) to mean 0..., we can't write a successor to 0..., because we would need to write a 1 at the end of the 0... that doesn't exist. But not only is the notation (0)1 not meaningful in normal mathematics (look, can I stop saying this? Everything I say relates to normal mathematics.)... Worse, it's explicit that there is no successor to 0. Why? Because the whole point of the reals is that they form a field, which means, amongst other things, that if a and b are reals, (a+b)/2 is also a real, which is between a and b. So two reals can never be adjacent, or one be the successor of the other. Why not save yourself the bother of all this - it's been done before. There must be 10,000 posts from or to Tony Orlow, who has done the same thing (at least to his own satisfaction) - why not try reading some of them? Brian Chandler === Subject: Re: The Computable Reals (beta) > The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. > If list(w) is a denumerable list, you're going to have to show what > element immediately follows the first element (0) in the lexographical > ordering, or at least how to compute it from its predecessor (0). > Otherwise you won't be able to show that the list is denumerable. > We are talking about infinite strings in the binary alphabet. We can > write the first string in the list as: 0...0, which is an infinite > succession of zeros. Ah, well, there's a big problem here. In normal mathematics, the > problem is that infinite means without end, you know, infinito, > endless, and this means that while an infinite string has _one_ end, > where it starts, it doesn't have a second end where it stops, because > it doesn't stop. So it is not possible to represent anything that > normal mathematics calls an infinite succession of zeros by writing > 0....0. We could recast infinite strings with both ends as infinite strings with the ends alternating at the first. For instance, 101(0)11011 could be mapped to 1 0 1 0 0 0 0 0 ... 1 1 0 1 1 1 0 ... that is, 1101100101010... I believe this would satisfy LV's notion of writing things and surely is acceptable in typical formulations. LV, is this rewriting acceptable to you? Does this capture everything you want to express with your strings? -paul- === Subject: Re: The Computable Reals (beta) Now, the successor of (0) is (0)1. So, again in normal mathematics, since there isn't a right end, while > we could write (0) to mean 0..., we can't write a successor to > 0..., because we would need to write a 1 at the end of the 0... > that doesn't exist. I don't think it is as clear cut as that. (0)1 can be given a very definite meaning (see the Hackenstrings posting for an example). (0)1 may not be the successor of anything (who knows, he does not define enough to have any real idea?) but you can define what it means to write 1 after an infinite number of zeros. Any collection of even unbounded strings can be given a precise and interesting numerical meaning and the result is, in some sense, normal mathematics. It may not be mainstream in that these extensions to the reals don't have (is seems) a great many uses, but they do come up all time in some fields (game theory being the obvious one). Of course, all such formalisms are susceptible to counting and diagonal arguments. Julio's problems are very general ones: (a) He leaves too many things undefined, so much of the discussion is reduced to guesswork and hand-waving. For example, if (0)1 is the successor of (0), what happened to (0)01, or even (0)(0)1? (b) He does not state what parts of sound, well-established arguments run counter to his (supposedly new) framework. If an establish proof is wrong or does not apply, you should be able to say exactly where and what is wrong or does not apply. (c) He does not do research to learn what other have done. Some of the things he is struggling with have been sorted out already. For example, how to give meaning to numbers like 0.(10)1. Anyway, what I wanted to say is that attaching meanings to strings like (0)1 is the least of his problems whereas defining successor is at the very heart of it. If he did that we could all go home... -- Ben. === Subject: Re: The Computable Reals (beta) > I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? > For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). > Okay then. Since every list(n) for n < w is a countable list, > can you show that list(w) is also countable? > If not, then you can't claim that list(w) is an exmple that > contradicts Cantor's theorem. > The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. > If list(w) is a denumerable list, you're going to have to show what > element immediately follows the first element (0) in the lexographical > ordering, or at least how to compute it from its predecessor (0). > Otherwise you won't be able to show that the list is denumerable. We are talking about infinite strings in the binary alphabet. We can > write the first string in the list as: 0...0, which is an infinite > succession of zeros. Ah, well, there's a big problem here. In normal mathematics, the > problem is that infinite means without end, you know, infinito, > endless, and this means that while an infinite string has _one_ end, > where it starts, it doesn't have a second end where it stops, because > it doesn't stop. So it is not possible to represent anything that > normal mathematics calls an infinite succession of zeros by writing > 0....0. That's a problem for normal mathematics maybe, and even that I don't buy anymore. In any case, the procedure I have outlined above is perfectly defined (we assume infinite strings as primitive objects there), as well as makes clear a distinction that indeed seems lost in your normal mathematics, the distinction between the sequences of digits and the infinite strings as such. As well as, on the more informal level, I have already said many times in this thread that an infinite list or string or infinite whatever, as far as it is not rather the limit of some sequence, it must have both ends, despite it's length is infinite. It's an unbounded sequence that has no right end, an infinite string has still both ends. It's the whole point of transfinite induction actually: the limit, or lambda case. Surely, another very basic notion that will take ~1000 post to get by the normal mathematician, isn't it? -LV === Subject: Re: The Computable Reals (beta) > Ah, well, there's a big problem here. In normal mathematics, the > problem is that infinite means without end, you know, infinito, > endless, and this means that while an infinite string has _one_ end, > where it starts, it doesn't have a second end where it stops, because > it doesn't stop. So it is not possible to represent anything that > normal mathematics calls an infinite succession of zeros by writing > 0....0. That's a problem for normal mathematics maybe, and even that I don't > buy anymore. Julio admits that everything he does is abnormal In any case, the procedure I have outlined above is perfectly defined Not until you have displayed without ellipsis one of your infinite strings. In standard mathematics, a sequential order must be order isomorphic to some subset of the set of integers, i.e., a set ordered so that BETWEEN any two given elements there are, at most, a finite number of elements. Since your notion of an infinite sequence differs, perhaps you could present us with a model satisfying your notion. Absent your presenting some equally clear model of julio-sequences. We must continue to hold them to be self-contradictory. > (we assume infinite strings as primitive objects there) The characters in YOUR 'infinite strings' are not sequentially ordered, as they must be for representation of real numbers. , as well as > makes clear a distinction that indeed seems lost in your normal > mathematics, the distinction between the sequences of digits and the > infinite strings as such. As well as, on the more informal level, I > have already said many times in this thread that an infinite list or > string or infinite whatever, as far as it is not rather the limit of > some sequence, it must have both ends, despite it's length is > infinite. It's an unbounded sequence that has no right end, an > infinite string has still both ends. Not in this world. > It's the whole point of > transfinite induction actually: the limit, or lambda case. Double talk! Surely, another very basic notion that will take ~1000 post to get by > the normal mathematician, isn't it? To get it by a normal human, including normal mathematicians, will take infinitely many posts. === Subject: Re: The Computable Reals (beta) > I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? > For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). > Okay then. Since every list(n) for n < w is a countable list, > can you show that list(w) is also countable? > If not, then you can't claim that list(w) is an exmple that > contradicts Cantor's theorem. > The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. > If list(w) is a denumerable list, you're going to have to show what > element immediately follows the first element (0) in the lexographical > ordering, or at least how to compute it from its predecessor (0). > Otherwise you won't be able to show that the list is denumerable. > We are talking about infinite strings in the binary alphabet. We can > write the first string in the list as: 0...0, which is an infinite > succession of zeros. Ah, well, there's a big problem here. In normal mathematics, the > problem is that infinite means without end, you know, infinito, > endless, and this means that while an infinite string has _one_ end, > where it starts, it doesn't have a second end where it stops, because > it doesn't stop. So it is not possible to represent anything that > normal mathematics calls an infinite succession of zeros by writing > 0....0. That's a problem for normal mathematics maybe, and even that I don't > buy anymore. In any case, the procedure I have outlined above is perfectly defined > (we assume infinite strings as primitive objects there), as well as > makes clear a distinction that indeed seems lost in your normal > mathematics, the distinction between the sequences of digits and the > infinite strings as such. As well as, on the more informal level, I > have already said many times in this thread that an infinite list or > string or infinite whatever, as far as it is not rather the limit of > some sequence, it must have both ends, despite it's length is > infinite. So if I were to start labelling the digits in your infinite string with integers, starting with 0 at the left end, what happens? Do I ever reach the right end? In which case what does the integer I label it with look like? Or do I somehow get stuck in the middle, in a sort of twilight zone I somehow can't get past? > It's an unbounded sequence that has no right end, an > infinite string has still both ends. It's the whole point of > transfinite induction actually: the limit, or lambda case. It is? What is the lambda case? Would this be Church's lambda calculus? > Surely, another very basic notion that will take ~1000 post to get by > the normal mathematician, isn't it? Are you asking me for an opinion? It appears to be the usual crank babble, actually. What _really_ is awesome is the extent to which everything you have said has been said before. And will I suppose be said again, ad infinitum. Brian Chandler === Subject: Re: The Computable Reals (beta) For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. Still: Maybe a limit case can work here? Or rather a > definition rethought in terms of transfinite recursion? What would you I'm afraid I still don't follow what you're trying to achieve, but never mind! I just thought if you've not met them before, perhaps you might be interested in the Surreal numbers invented by John Conway I believe. Maybe they'll give you some ideas, but if not they're quite interesting and fun in their own right... The reason I mention them is that their nature is that they are constructed recursively (or you might say inductively) - there are numbers born on days 1, 2, 3, 4,... i.e. the numbers that are constructed in successive generations from the younger numbers. The numbers born on finite days are just diadic rationals of the form m/2^n, and you would perhaps think the remaining real numbers are just a step away... and sure enough on day w (omega) all the remaining reals are constructed, along with the infinite w and infinitesimal 1/w. Statements about Surreal Numbers are typically proved using (transfinite) induction, which is natural given how the numbers are built up constructively from simpler numbers already constructed. Anyway, I thought this might be related to what you're trying to do... Mike. === Subject: Re: The Computable Reals (beta) I just thought if you've not met them before, perhaps you might be > interested in the Surreal numbers invented by John Conway I believe. do... I thought so too but the posted link to Hackenstrings[1] (which in many ways are even more similar) drew no comments from Julio. It too references the links to surreal numbers. [1] http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ -- Ben. === Subject: Re: The Computable Reals (beta) http://en.wikipedia.org/wiki/Transfinite induction Quote: > Let P(?) be a property defined for all ordinals ?. Suppose that > whenever P(?) is true for all ? < ?, then P(?) is also true. Then > transfinite induction tells us that P is true for all ordinals. Yes, I understand that. (I don't think you do, of course.) Wanderful. And, of course. I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Fine. No need for any induction - just consider any scheme of representing the (finite) set of binary strings of length n, and prove directly that this scheme works for any n. That's the easy bit. Now do you see how the definition of transfinite induction goes? It says Suppose that whenever P(a) is true for all a < w, then P(w) is also true. This is the *hypothesis* of transfinite induction. That means you have to *prove* it, *before* you can use transfinite induction. So you need to show that your list(w) (is that right?) is in fact a list of all infinite binary sequences, and this, of course, is what lots of people have told you is impossible. > Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Why do you keep saying we? Is there anyone else except you in this enterprise? > Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. Still: Maybe a limit case can work here? Or rather a > definition rethought in terms of transfinite recursion? What would you > think? I think that there is no list (if this has the standard meaning) of all (infinite) binary sequences, because there is a very simple proof that the existence of this list would lead to a contradiction. There is also an informal demonstration of why your intuition isn't going to work. Consider the set of binary strings of some (finite) length p. It's easy to see that they can be arranged in a list by lining them up, and using lexicographic order from the left end (so all the 0...xyz strings come before the 1...zyx strings), while incrementing from the right end (so for any string pqr...0 the next string is pqr...1 and so on). But we can do this because the strings have two ends. However, infinite strings have only one end. So we can arrange them successively by putting the end at the right, like this: ...000 ...001 ...010 ...011 and so on. Or we can order them lexicographically by putting the end at the left, like this: 000... . . . 010... . . . 100... and so on. But in the lexicographic ordering there are infinitely big gaps between any successive subsequences on the left (shown by the spaced dots . . .) This is an informal demonstration of the problem, not a proof, but it's a well-rehearsed one, because as I think you have also been told, more or less nothing you have said is new at all. You can search through the archives for worms, or indeed 000000 will throw up a lot of this stuff. > I think that rather than using transfinite induction, it would be > simpler for you to rely on the principle of manondulation, since you > don't seem too bothered about convincing anyone else of anything. I am not trying to convince anyone of anything, I was rather looking > for some support if not collaboration. Eresia, now I shall suppose. What form would support take? Lots of people have tried to help you with lots of things, but you haven't exactly responded in a very cooperative fashion, which is why you are largely reduced to futile bantering with Virgil. I don't know his motivation, but he will keep you going indefinitely if you choose to engage him. HTH Brian Chandler === Subject: Re: The Computable Reals (beta) Right. It just occurs to me you may have a genuine problem reading the Wikipedia definition. >http://en.wikipedia.org/wiki/Transfinite_induction > Quote: > Let P(a) be a property defined for all ordinals a. Suppose that > whenever P(a) is true for all a < b, then P(b) is also true. Then > transfinite induction tells us that P is true for all ordinals. If you carelessly read just the first two sentences, you might misunderstand this to mean: -----------NOT the principle of transfinite induction------------ (1) Let P(a) be a property defined for all ordinals a. (2) *IF* P(a) is true for all a < b,) **THEN IT IS THE CASE THAT** P(b) is also true. ------------------------------------------------------------------------- But this is not what it means. **[footnote] What it means is: --------------principle of transfinite induction------------------ (1) Let P(a) be a property defined for all ordinals a. *IF* (2) P(a) being true for all a < b always implies that P(b) is also true. *THEN* (by the principle of transfinite induction) (3) P is true for all ordinals. ------------------------------------------------------------------------- HTH Brian Chandler [footnote] ** It couldn't be, because of elementary counterexamples, like let P(a) be the property a < b. We are very familiar though with this (false) implication scheme, because cranks use it all the time. === Subject: Re: The Computable Reals (beta) > Now, and between me and you, I'd indeed claim that even irrational > sequences, and actually all computable sequences, are their in the > list, or rather a little step away: the transfinite step. > So are they there or not? Well, they're a little step away. What > does that mean, exactly? > The transfinite step. What is your mother tongue, ancient Greek? > My mother tongue is English - how is that relevant? I don't understand > (at all!) what you think you mean by transfinite step, so I'm asking > for a _mathematical_ explanation. >http://en.wikipedia.org/wiki/Transfinite_induction Quote: > Let P(.87) be a property defined for all ordinals .87. Suppose that > whenever P(.89) is true for all .89 < .87, then P(.87) is also true. Then > transfinite induction tells us that P is true for all ordinals. Yes, I understand that. (I don't think you do, of course.) Wanderful. And, of course. > I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. Still: Maybe a limit case can work here? Or rather a > definition rethought in terms of transfinite recursion? What would you > think? I would think that GIGO is as true of transfinite induction as of anything else. > I think that rather than using transfinite induction, it would be > simpler for you to rely on the principle of manondulation, since you > don't seem too bothered about convincing anyone _else_ of anything. I am not trying to convince anyone of anything, I was rather looking > for some support if not collaboration. Eresia, now I shall suppose. Why should anyone offer either support or collaboration unless they are convinced that your project has some chance of succeeding. And being so convinced would limit potential supporters to those who reject Cantor, the vast majority of whom appear to be incompetent to offer anything like effective support. === Subject: Re: The Computable Reals (beta) > Now, and between me and you, I'd indeed claim that even irrational > sequences, and actually all computable sequences, are their in the > list, or rather a little step away: the transfinite step. > So are they there or not? Well, they're a little step away. What > does that mean, exactly? > The transfinite step. What is your mother tongue, ancient Greek? > My mother tongue is English - how is that relevant? I don't understand > (at all!) what you think you mean by transfinite step, so I'm asking > for a mathematical explanation. >http://en.wikipedia.org/wiki/Transfinite induction Quote: > Let P(.87) be a property defined for all ordinals .87. Suppose that > whenever P(.89) is true for all .89 < .87, then P(.87) is also true. Then > transfinite induction tells us that P is true for all ordinals. Yes, I understand that. (I don't think you do, of course.) Wanderful. And, of course. I don't see > what transfinite induction has to do with your attempts to generate > the computable reals (Where we still don't really know what you mean > by this) -- in particular, if you are going to use transfinite > induction, you need to specify a property defined for all ordinals. > Can you? For all n < w (w the limit ordinal), list(n) is the list of all the > binary strings of length n (provable by finite induction). Now we want > to *extend* to a list(w) that is the infinite list of all the binary > strings of infinite length (length w). The reason why we are after > this extension is that we are after an exception-free system, so we > want our set of computables to be closed under all operations. Now, > how to go to write down such extended case? Intuitively, it's clear > how it looks like, at least in that the first entry must be (0), the > last entry must be (1), and all in between is lexicographically > ordered. Still: Maybe a limit case can work here? Or rather a > definition rethought in terms of transfinite recursion? What would you > think? BTW, there is also another path still to investigate: I think my construction might also be a step away from the p-adics, because the list itself defines the needed metric. The p-adics is an excellent target, as they have amazing properties with a huge impact on analysis and calculus too. The problem here is that the definitions I have found on the web are quite convoluted, so again I am lost as to how to fill in the gaps. The overall dream might be to get the computables and the p-adics at once, with a an interval-based exception-free system that exhausts computability. (For how improper, at least I hope you get what is my basic idea and goal.) -LV === Subject: Re: The Computable Reals (beta) > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > What julio seems to be carefully ignoring is that a set having a > sequence of numbers converging to some value does not mean that the set > contains that value. > So, for example, a sequence of rationals converging to the Champernowne > number does not itself contain the Champernowne number. > Exactly. > Julio, are you saying that your list of rationals that progressively > approximate the Champernowne-like irrational actually contains > an infinite-length binary fraction? æBecause I don't see one in > the list - all I see are finite-length rational fractions. > // 0.1 10 11 100 101 110 111 1000 ... (Champernowne) > c 00) æ0.() æ<=> æ< 0 , 0 , 1 , 0 , 1 > æ=> æ0+0/1+(0/1)/1 æ= > 0+0/1 æ=~ æ0 > c 01) æ0.(1) æ<=> æ< 0 , 0 , 1 , 1 , 1 > æ=> æ0+0/1+(1/1)/1 æ= > 0+1/1 æ=~ æ1 > c 02) æ0.(11) æ<=> æ< 0 , 0 , 1 , 3 , 3 > æ=> æ0+0/1+(1/1)/1 æ= > 0+1/1 æ=~ æ1 > c 03) æ0.(110) æ<=> æ< 0 , 0 , 1 , 6 , 7 > æ=> æ0+0/1+(6/7)/1 æ= > 0+6/7 æ=~ æ0.85714 > c 04) æ0.(1101) æ<=> æ< 0 , 0 , 1 , 13 , 15 > æ=> æ0+0/1+(13/15)/1 > = æ0+13/15 æ=~ æ0.86667 > c 05) æ0.(11011) æ<=> æ< 0 , 0 , 1 , 27 , 31 > æ=> æ0+0/1+(27/31)/1 > = æ0+27/31 æ=~ æ0.87097 > etc. > First, I have already said -- NOW TRICE -- that that's another > Virgil's INVENTION to inject noise into noise. Namely, what Virgil is > claiming I am ignoring is something I simply have NEVER said or > implied. And that is even apparent by just reading! > I hope that has finally got to the reader's attention, as I won't come > back to it for a forth time. > Now, and between me and you, I'd indeed claim that even irrational > sequences, and actually all computable sequences, are their in the > list, or rather a little step away: the transfinite step. > How to proof this how wouldn't know, as I don't quite know how to > formulate such a statement about infinite sequences in the very first > place. Still it is clear to me that we could at least leverage a > notion of limit here to define an extended case for transfinite > induction. > I am also looking into definitions by transfinite recursion, but here > it is more complicated as I have not yet been able to find any paper > on the web on the basic mechanics of it, and no help is coming from > this thread, quite the opposite. > But all of this we already know, don't we? > -LV > æ The thread has been helpful, it's told you that you cannot find such > a proof because a Turing machine only keeps an unbounded but finite > sequence on its output tape at all times. If you could derive a proof > that e.g. exactly 0.01(01) is in the set, you'd have a contradiction, > seeing how the Turing machine would then, at some point, store an > infinite number of symbols. This doesn't rely on Cantor's argument at > all (which was introduced for generality and useful for the more > complicated case of periodicals in the output). I think that you > should perhaps disprove these statements before spending too much on > that proof. > You simply keep missing the point, as it is since day one. > The above number is a rational number, and it's ALREADY covered by the > definition over FINITE INDUCTION I have given in version beta. Namely: > æ æ0.01(01) <=> < 0, 1, 4, 1, 3 > => 0+1/4+(1/3)/4 = 0+1/3 = 1/3 > What remains uncovered is the non-periodic (or, infinite-periodic) > expansions, the irrational numbers. Will you ever realize the > difference? > Then, as to the contradiction you seem to imply with Turing machines, > that just confirms that you don't know what you are talking about. > -LV > æ If you cannot get the point across, there are two people to blame, > not one. I have asked for clarifications and formal definitions, some > of which you have answered well, some of wish leaves me guessing, and > some simply ignored. > æ I thought your argument so far was that the Turing machine > TM may produce this set through recursion, or iteration, it doesn't > matter, it yields the enumerable set of finite binary sequences > interpretable as numbers in [0,1) which we've been talking about. (Of > course, you haven't said this explicitly, I have to guess to make it > formal.) > æ You then argue that you can produce an numbering (e.g. the above > <0,1,4,1,3>) for each rational in terms of your lists (i.e. the > enumerable set) such that using this number alone, you can produce an > arbitrarly good approximation. Assuming these steps can be formally > worked out (which appears easy), this shows that the rational is > working tape and, given the precision parameters, evaluates this > numbering to given precision. > æ What still doesn't follow, is that the rational is in the set, i.e. > none of your lists ever contains 0.01(01), i.e. the TM can never write > 0.01(01) on its tape, hence my objection. There is no objection: you just don't get it even after telling it > hundred times; and this, on a side, is not even a problem of mine, > while, on the other, is apparently not enough to stop you from again > putting words in my mouth. For what *I* am concerned, your feedback -- > as many others here -- is just and simply undesired, and actually a > SYSTEMATICALLY made IMPOSSIBLE. Too much talk, too much work. So now and again, just get lost. æ If you in the future try to get your work published... Neither you nor the hierarchy of asses you may be kissing have anything to do with true science and research. -LV === Subject: symmetries of differential equations does anyone know how goes the proof of Lie's theorem: An ordinary differential equation of order n has at most n+4 symmetries Or can anyone give a reference for this? === Subject: Re: probability question > On Sep 11, 12:09æam, The World Wide Wade On Sep 9, 11:51æpm, The World Wide Wade On Sep 9, 9:07æpm, The World Wide Wade I have a probablity question: > Suppose we throw a dice 100 times. What is the probability > that at > least 1 number appears exactly once. > I would be grateful for your help, > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it > would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =~ 8.4 * 10^-8 > Am I right? æWhat do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) > No, in fact you're low. Think about just the probability p_1 of > getting exactly one 1 in 100 rolls. The 1 can go into 100 slots, > each > time with probability 1/6. Once the 1 is placed, the other 99 slots > are randomly chosen from {2, ..., 6}. So p_1 = 100*(1/6)*(5/6)^99. > so the answer is .....1.4*E-8- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - > Yes,..the factor of 100 is needed. æ 1.4E-6 æThere are a hundred > ways ...so 100 (1/6)(5/6)^99 > The OP asked about *at least* one number appears only once, so I > think > we've covered all the basis with this answer. æHow the 99 are > distributed among the 5 remaining numbers, we could care less. > No certainly not. What about a 2 on the first roll and all 1's > thereafter? See my other post in this thread.- Hide quoted text - > - Show quoted text - > I don't understand. æThat case is covered by the 100*(1/6)*(5/6)^99 > calculation. First, 100*(1/6)*(5/6)^99 = 0.0000002414934694 can't be the final > answer, because it is the probability that a 1 occurs exactly once. > (It is also the probability that a 2 occurs exactly once, etc.) So how > could that be the answer? Hmmm....I think the OP wanted the calculation that a particular > number,x, appears exactly once. Not a particular number - some number > a)There are 100 slots that x can occur, thus a factor of 100 > b)The probability that x will occur at that slot is 1/6 > c)The probability that x will NOT occur at the other 99 slots is > (5/6)^99 > d)There are 6 possibilities for x Thus a*b*c*d= 100* (1/6)*[(5/6)^99]*6 = 1.4E-6 (an answer that Robert > Israel also gets) You only think you got the answer Robert and I did. Did you ever consider why that calculation is much more involved than yours? Your answer: 0.0000014489608167 Right answer: 0.00000144896079386 They're close numerically, but there's quite a bit of difference conceptually. Your answer is too high, because you are double counting. For example you are counting the rolls 1, 2, then 98 3's twice: once in the 1-computation and once in the 2-computation. > Second, how did you come up with 1.4E-6? Third, as I said earlier Let E_k be the event that k appears only > once in 100 rolls. You want P(E_1 U ... U E_6). Do you understand why > this is true? I don't understand your notation. It's completely standard in probability theory. > Fourth, I calculated P(E_1 U ... U E_6) using inclusion-exclusion. > This problem is more intricate than I think you appreciate. I don't think it's a complicated problem...true. Where is my flaw in > my calculation involving a,b,c,d above? > æOnce we have calculated the probability of having a > particular number appearing only once, the multiplicity of the rest of > the 99 doesn't matter. That statement makes no sense to me. The OP wanted to know What is the probability that at > least 1 number appears exactly once. Once that number has been > determined, the other 99 doesn't matter. If 1 appears once, it > doesn't matter if 2 (or 3,4,5,6) appears only once or not at all or 99 > times. We may have a diference in interpreting the OP's question. > æIf the OP had asked for the probability of a > number appearing only once and then gave conditions on the rest of the > 99 then we would further work to do. æBut the OP didn't say that each > number had to appear at least once. I think you are confused, but I can't put my finger on exactly what.- Hide > quoted text - - Show quoted text - === Subject: Re: probability question > I have a probablity question: > Suppose we throw a dice 100 times. What is the probability > that at > least 1 number appears exactly once. > I would be grateful for your help, > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it > would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =~ 8.4 * 10^-8 > Am I right? æWhat do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) > No, in fact you're low. Think about just the probability p 1 of > getting exactly one 1 in 100 rolls. The 1 can go into 100 slots, > each > time with probability 1/6. Once the 1 is placed, the other 99 slots > are randomly chosen from {2, ..., 6}. So p 1 = 100*(1/6)*(5/6)^99. > so the answer is .....1.4*E-8- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - > Yes,..the factor of 100 is needed. æ 1.4E-6 æThere are a hundred > ways ...so 100 (1/6)(5/6)^99 > The OP asked about *at least* one number appears only once, so I > think > we've covered all the basis with this answer. æHow the 99 are > distributed among the 5 remaining numbers, we could care less. > No certainly not. What about a 2 on the first roll and all 1's > thereafter? See my other post in this thread.- Hide quoted text - > - Show quoted text - > I don't understand. æThat case is covered by the 100*(1/6)*(5/6)^99 > calculation. > First, 100*(1/6)*(5/6)^99 = 0.0000002414934694 can't be the final > answer, because it is the probability that a 1 occurs exactly once. > (It is also the probability that a 2 occurs exactly once, etc.) So how > could that be the answer? Hmmm....I think the OP wanted the calculation that a particular > number,x, appears exactly once. Not a particular number - some number a)There are 100 slots that x can occur, thus a factor of 100 > b)The probability that x will occur at that slot is 1/6 > c)The probability that x will NOT occur at the other 99 slots is > (5/6)^99 > d)There are 6 possibilities for x Thus a*b*c*d= 100* (1/6)*[(5/6)^99]*6 = 1.4E-6 æ (an answer that Robert > Israel also gets) You only think you got the answer Robert and I did. Did you ever > consider why that calculation is much more involved than yours? Your answer: æ0.0000014489608167 > Right answer: 0.00000144896079386 They're close numerically, but there's quite a bit of difference > conceptually. Your answer is too high, because you are double > counting. For example you are counting the rolls 1, 2, then 98 3's > twice: once in the 1-computation and once in the 2-computation. > Second, how did you come up with 1.4E-6? > Third, as I said earlier Let E k be the event that k appears only > once in 100 rolls. You want P(E 1 U ... U E 6). Do you understand why > this is true? I don't understand your notation. It's completely standard in probability theory. > Fourth, I calculated P(E 1 U ... U E 6) using inclusion-exclusion. > This problem is more intricate than I think you appreciate. I don't think it's a complicated problem...true. æWhere is my flaw in > my calculation involving a,b,c,d above? > æOnce we have calculated the probability of having a > particular number appearing only once, the multiplicity of the rest of > the 99 doesn't matter. > That statement makes no sense to me. The OP wanted to know æWhat is the probability that at > least 1 number appears exactly once. æOnce that number has been > determined, the other 99 doesn't matter. æIf 1 appears once, it > doesn't matter if 2 (or 3,4,5,6) appears only once or not at all or 99 > times. æWe may have a diference in interpreting the OP's question. > æIf the OP had asked for the probability of a > number appearing only once and then gave conditions on the rest of the > 99 then we would further work to do. æBut the OP didn't say that each > number had to appear at least once. > I think you are confused, but I can't put my finger on exactly what.- Hide > quoted text - > - Show quoted text -- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - Looks like you're right. I was double counting. Sorry I wasted your time. === Subject: Re: probability question > On Sep 11, 3:21æpm, The World Wide Wade Suppose we throw a dice 100 times. What is the probability > that at > least 1 number appears exactly once. > I would be grateful for your help, > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it > would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =~ 8.4 * 10^-8 > Am I right? æWhat do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) > No, in fact you're low. Think about just the probability p_1 of > getting exactly one 1 in 100 rolls. The 1 can go into 100 slots, > each > time with probability 1/6. Once the 1 is placed, the other 99 slots > are randomly chosen from {2, ..., 6}. So p_1 = 100*(1/6)*(5/6)^99. > so the answer is .....1.4*E-8- Hide quoted text - > Yes,..the factor of 100 is needed. æ 1.4E-6 æThere are a hundred > ways ...so 100 (1/6)(5/6)^99 > The OP asked about *at least* one number appears only once, so I > think > we've covered all the basis with this answer. æHow the 99 are > distributed among the 5 remaining numbers, we could care less. > No certainly not. What about a 2 on the first roll and all 1's > thereafter? See my other post in this thread.- Hide quoted text - > I don't understand. æThat case is covered by the 100*(1/6)*(5/6)^99 > calculation. > First, 100*(1/6)*(5/6)^99 = 0.0000002414934694 can't be the final > answer, because it is the probability that a 1 occurs exactly once. > (It is also the probability that a 2 occurs exactly once, etc.) So how > could that be the answer? > Hmmm....I think the OP wanted the calculation that a particular > number,x, appears exactly once. > Not a particular number - some number > a)There are 100 slots that x can occur, thus a factor of 100 > b)The probability that x will occur at that slot is 1/6 > c)The probability that x will NOT occur at the other 99 slots is > (5/6)^99 > d)There are 6 possibilities for x Thus > a*b*c*d= 100* (1/6)*[(5/6)^99]*6 = 1.4E-6 æ (an answer that Robert > Israel also gets) > You only think you got the answer Robert and I did. Did you ever > consider why that calculation is much more involved than yours? > Your answer: æ0.0000014489608167 > Right answer: 0.00000144896079386 > They're close numerically, but there's quite a bit of difference > conceptually. Your answer is too high, because you are double > counting. For example you are counting the rolls 1, 2, then 98 3's > twice: once in the 1-computation and once in the 2-computation. > Second, how did you come up with 1.4E-6? > Third, as I said earlier Let E_k be the event that k appears only > once in 100 rolls. You want P(E_1 U ... U E_6). Do you understand why > this is true? > I don't understand your notation. > It's completely standard in probability theory. > Fourth, I calculated P(E_1 U ... U E_6) using inclusion-exclusion. > This problem is more intricate than I think you appreciate. > I don't think it's a complicated problem...true. æWhere is my flaw in > my calculation involving a,b,c,d above? > æOnce we have calculated the probability of having a > particular number appearing only once, the multiplicity of the rest of > the 99 doesn't matter. > That statement makes no sense to me. > The OP wanted to know æWhat is the probability that at > least 1 number appears exactly once. æOnce that number has been > determined, the other 99 doesn't matter. æIf 1 appears once, it > doesn't matter if 2 (or 3,4,5,6) appears only once or not at all or 99 > times. æWe may have a diference in interpreting the OP's question. > æIf the OP had asked for the probability of a > number appearing only once and then gave conditions on the rest of the > 99 then we would further work to do. æBut the OP didn't say that each > number had to appear at least once. > I think you are confused, but I can't put my finger on exactly what.- Hide > quoted text - Looks like you're right. I was double counting. Sorry I wasted your > time. I do not speak for Wade. He may agree with me. Supposing you know for a certainty that Wade's method is correct, then the time is not wasted. To cement your understanding look up the principal of inclusion and exclusion, and do some exercises with it. Do a web search. You will find many excellent introductions such as which I turned up in a minute. The PIE is a very powerful method. Once you familiarize yourself with it you will easily solve many questions that previously were difficult. -- Michael Press === Subject: Re: Why does everyone do it? > so all noncomputable reals are equal? > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > galathaea: prankster, fablist, magician, liar > Depends on how uncomputable they are. > An uncomputable that is computably negative shouldn't ever equal one > which is computably positive. > you're right > except that since there is no finite specification available > with which to extract such computable properties > the collection of such examples is null > I do not see that at all. > That a number is not computable does not mean that one can know nothing > at all about it, but only that it cannot be approximated to an > ARBITRARY degree of precision. There is nothing theoretically impossible > about knowing that a number can be approximated to within some fixed > positive epsilon but no further, as when say, one is given a number > between 1 and 2, but with no other properties. > can you give me an example of such a number? There are lots of them, pick one for yourself. how would i know that i've picked > one of these noncomputable numbers i am talking about > and not e/2 or pi/3 or whatever? just given a number chosen from [1, 2] doesn't guarantee that i notice that you have not given a response to the same rigor you hold others to if it wasn't apparent these are serious questions/points in effect this is one of the points that many who have been labelled cranks have made concerning lists and countability and diagonalisation only it is the form i prefer in that i think it better separates the metamathematical premises from the mathematical leibnizian identity is a foundation of the critique in that the semantics of what properties are is fundamental to what mathematics can talk about if all noncomputable reals are not equal in what sense can one apply this idea of identity? to paraphrase bell it's all about the speakable and unspeakable in mathematics -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: how to solve homogeneous linear equations with variable coefficients I have a question based on homogeneous linear system. a11f11(w)+a12f12(w)...+a1if1i(w)=0 a21f21(w)+a22f22(w)...+a2if2i(w)=0 .. ai1fi1(w)+ai2fi2(w).....+aiifii(w)=0 i equations, fij is the coefficient, however it is a function of w. How to solve aij? f(w) relations are known as polynomial fij=c1w+c2w^2+c3w^3.... I used det(fij(w))=0 to get w first, then obtain fij(w) then get aij, which is not very efficient. Any other method? Especially since Mathematica can only process 10x10 symbolic matrix, is there any way to solve this with Mathematica? thank you! === Subject: Again on Lobachevskij the equations of the geodesics computed by maple for the Lobachevskij plane (semi-plane (u,v) with v > 0) are diff(u(t), t, t)-2*diff(u(t), t)*diff(v(t), t)/v = 0, diff(v(t), t, t)+diff(u(t), t)2/v-diff(v(t), t)2/v = 0. They're even available in the help for geodesics. I don't understand why the geodesic line u = 0, v = t, t > 0 does not verify the system. === Subject: Re: Again on Lobachevskij u=0 satisfies the first . v=t is not geodesic . === Subject: Re: Again on Lobachevskij > u=0 satisfies the first . > v=t is not geodesic . u=0, v=t, t>0 is a vertical straight line and IS a geodesic line for sure. Bye, Nicola === Subject: Re: Again on Lobachevskij [David Hilbert] the equations of the geodesics computed by maple for > the Lobachevskij plane (semi-plane (u,v) with v > 0) are > diff(u(t), t, t)-2*diff(u(t), t)*diff(v(t), t)/v = 0, diff(v(t), t, t)+diff(u(t), t)2/v-diff(v(t), t)2/v = 0. They're even available in the help for geodesics. > I don't understand why the geodesic line > u = 0, v = t, t > 0 does not verify the system. u=0 satisfies the first . > v=t is not geodesic . However, u(t) = 0 v(t) = A*exp(b*t) satisfies both equations, and is the same curve as the previous one, just parametrized differently. Jim Burns === Subject: Re: Again on Lobachevskij > However, u(t) = 0 > v(t) = A*exp(b*t) satisfies both equations, and is the same curve > as the previous one, just parametrized differently. Hi Jim, thank you. But this means that the geodesics equations are not valid for any parametrization? Which kind of parametrization one has to use to check if a given curve is a geodesics line? Bye, Nicola === Subject: Re: Again on Lobachevskij > However, > u(t) = 0 > v(t) = A*exp(b*t) > satisfies both equations, and is the same curve > as the previous one, just parametrized differently. > thank you. But this means that the geodesics equations > are not valid for any parametrization? Which kind of > parametrization one has to use to check if a given > curve is a geodesics line? I am not very familiar with geodesic equations and all that. However, we can see that changing the parametrization would change the geodesic equations. Suppose we define a new parameter t' with t = f(t'). We could set u(t) = u(f(t')) = U(t'), which is not a problem, but then we would have to make u'(t) = f'(t')U(t'). The only way I know of to check if an unparametrized curve is a geodesic is to solve the geodesic equations, create equations using the solutions without the parameter, and check against that. As I said, I'm not very familiar with all that. I wouldn't be at all surprised if there were much better techniques available. Jim Burns === Subject: Factorization into irreducibles in a ring of integers D (Stewart and Tall) Stewart and Tall proves in chapter 4 that for any ring of integers D, factorization into irreducibles is possible in D. The proof of this uses countable choice, is this necessary? === Subject: Re: Factorization into irreducibles in a ring of integers D (Stewart and Tall) > Stewart and Tall proves in chapter 4 that for any ring of integers D, > factorization into irreducibles is possible in D. The proof of this > uses countable choice, is this necessary? No. I presume D is a ring of integers in a number field K. You can use induction on |N(a)| where N is the form from K to Q. Victor Meldrew I don't believe it! === Subject: Re: Factorization into irreducibles in a ring of integers D (Stewart and Tall) > Stewart and Tall proves in chapter 4 that for any ring of integers D, > factorization into irreducibles is possible in D. The proof of this > uses countable choice, is this necessary? No, by contraposition take a maximal element of the family F of principal ideals (a) where a is reducible non-unit in D. The existence of such a ideal maximal in F is guaranteed by the fact that D is Noetherian. That means that Noetherian induction does the job here. -- Best wishes, J. === Subject: Re: Factorization into irreducibles in a ring of integers D (Stewart and Tall) posting-account=NNXu6AoAAABAiCjMmqsEQMWexPYbGhY2 Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Sep 11, 10:25 pm, Jannick Asmus Stewart and Tall proves in chapter 4 that for any ring of integers D, > factorization into irreducibles is possible in D. The proof of this > uses countable choice, is this necessary? No, by contraposition take a maximal element of the family F of > principal ideals (a) where a is reducible non-unit in D. The existence > of such a ideal maximal in F is guaranteed by the fact that D is > Noetherian. That means that Noetherian induction does the job here. -- > Best wishes, > J. Hang on, what is Noetherian induction? === Subject: Re: Factorization into irreducibles in a ring of integers D (Stewart and Tall) > Hang on, what is Noetherian induction? http://planetmath.org/encyclopedia/PartialWellOrder.html - scroll down a bit. -- Best wishes, J. === Subject: Re: Factorization into irreducibles in a ring of integers D (Stewart and Tall) posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) On Sep 11, 2:25 pm, Jannick Asmus Stewart and Tall proves in chapter 4 that for any ring of integers D, > factorization into irreducibles is possible in D. The proof of this > uses countable choice, is this necessary? No, by contraposition take a maximal element of the family F of > principal ideals (a) where a is reducible non-unit in D. The existence > of such a ideal maximal in F is guaranteed by the fact that D is > Noetherian. That means that Noetherian induction does the job here. i'm sorry if this off base but isn't well-founded induction equivalent to countable choice? -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: Factorization into irreducibles in a ring of integers D (Stewart and Tall) > On Sep 11, 2:25 pm, Jannick Asmus Stewart and Tall proves in chapter 4 that for any ring of > integers D, factorization into irreducibles is possible in D. The > proof of this uses countable choice, is this necessary? > No, by contraposition take a maximal element of the family F of > principal ideals (a) where a is reducible non-unit in D. The > existence of such a ideal maximal in F is guaranteed by the fact > that D is Noetherian. > That means that Noetherian induction does the job here. i'm sorry if this off base but isn't well-founded induction > equivalent to countable choice? Hmm, I can only see that Noetherian induction is equivalent to the ascending chain condition assuming the axiom of countable choice (which is basically needed for one implication only). -- Best wishes, J. === Subject: Re: a rather pedestrian sum > So I'll just thank you for >pointing out the errors, which were based on the > graph. As you no doubt have checked, >the graph is remarkable in that at first glance, >and second, it looks to have the property I > mentioned, > I never looked at the graph. Instead I _thought_ > about it. And > thinking about it gave me a much more accurate > picture of > what's going on than you got by looking at the actual > physical > picture. > The moral: You can't tell whether something's true or > not > by looking at a picture. The picture can give you a > hint, > but the hint you get from the picture may be wrong. Here's the story. First think about the graph of the > function > g(x) = exp(-x^2). That's a bell-shaped curve, with a > peak at > the origin, and dying to zero very fast as you move > away > from the origin. Now what is your function S? It's simply S(x) = -g(x-2) + g(x-3) - g(x-5) + g(x-7) - g(x-11) > 1) + ... (I was off by a minus sign in my first post because I > was > thinking the sum was +-+-+-... instead of -+-+-+... > .) What does this look like, say, for x near 3? If x is > close > to three then the most important term in the sum is > going to be the term g(x-3). The term g(x-3) does > indeed > have a maximum value of +1 at x = 3; the derivative > of that term is zero at x = 3. For x near 3 the term g(x-3) is the most important > one; > all the other terms are fairly small. But the sum of > the > other terms is not zero near x=3, or at least we have > no reason to think it's exactly 0 (and in fact it's > not), > it's just small. So near x - 3 we have something that > looks a lot like the function g(x-3), but not quite. > The value at x = 3 is going to be close to 1, there's > going to be a maximum _nearby_, etc. Same thing for the zeroes between pairs of primes. > Say x = 6. If x = 6 the sum of the two terms > -g(x-5) + g(x-7) is exactly 0. But those other > terms are still there, although they're small. > So S(6) is probably pretty small, and there's > probably a zero of S somewhere _near_ x = 6. > A function that actually _did_ have > exactly the > properties you announced would be interesting, but > getting > a function that _almost_ has those properties is much > too > easy to be interesting: / OK. I *acknowledged* your point about pictures being misleading. I do also appreciate the outline of your approach to function. So much so that I am going to take some time this evening to review your note and make sure I do understand it. / As for the inherent interest of the function, I said in my second note that if you form a sum involving primes, you would expect prime-related behavior. As you said, unless it's exact, it's not very exciting. Maybe interesting in the sense that I haven't seen it in any books relating to, say, trig series or the like, because this is really a family of functions with related properties--primes aside. dt === Subject: Complexities: Boolean circuit/formula/MSP posting-account=8WKDHAoAAACYqN5DFDzDzFfFo3Bkg_vB Gecko/20080715 Ubuntu/7.10 (gutsy) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Does anyone know of a monotone Boolean function with reasonable Boolean circuit complexity, but high monotone span program size and === Subject: Permutation groups of maximal minimal index Reply-to: weu_rznvy-hfrarg@lnubb.pbz.invalid If G is a non-trivial subgroup of S_n, then by looking at the permutation actions of G on the sets of cosets of the subgroups of its restrictions to its orbits (whew!), it's not hard to see that G must have a proper subgroup of index <= n. Is it possible to completely characterize those subgroups G of S_n none of whose proper subgroups has index < n ? The argument above shows that any such G must be primitive, but what else can be said? (Obviously primitivity is insufficient; consider S_n itself for n > 2.) One obvious class is the simple groups G, for the minimal n such that G <= S_n. I took a quick stab at trying to find another example, or proving that there couldn't be one, but didn't get much further than convincing myself via the O'Nan-Scott Theorem that G is either cyclic of prime order, or has an unsolvable socle. -- Jim Heckman === Subject: Re: Permutation groups of maximal minimal index posting-account=-PngCgkAAAD2yUjosqWv1Nf1lkqWP4lp rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) If G is a non-trivial subgroup of S_n, then by looking at the > permutation actions of G on the sets of cosets of the subgroups of > its restrictions to its orbits (whew!), it's not hard to see that G > must have a proper subgroup of index <= n. Is it possible to completely characterize those subgroups G of S_n > none of whose proper subgroups has index < n ? The argument above > shows that any such G must be primitive, but what else can be said? > (Obviously primitivity is insufficient; consider S_n itself for n > 2.) One obvious class is the simple groups G, for the minimal n such > that G <= S_n. I took a quick stab at trying to find another > example, or proving that there couldn't be one, but didn't get much > further than convincing myself via the O'Nan-Scott Theorem that G > is either cyclic of prime order, or has an unsolvable socle. > -- > Jim Heckman I think simple groups acting on cosets of subgroups of minimal index (which includes cyclic groups of prime order) are the only examples. Looking at the O'Nan-Scott Theorem, for affine groups it is easy to see that cyclic groups of prime order are the only examples. Groups of product type and also groups of twisted wreath product type have degree f^m for some m>1 with a transitive permutation group of degree m as a quotient, so there is a subgroup of index m1 and SxT for any nontrivial subgroup of S otherwise. That leaves almost simple groups with structure S.H with H<=Out(S). If H>1 then a subgroups of prime index clearly has index less than the degree of the group. So we must have H=1 and the group is nonabelian simple. Derek Holt. === Subject: Re: Permutation groups of maximal minimal index Reply-to: weu_rznvy-hfrarg@lnubb.pbz.invalid On 13-Sep-2008, Derek Holt permutation actions of G on the sets of cosets of the subgroups of > its restrictions to its orbits (whew!), it's not hard to see that G > must have a proper subgroup of index <= n. Is it possible to completely characterize those subgroups G of S_n > none of whose proper subgroups has index < n ? The argument above > shows that any such G must be primitive, but what else can be said? > (Obviously primitivity is insufficient; consider S_n itself for n > 2.) [...] > I think simple groups acting on cosets of subgroups of minimal index > (which includes cyclic groups of prime order) are the only examples. time to digest your reply right now, but I'll try to get back to it soon, and post any follow-up questions I might have. -- Jim Heckman === Subject: Re: Have Scientist ever seen Anti Matter? posting-account=x2WXVAkAAACheXC-5ndnEdz_vL9CA75q Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > I am a scientist and I saw antimatter in our > laboratory yesterday. What color was it? -- Rich === Subject: Re: Have Scientist ever seen Anti Matter? > I am a scientist and I saw antimatter in our > laboratory yesterday. What color was it? -- > Rich It was meagre but hollow. ------------ And now a word from our sponsor ------------------ Do your users want the best web-email gateway? Don't let your customers drift off to free webmail services install your own web gateway! -- See http://netwinsite.com/sponsor/sponsor_webmail.htm ---- === Subject: Re: Have Scientist ever seen Anti Matter? <48c9f19c$1@news.auckland.ac.nz> posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) Universe is about half of antimatter in the Alfven cosmology, which is based upon plasma physics from the lab; their is no light & antilight, though; antimatter looks like matter! thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch_brown.html http://wlym.com === Subject: Re: Have Scientist ever seen Anti Matter? posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) I am a scientist and I saw antimatter in our > ælaboratory yesterday. What color was it? The anti-color of the original matter. If the matter is red, the anti-matter is cyan. If the matter is green, the anti-matter is magenta. If the matter is blue, the anti-matter is yellow. -- > Rich === Subject: Re: Spiral problem posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) > I didn't follow Angus's first reply but my thoughts are given below and I > think it's all clear now. > How many 17 year olds can do that in a few minutes? I don't know. I couldn't have when I was 17. And when I encountered this, I had a Mac and Turbo Pascal, >resources I didn't have at 17. But if this thing had been covered in class, I certainly >would expect a 17 year old to be able to handle it. It's >not very hard once you understand the principles. Yes you're probably right there. I never encountered anything like this at > 17. It was all factor this, solve this equation, integrate this, do > something with a triangle or circle, differentiate that etc. In my case, I was trying to create a formula to give me the >x,y grid coordinates of an arbitrary number so I could jump >to any point on the spiral without having to step trough all >the preceeding points. Does such a formula exist? I should have said algorithm, but yes. > If (0,0) is the 1 in the centre of my spiral then > what formula or algorithm can give me the number at any position? Here's one for Excel using VBA. Keep in mind that not all coordinate systems are the same. Excel uses columns and rows for x,y and only permits positive values. Furthermore, the y coordinate increases from top to bottom. Many graphic systems are similar. Changing coordinate systems is just a matter of changing signs in the coordinate calculations. Sub Ulam() 'because this is usually referred to as Ulam's spiral '(where primes are highlighted to observe the interesting 'patterns) ' ' Ulam Macro ' Macro recorded 9/11/2008 by Mensanator 'set origin to cell selected ActiveCell.Value = 1 'get coordinates of origin Xorg = ActiveCell.Row Yorg = ActiveCell.Column 'draw spiral out to 99 For target = 2 To 99 'determine which squares target is between ith = Int(target ^ 0.5) square 1 = ith ^ 2 square 2 = (ith + 1) ^ 2 'find coordinates of the two squares depending on whether 'the first is even or odd If (square 1 Mod 2) = 0 Then 'it's even square 1row = Xorg - (ith) / 2 square 1col = Yorg - (ith) / 2 + 1 square 2row = Xorg + (ith) / 2 square 2col = Yorg + (ith) / 2 Else 'it's odd square 2row = Xorg - (ith - 1) / 2 - 1 square 2col = Yorg - (ith - 1) / 2 square 1row = Xorg + (ith - 1) / 2 square 1col = Yorg + (ith - 1) / 2 End If 'is the target one of the squares or is it between them? Select Case target Case square 1 'it's the 1st square, so we already know the coordinates targetrow = square 1row targetcol = square 1col Case square 2 'likewise if it happens to be the 2nd square targetrow = square 2row targetcol = square 2col Case Else 'it's between them, but how far? If (target - square 1) < (square 2 - square 1) / 2 Then 'less than halfway 'is square 1 even or odd? If (square 1 Mod 2) = 0 Then 'it's even targetrow = square 1row + (target - square 1 - 1) targetcol = square 1col - 1 Else 'it's odd targetrow = square 1row - (target - square 1 - 1) targetcol = square 1col + 1 End If Else 'more (or equal) than halfway '(exactly halway treated as offset from square 2) If (square 2 Mod 2) = 0 Then 'it's even targetrow = square 2row targetcol = square 2col + (square 2 - target) Else 'it's odd targetrow = square 2row targetcol = square 2col - (square 2 - target) End If End If End Select 'target coordinates now known, so stuff target into cell Cells(targetrow, targetcol).Value = target Next End Sub > I thought about trying to write a program to fill up a very large array with > the spiral and get the answer but didn't try because I knew there had to be > a simpler way. Excel is handy for this. It's like having a big empty array that costs nothing. Thw fun is filling it. This program would have solved it if you set the loop big enough (making sure you set the origin deep enough into the worksheet). But as we see, it would be overkill as it's solveable without the array. The array generator is handy if you actually want to highlight the primes and see Ulam's spiral. Look it up in Wikipedia if you're interested. For fun you might ponder the inverse algorithm: what is the number found in a given set of coordinates? >Of course, you have to be careful of boundary cases, such as the >given number being EXACTLY on the lower left corner, such that the >number above it is on the same ring but the number below is on a >my formula, but I couldn't repeat them back to you now. Yes I noticed that some cases would need special treatment after I read your > reply and drew the spiral up to 81. > So let's say I wanted the numbers above and below 23. > The horizontal run which includes 23 must include 5^2 > And 5^2 is 2 more than 23. > Above left of 5^2 must be 3^2 which is 1 to the left. > Taking off the other 1 gives 8 as the number above 23. > Below right of 5^2 must be 7^2 which is 1 to the right. > So take off 2 plus the other 1 and get 46 > The square at the end of the run which includes 2007 must be 45^2, 2007 > is > 18 to the left. > Above left must be 43^2 so the number above is 17 to the left of that > which > is 1832. > Below right of 45^2 is 47^2 so the number below 2007 must be 47^2 -19 > which > is 2190 > So the answer is 1832 + 2190 = 4022 >No problem. > === Subject: Re: Spiral problem > I didn't follow Angus's first reply but my thoughts are given below and I > think it's all clear now. > How many 17 year olds can do that in a few minutes? >I don't know. I couldn't have when I was 17. >And when I encountered this, I had a Mac and Turbo Pascal, >resources I didn't have at 17. >But if this thing had been covered in class, I certainly >would expect a 17 year old to be able to handle it. It's >not very hard once you understand the principles. Yes you're probably right there. I never encountered anything like this at > 17. It was all factor this, solve this equation, integrate this, do > something with a triangle or circle, differentiate that etc. >In my case, I was trying to create a formula to give me the >x,y grid coordinates of an arbitrary number so I could jump >to any point on the spiral without having to step trough all >the preceeding points. Does such a formula exist? I should have said algorithm, but yes. > If (0,0) is the 1 in the centre of my spiral then > what formula or algorithm can give me the number at any position? Here's one for Excel using VBA. Keep in mind that not all coordinate > systems are the same. Excel uses columns and rows for x,y and only > permits positive values. Furthermore, the y coordinate increases from > top to bottom. Many graphic systems are similar. Changing coordinate > systems is just a matter of changing signs in the coordinate > calculations. Sub Ulam() [...] Ulam spiral. 17 16 15 14 13 18 5 4 3 12 19 6 1 2 11 20 7 8 9 10 21 22 23 24 25 ... The diagonal ray starting at 1 and extending downward to the right are the odd squares. Find k such that (2k - 1)^2 < n <= (2k + 1)^2. Find q, r such that (2k + 1)^2 - n = q.2k + r, 0 <= r < 2k. Set a = k - r. Then depending on whether q is 0, 1, 2, or 3 the lattice point (x,y) is respectively (x, y) = (a, -k), (-k, -a), (-a, k), (k, a). Here is a C function for finding the lattice point that holds a given natural. #include #include #include typedef struct { int xc; int yc; int n; } _lp; typedef _lp lattice_point[1]; /* To find the coordinates of the lattice point occupied by n. */ void n_to_lattice_coord(lattice_point lp) { int n; int k; int t; int twok; int sq; div_t quotrem; n = lp->n; /* Find k such that (2k - 1)^2 < n <= (2k + 1)^2. */ t = ceill(sqrt(n)); if((t % 2) == 0) t += 1; sq = t * t; twok = t-1; k = twok/2; quotrem = div(sq - n, twok); switch(quotrem.quot) { case 0: lp->xc = k - quotrem.rem; lp->yc = -k; break; case 1: lp->xc = -k; lp->yc = -k + quotrem.rem; break; case 2: lp->xc = -k + quotrem.rem; lp->yc = k; break; case 3: lp->xc = k; lp->yc = k - quotrem.rem; break; default: fprintf(stderr, Case statement out of bounds); exit(1); } printf(Lattice cooridinates for %d = (%d, %d)n, n, lp->xc, lp->yc); } -- Michael Press === Subject: Re: Spiral problem posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 FunWebProducts; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) spider-mtc-tf05.proxy.aol.com[400C70A5] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad05.proxy.aol.com[400C74C7] (Traffic-Server/6.1.5 [uScM]) > I didn't follow Angus's first reply but my thoughts are given below and I > think it's all clear now. > How many 17 year olds can do that in a few minutes? >I don't know. I couldn't have when I was 17. >And when I encountered this, I had a Mac and Turbo Pascal, >resources I didn't have at 17. >But if this thing had been covered in class, I certainly >would expect a 17 year old to be able to handle it. It's >not very hard once you understand the principles. > Yes you're probably right there. I never encountered anything like this at > 17. It was all factor this, solve this equation, integrate this, do > something with a triangle or circle, differentiate that etc. >In my case, I was trying to create a formula to give me the >x,y grid coordinates of an arbitrary number so I could jump >to any point on the spiral without having to step trough all >the preceeding points. > Does such a formula exist? I should have said algorithm, but yes. > If (0,0) is the 1 in the centre of my spiral then > what formula or algorithm can give me the number at any position? Here's one for Excel using VBA. Keep in mind that not all coordinate > systems are the same. Excel uses columns and rows for x,y and only > permits positive values. Furthermore, the y coordinate increases from > top to bottom. Many graphic systems are similar. Changing coordinate > systems is just a matter of changing signs in the coordinate > calculations. Sub Ulam() [...] Ulam spiral. 17 16 15 14 13 > 18 5 4 3 12 > 19 6 1 2 11 > 20 7 8 9 10 > 21 22 23 24 25 ... The diagonal ray starting at 1 and extending > downward to the right are the odd squares. > Find k such that (2k - 1)^2 < n <= (2k + 1)^2. > Find q, r such that (2k + 1)^2 - n = q.2k + r, 0 <= r < 2k. > Set a = k - r. Then depending on whether q is 0, 1, 2, or 3 > the lattice point (x,y) is respectively > (x, y) = (a, -k), (-k, -a), (-a, k), (k, a). Here is a C function for finding the lattice > point that holds a given natural. #include typedef struct > { > int xc; > int yc; > int n;} _lp; typedef _lp lattice_point[1]; /* To find the coordinates of the lattice point occupied by n. */ void n_to_lattice_coord(lattice_point lp) > { > int n; > int k; > int t; > int twok; > int sq; > div_t quotrem; n = lp->n; /* Find k such that (2k - 1)^2 < n <= (2k + 1)^2. */ > t = ceill(sqrt(n)); > if((t % 2) == 0) t += 1; > sq = t * t; > twok = t-1; > k = twok/2; > quotrem = div(sq - n, twok); > switch(quotrem.quot) > { > case 0: > lp->xc = k - quotrem.rem; > lp->yc = -k; > break; > case 1: > lp->xc = -k; > lp->yc = -k + quotrem.rem; > break; > case 2: > lp->xc = -k + quotrem.rem; > lp->yc = k; > break; > case 3: > lp->xc = k; > lp->yc = k - quotrem.rem; > break; > default: > fprintf(stderr, Case statement out of bounds); > exit(1); > } > printf(Lattice cooridinates for %d = (%d, %d)n, n, lp->xc, lp->yc); } Interesting. I take it you didn't test this code, otherwise you would have noticed the divide-by-0 error when asking for the lattice point of n=1. You're also assuming that the origin is 0,0 and that up is increasing y, which wouldn't work in Excel and many graphic systems. Here's your program translated to Python and uses turtle grapics to draw the spiral, plotting only n that are prime. Of course, if plotting consecutive n, there is an actual proper turtle algorithm you could use instead. Although this demo plots 10000 consecutive points, by using the lattice point algorithm you easily change it to plot 10000 primes instead. Turtle graphics has one advantage, the center of a created window is always 0.0,0.0 with top equal to +y, so no coordinate system adjustment is necessary. It DOES, though, have a HUGE disadvantage. That being the origin is 0.0,0.0. In other words, coordinates are FLOATING POINT numbers, not integers. As such, depending on the size of the window, goto() is subject to floating point rounding errors ruining the spiral image. This doesn't happen when an actual turtle algorithm is used. # Python import math import turtle import gmpy def ulam(n): if n<2: return (0,0) t = int(math.ceil(math.sqrt(n))) if (t % 2)==0: t += 1 sq = t**2 twok = t-1 k = twok/2 quotrem = divmod(sq-n,twok) if quotrem[0]==0: lpxc = k - quotrem[1] lpyc = -k elif quotrem[0]==1: lpxc = -k lpyc = -k + quotrem[1] elif quotrem[0]==2: lpxc = -k + quotrem[1] lpyc = k elif quotrem[0]==3: lpxc = k lpyc = k - quotrem[1] else: print out of bounds return (lpxc,lpyc) turtle.setup(width=200, height=200, startx=100, starty=100) turtle.tracer(False) origin = turtle.position() turtle.up() turtle.goto(-80,80) turtle.write(Ulam's Spiral) for n in xrange(10000): p = ulam(n) if gmpy.is_prime(n): turtle.color(red) else: turtle.color(white) turtle.up() turtle.goto(origin[0]+p[0],origin[1]+p[1]) turtle.down() turtle.forward(1) turtle.done() -- > Michael Press === Subject: Re: Spiral problem > I didn't follow Angus's first reply but my thoughts are given below and I > think it's all clear now. > How many 17 year olds can do that in a few minutes? >I don't know. I couldn't have when I was 17. >And when I encountered this, I had a Mac and Turbo Pascal, >resources I didn't have at 17. >But if this thing had been covered in class, I certainly >would expect a 17 year old to be able to handle it. It's >not very hard once you understand the principles. > Yes you're probably right there. I never encountered anything like this at > 17. It was all factor this, solve this equation, integrate this, do > something with a triangle or circle, differentiate that etc. >In my case, I was trying to create a formula to give me the >x,y grid coordinates of an arbitrary number so I could jump >to any point on the spiral without having to step trough all >the preceeding points. > Does such a formula exist? > I should have said algorithm, but yes. > If (0,0) is the 1 in the centre of my spiral then > what formula or algorithm can give me the number at any position? > Here's one for Excel using VBA. Keep in mind that not all coordinate > systems are the same. Excel uses columns and rows for x,y and only > permits positive values. Furthermore, the y coordinate increases from > top to bottom. Many graphic systems are similar. Changing coordinate > systems is just a matter of changing signs in the coordinate > calculations. > Sub Ulam() [...] Ulam spiral. 17 16 15 14 13 > 18 5 4 3 12 > 19 6 1 2 11 > 20 7 8 9 10 > 21 22 23 24 25 ... The diagonal ray starting at 1 and extending > downward to the right are the odd squares. > Find k such that (2k - 1)^2 < n <= (2k + 1)^2. > Find q, r such that (2k + 1)^2 - n = q.2k + r, 0 <= r < 2k. > Set a = k - r. Then depending on whether q is 0, 1, 2, or 3 > the lattice point (x,y) is respectively > (x, y) = (a, -k), (-k, -a), (-a, k), (k, a). [...] > Interesting. I take it you didn't test this code, otherwise you > would have noticed the divide-by-0 error when asking for the > lattice point of n=1. Tested, but not at n=1. Sorry. > You're also assuming that the origin is 0,0 and that up is increasing > y, which wouldn't work in Excel and many graphic systems. Yes. Adjust appropriately for coordinate system. I posted in an attempt to offer a mathematical formula. > Here's your program translated to Python and uses turtle grapics to > draw the spiral, plotting only n that are prime. Of course, if > plotting > consecutive n, there is an actual proper turtle algorithm you could > use instead. Although this demo plots 10000 consecutive points, by > using the lattice point algorithm you easily change it to plot 10000 > primes instead. Turtle graphics has one advantage, the center of a created window > is always 0.0,0.0 with top equal to +y, so no coordinate system > adjustment is necessary. It DOES, though, have a HUGE disadvantage. That being the origin is > 0.0,0.0. In other words, coordinates are FLOATING POINT numbers, > not integers. As such, depending on the size of the window, goto() > is subject to floating point rounding errors ruining the spiral > image. This doesn't happen when an actual turtle algorithm is used. # Python > import math > import turtle > import gmpy def ulam(n): > if n<2: return (0,0) > t = int(math.ceil(math.sqrt(n))) > if (t % 2)==0: t += 1 > sq = t**2 > twok = t-1 > k = twok/2 > quotrem = divmod(sq-n,twok) > if quotrem[0]==0: > lpxc = k - quotrem[1] > lpyc = -k > elif quotrem[0]==1: > lpxc = -k > lpyc = -k + quotrem[1] > elif quotrem[0]==2: > lpxc = -k + quotrem[1] > lpyc = k > elif quotrem[0]==3: > lpxc = k > lpyc = k - quotrem[1] > else: > print out of bounds > return (lpxc,lpyc) turtle.setup(width=200, height=200, startx=100, starty=100) > turtle.tracer(False) > origin = turtle.position() > turtle.up() > turtle.goto(-80,80) > turtle.write(Ulam's Spiral) > for n in xrange(10000): > p = ulam(n) > if gmpy.is_prime(n): > turtle.color(red) > else: > turtle.color(white) > turtle.up() > turtle.goto(origin[0]+p[0],origin[1]+p[1]) > turtle.down() > turtle.forward(1) > turtle.done() Neat. -- Michael Press === Subject: Re: Spiral problem snip > Does such a formula exist? If (0,0) is the 1 in the centre of my spiral then > what formula or algorithm can give me the number at any position? > I thought about trying to write a program to fill up a very large array with > the spiral and get the answer but didn't try because I knew there had to be > a simpler way. > The numbers directly below the one seem be the sum of an arithmetic progression. The number in the n'th row below one is 1 + sum [k = 0 to n-1] (8k + 7) = 1 + 7n + 8 * sum[k=1 to n-1] k = 1 + 7n + 8(n-1)n/2 = 1 + 7n + 4n^2 -4n = 1 + 3n + 4n^2 this number. The numbers directly above the one in the n'th row are 1 + sum [k=0 to n-1] (8k+3) = 4n^2 - n + 1 Finding a number in a particular position will take a little more work, or maybe a lot more work. cheers hanford === Subject: What is the name of a polynomial with the coefficient of the highest order being 1? posting-account=QQ-KfAoAAABDVwrVNFoaFFcBU7ec1cHk Gecko/20080702 Iceweasel/2.0.0.16 (Debian-2.0.0.16-0etch1),gzip(gfe),gzip(gfe) I'm wondering how people call a polynomial of the following form? x^n+a_{n-1} x^{n-1} +...+a_1 x^1 +a_0 Peng === Subject: Re: What is the name of a polynomial with the coefficient of the highest order being 1? I'm wondering how people call a polynomial of the following form? x^n+a_{n-1} x^{n-1} +...+a_1 x^1 +a_0 Monic. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: What is the name of a polynomial with the coefficient of the highest order being 1? posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) I'm wondering how people call a polynomial of the following form? x^n+a {n-1} x^{n-1} +...+a 1 x^1 +a 0 > That is a monic polynomial. -- m === Subject: Re: What is the name of a polynomial with the coefficient of the highest order being 1? posting-account=yxbZkgkAAABQBvyYeebYQ-PAvi0uT3tG Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > I'm wondering how people call a polynomial of the following form? > x^n+a {n-1} x^{n-1} +...+a 1 x^1 +a 0 That is a monic polynomial. And you can derive such a polynomial by dividing all the terms by the a {n} coefficient of x^n if a {n} is not 1. (But I assume that you already knew that.) I was just reading (in Unknown Quantity by John Derbyshire) about a theorem by Erland Bring (1786) which states that any quintic equation can be reduced to one of the form x^5 + px + q = 0, eliminating the middle terms. === Subject: Re: Maximum gap between primes? <9Qoxk.179209$oT7.48941@newsfe10.ams2> posting-account=8a094goAAACSgd33IDG1GOMm_lzp9sfY Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 9, 10:54æpm, Gerry Myerson Also, has it been proven that there is a one-to-one correspondence > between natural numbers and g(p) - in other words, that there really > is aprimegapof every length starting with 1 and growing by 1? That > somewhere in the sequence, g(p) is, say, 7,312? You are very confused. Yes, that is certainly true, at least regarding the correspondence issue that you explain below. First of all, there is no proof that for every even number > there is agapbetween consecutive primes equal exactly > to that number. It is conjectured that this is true, and no > one seriously doubts that it is true - in fact, it is conjectured > that for every even number k there are infinitely many primes p > such that the nextprimeis p + k - but as of yet there is > no proof. What is proved is that there is agapof length > AT LEAST k (and, it follows, infinitely many such gaps). But that has nothing to do with one-to-one correspondence, > at least not in the way that that phrase is always used in > mathematics. Two sets are said to be in one-to-one correspondence > if there is any way of pairing off the two sets so nothing is left out > in either one. So for instance there is a one-to-one correspondence > between the natural numbers and theprimenumbers, even though > theprimenumbers don't give all the natural numbers; simply pair > 1 with 2, 2 with 3, 3 with 5, 4 with 7, 5 with 11, and so on, > and each natural will be paired with its own personalprime, > with nothing left out in either set. -- This is what is confusing to me, given what I think were previous explanations saying that the natural numbers are infinite (I learned that a long time ago) but that the prime gaps are finite but unbounded. I'm not speaking about pairing up the prime numbers with the natural numbers but the prime gaps with the natural numbers - pairing up the natural number 1 with the prime gap of 1 (such as the numerical value of the gap between 11 and 13), the number 2 with the prime gap of 2, and so on. If prime gaps can be paired off with natural numbers with nothing left out of either set, shouldn't they both be infinite or finite but unbounded, instead of each being categorized differently? Or is there still something difference between the two sets? I hope I have made enough sense to at least have asked the question correctly, but let me know if I haven't. Shepherdmoon === Subject: Re: Maximum gap between primes? > This is what is confusing to me, given what I think were previous > explanations saying that the natural numbers are infinite (I learned > that a long time ago) but that the prime gaps are finite but > unbounded. The prime gaps are natural numbers. Each individual natural number is finite, so each prime gap is finite. The natural numbers, as a set, are unbounded, and, as has been shown earlier in this thread, the prime gaps, as a set, are also unbounded. The set of all natural numbers is an infinite set, each of whose members is finite; the set of all prime gaps is an infinite set, each of whose members is finite. [In case elaboration is needed on any of the above: [Each individual natural number is finite that is, the number 17 [is certainly finite, and so is the number 43, and the number [574896599485763542243; whatever natural number you may [be thinking of right now, it's finite. [The natural numbers, as a set, are unbounded, that is, there is [no bound you can name that is bigger than every natural number. [The natural numbers aren't bounded by 92, because there's 93; [they aren't bounded by 5896876453, because there's 5896876454, [and so on; they just aren't bounded; they're unbounded.] > I'm not speaking about pairing up the prime numbers with the natural > numbers but the prime gaps with the natural numbers - pairing up the > natural number 1 with the prime gap of 1 (such as the numerical value > of the gap between 11 and 13), the number 2 with the prime gap of 2, > and so on. If prime gaps can be paired off with natural numbers with nothing left > out of either set, shouldn't they both be infinite or finite but > unbounded, instead of each being categorized differently? Or is there > still something difference between the two sets? They are both infinite sets, meaning they each have infinitely many members. Each individual member of each set is finite. Taken as a whole, each set is unbounded. As far as these properties go, the sets are not different. As far as exactly which numbers are members, the sets are different. The way you count them, the prime gaps are 0, 1, 3, 5, 7, 9, etc., so 0 is a prime gap but not a natural number, and 2 is a natural number but not a prime gap. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Factoring and GCD I am not a number theorist, so apologies if this is a stupid question. Suppose we have a very large number m in N, for which we don't know its factorization. Obviously if GCD(m, p_i) = p_i, for some prime p_i < m, then p_i is a factor of m, hence, theoretically, by going through a loop which calculates GCD(m, p_i) for all all primes p_i < m, we can conceivably unveil its factorization, depending on whether we get GCD(m, p_i) = 1 or p_i. Wherein lies the practical difficulty with this process for large unknown m? The reason I am asking is because my impression is that the GCD algorithm is fairly fast, so what makes the above algorithm impractical? -- I.N. Galidakis === Subject: Re: Factoring and GCD > I am not a number theorist, so apologies if this is a stupid question. Suppose we have a very large number m in N, for which we don't know its > factorization. Obviously if GCD(m, p_i) = p_i, for some prime p_i < m, then p_i is a factor of > m, hence, theoretically, by going through a loop which calculates GCD(m, p_i) > for all all primes p_i < m, we can conceivably unveil its factorization, > depending on whether we get GCD(m, p_i) = 1 or p_i. Wherein lies the practical difficulty with this process for large unknown m? The reason I am asking is because my impression is that the GCD algorithm is > fairly fast, so what makes the above algorithm impractical? > Primarily, the interesting large numbers to factor are semi-prime, i.e., they are the product of the two prime numbers, and they also tend to be about the same size, so we're looking for factors around sqrt(m). The smallest RSA factoring challenge is a mere 1.5 googol. Calculating merely the primes less than the square root--about 10^50--would be impractical, let alone the GCD step. === Subject: Re: Factoring and GCD days. My association with the Department is that of an alumnus. >I am not a number theorist, so apologies if this is a stupid question. Suppose we have a very large number m in N, for which we don't know its >factorization. Obviously if GCD(m, p_i) = p_i, for some prime p_i < m, then p_i is a factor of >m, hence, theoretically, by going through a loop which calculates GCD(m, p_i) >for all all primes p_i < m, we can conceivably unveil its factorization, >depending on whether we get GCD(m, p_i) = 1 or p_i. Wherein lies the practical difficulty with this process for large unknown m? The reason I am asking is because my impression is that the GCD algorithm is >fairly fast, so what makes the above algorithm impractical? GCD testing is fast, but is no faster than doing a single division-with-remainder (think about ->how<- one finds gcds). So what you propose is essentially, doing trial division of m by each prime p_i less than m (in fact, it is enough to check every prime less than or equal to sqrt(m), because if m is not a prime, then it has a factor that is no more than sqrt(m)). And that is a lot: it is O(sqrt(m)), and that is exponential in the length of the input (which is log_2(m)). That said: most of our good factoring methods do actually rely on computing gcds. But what one does is come up with numbers n (in complicated ways, with heuristics that suggest that they are good candidates), and commpute gcd(n,m) and hope that you get something between 1 and m (that will suffice to find a factor). -- magidin-at-member-ams-org === Subject: Re: Factoring and GCD > I am not a number theorist, so apologies if this is a stupid question. > Suppose we have a very large number m in N, for which we don't know its > factorization. > Obviously if GCD(m, p_i) = p_i, for some prime p_i < m, then p_i is a factor > of m, hence, theoretically, by going through a loop which calculates GCD(m, > p_i) for all all primes p_i < m, we can conceivably unveil its factorization, > depending on whether we get GCD(m, p_i) = 1 or p_i. > Wherein lies the practical difficulty with this process for large unknown m? > The reason I am asking is because my impression is that the GCD algorithm is > fairly fast, so what makes the above algorithm impractical? GCD testing is fast, but is no faster than doing a single > division-with-remainder (think about ->how<- one finds gcds). So what > you propose is essentially, doing trial division of m by each prime > p_i less than m (in fact, it is enough to check every prime less than > or equal to sqrt(m), > because if m is not a prime, then it has a factor > that is no more than sqrt(m)). And that is a lot: it is O(sqrt(m)), > and that is exponential in the length of the input (which is > log_2(m)). That said: most of our good factoring methods do actually rely on > computing gcds. But what one does is come up with numbers n (in > complicated ways, with heuristics that suggest that they are good > candidates), and commpute gcd(n,m) and hope that you get something > between 1 and m (that will suffice to find a factor). first step on the GCD algorithm is: m = p_i*q + r, hence I might as well be going through the loop doing direct divide checks p_i|m. My question now is, how 'fast' is such a single divide check for such a case of large unknown m and (a large) prime p? In other words, if m is of the order of ~10^100, say, and p is largish (say around 10^23), is the operation of finding q and r in m = p*q + r 'fast' in the sense that it always 'completes' in reasonable time? Can I force ANY computer to eventually get stuck for an eon searching for q and r for sufficiently chosen large m and p? -- I.N. Galidakis === Subject: Re: Factoring and GCD days. My association with the Department is that of an alumnus. [...] >well: The first step on the GCD algorithm is: >m = p_i*q + r, hence I might as well be going through the loop doing >direct divide checks p_i|m. My question now is, how 'fast' is such a >single divide check for such a case of large unknown m and (a large) >prime p? Neil Koblitz' A Course in Number Theory and Cryptography has the bit-complexity analysis. Dividing a number a by a number b (b < a) takes about O(log^2 a) bit operations. So for given m and a, a In other words, if m is of the order of ~10^100, say, and p >is largish (say around 10^23), is the operation of finding q and r in >m = p*q + r 'fast' in the sense that it always 'completes' in >reasonable time? Yes; it is polynomial in log(m), hence deemed reasonable. >Can I force ANY computer to eventually get stuck >for an eon searching for q and r for sufficiently chosen large m and >p? Sufficiently large numbers may cause overflows. And reasonable does not mean sometime before the sun burns out; but the process is extremely straightforward: all you have to do is successive subtractions, after all. It won't get stuck because it cannot find q and r, it will simply perhaps take a very long time to be done doing some very straightforward things, so long that ->you<- may give up, or the hardware may fail. -- magidin-at-member-ams-org === Subject: Re: Factoring and GCD [...] > well: The first step on the GCD algorithm is: > m = p_i*q + r, hence I might as well be going through the loop doing > direct divide checks p_i|m. My question now is, how 'fast' is such a > single divide check for such a case of large unknown m and (a large) > prime p? Neil Koblitz' A Course in Number Theory and Cryptography has the > bit-complexity analysis. Dividing a number a by a number b (b < a) > takes about O(log^2 a) bit operations. So for given m and a, a (regardless of whether a is prime or not) you expect it to take > O(log^2 m) steps. Finding the gcd will take at most O(log m) such > divisions, so the Euclidean algorithm will take about O(log^3 m) bit > operations. You can even do the extended Euclidean algorithm in > O(log^3 m) operations. > In other words, if m is of the order of ~10^100, say, and p > is largish (say around 10^23), is the operation of finding q and r in > m = p*q + r 'fast' in the sense that it always 'completes' in > reasonable time? Yes; it is polynomial in log(m), hence deemed reasonable. > Can I force ANY computer to eventually get stuck > for an eon searching for q and r for sufficiently chosen large m and > p? Sufficiently large numbers may cause overflows. And reasonable does > not mean sometime before the sun burns out; but the process is > extremely straightforward: all you have to do is successive > subtractions, after all. It won't get stuck because it cannot find q > and r, it will simply perhaps take a very long time to be done doing > some very straightforward things, so long that ->you<- may give up, or > the hardware may fail. -- I.N. Galidakis === Subject: Re: Factoring and GCD > I am not a number theorist, so apologies if this is a stupid question. Suppose we have a very large number m in N, for which we don't know its > factorization. Obviously if GCD(m, p_i) = p_i, for some prime p_i < m, then p_i is a > factor of > m, hence, theoretically, by going through a loop which calculates GCD(m, > p_i) > for all all primes p_i < m, we can conceivably unveil its factorization, > depending on whether we get GCD(m, p_i) = 1 or p_i. Wherein lies the practical difficulty with this process for large unknown > m? The reason I am asking is because my impression is that the GCD algorithm > is > fairly fast, so what makes the above algorithm impractical? > There are lots of primes. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Simple topology question Gecko/20080715 Epiphany/2.20 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) How might I go about showing that the punctured plane (R^2) and an annulus, say, a < |z| < b where |z|=(x)^2 + (y)^2, are diffeomorphic? for any help. === Subject: Re: Simple topology question > How might I go about showing that the punctured plane (R^2) Strange notation. I'll work with R^2 {0}. > and an > annulus, say, a < |z| < b where |z|=(x)^2 + (y)^2, No, surely you want |z|^2 = x^2 + y^2. > are diffeomorphic? > for any help. For a nice enough f : (0, oo) -> (a, b), the map z -> f(|z|)*(z/|z|) should work. === Subject: Re: Mathematics: how to start again <48C8F228.8C479DB9@tesco.net> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) Consider the segment [0,1], and to each point we assign a probability of 1/2 that the point exists. Consider the points, as stated, and not the expected length. Then, there is a nonzero probability that the whole thing is totally connected, and a nonzero probability that the whole thing is totally disconnected. The only problem is that this solution requires existential indeterminacy - therefore, the solution itself is wholly questionable as to whether it is even math or not. This is indeterminate, and so why am I posting in sci.math ? Because it might be math ! It also might be nonsense. Again, that's indeterminate. I believe that was stated properly - but you should also probably not listen to what I say. Im sure there is a perfectly logical answer to the question, as opposed to mine which is indeterminate as to whether it is logical or not. === Subject: Re: Mathematics: how to start again > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. > Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > The only problem is that this solution requires existential > indeterminacy - therefore, the solution itself is wholly questionable > as to whether it is even math or not. This is indeterminate, and so > why am I posting in sci.math ? Because it _might_ be math ! > It also might be nonsense. Again, that's indeterminate. > I believe that was stated properly - but you should also probably not > listen to what I say. Im sure there is a perfectly logical answer to > the question, as opposed to mine which is indeterminate as to whether > it is logical or not. Good grief. You are working much too hard. Forget probabilities and just look at the definitions. It should be obvious. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Mathematics: how to start again posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. > Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > The only problem is that this solution requires existential > indeterminacy - therefore, the solution itself is wholly questionable > as to whether it is even math or not. This is indeterminate, and so > why am I posting in sci.math ? Because it might be math ! > It also might be nonsense. Again, that's indeterminate. > I believe that was stated properly - but you should also probably not > listen to what I say. Im sure there is a perfectly logical answer to > the question, as opposed to mine which is indeterminate as to whether > it is logical or not. Good grief. You are working much too hard. æForget probabilities and just look at the > definitions. æIt should be obvious. Well, if one is doing mathematics proper, then the empty set is a good candidate. I've been trying to train myself to think differently and so the most obvious answer to me really is spacetime. But I cannot claim that I am doing math, only that it might be math. Im not designing airplanes or anything so I have the freedom to think about silly things like that. === Subject: Re: Mathematics: how to start again > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. > Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > The only problem is that this solution requires existential > indeterminacy - therefore, the solution itself is wholly questionable > as to whether it is even math or not. This is indeterminate, and so > why am I posting in sci.math ? Because it _might_ be math ! > It also might be nonsense. Again, that's indeterminate. > I believe that was stated properly - but you should also probably not > listen to what I say. Im sure there is a perfectly logical answer to > the question, as opposed to mine which is indeterminate as to whether > it is logical or not. > Good grief. > You are working much too hard. æForget probabilities and just look at the > definitions. æIt should be obvious. > Well, if one is doing mathematics proper, then the empty set is a good > candidate. The empty space works, but there is also a nonempty solution. It's unique, up to homeomorphism. > I've been trying to train myself to think differently and so the most > obvious answer to me really is spacetime. But I cannot claim that I am > doing math, only that it might be math. Im not designing airplanes or > anything so I have the freedom to think about silly things like that. If you are looking at anything other than the simple mathematical definitions of the terms, then you are looking in the wrong place. When all else fails, start by writing down the definitions. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Mathematics: how to start again posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. > Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > The only problem is that this solution requires existential > indeterminacy - therefore, the solution itself is wholly questionable > as to whether it is even math or not. This is indeterminate, and so > why am I posting in sci.math ? Because it might be math ! > It also might be nonsense. Again, that's indeterminate. > I believe that was stated properly - but you should also probably not > listen to what I say. Im sure there is a perfectly logical answer to > the question, as opposed to mine which is indeterminate as to whether > it is logical or not. > Good grief. > You are working much too hard. æForget probabilities and just look at the > definitions. æIt should be obvious. > Well, if one is doing mathematics proper, then the empty set is a good > candidate. The empty space works, but there is also a nonempty solution. æIt's > unique, up to homeomorphism. I've been trying to train myself to think differently and so the most > obvious answer to me really is spacetime. But I cannot claim that I am > doing math, only that it might be math. Im not designing airplanes or > anything so I have the freedom to think about silly things like that. If you are looking at anything other than the simple mathematical > definitions of the terms, then you are looking in the wrong place. > When all else fails, start by writing down the definitions. If one is doing mathematics, yes. I agree. Im pursuing something else, something that might be math, but might not be math. If you have zero bananas and zero oranges, then bananas are oranges ? Is zero bananas even a banana ? Relying on definitions cannot resolve this. I want to create an algebra where you have an operator which is indeterminately either addition or multiplication. I'm not sure if this is really defineable in the traditional sense of the word. Indeterminacy is built into the definition, which seems very akward and even bizarre. In fact, it reinforces what I said that it may or may not be math. In my (controversial) view, math is like a big thick book. There is a very thick portion full of logical structures which deals with things that are said to exist. On the very last page somewhere, you have this point sized singularity which is nonexistence. So you have two main sections. In my view, there is AND is not a third section of that book which is very large which may or may not be there, and it is based on existential indeterminacy. Im trying to understand that third chapter, which may or may not be there. One strange thing is that I posted two different solutions to the OP's question. One based on existential indeterminacy, and one based on unadulterated mathematics. Both solutions make sense, and as one would expect - the empty set had a hand in this. To me, this merely reinforces the view that indeterminacy is indeterminate, hence the third chapter of the book. === Subject: Re: Mathematics: how to start again > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. > Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > The only problem is that this solution requires existential > indeterminacy - therefore, the solution itself is wholly questionable > as to whether it is even math or not. This is indeterminate, and so > why am I posting in sci.math ? Because it _might_ be math ! > It also might be nonsense. Again, that's indeterminate. > I believe that was stated properly - but you should also probably not > listen to what I say. Im sure there is a perfectly logical answer to > the question, as opposed to mine which is indeterminate as to whether > it is logical or not. > Good grief. > You are working much too hard. æForget probabilities and just look at the > definitions. æIt should be obvious. > Well, if one is doing mathematics proper, then the empty set is a good > candidate. > The empty space works, but there is also a nonempty solution. æIt's > unique, up to homeomorphism. > I've been trying to train myself to think differently and so the most > obvious answer to me really is spacetime. But I cannot claim that I am > doing math, only that it might be math. Im not designing airplanes or > anything so I have the freedom to think about silly things like that. > If you are looking at anything other than the simple mathematical > definitions of the terms, then you are looking in the wrong place. > When all else fails, start by writing down the definitions. > If one is doing mathematics, yes. I agree. This is sci.math, after all. > Im pursuing something else, something that might be math, but might > not be math. > If you have zero bananas and zero oranges, then bananas are oranges ? > Is zero bananas even a banana ? Relying on definitions cannot resolve > this. Wrong. There is indeed a nonempty solution, and it follows directly from the definitions. [ nonsense snipped ] > One strange thing is that I posted two different solutions to the OP's > question. One based on existential indeterminacy, and one based on > unadulterated mathematics. Both solutions make sense, and as one would > expect - the empty set had a hand in this. To me, this merely > reinforces the view that indeterminacy is indeterminate, hence the > third chapter of the book. You posted one solution; the empty topological space. There is, however, a nonempty solution. Definition. A _component_ of a topological space (X,T) is a nonempty subset of X that is both open and closed. Definition. A space is _connected_ if its only component is the space itself. Definition. A space is _totally disconnected_ if each of its components consists of a single point. Now let's suppose that (X,T) is a nonempty space that is both connected and totally disconnected. Question 1. How many components does the space have? Answer 1. One, because it is connected. Question 2. How many points are in each component? Answer 2. One, because the space is totally disconnected. Question 3. How many points are in the space altogether? Answer 3. One, by answers 1 and 2. Indeed, every singleton space is clearly a solution. No hocus-pocus or handwaving is required. The definitions alone are sufficient. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Mathematics: how to start again posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > [ nonsense snipped ] Technically it is indeterminate whether it is nonsense or not, so you're half right on that. But that doesn't really do justice to the other half. === Subject: Re: Mathematics: how to start again posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. > Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > The only problem is that this solution requires existential > indeterminacy - therefore, the solution itself is wholly questionable > as to whether it is even math or not. This is indeterminate, and so > why am I posting in sci.math ? Because it might be math ! > It also might be nonsense. Again, that's indeterminate. > I believe that was stated properly - but you should also probably not > listen to what I say. Im sure there is a perfectly logical answer to > the question, as opposed to mine which is indeterminate as to whether > it is logical or not. Good grief. You are working much too hard. æForget probabilities and just look at the > definitions. æIt should be obvious. > Well, if one is doing mathematics proper, then the empty set is a good > candidate. > The empty space works, but there is also a nonempty solution. æIt's > unique, up to homeomorphism. > I've been trying to train myself to think differently and so the most > obvious answer to me really is spacetime. But I cannot claim that I am > doing math, only that it might be math. Im not designing airplanes or > anything so I have the freedom to think about silly things like that. > If you are looking at anything other than the simple mathematical > definitions of the terms, then you are looking in the wrong place. > When all else fails, start by writing down the definitions. > If one is doing mathematics, yes. I agree. This is sci.math, after all. Im pursuing something else, something that might be math, but might > not be math. > If you have zero bananas and zero oranges, then bananas are oranges ? > Is zero bananas even a banana ? Relying on definitions cannot resolve > this. Wrong. æThere is indeed a nonempty solution, and it follows directly from > the definitions. If zero is a number, then zero bananas is a banana. Then also, zero oranges is really an orange. So, nothingness is both an orange, and a banana. What good are definitions when a banana is an orange ? If zero is not a number, then zero bananas is not a banana, it's just nothingness. But then you have to say that zero is not a number. I would resolve this by saying that we dont know if zero bananas is a banana. It is indeterminate. But for that to work, you need existential indeterminacy. You do not know if banana exists or not, when there are zero bananas. Existence of banana is indeterminate. That's why I posted the carefully crafted caveats explaining that it may or may not be math. Existence exists, Nonexistence does not exist, Existential indeterminacy might exist. === Subject: Re: Mathematics: how to start again posting-account=EryW6woAAAB3US2vhkzCS5BtRRNBkPAq Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) ... > If zero is a number, then zero bananas is a banana. > Then also, zero oranges is really an orange. ... 0 != 1, so zero bananas != a banana but, zero bananas is as well zero oranges. zero anything is as well zero anything else. ;) === Subject: Re: Mathematics: how to start again > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. > Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > The only problem is that this solution requires existential > indeterminacy - therefore, the solution itself is wholly questionable > as to whether it is even math or not. This is indeterminate, and so > why am I posting in sci.math ? Because it _might_ be math ! > It also might be nonsense. Again, that's indeterminate. > I believe that was stated properly - but you should also probably not > listen to what I say. Im sure there is a perfectly logical answer to > the question, as opposed to mine which is indeterminate as to whether > it is logical or not. > Good grief. > You are working much too hard. æForget probabilities and just look at the > definitions. æIt should be obvious. > Well, if one is doing mathematics proper, then the empty set is a good > candidate. > The empty space works, but there is also a nonempty solution. æIt's > unique, up to homeomorphism. > I've been trying to train myself to think differently and so the most > obvious answer to me really is spacetime. But I cannot claim that I am > doing math, only that it might be math. Im not designing airplanes or > anything so I have the freedom to think about silly things like that. > If you are looking at anything other than the simple mathematical > definitions of the terms, then you are looking in the wrong place. > When all else fails, start by writing down the definitions. > If one is doing mathematics, yes. I agree. > This is sci.math, after all. > Im pursuing something else, something that might be math, but might > not be math. > If you have zero bananas and zero oranges, then bananas are oranges ? > Is zero bananas even a banana ? Relying on definitions cannot resolve > this. > Wrong. æThere is indeed a nonempty solution, and it follows directly from > the definitions. > If zero is a number, then zero bananas is a banana. Not so. Zero is a number, but zero bananas is not a banana. [ nonsense snipped ] -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Mathematics: how to start again posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. > Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > The only problem is that this solution requires existential > indeterminacy - therefore, the solution itself is wholly questionable > as to whether it is even math or not. This is indeterminate, and so > why am I posting in sci.math ? Because it might be math ! > It also might be nonsense. Again, that's indeterminate. > I believe that was stated properly - but you should also probably not > listen to what I say. Im sure there is a perfectly logical answer to > the question, as opposed to mine which is indeterminate as to whether > it is logical or not. > Good grief. > You are working much too hard. æForget probabilities and just look at the > definitions. æIt should be obvious. > Well, if one is doing mathematics proper, then the empty set is a good > candidate. The empty space works, but there is also a nonempty solution. æIt's > unique, up to homeomorphism. I've been trying to train myself to think differently and so the most > obvious answer to me really is spacetime. But I cannot claim that I am > doing math, only that it might be math. Im not designing airplanes or > anything so I have the freedom to think about silly things like that. If you are looking at anything other than the simple mathematical > definitions of the terms, then you are looking in the wrong place. > When all else fails, start by writing down the definitions. > If one is doing mathematics, yes. I agree. > This is sci.math, after all. > Im pursuing something else, something that might be math, but might > not be math. > If you have zero bananas and zero oranges, then bananas are oranges ? > Is zero bananas even a banana ? Relying on definitions cannot resolve > this. > Wrong. æThere is indeed a nonempty solution, and it follows directly from > the definitions. > If zero is a number, then zero bananas is a banana. Not so. æZero is a number, but zero bananas is not a banana. [ nonsense snipped ] -- > Dave Seaman > Third Circuit ignores precedent in Mumia Abu-Jamal ruling. > - Hide quoted text - - Show quoted text - Certainly you would agree that we have existence, and that there is also something called nonexistence. Considering my big book analogy of mathematics - you would have to wonder about the boundary between that which exists and that which does not. Existential indeterminacy, as I have described, accomplishes that. It makes sense that such a boundary may or may not exist, because it makes no sense that nonexistence could even have a boundary, but it does, and that boundary is this world where indeterminacy lives. With time & pressure - I will eventually prove it. === Subject: Re: Mathematics: how to start again posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) What interests me about randomness is that you have an outcome space, and random output is generated by some process which is basically functionless. You can easily argue that the function which generates the random numbers cannot exist, because if it did then that would make your random output non-random. If random numbers could be generated by some function, then the output cannot possibly be considered random because it would then be deterministic. So they trot in the random variables and wave a magic wand and call it random. But that does nothing to explain the fact that you have a domain, and a range, and there is a process which gets you from point A to point B, and in the current context of math proper you would be talking about a function which does not exist. You have a black box. You insert your output space and it spits out a random value, but the black box which performs this magical feat does not contain a function of any kind. There can be no known process or function, otherwise the output is deterministic. So, in that black box you must have a nonexistent function. Does that really make more sense than what I have been claiming ? I dont think so. === Subject: Re: Mathematics: how to start again posting-account=EryW6woAAAB3US2vhkzCS5BtRRNBkPAq Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > On Sep 11, 5:25æam, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. ... Now, I remember these words connected and disconnected in graph theory, it means reachable (?!) from a node to another. I doubt the usage of the word, 'and', shouldn't it be 'or'? There are nonzero probabilities both the way, but once it is connected, then it is not disconnected. For example, If we toss a coin there is a nonzero probability to get a head and the same is true for tail, but once it is a head, then it can not be tail. === Subject: Re: Mathematics: how to start again > On Sep 11, 5:25 am, Frederick Williams Give an example of a topological space that is both connected and totally > disconnected. > I don't know what mathematically that means, but is it possible > something to be both connected and disconnected..?? If it is connected > then it is not disconnected, or, if it is disconnected then there is > no connection. :-) > Consider the segment [0,1], and to each point we assign a probability > of 1/2 that the point exists. Consider the points, as stated, and not > the expected length. Then, there is a nonzero probability that the whole thing is totally > connected, and a nonzero probability that the whole thing is totally > disconnected. > ... Now, I remember these words connected and disconnected in graph > theory, it means _reachable_ (?!) from a node to another. William Elliot wasn't asking about graphs but about topological spaces > I doubt the usage of the word, 'and', shouldn't it be 'or'? Had WE asked 'Give an example of a topological space that is both connected or totally disconnected.' the word 'both' would have read most oddly. Had he asked 'Give an example of a topological space that is connected or totally disconnected.' the problem would have been trivial. So, no, it shouldn't be 'or'. > There are > nonzero probabilities both the way, but once it is connected, then it > is not disconnected. The jargon can be looked up on Wikipedia. -- He is not here; but far away The noise of life begins again And ghastly thro' the drizzling rain On the bald street breaks the blank day. === Subject: Statistics posting-account=aZAW_QoAAABHaPf02XTsrle7hWr4wpvx AppleWebKit/525.19 (KHTML, like Gecko) Version/3.1.2 Safari/525.21,gzip(gfe),gzip(gfe) Need to pick a simple topic for a short project. Must use true data from true sources. Topics may include but no limited to Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression. === Subject: Re: Statistics > Need to pick a simple topic for a short project. Must use true data > from true sources.... You may get helpful suggestions from the or news group. Ken Pledger. === Subject: Re: i want to make a science project > hi, > can u please suggest a science project. > i want a project of college level.i m doing b sc final year. > project may be on any topic. > u can also suggest a science project for a secondary level student. Research spelling and punctuation. After that you can tackle the theory of the electron. -- Michael Press === Subject: Re: i want to make a science project Distribution: world > hi, > can u please suggest a science project. > i want a project of college level.i m doing b sc final year. > project may be on any topic. > u can also suggest a science project for a secondary level student. You know what the inevitable consequence of spelling flames is, don't you? >I suggest you learn to write before you try anting too hard for you. anything >For your science project, go back to grade school and learn simple basic >things like punctuation. After accomplishing that, which you should have >done years ago, we can proceed to more advance stuff like the basics of advanced >speach such as enunciation. speech For extra credit, point out *my* spelling errors. -- Michael F. Stemper #include Time flies like an arrow. Fruit flies like a banana. === Subject: Re: i want to make a science project Distribution: world posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Sep 11, 10:48 am, mstem...@walkabout.empros.com (Michael Stemper) > hi, > can u please suggest a science project. > i want a project of college level.i m doing b sc final year. > project may be on any topic. > u can also suggest a science project for a secondary level student. You know what the inevitable consequence of spelling flames is, > don't you? I did not interpret William Elliot's message as a spelling flame, but rather as a sloppiness, laziness and don't-give-a damn flame. I agree with him. If the OP cannot be bothered to even try to communicate properly, why should anyone waste time responding with helpful information? R.G. Vickson I suggest you learn to write before you try anting too hard for you. anything For your science project, go back to grade school and learn simple basic >things like punctuation. After accomplishing that, which you should have >done years ago, we can proceed to more advance stuff like the basics of advanced speach such as enunciation. speech For extra credit, point out *my* spelling errors. -- > Michael F. Stemper > #include Fruit flies like a banana. === Subject: Re: i want to make a science project Distribution: world >hi, >can u please suggest a science project. >i want a project of college level.i m doing b sc final year. >project may be on any topic. Why, are you majoring in science? -- Michael F. Stemper #include Time flies like an arrow. Fruit flies like a banana. === Subject: Re: i want to make a science project posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) If you must indulge in spelling/grammar flames (well-deserved in this case), make sure your own spelling/grammar is immaculate. > I suggest you learn to write before you try anting too hard for you. anting? > done years ago, we can proceed to more advance stuff like the basics of advanced > speach such as enunciation. speech You might also have pointed out the OP's omitted apostrophe. Victor Meldrew I don't believe it! === Subject: Re: i want to make a science project posting-account=gpERugkAAAB5_qKVhbO9UpGpOXFNrIYf 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > hi, > can u please suggest a science project. > i want a æproject of college level.i m doing b sc final year. > project may be on any topic. > u can also suggest a science project for a secondary level student. Go to the city dump and get thousands of old camp lanterns, pull out their elements (made of Thorium) and then get some smoke detectors (Americurium) as your neutron seed. Irradiate yourself for fund. Socks === Subject: Re: i want to make a science project > Go to the city dump and get thousands of old camp > lanterns, pull out their elements (made of Thorium) Not any more. Old camp lanterns with Welsbach mantles containing Thorium were discontinued long time ago. Modern lanterns with Welsbach mantles now use yttrium/zirconium oxides. They are not as efficient as Thorium mantles, but they are not radioactive either. Besides, even if she found Thorium mantles, after the mantle flares up for the first time, it burns up, leaving only a crusty residue, which will convert to ash/dust if you knock-off the lantern ;o) > and then get some smoke detectors (Americurium) as > your neutron seed. Irradiate yourself for fund. > Socks -- I.N. Galidakis === Subject: Re: i want to make a science project posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > hi, > can u please suggest a science project. > i want a æproject of college level.i m doing b sc final year. > project may be on any topic. > u can also suggest a science project for a secondary level student. Build a bigger one of these http://en.wikipedia.org/wiki/Large Hadron Collider You are welcome! === Subject: Re: i want to make a science project posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) hi, > can u please suggest a science project. > i want a æproject of college level.i m doing b sc final year. > project may be on any topic. > u can also suggest a science project for a secondary level student. Build a bigger one of thesehttp://en.wikipedia.org/wiki/Large Hadron Collider You are welcome! Seriously - it seems to me that you should be asking your professor for examples of projects he/she deems worthy. I am not sure anyone here can make good suggestions with the information we currently have. Have you asked your instructor(s)? ~A === Subject: Terminology: another word for azimuthal posting-account=x5CZsAoAAADQB5YGnbi790y2igTGpwCe Gecko/20061201 Firefox/2.0.0.16 (Ubuntu-feisty),gzip(gfe),gzip(gfe) Spherical coordinates are radial (how far from the centre), azimuthal (measured over 2*pi or 360Á around a reference great circle), and polar (between +90Á and -90Á along arcs normal to the reference great circle). Examples of azimuthal and polar are longitude and latitude, or right ascension and declination. I am editing a text and would like to replace the word azimuthal, provided there is an alternative that would be recognized by most readers. The reason is that I want to avoid confusion between the terms azimuthal, which refers to spherical coordinate systems in general, and azimuth, which is a specific azimuthal coordinate (measured around the observer's horizon). Is there such an alternative term? === Subject: Re: Terminology: another word for azimuthal Spherical coordinates are radial (how far from the centre), azimuthal (measured over 2*pi or 360Á around a reference great circle), and polar (between +90Á and -90Á along arcs normal to the reference great circle). Examples of azimuthal and polar are longitude and latitude, or right ascension and declination. I am editing a text and would like to replace the word azimuthal, provided there is an alternative that would be recognized by most readers. The reason is that I want to avoid confusion between the terms azimuthal, which refers to spherical coordinate systems in general, and azimuth, which is a specific azimuthal coordinate (measured around the observer's horizon). Is there such an alternative term? Are you going to replace polar as well (to avoid confusion because pole is a specific polar coordinate)? A word to the wise: additional terms create more confusion, not less. You are not going to replace any terms, you can only add more. Britain decides to go metric, the USA doesn't. When it's 30 degrees it's a hot day here and a cold day in the USA and the public refuses to give up the pint, which is why I buy milk for my coffee in quantities of 1.136 litres (a quart, which is NOT a quarter of a US gallon) . A Roman mile was 1000 paces, an imperial ton is 2240 lbs, a baker's dozen is 13, but not at my local bakery. Within a radius of 1.6 km of my home the speed limit is 30 miles per hour. The Canadian speed limit is 100 km/hour for miles and miles. The mile will remain because the USA has been laid out on a grid, the roads are in place. Start using French if you want to replace English terms, people simply won't buy your text. === Subject: Re: Terminology: another word for azimuthal >Spherical coordinates are radial (how far from the >centre), azimuthal (measured over 2*pi or 360=B0 >around a reference great circle), and polar (between >+90=B0 and -90=B0 along arcs normal to the reference >great circle). Examples of azimuthal and polar are >longitude and latitude, or right ascension and >declination. I am editing a text and would like to replace the word >azimuthal, provided there is an alternative that >would be recognized by most readers. The reason is >that I want to avoid confusion between the terms >azimuthal, which refers to spherical coordinate >systems in general, and azimuth, which is a specific >azimuthal coordinate (measured around the >observer's horizon). Is there such an alternative >term? Is polar recognized as an angle by most readers? And should polar be accompanied with azimuthal? Why not e.g. equatorial instead? You could of course replace both terms by e.g. longitudal and latitudal, or something similar. Of course longitude/latitude also refers to one of several cases of specific coordinates, but at least people in general are familiar with these terms: fewer people know what azimuth is than those who know what longitude is. -- ---------------------------------------------------------------- Paul Schlyter, Grev Turegatan 40, SE-114 38 Stockholm, SWEDEN e-mail: pausch at stjarnhimlen dot se WWW: http://stjarnhimlen.se/ === Subject: Re: Terminology: another word for azimuthal posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Spherical coordinates are radial (how far from the > centre), azimuthal (measured over 2*pi or 360Á > around a reference great circle), and polar (between > +90Á and -90Á along arcs normal to the reference > great circle). Examples of azimuthal and polar are > longitude and latitude, or right ascension and > declination. I am editing a text and would like to replace the word > azimuthal, provided there is an alternative that > would be recognized by most readers. The reason is > that I want to avoid confusion between the terms > azimuthal, which refers to spherical coordinate > systems in general, and azimuth, which is a specific > azimuthal coordinate (measured around æthe > observer's horizon). Is there such an alternative > term? The key is the phrase *by most readers*. I'd have to say that there probably isn't such a term, and it's regrettable because your conundrum is a very common one. Depending on what field you work in or what country you come from, the conventions and names can be different. My suggestion would be perhaps not to worry so much about naming things if it is not necessary. If you can define phi and give a good illustration of it, then simply use the letter phi and not the name (which is probably simpler and more compact anyways). HTH, M === Subject: Re: Terminology: another word for azimuthal posting-account=x5CZsAoAAADQB5YGnbi790y2igTGpwCe Gecko/20061201 Firefox/2.0.0.16 (Ubuntu-feisty),gzip(gfe),gzip(gfe) conclusion. As for polar I do, in fact, intend to use transverse instead. I won't use equatorial or longitud(in)al for the same reason that I hesitate to use azimuthal: these terms all refer to specific coordinate systems while I wish to convey a general notion that includes all the === Subject: Re: Terminology: another word for azimuthal posting-account=zga2wgoAAAD_6fmi3XyA1bMyNINP0zBK WorldLynx),gzip(gfe),gzip(gfe) > conclusion. As for polar I do, in fact, intend to > use transverse instead. I won't use equatorial > or longitud(in)al for the same reason that I > hesitate to use azimuthal: these terms all refer > to specific coordinate systems while I wish to > convey a general notion that includes all the Sorry for the glitch on my first post-- I apparently hit a key that I wasn't supposed to! So, I'll try again. When I used to teach practical astronomy and orbital mechanics, before getting to the specific coordinates of longitude, latitude, azimuth, and the like, I used generic terms such as the following: fundamental circle secondary circle pole of the fundamental circle Then we invented names for the various angular distances representing the desired coordinates. One student used fundis and secdis ; another suggested dafunc (meaning: distance along fundamental circle) and dasecc. Other names were tried, but I don't recall that any of them lasted very long, and then just in my class. Maybe this might give you some ideas. Anyway, best wishes in your activities! Grover Hughes === Subject: Re: Terminology: another word for azimuthal posting-account=zga2wgoAAAD_6fmi3XyA1bMyNINP0zBK WorldLynx),gzip(gfe),gzip(gfe) > conclusion. As for polar I do, in fact, intend to > use transverse instead. I won't use equatorial > or longitud(in)al for the same reason that I > hesitate to use azimuthal: these terms all refer > to specific coordinate systems while I wish to > convey a general notion that includes all the When I used to teach courses in practical astronomy and orbital mechanics, the generic terms I used before getting into the specifics of longitude, latitude, azimuth, and so on were: fundamental circle pole of the fundamental circle === Subject: Re: Finding all paths between 2 nodes undirected multigraph posting-account=cHhEbwkAAACkacPfIKvF7-VAmdyJ8Ede .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; TerraClient 1.0),gzip(gfe),gzip(gfe) I have a undirected multigraph i.e there can be more than 1 edge > between 2 vertices in the undirected graph. æ0 --a-- 1 --b-- 2 > æ| c | d | Vertices: 0, 1, 2 > Edges: > æ 0 - 1, Edge a > æ 0 - 1, Edge c > æ 1 - 2, Edge b > æ 1 - 2, Edge d Suppose I want to find all the paths between vertices 0 and 2. The > answer is: [a,b], [a,d], [c,b], [c,d]. In my implementation, I use a 'pathList' (to store each discovered > path) and a stack-based DFS. In my DFS implementation: stack.push( target(starting vertex) > while (!stack.empty()) > { > æ æ edge e = stack.pop() > æ æ v = target(e) > æ æ if (color(v) == WHITE) > æ æ{ > æ æ æ // tree edge > æ æ æ stack.push( adj edges(v) ) > æ æ æ pathList.append(e) > æ æ æ if (v == target vertex) > æ æ æ æ æ break; > æ æ} > æ æelse if (color(v) == GRAY) || (color(v) == BLACK) > æ æ æ æcontinue;} color(v) = BLACK How can I extend this skeleton to allow for continuing the exploration > without restarting from the DFS walk from the start vertex? Instead, I > should be able to backtrack from the target vertex back and pop out > the unwanted stack and pathList entries. Not sure if you are committed to this algorithm. If not, you could use recursion. The path from source vertex S to target vertex T is path from S to an intermediate vertex I concatenated with the path from I to T. So you would write a function Path path (Vertex S, Vertex T) { if (S == T) return Path(); // empty path assumes no self-cycles // Accumulate all white edges sourced by S. // If there are no white edges, then that means // you are trapped, so... return Path(); // again empty path // For each such edge e, // 1. mark it visited // 2. determine target vertex I of e // 3. pi = path(I, T) // if (pi.is empty()) // pf = Path(); // empty path // else // pf = e + pi // mark all e as unvisited // return pf. } I haven't checked this code, so it is most likely incorrect, but this is basic idea. -Le Chaud Lapin- === Subject: The circumference of the Universe posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) If the universe has uniform curvature, then it is easy to relate its circumference to its curvature: if finite, it is positively curved, if countably infinite, it is flat, and a negative or hyperbolic curvature gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we would have to say that the circumference would be the greater of those. So if we were elliptical in one direction and hyperbolic in the other, we would have like a hyperboloid of one sheet (albeit in 4 dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the universe being projective (although perhaps there could not be a big bang then) and the curvature doesn't matter as all quadratics and the flat surface are projectively equivalent. But distance would then have no fixed meaning wouldn't it, giving a very strange kind of physics - Mach's principle! Andrew Usher === Subject: Re: The circumference of the Universe posting-account=504E-QkAAAA2v90r8nGnJKpfySa_yBSU 5.1),gzip(gfe),gzip(gfe) > If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! Andrew Usher Think orders of magnitude larger, but don't give up your day job. Harry C. === Subject: Re: The circumference of the Universe Since, long ago, the circumference of the universe divided by c was less than the current age of the universe, is it possible that one of the distant galaxies we see is really our own galaxy, as it appeared 10 billion years ago or something, as its light wrapped around the early universe? If so, would we know it? === Subject: Re: The circumference of the Universe posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > Think orders of magnitude larger, but don't give up your day job. Idiot. Andrew Usher === Subject: Re: The circumference of the Universe posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) > If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! Andrew Usher Round 4th dimension === Subject: Re: The circumference of the Universe > If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! Andrew Usher No Center http://www.astro.ucla.edu/~wright/nocenter.html http://www.astro.ucla.edu/~wright/infpoint.html Also see Ned Wright's Cosmology Tutorial http://www.astro.ucla.edu/~wright/cosmolog.htm http://www.astro.ucla.edu/~wright/cosmology_faq.html http://www.astro.ucla.edu/~wright/CosmoCalc.html WMAP: Foundations of the Big Bang theory http://map.gsfc.nasa.gov/m_uni.html WMAP: Tests of Big Bang Cosmology http://map.gsfc.nasa.gov/m_uni/uni_101bbtest.html === Subject: Re: The circumference of the Universe posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! Andrew Usher No Center > http://www.astro.ucla.edu/~wright/nocenter.html > http://www.astro.ucla.edu/~wright/infpoint.html Also see Ned Wright's Cosmology Tutorial > http://www.astro.ucla.edu/~wright/cosmolog.htm > http://www.astro.ucla.edu/~wright/cosmology_faq.html > http://www.astro.ucla.edu/~wright/CosmoCalc.html WMAP: Foundations of the Big Bang theory > http://map.gsfc.nasa.gov/m_uni.html WMAP: Tests of Big Bang Cosmology > http://map.gsfc.nasa.gov/m_uni/uni_101bbtest.html Sam Wormley misses the point again. 1. I was talking about hypothetical worlds only, not the real universe. 2. There's no center in any model I mentioned, either. Andrew Usher === Subject: Re: The circumference of the Universe posting-account=HZYXOQoAAAB0CZtsRCtABgys4tHYIT8J 3.2.0; .NET CLR 1.1.4322; InfoPath.2),gzip(gfe),gzip(gfe) > If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! > In fact if you could go roung the Universe this would be reflected in the 2.7K radiation. Its pattern would be repeating. This has been looked for but not found. - Ian Parker === Subject: Re: The circumference of the Universe posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > In fact if you could go roung the Universe this would be reflected in > the 2.7K radiation. Its pattern would be repeating. This has been > looked for but not found. How would this work? I would think that if the Universe were spherical all the images of a point would coincide. Andrew Usher === Subject: Re: The circumference of the Universe posting-account=HZYXOQoAAAB0CZtsRCtABgys4tHYIT8J 3.2.0; .NET CLR 1.1.4322; InfoPath.2),gzip(gfe),gzip(gfe) In fact if you could go roung the Universe this would be reflected in > the 2.7K radiation. Its pattern would be repeating. This has been > looked for but not found. How would this work? I would think that if the Universe were spherical > all the images of a point would coincide. > When you are looking at 2.7K you are looking at a smaller Universe some 350,000 years old. We in fact see a Universe which will expand to some 80GPa (NOT the ~4GPa of 13.7 billion LY). Ok this is a little bit hard to visualise. If a Universe were a sphere we could go round it. If it had a radius (say) of 2GPa we would see the Universe repeated in the 2.7K radiation. We can conclude, since WMAP shows no repeating structures that the Universe is at least 80GPa in size and has open topology on that scale. Sam Wormley has given some excellent references which I think help you to visualize this. A little difficult I know. - Ian Parker === Subject: Re: The circumference of the Universe posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > In fact if you could go roung the Universe this would be reflected in > the 2.7K radiation. Its pattern would be repeating. This has been > looked for but not found. How would this work? I would think that if the Universe were spherical > all the images of a point would coincide. When you are looking at 2.7K you are looking at a smaller Universe > some 350,000 years old. We in fact see a Universe which will expand to > some 80GPa (NOT the ~4GPa of 13.7 billion LY). Ok this is a little bit > hard to visualise. If a Universe were a sphere we could go round it. > If it had a radius (say) of 2GPa we would see the Universe repeated in > the 2.7K radiation. We can conclude, since WMAP shows no repeating > structures that the Universe is at least 80GPa in size and has open > topology on that scale. The symbol for parsecs in pc; Pa is pascals. Anyway, I certainly can visualise it; it's obvious that all images of the same object would coincide. Andrew Usher === Subject: The circumference of the Universe posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) If the universe has uniform curvature, then it is easy to relate its circumference to its curvature: if finite, it is positively curved, if countably infinite, it is flat, and a negative or hyperbolic curvature gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we would have to say that the circumference would be the greater of those. So if we were elliptical in one direction and hyperbolic in the other, we would have like a hyperboloid of one sheet (albeit in 4 dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the universe being projective (although perhaps there could not be a big bang then) and the curvature doesn't matter as all quadratics and the flat surface are projectively equivalent. But distance would then have no fixed meaning wouldn't it, giving a very strange kind of physics - Mach's principle! Andrew Usher === Subject: Re: The circumference of the Universe posting-account=p0JNqwkAAAChY16-5zbk2O2xWfBB6K-z Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! Andrew Usher Now, are we speaking of the circumference at our causal horizon, or > the circumference of the space which is external to the causal horizon? I was talking about the circumference at infinity (mathematical coordinates). I was not discussing the real universe, but hypothetical geometries. Andrew Usher === Subject: Re: The circumference of the Universe posting-account=HZYXOQoAAAB0CZtsRCtABgys4tHYIT8J 3.2.0; .NET CLR 1.1.4322; InfoPath.2),gzip(gfe),gzip(gfe) If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! Andrew Usher Now, are we speaking of the circumference at our causal horizon, or > the circumference of the space which is external to the causal horizon?- Hide quoted text - > I am talking about the minimum circuferance (80GPa on WMAP) not the event horizon which is, of course, 13.7 billion LY. - Ian Parker === Subject: Re: The circumference of the Universe > If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. > Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. > But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! > Andrew Usher > Now, are we speaking of the circumference at our causal horizon, or > the circumference of the space which is external to the causal horizon?- Hide quoted text - > I am talking about the minimum circuferance (80GPa on WMAP) not the > event horizon which is, of course, 13.7 billion LY. - Ian Parker Giga Pascals? === Subject: Re: The circumference of the Universe posting-account=HZYXOQoAAAB0CZtsRCtABgys4tHYIT8J 3.2.0; .NET CLR 1.1.4322; InfoPath.2),gzip(gfe),gzip(gfe) If the universe has uniform curvature, then it is easy to relate its > circumference to its curvature: if finite, it is positively curved, if > countably infinite, it is flat, and a negative or hyperbolic curvature > gives an uncountably great perimeter. > Now if we had different curvatures in different directions, then we > would have to say that the circumference would be the greater of > those. So if we were elliptical in one direction and hyperbolic in the > other, we would have like a hyperboloid of one sheet (albeit in 4 > dimensions) and that again has an uncountable circumference. > But Projective geometry is self-consistent so we can imagine the > universe being projective (although perhaps there could not be a big > bang then) and the curvature doesn't matter as all quadratics and the > flat surface are projectively equivalent. But distance would then have > no fixed meaning wouldn't it, giving a very strange kind of physics - > Mach's principle! > Andrew Usher > Now, are we speaking of the circumference at our causal horizon, or > the circumference of the space which is external to the causal horizon?- Hide quoted text - I am talking about the minimum circuferance (80GPa on WMAP) not the > event horizon which is, of course, 13.7 billion LY. æ - Ian Parker æ æGiga Pascals?- Hide quoted text - - Show quoted text - I meant Parsecs. - Ian Parker === Subject: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) I don't know how to solve this problem in lattice theory: http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice Maybe you can solve? === Subject: Re: A conjecture in lattice theory > I don't know how to solve this problem in lattice theory: > http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice Maybe you can solve? | I will call center Z(mathfrak{A}) of a bounded lattice | mathfrak{A} the sublattice of all complemented elements of a | mathfrak{A} . The center of a bounded lattice A, Z(A) = Z = { x | some y with x + y = 1, xy = 0 }. . . where 1 = top and 0 = bottom. Is Z a sublattice? If a,b in Z, then some x,y with a + x = 1 = b + y, ax = 0 = by. a + b + x + y = 1 + 1 = 1; ab(x + y) = a0 + b0 = 0 + 0 = 0. Ok, Z is closed under sup and inf. | I will call a bounded lattice mathfrak{A} a lattice with | seaparable center when | forall x,yinmathfrak{A}: (xcap y=0Rightarrow exists Xin | Z(mathfrak{A}):(xsubseteq Xwedge Xcap y = 0)) . Yicks, TeX is such a mess. for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0). Is that correct translation? | Equivalently a bounded lattice with separable center is such a bounded | lattice mathfrak{A} that | forall x,yinmathfrak{A}:(xcap y=0Rightarrowexists X,Yin | Z(mathfrak{A}):(xsubseteq Xwedge ysubseteq Ywedge Xcap Y = 0)) for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0) Is that correct translation? . | Conjecture There exist bounded lattices which are not with separable | center. 1 | a / x y / 0 Z = {0,1}, xy = 0 === Subject: Re: A conjecture in lattice theory I don't know how to solve this problem in lattice theory: > http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice | I will call center Z(mathfrak{A}) of a bounded lattice > | mathfrak{A} the sublattice of all complemented elements of a > | mathfrak{A} . The center of a bounded lattice A, > Z(A) = Z = { x | some y with x + y = 1, xy = 0 }. > . . where 1 = top and 0 = bottom. Is Z a sublattice? > If a,b in Z, then some x,y with a + x = 1 = b + y, ax = 0 = by. > a + b + x + y = 1 + 1 = 1; ab(x + y) = a0 + b0 = 0 + 0 = 0. Whoops, I've assume A is distributive. Worse than that, it's nonthink. Assuming A is distributive a + b + xy = (a + b + x)(a + b + y) = 1 (a + b)xy = axy + bxy = 0; a + b in Z ab + x + y = (a + x + y)(b + x + y) = 1 ab(x + y) = abx + aby = 0; ab in Z > Ok, Z is closed under sup and inf. Provided A is distributive. Hm, stronger should be possible. > | I will call a bounded lattice mathfrak{A} a lattice with > | seaparable center when > | forall x,yinmathfrak{A}: (xcap y=0Rightarrow exists Xin > | Z(mathfrak{A}):(xsubseteq Xwedge Xcap y = 0)) . Yicks, TeX is such a mess. for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0). Is that correct translation? | Equivalently a bounded lattice with separable center is such a bounded > | lattice mathfrak{A} that > | forall x,yinmathfrak{A}:(xcap y=0Rightarrowexists X,Yin > | Z(mathfrak{A}):(xsubseteq Xwedge ysubseteq Ywedge Xcap Y = 0)) for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0) Is that correct translation? > . > | Conjecture There exist bounded lattices which are not with separable > | center. 1 > | > a > / > x y > / > 0 Z = {0,1}, xy = 0 === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > I don't know how to solve this problem in lattice theory: >http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice | I will call center Z(mathfrak{A}) of a bounded lattice > | mathfrak{A} the sublattice of all complemented elements of a > | mathfrak{A} . My mistake also. I should say bounded distributive lattice instead of bounded lattice. > Is Z a sublattice? > If a,b in Z, then some x,y with a + x = 1 = b + y, ax = 0 = by. > a + b + x + y = 1 + 1 = 1; ab(x + y) = a0 + b0 = 0 + 0 = 0. Whoops, I've assume A is distributive. > Worse than that, it's nonthink. Assuming A is distributive > a + b + xy = (a + b + x)(a + b + y) = 1 > (a + b)xy = axy + bxy = 0; a + b in Z ab + x + y = (a + x + y)(b + x + y) = 1 > ab(x + y) = abx + aby = 0; ab in Z Ok, Z is closed under sup and inf. Provided A is distributive. Hm, stronger should be possible. You can assume that it is distributive. > | Conjecture There exist bounded lattices which are not with separable > | center. 1 > | > a > / > x y > / > 0 Z = {0,1}, xy = 0 Should say There exist bounded distributive lattices which are not with separable center. Oops, is the William Elliot's lattice distributive? === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > I don't know how to solve this problem in lattice theory: >http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice I have corrected the problem statement adding the word distributive where necessary. See http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice The problem is open again. === Subject: Re: A conjecture in lattice theory I have corrected the problem statement adding the word distributive > where necessary. See > http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice The problem is open again. > It's an open and shut problem. Some of your research work interests me. As it's in pdf formate, I can't read it. Do you have (yuck) TeX version, or (yah) ASCII version? Let A be a bounded lattice. The center of A, Z = Z(A) = { x | some y with x + y = 1, xy = 0 }. When A is distributive, then Z(A) is a sublattice. If a,b in Z, then some x,y (also in Z) with . . a + x = 1 = b + y, ax = 0 = by Thus a + b, ab in Z, for . . a + b + xy = (a + b + x)(a + b + y) = 1 . . (a + b)xy = axy + bxy = 0; a + b in Z . . ab + x + y = (a + x + y)(b + x + y) = 1 . . ab(x + y) = abx + aby = 0; ab in Z Here's an example of a bounded lattice with a center that's not a sublattice. 1 / | r a s | / | x y / / 0 Z = { 0, x,y, r,s, 1 }; x + y = a not in Z. Open problem. If A is a bounded lattice, is Z(A) with the inherited order a lattice? When it is, is it distributive? No, Z above isn't distributive. Within Z . . r(x + y) = r1 = r . . rx + ry = x + 0 = x -- Speculation Let L be a lattice and Z(a,b) = Z({ x | a <= x <= b }). Can L be described by the centers? What significance does a center hold? What significance does a maximal center have? Let L = R. Then Z(a,b) = { a,b } or nulset and every two element center is maximal. -- Exercise. A complemented lattice is distributive iff complements are unique. -- The center Z is separable when . . for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0). equivalently . . for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0) Conjecture There exist bounded distributive lattices which are not with separable center. 1 | a / x y / 0 Z = {0,1}, xy = 0 This has two linear sublattices. Thus to check for distributivity, only expressions with both x and y need to be checked. That is. . . u(x + y), u = 0, x,y, a, 1 . . x(u + y) . . y(u + x) Which is two cases, taking symmetry into account. Subcases are u = x; u = a,1; u = 0. Easier is to notice that it doesn't have either of the forbidden sublattices. ---- === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Some of your research work interests me. > As it's in pdf formate, I can't read it. > Do you have (yuck) TeX version, or (yah) ASCII version? William Elliot, I will send you email with my research in the format HTML with images (for formulas). Does anybody also have problem with reading PDF? If yes, I may consider publishing my work in HTML format. (To do this is desirable to write a multiformat-conversion script in Schema (the scripting language of TeXmacs, http://www.texmacs.org), a programming language I don't know.) You can however read PDF with the following free program, please download it: http://www.adobe.com/go/gntray_dl_get_reader === Subject: Re: A conjecture in lattice theory Some of your research work interests me. > As it's in pdf formate, I can't read it. > Do you have (yuck) TeX version, or (yah) ASCII version? William Elliot, I will send you email with my research in the format > HTML with images (for formulas). That will not suffice as I do not have graphic capacity. > Does anybody also have problem with reading PDF? If yes, I may consider > publishing my work in HTML format. (To do this is desirable to write a > multiformat-conversion script in Schema (the scripting language of > TeXmacs, http://www.texmacs.org), a programming language I don't know.) > The prefered math writing software is TeX. > You can however read PDF with the following free program, please > download it: > http://www.adobe.com/go/gntray_dl_get_reader It's do large to download. pdf2txt@adobe.com will convert pdf files to txt files. It works well for simple written material but not for math formulas, diagrams, etc. The problem of showing that a complement lattice with unique complements is distributive, is harder than I expected. Have you any suggestions? === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Some of your research work interests me. > As it's in pdf formate, I can't read it. > Do you have (yuck) TeX version, or (yah) ASCII version? However I doubt whether a man is able to read these messy formulas in LaTeX format without special software converting LaTeX to graphical presentation. > The prefered math writing software is TeX. It _was_ TeX in the past. Now the preferred math writing software is TeXmacs, a WYSIWYG math texts editor, see http://www.texmacs.org TeXmacs has the ability to export to LaTeX (and several other formats). > The problem of showing that a complement lattice with unique > complements is distributive, is harder than I expected. Have > you any suggestions? No. I'm not interested in this problem. I'm interested only about lattices which are already known to be distributive. === Subject: Re: A conjecture in lattice theory Some of your research work interests me. > As it's in pdf formate, I can't read it. > Do you have (yuck) TeX version, or (yah) ASCII version? However I doubt whether a man is able to read these messy formulas in > LaTeX format without special software converting LaTeX to graphical > presentation. The prefered math writing software is TeX. It _was_ TeX in the past. Now the preferred math writing software is > TeXmacs, a WYSIWYG math texts editor, see http://www.texmacs.org TeXmacs has the ability to export to LaTeX (and several other > formats). > The advantage of TeX is that it's files are simple ascii files. Is TeXmacs like that? > The problem of showing that a complement lattice with unique > complements is distributive, is harder than I expected. Have > you any suggestions? No. I'm not interested in this problem. I'm interested only about > lattices which are already known to be distributive. It's useful to show when a lattice isn't distributive. ---- === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > The prefered math writing software is TeX. It _was_ TeX in the past. Now the preferred math writing software is > TeXmacs, a WYSIWYG math texts editor, see http://www.texmacs.org TeXmacs has the ability to export to LaTeX (and several other > formats). The advantage of TeX is that it's files are simple ascii files. > Is TeXmacs like that? Yes, both TeXmacs texts and TeXmacs styles are plain ASCII. These are readable and probably more readable that TeX. === Subject: Re: A conjecture in lattice theory The prefered math writing software is TeX. > It _was_ TeX in the past. Now the preferred math writing software is > TeXmacs, a WYSIWYG math texts editor, see http://www.texmacs.org > TeXmacs has the ability to export to LaTeX (and several other > formats). The advantage of TeX is that it's files are simple ascii files. > Is TeXmacs like that? Yes, both TeXmacs texts and TeXmacs styles are plain ASCII. > These are readable and probably more readable that TeX. > Hum. What happened to the list of your papers? The web site for them isn't in this thread. ---- === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Hum. What happened to the list of your papers? > The web site for them isn't in this thread. http://www.mathematics21.org/algebraic-general-topology.html === Subject: Re: A conjecture in lattice theory The web site for them isn't in this thread. http://www.mathematics21.org/algebraic-general-topology.html > Here's a list of your works along with some of your hand waving bravado. It seems that your research is yet incomplete, thus immature for climaing a Nobel prize. You are too impatient. The list is presented here in order of readablity and interest. I will review the first two first becauase they are in readable ascii for the most part but in need of editing for typos. As notions vary, it'd be helpful to include some introductory material such as are you considering binary relaltions on a single set S, that is a subset of S^2, or of binary relation between two sets, X,Y. ie a subset of XxY? The other point to clarify is what is the domain of a relation and what is a restriction of a relation? Knowing that, I will review the first and as time allows the second paper. Have I questions, clarifications, corrections or comments to make, I will place them in a new thread Ordering Binary Relations. [10]Vertical order of binary relations Defined a partial order relation between binary relations f and g by the formula g = f|[dom g]. [11]Theorem about binary relation limited to a set Theorem (with proof), expanding and generalizing the statement: For two binary relations f and g the formula g = f|[dom g]is equivalent to conjunction of g being a subset of f and the formula g g-1 = f g-1 (or equivalently g g-1 = g f^-1). -- Set Theoretic Filters (PDF, very preliminary draft) Considers the lattice of set theoretic filters. Filters on Posets (PDF, very preliminary draft) Partially Ordered Categories with Inverses (PDF) Defined partially ordered category with inverses of morphisms. For such categories defined monovalued morphisms and entirely defined morphisms. Funcoids and Reloids (PDF, draft) Consider generalizations of proximity spaces and uniform spaces. Generalized Continuousness (PDF, draft) Defines continuousness algebraically hiding old epsilon-delta notion under a smart algebra. Generalizes continuousness, uniform continuousness, and proximity-continuousness in one formula. Connectedness of funcoids and reloids (PDF, draft) Defined the notion of connectedness for funcoids and reloids. Shown how connectedness of funcoids is related with connectedness of reloids. Convergence of funcoids (PDF, draft) Defined the notion of convergence and limit for funcoids. Open Problems in AGT (PDF) List unsolved problems and conjectures in the field of AGT. -- Achievements and advantages of AGT * general topology expressed in simple algebraic operations * simplicity of operating with infinities, as infinities now can be comprehended as something whole, not a mess of parts * two-three line proofs of some old pages length analysis theorems * multivalued functions are now so simple to study as single valued * frees analysis from its messy epsilon-delta notation * analysis of non-continuous functions * partially formally unifies math analysis and discrete mathematics * not limited in any way to metrizable spaces and countable sets -- AGT isn't a continuation of former functional analysis research, it is a new beginning almost from scratch requiring little beyond first level courses to understand it. This new research field both generalizes former analysis and gives new theorems/concepts not having analogs in old theories. Several different theorems of analysis often collapse into one AGT equation of which they are obvious consequences. AGT is very abstract, indeed even the current level of AGT knowledge often allowed me to find simple solutions of practical tasks (such as calculations of infinite sums). I have not yet reached the level of integrals in the synthesis research. AGT is a kinda thinking with equations. No real numbers analysis expressiveness with visual images preserved. That is not needed anyway as the equations of AGT are even more clear than graphics of old analysis. AGT is simple, natural, and beautiful. Note that Algebraic General Topology being a generalization of General Topology has nothing in common (except of the name) with Algebraic Topology. Math synthesis is a generalization of functional analysis. -- [23]Algebraic General Topology at WikInfo. [25]My homepage [26]My math page 10. http://www.mathematics21.org/misc/vertical-order.html 11. http://www.mathematics21.org/misc/limiting-binary-relations-theorem.html 23. http://www.wikinfo.org/index.php/Algebraic_general_topology 25. http://portonvictor.org/ 26. http://www.mathematics21.org/index.html ---- === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Hum. What happened to the list of your papers? > The web site for them isn't in this thread. http://www.mathematics21.org/algebraic-general-topology.html Here's a list of your works along with some of your hand waving > bravado. It seems that your research is yet incomplete, thus > immature for climaing a Nobel prize. You are too impatient. Not Nobel Prize (there are no Nobel Prize for math works) but Abel Prize. In the prize rules there are no requirement that the research should be complete. They say that Abel Prize is for (among other) those who are opened new major research areas. My Algebraic General Topology is opening up a new major research area. I hope this may be enough to receive Abel Prize. And philosophically: Can a research be ever completed? Isn't it infinite. The biggest gap in my research is yet missing research of objects similar to compact spaces. Is missing it enough to prevent me from Abel Prize? ;-) BTW, I yet plea to nominate me for Abel Prize: http://www.mathematics21.org/abel-prize.html You say that I'm to impatient. My impatience is caused mainly by the desire to receive money to be able to fire from the daily job and dedicate myself things I deem more important such as math research. So it is somehow hard for me to to the research before receiving Abel Prize. This makes the things reverse: first prize and then research :-) > ascii for the most part but in need of editing for typos. As > notions vary, it'd be helpful to include some introductory material > such as are you considering binary relaltions on a single set S, > that is a subset of S^2, or of binary relation between two sets, > X,Y. ie a subset of XxY? The other point to clarify is what > is the domain of a relation and what is a restriction of a relation? I'm going to write a book Filters on Posets (which will replace the materials probably will be put to that my book to be able to extend its size to more that about 100 pages, as publishing companies > Knowing that, I will review the first and as time allows the second paper. > Have I questions, clarifications, corrections or comments to make, I will > place them in a new thread Ordering Binary Relations. Better you'd also email me (porton@narod.ru) as I'm not sure that I will read the thread Ordering Binary Relations. Why you call the thread Ordering Binary Relations? I don't understand how this is related with ordering binary relations. First and second papers are Set Theoretic Filters and Filters on Posets? The second is a newer but less complete version of the first. Think about Filters on Posets as a development version of more stable Set Theoretic Filters. I have already said that I am going to write a book Filters on === Subject: Re: A conjecture in lattice theory ascii for the most part but in need of editing for typos. As > notions vary, it'd be helpful to include some introductory material > such as are you considering binary relaltions on a single set S, > that is a subset of S^2, or of binary relation between two sets, > X,Y. ie a subset of XxY? The other point to clarify is what > is the domain of a relation and what is a restriction of a relation? I'm going to write a book Filters on Posets (which will replace the > materials probably will be put to that my book to be able to extend > its size to more that about 100 pages, as publishing companies Knowing that, I will review the first and as time allows the second paper. > Have I questions, clarifications, corrections or comments to make, I will > place them in a new thread Ordering Binary Relations. Better you'd also email me (porton@narod.ru) as I'm not sure that I > will read the thread Ordering Binary Relations. > The advantage of posting here is that others may participate in the thread. Will you answer my questions? Are binary relations a subset of S^2 or of XxY ? What's an exact definition of domain and restriction? Wikipedia definitions aren't as exact as I'd like. > Why you call the thread Ordering Binary Relations? I don't > understand how this is related with ordering binary relations. > I'm starting with your two papers on binary relations as mentioned in my previous post and also mention in your web site in the misc. section of your works. After that, I will get to the next papers on filters which protent to be interesting, if presented in the propriate formate. I'll skip the paper on catagory theory as catagory theory doesn't interest me. > First and second papers are Set Theoretic Filters and Filters on > Posets? The second is a newer but less complete version of the first. > Think about Filters on Posets as a development version of more > stable Set Theoretic Filters. I have already said that I am going to write a book Filters on > Well, until then, I've your papers to read. === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Better you'd also email me (porton@narod.ru) as I'm not sure that I > will read the thread Ordering Binary Relations. The advantage of posting here is that others may participate in the > thread. Will you answer my questions? Are binary relations a subset of > S^2 or of XxY ? What's an exact definition of domain and restriction? > Wikipedia definitions aren't as exact as I'd like. One variant of understanding binary relations, is to consider it as sets of pairs of elements of a certain (fixed) set (which may be called the universal set). This definition of binary relation well suits for my Theorem about limiting binary relations. The domain of a binary relation f is dom f={a | (a;b)in f}. The restriction of a binary relation f to a set A is f|_{A}={(a;b)in f | ain A}. > Why you call the thread Ordering Binary Relations? I don't > understand how this is related with ordering binary relations. I'm starting with your two papers on binary relations as mentioned > in my previous post and also mention in your web site in the misc. section > of your works. Now I understood that you a telling about Misc section of my site, not the General Topology session. Misc session does not deserve Abel Prize. My serious works are in General Topology section: http://www.mathematics21.org/algebraic-general-topology.html However recently I have invented how my Theorem about binary relation limited to a set probably may be used in Algebraic General Topology. (Not now a time to speak about this however.) > After that, I will get to the next papers on filters which protent to be > interesting, if presented in the propriate formate. I'll skip the paper > on catagory theory as catagory theory doesn't interest me. My research is not category theory based like some other generalizations of point-set topology. It however significantly use category theory to define the notion of continuousness: http://www.mathematics21.org/binaries/continuousness.pdf If you want to deal with anything related to continuousness in my theory you need basic category theory. (I do not use advanced category theory indeed, only the basic.) === Subject: Re: A conjecture in lattice theory Don't you mean f = { a | some b with (a,b) in f } ? When f subset XxY, dom f = p1(f), where p1 is the first or X projection of XxY. > The restriction of a binary relation f to a set A is > f|_{A}={(a;b)in f | ain A}. f|A = f / AxY, / is intersection. > Now I understood that you a telling about Misc section of my site, > not the General Topology session. First things first. Besides it was readable while your others are not. Well mostly readable. It is missing many important math symbols. Most of which I've been able to desern. Here's a section I cannot understand Will you fill in the missing symbols? Proposition. f g <=> f* g* <=> f* g*. Theorem. f g iff x:( f*(x) = g*(x) f*(x) = ). Theorem. Vertical order is a meet-semilattice (that is has infimum of any two binary relations). Proof. This follows from f g <=> f* g* and that any two elements of the set of functions, whose images are sets of sets, have the infimum (namely set theoretic intersection). The operation * is a partial order isomorphism and so maps a semilattice to semilattice. Vertical infimum of two binary relations f and g can be so restored by the formula (f g)* = f* g*. End of proof.. Except for this snafu, I expect to quickly finish with your two misc. papers. The results I will post within a day or two after your reply. It will be in the thread Ordering Binary Relations. Look for it. You'll see major revisions and proofs in much greater detail and in a simpler fashion because of the use of algebraic set theory. ---- === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > The domain of a binary relation f is dom f={a | (a;b)in f}. Don't you mean f = { a | some b with (a,b) in f } ? Yes, I mean this. > When f subset XxY, dom f = p1(f), where > p1 is the first or X projection of XxY. The restriction of a binary relation f to a set A is > f|_{A}={(a;b)in f | ain A}. f|A = f / AxY, / is intersection. Yes. > First things first. Besides it was readable while your others are not. > Well mostly readable. It is missing many important math symbols. Most > of which I've been able to desern. Here's a section I cannot understand > Will you fill in the missing symbols? Oh, now I noticed that my XML processor was wrongly configured. So there were some missing symbols in http://www.mathematics21.org/misc/vertical-order.html and http://www.mathematics21.org/misc/limiting-binary-relations-theorem.html I have corrected these, now the Unicode symbols are in their places. === Subject: Re: A conjecture in lattice theory Victor Porton a .8ecrit : > Hum. What happened to the list of your papers? > The web site for them isn't in this thread. > http://www.mathematics21.org/algebraic-general-topology.html > Here's a list of your works along with some of your hand waving > bravado. It seems that your research is yet incomplete, thus > immature for climaing a Nobel prize. You are too impatient. Not Nobel Prize (there are no Nobel Prize for math works) but Abel > Prize. In the prize rules there are no requirement that the research should > be complete. They say that Abel Prize is for (among other) those who > are opened new major research areas. If your english and your reading abilities are par to your math abilities... Abel prize is for those who *have* opened etc., i.e for life works, time-recognized ourstanding achievements, etc. (see what , http://en.wikipedia.org/wiki/Abel_Prize ; the achievements of the later lureates are quite impressive too) So, even if you really were opening new domains, a try for a Fields medal would be a better guess. As for we nominating you, why dont you ask noomination for US presidency (or, better, for world presidency)? Your chances would actually be better, as some of us are votiers(or at least future voters)... === Subject: Re: A conjecture in lattice theory <48cce82b$0$7098$7a628cd7@news.club-internet.fr> posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Sep 14, 1:32 pm, Denis Feldmann be complete. They say that Abel Prize is for (among other) those who > are opened new major research areas. If your english and your reading abilities are par to your math > abilities... Abel prize is for those who *have* opened etc., i.e for It was a misspelling. I know English grammar rules about constructs like have opened. Sorry for a misspelling. === Subject: Re: A conjecture in lattice theory Victor Porton a .8ecrit : > On Sep 14, 1:32 pm, Denis Feldmann Victor Porton a .8ecrit : > In the prize rules there are no requirement that the research should > be complete. They say that Abel Prize is for (among other) those who > are opened new major research areas. > If your english and your reading abilities are par to your math > abilities... Abel prize is for those who *have* opened etc., i.e for It was a misspelling. I know English grammar rules about constructs > like have opened. Sorry for a misspelling. Good. Did you mispelled Abel prise for Fields medal too? And why not postullate for world president? === Subject: Re: A conjecture in lattice theory posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > I don't know how to solve this problem in lattice theory: >http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice > Yicks, TeX is such a mess. for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0). Is that correct translation? Yes. > for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0) Is that correct translation? Yes. > | Conjecture There exist bounded lattices which are not with separable > | center. 1 > | > a > / > x y > / > 0 Z = {0,1}, xy = 0 Certainly, I would to think about finite lattices myself before sending this conjecture to others. I used to use to think only about infinities, sorry. The problem was solved too easily. Should we keep it in Open Problem Garden or I should delete it? See http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice === Subject: Re: A conjecture in lattice theory Garden or I should delete it? Here's the problem that interests me. Let A be a bounded lattice. Let Z = { x | some y with x + y = 1, xy = 1 } . . = the set of all complemented elements of A. When A is distributive, then Z is a sublattice as I showed in another post. Is Z a sublattice when A isn't distributive? Why? -- A has separable center when > for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0). > for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0) | Conjecture There exist bounded lattices which are not with separable > | center. 1 > | > a > / > x y > / > 0 Z = {0,1}, xy = 0 > Certainly, I would to think about finite lattices myself before > sending this conjecture to others. I used to use to think only about > infinities, sorry. The problem was solved too easily. Should we keep it in Open Problem > Garden or I should delete it? See > http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice > === Subject: Re: A conjecture in lattice theory posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) Certainly, I would to think about finite lattices myself before > sending this conjecture to others. I used to use to think only about > infinities, sorry. The problem was solved too easily. Do you still want your Abel prize? Victor Meldrew I don't believe it! === Subject: rotation matrix posting-account=jYaw-QoAAABPGejvngHh2e8bmZF2Qagm Gecko/20070223 Fedora/1.5.0.10-1.fc5 Firefox/1.5.0.10,gzip(gfe),gzip(gfe) hi, I model a rectangle in matlab like : | 1 0 | | 40| | 0 1 | *| x y | < |30| | -1 0 | |-20| | 0 -1 | |-10| So this is a rectangle of which corners are, (20,10), (20,30), (40,10) and (40,30). Question is , if I want to rotate this rectangle around the center what matrix will I have to add to this inequality in terms of theta, the rotation angle? === Subject: Re: rotation matrix posting-account=cvz5-QoAAABVNzogw177Plx_25TguPUZ CLR 1.0.3705; .NET CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) > hi, > I model a rectangle in matlab like : | 1 æ0 æ | æ æ æ æ æ æ æ æ æ| 40| > | æ0 æ1 æ| æ*| x y | æ< æ |30| > | -1 æ0 æ| æ æ æ æ æ æ æ æ æ|-20| > | æ0 æ-1 | æ æ æ æ æ æ æ æ æ|-10| So this is a rectangle of which corners are, (20,10), (20,30), (40,10) > and (40,30). > Question is , if I want to rotate this rectangle around the center > what matrix will I have to add to this inequality in terms of theta, > the rotation angle? To rotate an angle t about a point (xc, yc) replace |x y| in your equation with |x y | is replaced with (C * |(x- x0) (y - y0)| - |x0 y0|) where C = | cos(t) sin(t)| |-sin(t) cos(t)| - MO === Subject: Re: rotation matrix posting-account=HZYXOQoAAAB0CZtsRCtABgys4tHYIT8J 3.2.0; .NET CLR 1.1.4322; InfoPath.2),gzip(gfe),gzip(gfe) > hi, > I model a rectangle in matlab like : | 1 æ0 æ | æ æ æ æ æ æ æ æ æ| 40| > | æ0 æ1 æ| æ*| x y | æ< æ |30| > | -1 æ0 æ| æ æ æ æ æ æ æ æ æ|-20| > | æ0 æ-1 | æ æ æ æ æ æ æ æ æ|-10| So this is a rectangle of which corners are, (20,10), (20,30), (40,10) > and (40,30). > Question is , if I want to rotate this rectangle around the center > what matrix will I have to add to this inequality in terms of theta, > the rotation angle? You first of all have to select a point to rotate around. The matrix you then want is Cox x Sin x -Sin x Cos x - Ian Parker === Subject: Help with integral I have encountered this integral int[ x^2/(exp(x)-1)dx ] with the bounds [0,Inf], but i am not sure how to attack it. It seems that if the power was 1 or 3 it is a standard integral, but this one I have not been able to look up. Can anyone help me? Anders Lund === Subject: Re: Help with integral >I have encountered this integral >int[ x^2/(exp(x)-1)dx ] with the bounds [0,Inf], but i am not sure how >to attack it. It seems that if the power was 1 or 3 it is a standard >integral, but this one I have not been able to look up. >Can anyone help me? Anders Lund Others have answered your question. In case it's of interest, the following provide related information: See (37). See (1). === Subject: Re: Help with integral > I have encountered this integral > int[ x^2/(exp(x)-1)dx ] with the bounds [0,Inf], but i am not sure how > to attack it. It seems that if the power was 1 or 3 it is a standard > integral, but this one I have not been able to look up. > Can anyone help me? hypothesis, he states that, when Re(s) > 1, zeta(s).Gamma(s) = int_0^{oo} x^{s - 1}/(e^x - 1) dx (*) So, your integral is equal to zeta(3).Gamma(3) = sum_n 2/n^3. Whittaker and Watson's Modern Analysis contains a short and elementary proof of (*) which, as far as I know, is not available in later textbooks. The tough part of the proof consists, as it often happens, in justifying that two limits commute. Jose Carlos Santos === Subject: Re: Help with integral > I have encountered this integral > int[ x^2/(exp(x)-1)dx ] with the bounds [0,Inf], but i am not sure how > to attack it. It seems that if the power was 1 or 3 it is a standard > integral, but this one I have not been able to look up. > Can anyone help me? > hypothesis, he states that, when Re(s) > 1, > zeta(s).Gamma(s) = int_0^{oo} x^{s - 1}/(e^x - 1) dx (*) > So, your integral is equal to > zeta(3).Gamma(3) = sum_n 2/n^3. > Whittaker and Watson's Modern Analysis contains a short and elementary > proof of (*) which, as far as I know, is not available in later > textbooks. ??? There are certainly short and elementary proofs in plenty of > texts. I'm wondering if you're unaware of these expositions or > you are and the proof in W&W is significantly shorter and more > elementary. That's what I claim. > The tough part of the proof consists, as it often happens, in > justifying that two limits commute. It's not that hard. We need to show that int_0^infinity sum f_n = sum int_0^infinity f_n. But given epsilon > 0 it's easy to show that there exists > A such that int_A^infinity sum |f_n| < epsilon, and > the sum converges uniformly on [0,A]. Yes, that's the standard elementary way of doing it, the non-elementary way being through the use of the Lebesgue dominated convergence theorem. However, I prefer the proof by Whittaker & Watson (I shall use int to denote the integral from 0 to +oo): sum_{n <= N} Gamma(s)/n^s = int sum_{n <= N} t^{s - 1}e^{-nt} dt = int t^{s - 1}/(e^t - 1) dt - int t^{s - 1}e^{-Nt}/(e^t - 1) dt and so it will be enough to prove that lim_N int t^{s - 1}e^{-Nt}/(e^t - 1) dt = 0. But |int t^{s - 1}e^{-Nt}/(e^t - 1) dt| <= <= int t^{Re(s) - 2)}e^{-Nt} dt (because t > 0 => e^t - 1 > t) = Gamma(Re(s) - 1)/N^{Re(s) - 1) and the limit of this expression (as N growths) is obviously 0. Jose Carlos Santos === Subject: Re: Help with integral posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > I have encountered this integral > int[ x^2/(exp(x)-1)dx ] with the bounds [0,Inf], but i am not sure how > to attack it. It's related to the Riemann zeta function. Write 1/(e^x-1) as a series e^{-x} + e^{-2x} + ... and integrat termwise. Victor Meldrew I don't believe it! === Subject: Re: Algorithm collection > I created a collection of algorithms which is available under > http://seed7.sourceforge.net/algorith/algorith.htm People who use recursion to calculate Fibonacci numbers shouldn't be > allowed > on sci.math. Or the Internet, for that matter. > (In case you don't know why, try your algorithm to calculate the 100th > Fibonacci number.) > I can happily use recursion to calculate the 1000000th Fibonacci number. Well, you wouldn't want to using the OP's algorithm. Unless you're prepared >to wait for the program to complete the 10^208987 function calls, of course. >Better start making some fresh coffee :-) >If the OP were interested in the value of Fn he would probably use Binet >formula. I assume he presents the algorithm as an illustration of recursion, >and as such it's, well, pretty bad... He could use the Binet formula, though I wouldn't recommend it. Or he could use any of the many identities for a recursion that uses O(log n) steps, e.g. in pseudocode # fib(n) - returns a pair containing the n'th # and the (n-1)th numbers in the fibonacci sequence function fib(n) if (n <= 0) return (0,0) if (n == 0) return (1,0) (hi,lo) = fib((n+1)/2) if (mod(n,2)) return (hi^2+lo^2,2*hi*lo-lo^2) else return (hi^2+hi*lo,hi^2+lo^2) Richard Harter, cri@tiac.net http://home.tiac.net/~cri, http://www.varinoma.com Save the Earth now!! It's the only planet with chocolate. === Subject: Re: A game (Leroy probably already considered it :-) > I would bet that you can't. ;-) OK, you won, but a flu ruins my genius at the moment :-) > They have the form a_n/n! where {a_n} > is a sequence of positive integers, but it's not in the online > encyclopedia, so I think you need to do something about that! Starting > with n = 0 the sequence goes 1, 1, 2, 5, 16, 62, 286, 1519, 9184, > 62000, 463964, 3800684, 33911424, ... You can simply submit it, then it's there :-) (No credit for me needed) -- Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de Er-a svo gott sem gott kve[CapitalYAcute]a .9al alda sonum, §v.92 a[CapitalYAcute] f.berra veit er fleira drekkur s.92ns til ge[CapitalYAcute]s gumi. === Subject: Re: A game (Leroy probably already considered it :-) <6iv1nvFjo1aU1@mid.dfncis.de> posting-account=sFP0HgkAAADJMwhdrXAaC5VX7Tc3BtzY .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > They have the form a_n/n! where {a_n} > is a sequence of positive integers, but it's not in the online > encyclopedia, so I think you need to do something about that! Starting > with n = 0 the sequence goes 1, 1, 2, 5, 16, 62, 286, 1519, 9184, > 62000, 463964, 3800684, 33911424, ... You can simply submit it, then it's there :-) > (No credit for me needed) I'll look into it. --- J K Haugland http://home.no.net/zamunda === Subject: Re: Simple doubt on continuity of multivarible function I plotted the surface of the function f(x,y) = xy/(x^2 + y^2), when x,y /= 0 f(0,0) = 0 using matlab to see how the surface look like. My intuition was if the surface has some discontinuity then it should be reflected either in function keeping x constant or y constant. :( (If anybody interested in how does the surface look like: http://people.csa.iisc.ernet.in/sujit/images/XY_Surf.jpg) Anyways, Let F = {f(x_0,y),f(x,y_0) | x_0, y_0 in R } is collection of functions from R to R either by keeping x or y constant. Suppose F is equicontinuous, will then f be continuous? I guess it should. Mostly it should work like this, given epsilon, find delta which will keep function in F within epsilon/2 at any point. Now use sqrt delta at point x,y for f which will ensure that function will be within epsilon difference from f(x,y) in ball of radius sqrt delta arnd (x,y). Am I right? Or some pitfall in the argument? --- Sujit Gujar. IISc Bangalore. Web: http:/people.csa.iisc.ernet.in/sujit === Subject: =?ISO-8859-1?Q?Reklam_och_Annonsering_=2D_Media_Produktion_f=F6r_Markn?= =?ISO-8859-1?Q?adsf=F6ring?= posting-account=wh3luAoAAAANfybtFuhMdn3ISHGbCf_r 5.0),gzip(gfe),gzip(gfe) International Visual Art & Media - Advertisement Sales / Promotion / PR Promos Arctic Extreme Sports - Board of Hollywood NetBook». Further Information: We are aheding this produce Arctic Extreme Sports for television broadcast, media, maps, sports products, and promotion material since 2008-09-01 In Northern Lapland, Norway, Finland and Sweden, Market Beneficiaries: ? International Media & Marketing Policy ? Broadcasting to seventeen countries ? Brand Integration ? Corporate Profile ? Agency & Lobbying Network ? 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Retail Order Form Hollywood NetBook» Retail Order Form http://publishing.yudu.com/Freedom/Akwem/Upphandling Keywords: Arctic Extreme Sports, documentary film produce, elite sports, exclusive events, adventure, satellite map, mountainbike adventure, Lapland Adventure, PR, PR produce, marketing, tourism integration, corporate events, corporate conference, media & film, sports PR, walking, health, lifestyle, exclusive travelling, travel agency,charter tourism, charter travelling, buscharter, high mountain excursion, river-raffting, programme, company, sapmi, challenges, hunting, eco tourism, Eu visit, holiday, exclusive holiday, salmon fishing, flyfishing, outdoors, wildlife,outdoor lifestyle, outdoor entertainment, safari, safaris, lapland safari, development cooperation, corporate activities,active, activity, promotional media product, promotion producer, promo, market sharing, long-term market, patent,trademark, trademark integration, trademark sales, trading, trade, world traveller, forum, professional sportsmen, advertising, reklam, classified advertisement, ads, ads sales, advertisement sales, marketing and advertisements, South Africa marketing, Alaska marketing, Siberia marketing, Alps marketing, Scandinavian marketing, Europ.8e marketing, South America marketing, Japan marketing, Ireland marketing, Scotland marketing, United kingdom marketing, Sweden marketing, Norway marketing, Finland marketing, Russia marketing, Televeision marketing, Television advertisement, TV advertising, Adventure & Travel Media Project, North Scandinavia === Subject: Infinite Binary Strings: A Question Given a line segment AB, and a point P arbitrarily chosen upon it, one can ask which half of AB P lies on, left or right, then having selected the half interval P lies on we can ask which half of that interval P lies upon, and so on repeatedly. If we happen to have chosen a point P such that AP is incommensurable with AB, the point P will never lie exactly at the end of any half interval. (It will never lie at the end of any fractional interval of the line segment.) So the point P produces an infinite, and aperiodic, infinite string eg LRRLLLR...... Is the converse true? That is, does an infinite, aperiodic binary string pick out a precise point on AB? Comon sense, perhaps, would tell us that you cannot get to a point by this repeated narrowing down -- it's intervals all the way down. Mathematics seems to be telling us that, by somehow treating the infinite sequence of narrowings down as a whole, an infinite binary string would indeed determine a precise point on AB. My question is: which bit of mathematics is it, exactly, that is telling us that this is so? Leon === Subject: Re: Infinite Binary Strings: A Question posting-account=UJeUTgkAAADYai-ULU41ORCvNnkXmdRu Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) | Is the converse true? That is, does an infinite, aperiodic binary |string pick out a precise point on AB? Comon sense, perhaps, would tell us |that you cannot get to a point by this repeated narrowing down -- it's |intervals all the way down. I asked a pre-calc class essentially this question (though not assuming that the sequence of nested intervals whose lengths goes to 0 was necessarily this kind of sequence of binary intervals). I wanted them to think a bit about it and explore their own intuition on it. I finally got one of my students (one of the A students) to hazard a guess, and her impression was that it wasn't enough to specify a single point. I am, like a lot of philosophers, more interested in intuitions than most mathematicians are. I think that in this case you have identified a certain counterintuitive quality in the standard construction of the real line. Counterintuitive elements are at least worth spelling out more specifically. Taking seriously that something is counterintuitive doesn't mean of course assuming that there's something wrong with it. I find the interplay between reasoning and intuition interesting. There's a sense in which an intuition can be dispelled by further examination. But even in some cases where it's very clear that the thing which seemed counterintuitive is correct, a certain air of counterintuitiveness remains. The counterintuitive quality of space-filling curves, for example, seems very stubborn for me. I think I understand how they're constructed pretty well, and it's very nearly dispelled the perception that something is strange about them, but not 100% somehow. In 1800 there were serious discussions among some of the most prominent mathematicians of the time (like Gauss) about some of the basic concepts in real analysis. It was quite a long time before the current paradigm (to use a somewhat overused word) became settled. It was a very interesting process, not to be underestimated. This intuition that functions should be given by formulas, for example, was attractive to some for a long time. There was a perception by some that functions given by a formula and curves given mechanically should be treated as distinct concepts. Fourier series tested these intuitions, since it seemed to say that even some very arbitrarily given function, even one just given mechanically by drawing a curve, could be expressed in a sense by an infinite series. |Mathematics seems to be telling us that, by |somehow treating the infinite sequence of narrowings down as a whole, an |infinite binary string would indeed determine a precise point on AB. My |question is: which bit of mathematics is it, exactly, that is telling us |that this is so? There are two ingredients. The existence of a number in the middle of the nested sequence is due to one of the completeness axioms for the real line. The uniqueness is due to Archimedes' principle (also called Eudoxus' principle). Whether these are treated as axioms or as theorems depends on how you develop the theory. There are various ways of developing it, but they all give you the same end result, so it's not necessary to choose just one. Archimedes principle says that if x>0 and y is some real, then n*x > y for some integer n. Here we define n*x inductively to be x+...+x where there are n x's. If two distinct reals a,b were in all of the nested intervals, say a1, or b-a>1/n. But past a certain point all of the intervals are shorter than 1/n, so this is impossible. The completeness principles implying that there is a real lying inside all of the nested intervals are perhaps more fundamental. For these I think I'll just point you to a couple of Wikipedia entries. The first is for order completeness, and the other is for completeness as a metric space: http://en.wikipedia.org/wiki/Completeness_(order_theory) http://en.wikipedia.org/wiki/Complete_(topology) If you digest the reasoning showing that the existence of a unique real lying in the intersection of the intervals follows from one of these sets of axioms, it should help to dispel the sense of counterintuitiveness about it. One of the ways of developing the theory that is maybe one of the simplest is just to define a real number to be a compatible collection of rational intervals containing ones of arbitrarily short length, where two reals are considered equal if their intervals are mutually compatible. Compatible means that any finite intersection is nonempty. That way the fact that a nested collection of intervals like yours defines a real follows relatively directly from the definition. (And this highlights the fact that it is a matter of definition.) On the other hand, I think for some students a more axiomatic approach may do more to make the theory seem intuitive. The completeness axioms (any of them) can be informally described as saying that the real line doesn't have holes in it, and the Archimedean principle says that it doesn't include infinitesimals. Despite what some people say, I don't think most people's intuitions about space or numbers usually lean toward active denial of either property. This can be introduced in the context of Euclidean geometry. It's one of the aspects of the theory that took the longest to appreciate. For centuries it wasn't realized that it was needed. The points on the plane that can be constructed by straightedge-and-compass from a given line segment satisfy the rest of the axioms, though, so you need something more to show that 20 degree angles exist and so on. If we have a ray XY and draw a circle around X, the points Z on the circle for which the angle YXZ is less than 20 degrees and the points Z for which YXZ is greater than 20 degrees would disconnect the circle if there weren't also points which form an angle of exactly 20 degrees. Keith Ramsay === Subject: Re: Infinite Binary Strings: A Question | Is the converse true? That is, does an infinite, aperiodic >binary >|string pick out a precise point on AB? Comon sense, perhaps, would >tell us >|that you cannot get to a point by this repeated narrowing down -- >it's >|intervals all the way down. Wow. That probably covers it. After I first read this, I was going to say this is my stop, go away and absorb the lesson. This is still probably what I should still say -- certainly there is plenty for me to think about. After a few comments in the text, I might try and say express the niggling feeling that it doesn't quite cover it. >I asked a pre-calc class essentially this question (though >not assuming that the sequence of nested intervals whose >lengths goes to 0 was necessarily this kind of sequence of >binary intervals). I wanted them to think a bit about it >and explore their own intuition on it. I finally got one of >my students (one of the A students) to hazard a guess, >and her impression was that it wasn't enough to specify a >single point. I am, like a lot of philosophers, more interested in >intuitions than most mathematicians are. I think that in >this case you have identified a certain counterintuitive >quality in the standard construction of the real line. >Counterintuitive elements are at least worth spelling out >more specifically. Taking seriously that something is >counterintuitive doesn't mean of course assuming that >there's something wrong with it. Understood. >I find the interplay between reasoning and intuition >interesting. There's a sense in which an intuition can >be dispelled by further examination. But even in some >cases where it's very clear that the thing which seemed >counterintuitive is correct, a certain air of >counterintuitiveness remains. The counterintuitive >quality of space-filling curves, for example, seems >very stubborn for me. I think I understand how they're >constructed pretty well, and it's very nearly dispelled >the perception that something is strange about them, >but not 100% somehow. That's interesting. >In 1800 there were serious discussions among some of the >most prominent mathematicians of the time (like Gauss) >about some of the basic concepts in real analysis. It was >quite a long time before the current paradigm (to use a >somewhat overused word) became settled. It was a very >interesting process, not to be underestimated. This intuition that functions should be given by formulas, >for example, was attractive to some for a long time. There >was a perception by some that functions given by a formula >and curves given mechanically should be treated as >distinct concepts. Fourier series tested these intuitions, >since it seemed to say that even some very arbitrarily >given function, even one just given mechanically by >drawing a curve, could be expressed in a sense by an >infinite series. I can recall trying to wade through some of the mathematical parts of Penrose's Road to Reality. There's an extended section about the continuity of functions, especially in relation to complex number. So I can roughly follow you here. >|Mathematics seems to be telling us that, by >|somehow treating the infinite sequence of narrowings down as a whole, >an >|infinite binary string would indeed determine a precise point on AB. >My >|question is: which bit of mathematics is it, exactly, that is telling >us >|that this is so? There are two ingredients. The existence of a number in the >middle of the nested sequence is due to one of the >completeness axioms for the real line. The uniqueness is >due to Archimedes' principle (also called Eudoxus' >principle). Whether these are treated as axioms or as >theorems depends on how you develop the theory. There are >various ways of developing it, but they all give you the >same end result, so it's not necessary to choose just one. Archimedes principle says that if x>0 and y is some real, >then n*x > y for some integer n. Here we define n*x >inductively to be x+...+x where there are n x's. If two >distinct reals a,b were in all of the nested intervals, >say a1, >or b-a>1/n. But past a certain point all of the intervals >are shorter than 1/n, so this is impossible. The completeness principles implying that there is a real >lying inside all of the nested intervals are perhaps more >fundamental. For these I think I'll just point you to a >couple of Wikipedia entries. The first is for order >completeness, and the other is for completeness as a >metric space: http://en.wikipedia.org/wiki/Completeness_(order_theory) >http://en.wikipedia.org/wiki/Complete_(topology) If you digest the reasoning showing that the existence of >a unique real lying in the intersection of the intervals >follows from one of these sets of axioms, it should help >to dispel the sense of counterintuitiveness about it. OK. To do >One of the ways of developing the theory that is maybe one >of the simplest is just to define a real number to be a >compatible collection of rational intervals containing ones >of arbitrarily short length, where two reals are considered >equal if their intervals are mutually compatible. Compatible >means that any finite intersection is nonempty. That way >the fact that a nested collection of intervals like yours >defines a real follows relatively directly from the >definition. (And this highlights the fact that it is a >matter of definition.) > OK. Slightly disturbed that I'm talking about spatial intervals, whereas this is about numerical intervals, apparently. It struck me that we define the real numbers so as to be space-like, then we turn round and analyze space with notions of compactness and the like and show that it's number like, which seems slightly Irish, but will probably all come out in the wash when I think about these things properly. >On the other hand, I think for some students a more >axiomatic approach may do more to make the theory seem >intuitive. The completeness axioms (any of them) can be >informally described as saying that the real line doesn't >have holes in it, and the Archimedean principle says that >it doesn't include infinitesimals. Despite what some >people say, I don't think most people's intuitions about >space or numbers usually lean toward active denial of >either property. Certainly not for space. >This can be introduced in the context of Euclidean geometry. >It's one of the aspects of the theory that took the longest >to appreciate. For centuries it wasn't realized that it >was needed. The points on the plane that can be constructed >by straightedge-and-compass from a given line segment >satisfy the rest of the axioms, though, so you need something >more to show that 20 degree angles exist and so on. If we >have a ray XY and draw a circle around X, the points Z on >the circle for which the angle YXZ is less than 20 degrees >and the points Z for which YXZ is greater than 20 degrees >would disconnect the circle if there weren't also points >which form an angle of exactly 20 degrees. I confess I hadn't of the impossibility of trisecting 60 degrees as casting doubt on the existence of the point at 20 degrees, but that's probably me not thinking through the consequences of the axiomatic method. >Keith Ramsay There are two, I think connected, worries I still have, that I'm not sure are squarely addressed. I'm not sure they are cogent, and they may indeed dissipate by the appropiate study as you've suggested. But I may as well express them. They suggest to me an asymmetry between the infinite binary strings on the one hand and the points on the line segment on the other, almost regardless of how the reals are defined. One concern, which I've expressed to another poster, is with the idea of an arbitrary infinite binary string. Whether those concerns are real or not, I suppose I'm taking it for granted that there's nothing problematic about selecting an arbitrary point on a line segment. I suppose I think of this in terms of just plumping for one, with an infinitely thin pencil point. The second concern is harder to express. If the irrationals are to sit alongside the reals on the number line, it seems reasonable to expect that they should have, in common with the rationals, a precise numerical magnitude, by which I mean that if we take a line segment as a REPRESENTATION of the number line (in the unit interval, say), then an irrational number should have upon it an exact length which ought to be derivable from the number, i.e. from the infinite, aperiodic binary string in the case of an irrational number. Suppose we inscribe an infinite binary tree in a triangle as follows. We let the distance between successive node levels be half the distance between the previous node levels. Instead of left sloping path steps we will have downward paths, and right-sloping path steps will be at 45 degrees. The root node is at O, with A at unit distance vertically below. And at right anles to OA we have a unit length AB going off to the right. The first 0-step goes to the midpoint of OA, the 1-step to the midpoint of OB. The next steps go half that distance again leading to 4 equidistant nodes, and so on. Every infinite binary string has a representation as a point P on AB, with a precise length AP. As before. an irrational point P will generate an infinite binary string since a line drawn from O to P will produce an infinite sequence of minima as it traverses through adjacent nodes at each level of the tree. The question is, how does a string determine a particular point on AB? I still have the feeling that a string being able to do this should not be a matter of definition. In the case of periodic strings this is easy to do. After following any initial finite path, we then join up the beginnings and ends of the periodic blocks and project them in a straight line to AB. (Equivalent to summing a geometic series.) How can an aperiodic string achieve this? Is there some non-linear way of projecting to AB? How can the binary string for pi - 3, say, be thought of as picking out that precise length without our already knowing where that point is (non-representationally, so to speak)? If this is rubbish, just tell me. I'll go back to the books and the Wiki. leon. === Subject: Re: Infinite Binary Strings: A Question > The question > is, how does a string determine a particular point on AB? I still have the > feeling that a string being able to do this should not be a matter of > definition. The thing that is a matter of definition is not so much 'the string being able to determine a point'. It's the fact that there /is/ a point for it to determine that's a matter of definition (namely of the real number set). The real number set is defined so as to include a limit for every sequence that 'geometrically seems to converge' (Cauchy sequence), so that every seemingly convergent sequence has something to converge to. > In the case of periodic strings this is easy to do. After > following any initial finite path, we then join up the beginnings and ends > of the periodic blocks and project them in a straight line to AB. > (Equivalent to summing a geometic series.) How can an aperiodic string > achieve this? Is there some non-linear way of projecting to AB? How can the > binary string for pi - 3, say, be thought of as picking out that precise > length without our already knowing where that point is > (non-representationally, so to speak)? The string is /not/ thought of as picking out a precise length - at least not in one stroke. The precise length is 'a sensible something' to the same extent that the string is. What /is/ a length? a real number - ok. How do you specify such a real number? giving a sequence of rationals converging to the number is one way to specify such a number. Real numbers are in many respects quite similar to infinite strings. > If this is rubbish, just tell me. I'll go back to the books and the Wiki. No, it's not rubbish, but books and wiki won't harm you anyway. -- Herman Jurjus === Subject: Re: Infinite Binary Strings: A Question > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such > that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Yes. > Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? Each truncation of the infinite aperiodic string to n digits produces a rational number, The sequence of numbers produced as n goes from 1 to infinity defines a Cauchy sequence, and hence defines a Real number. Leon === Subject: Re: Infinite Binary Strings: A Question > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such > that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Yes. > Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? Each truncation of the infinite aperiodic string to n digits produces a >rational number, The sequence of numbers produced as n goes from 1 to >infinity defines a Cauchy sequence, and hence defines a Real number. Still, one cannot help projecting them. leon === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such > that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Yes. Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? Each truncation of the infinite aperiodic string to n digits produces a > rational number, The sequence of numbers produced as n goes from 1 to > infinity defines a Cauchy sequence, and hence defines a Real number. You give a straight yes above, but here all we have is a definition by a limit, so that the real number here has still not been strongly constructed, as we keep missing an extended case. In simpler and hopefully less improper words: such a definition does not make justice to the *existence* of infinite strings, which is the culprit of the OP. FWIW, it is my intuition that, to overcome the empasse, our infinite strings, the points on the line, should rather be taken as primitive objects on their own right. -LV === Subject: Re: Infinite Binary Strings: A Question > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such > that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Yes. > Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? Each truncation of the infinite aperiodic string to n digits produces a > rational number, The sequence of numbers produced as n goes from 1 to > infinity defines a Cauchy sequence, and hence defines a Real number. You give a straight yes above, but here all we have is a definition > by a limit, so that the real number here has still not been strongly > constructed, as we keep missing an extended case. Even for the most strict of constructionists the constructibility of a real it is only needed that given some positive constructible rational epsilon one can construct a rational within epsilon of the real. No extended case needed. In simpler and > hopefully less improper words: such a definition does not make justice > to the *existence* of infinite strings, which is the culprit of the > OP. FWIW, it is my intuition that, to overcome the empasse, our > infinite strings, the points on the line, should rather be taken as > primitive objects on their own right. In pure geometry, those real points ARE taken as primitive, but in modeling the real line by the set of real numbers, the real numbers are taken as primitive. === Subject: Re: Infinite Binary Strings: A Question David C. Ullrich a .8ecrit : > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such > that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? > Yes. > Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? > Each truncation of the infinite aperiodic string to n digits produces a > rational number, The sequence of numbers produced as n goes from 1 to > infinity defines a Cauchy sequence, and hence defines a Real number. > You give a straight yes above, but here all we have is a definition > by a limit, so that the real number here has still not been strongly > constructed, as we keep missing an extended case. In simpler and > hopefully less improper words: such a definition does not make justice > to the *existence* of infinite strings, which is the culprit of the > OP. FWIW, it is my intuition that, to overcome the empasse, our > infinite strings, the points on the line, should rather be taken as > primitive objects on their own right. Why in the world do you think that anyone cares about > your feelings or intutions about any of this? You simply > don't know what you're talking about. Wrong : he is a troll and know his craft well > -LV David C. Ullrich Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) === Subject: Re: Infinite Binary Strings: A Question ... he is a troll and know his craft well > troll := asshole full of ? B. === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such > that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? > Yes. > Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? > Each truncation of the infinite aperiodic string to n digits produces a > rational number, The sequence of numbers produced as n goes from 1 to > infinity defines a Cauchy sequence, and hence defines a Real number. You give a straight yes above, but here all we have is a definition >by a limit, so that the real number here has still not been strongly >constructed, as we keep missing an extended case. In simpler and >hopefully less improper words: such a definition does not make justice >to the *existence* of infinite strings, which is the culprit of the >OP. FWIW, it is my intuition that, to overcome the empasse, our >infinite strings, the points on the line, should rather be taken as >primitive objects on their own right. Why in the world do you think that anyone cares about > your feelings or intutions about any of this? You simply > don't know what you're talking about. You just confirm youself a pernicious troll, and overall simply an idiot. As an idiot, of course you won't get it until it's finished. -LV === Subject: Re: Infinite Binary Strings: A Question Why in the world do you think that anyone cares about > your feelings or intutions about any of this? You simply > don't know what you're talking about. You just confirm youself a pernicious troll, and overall simply an idiot. As an idiot, of course you won't get it until it's finished. -LV ********************* If you are referring to Ullrich, he is an anti-troll; his posts are short, sharp and correct rather than long, vague and wrong as are most trolls. When he makes a mistake, he acknowledges it, not trollish at all. He is certainly not an idiot. He can be rude, but that's probably because he doesn't give a flying about whether the people he responds to like him or not. === Subject: Re: Infinite Binary Strings: A Question > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? To put it a bit simplistically: At the end of the 19th century, it was Dedekind that wanted this 'to be so', and he was thrilled when he heard about a guy called Cantor, who had a theory providing just that: set theory. Nowadays, it's a necessary consequence of the way the real number line is -defined- using set theory (either via Dedekind cuts or via Cauchy sequences, see for example http://en.wikipedia.org/wiki/Construction_of_real_numbers). In short, the answer to your question is: the standard -definition- of the real number set is the bit of mathematics that you're looking for. -- Herman Jurjus === Subject: Re: Infinite Binary Strings: A Question > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? To put it a bit simplistically: >At the end of the 19th century, it was Dedekind that wanted this 'to be >so', and he was thrilled when he heard about a guy called Cantor, who >had a theory providing just that: set theory. Nowadays, it's a necessary consequence of the way the real number line >is -defined- using set theory (either via Dedekind cuts or via Cauchy >sequences, see for example >http://en.wikipedia.org/wiki/Construction_of_real_numbers). In short, the answer to your question is: the standard -definition- of >the real number set is the bit of mathematics that you're looking for. But isn't what's at stake here more general than the issue of the definition of the real number? And I'm still struck by the feeling that the idea of this binary sequence of lefts and rights determining a point is unconvincing. Of course I need to say why. Part of what I have in mind is that an aperiodic string is in general, if not always, chaotic or unpredictable. By that I suppose I should mean something specific to the effect that there is no significantly shorter way to determine the k-th digit (for some arbitrary, perhaps large k) than to compute the string up to the k-th digit, or something like that. At any rate, the last digit computed so far flips erratically between 0 and 1 as the string is explicated. Where is the precision in such a concept? I also have in mind an argument which I couldn't lay out in a couple of paragraphs but I can summarize here. (I might set it out in a separate thread if I could get more confident about it.) I think there is a difficulty with the notion of an arbitray infinite binary string, understood as a point in a uniform combinatorial space of 2^infinity possibilities. I believe an infinite binary string has to come from somewhere, has to have something that produces it. In the present context that is the arbitrary point P. If we had chosen the particular point P such that AB/AP = pi, then the string we produce for this ratio by applying a fromula for pi has its anchorage in the definite position this point has on the line, so that geometry serves a kind of GPS for this string. (It is not necessary to believe that pi is a geometrical notion -- it is sufficient that it is believed to be a precise ratio, which can then certainly be represented as a point on a line.) In short, I don't believe that the infinite binary string associated with an arbitrary point on the line segment has an independent existence apart from that point. On that basis, it cannot determine anything. If that's not complete bollocks, I'd be glad Leon === Subject: Re: Infinite Binary Strings: A Question > Part of what I have in mind is that an aperiodic string is in > general, if not always, chaotic or unpredictable. By that I suppose I > should mean something specific to the effect that there is no significantly > shorter way to determine the k-th digit (for some arbitrary, perhaps large > k) than to compute the string up to the k-th digit, or something like that. > At any rate, the last digit computed so far flips erratically between 0 and > 1 as the string is explicated. Where is the precision in such a concept? The answer to that last question is: there is none, and none is needed. Set theory shows you how we can get away with -not- providing a precise concept. It could be that we have two models of ZFC, one model assesses some string as valid, and the other doesn't. And whenever that's the case, one model will also have a real number set different from the one in the other model. > I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. In say, ZFC/FOL, they are 'produced' by some (fictitious) model (of ZFC). They're -assumed- to be there, just like the Euclidean axioms assumed points and lines to be there. But perhaps your question is -why- it's reasonable to assume this, or why it's so necessary or handy for mathematics to have this available? -- Herman Jurjus === Subject: Re: Infinite Binary Strings: A Question reply-type=response > Part of what I have in mind is that an aperiodic string is in > general, if not always, chaotic or unpredictable. By that I suppose I > should mean something specific to the effect that there is no > significantly > shorter way to determine the k-th digit (for some arbitrary, perhaps > large > k) than to compute the string up to the k-th digit, or something like > that. > At any rate, the last digit computed so far flips erratically between 0 > and > 1 as the string is explicated. Where is the precision in such a concept? The answer to that last question is: there is none, and none is needed. > Set theory shows you how we can get away with -not- providing a precise > concept. > It could be that we have two models of ZFC, one model assesses some string > as valid, and the other doesn't. And whenever that's the case, one model > will also have a real number set different from the one in the other > model. > Now hold on there. The question is whether it is constructible in ZFC, not whether it is valid. The OP is correct when he says that if the last bit flips erratically (which I will take to mean randomly) then there is no basis for constructing a number in this manner in ZF alone, as you need to make a simultaneous choice between a countably infinite sets of {0,1}. Choice does however allow you to postulate arbitrary binary strings, even if we can't construct them. > I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is > a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. You can only construct countably many infinite binary strings in ZFC. They are produced by the operations of power set, union, etc. The others you can't produce, even with the axiom of choice; they have nowhere to come from. In say, ZFC/FOL, they are 'produced' by some (fictitious) model (of ZFC). > They're -assumed- to be there, just like the Euclidean axioms assumed > points and lines to be there. But perhaps your question is -why- it's reasonable to assume this, or why > it's so necessary or handy for mathematics to have this available? -- > Herman Jurjus === Subject: Re: Infinite Binary Strings: A Question <48ce2837$0$27211$ba620dc5@text.nova.planet.nl> <48ce62e1$0$28215$afc38c87@news.optusnet.com.au> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 15, 10:28æam, Peter Webb > Part of what I have in mind is that an aperiodic string is in > general, if not always, chaotic or unpredictable. By that I suppose I > should mean something specific to the effect that there is no > significantly > shorter way to determine the k-th digit (for some arbitrary, perhaps > large > k) than to compute the string up to the k-th digit, or something like > that. > At any rate, the last digit computed so far flips erratically between 0 > and > 1 as the string is explicated. Where is the precision in such a concept? The answer to that last question is: there is none, and none is needed. > Set theory shows you how we can get away with -not- providing a precise > concept. > It could be that we have two models of ZFC, one model assesses some string > as valid, and the other doesn't. And whenever that's the case, one model > will also have a real number set different from the one in the other > model. Now hold on there. The question is whether it is constructible in ZFC, not > whether it is valid. The OP is correct when he says that if the last bit flips erratically > (which I will take to mean randomly) then there is no basis for > constructing a number in this manner in ZF alone, as you need to make a > simultaneous choice between a countably infinite sets of {0,1}. Choice does > however allow you to postulate arbitrary binary strings, even if we can't > construct them. > I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is > a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. You can only construct countably many infinite binary strings in ZFC. They > are produced by the operations of power set, union, etc. The others you > can't produce, even with the axiom of choice; they have nowhere to come > from. What do you mean by 'produce', exactly? -- m === Subject: Re: Infinite Binary Strings: A Question Part of what I have in mind is that an aperiodic string is in > general, if not always, chaotic or unpredictable. By that I suppose I > should mean something specific to the effect that there is no > significantly > shorter way to determine the k-th digit (for some arbitrary, perhaps > large > k) than to compute the string up to the k-th digit, or something like > that. > At any rate, the last digit computed so far flips erratically between > 0 and > 1 as the string is explicated. Where is the precision in such a concept? > The answer to that last question is: there is none, and none is needed. > Set theory shows you how we can get away with -not- providing a > precise concept. > It could be that we have two models of ZFC, one model assesses some > string as valid, and the other doesn't. And whenever that's the case, > one model will also have a real number set different from the one in > the other model. > Now hold on there. The question is whether it is constructible in ZFC, > not whether it is valid. It can be present in one model, and absent from another. Hence, the axioms of ZFC in itself do not uniquely determine one (extension of the) concept 'infinite binary string'. (And, as said, fortunately you don't need one, anyway.) > The OP is correct when he says that if the last bit flips erratically > (which I will take to mean randomly) then there is no basis for > constructing a number in this manner in ZF alone, [...] Sure, but that's not what i was talking about. > Choice > does however allow you to postulate arbitrary binary strings, even if we > can't construct them. And what does that mean, exactly? How could you 'postulate arbitrary binary strings'? Does ZFC manage to do that? And in a unique and unambiguous way? > You can only construct countably many infinite binary strings in ZFC. > They are produced by the operations of power set, union, etc. The others > you can't produce, even with the axiom of choice; they have nowhere to > come from. Sure; my hunch was that that would not satisfy the OP, but now that you mention it, i could very well be wrong. Mr. Street? -- Herman Jurjus === Subject: Re: Infinite Binary Strings: A Question I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is > a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. In the present context > that is the arbitrary point P. If we had chosen the particular point P > such > that AB/AP = pi, then the string we produce for this ratio by applying a > fromula for pi has its anchorage in the definite position this point has > on > the line, so that geometry serves a kind of GPS for this string. (It is > not > necessary to believe that pi is a geometrical notion -- it is sufficient > that it is believed to be a precise ratio, which can then certainly be > represented as a point on a line.) In short, I don't believe that the > infinite binary string associated with an arbitrary point on the line > segment has an independent existence apart from that point. On that basis, > it cannot determine anything. If that's not complete bollocks, I'd be glad There really is no problem with arbitrary infinite binary strings. For any given line segment, you can use such a string to reach many points on the line segment. In the case of a line segment like [0,1], the infinite binary string for 1/pi is completely determined but there are 2^ALEPH_0 locations on [0,1] whose infinite binary strings cannot be completely determined in any finite way. Those strings still exist but they have no finite representation and no physical instantiation. The total string for 1/pi has no physical instantiation but the formula for 1/pi can serve as an algorithm for computing it. Using the string for 1/pi, the location 1/pi can be reached by going left or light, in each successive half-size segment, at time t=1-1/k, for the kth character in 1/pi's string. At time t=1 you will be at the location 1/pi. ALEPH_0 is so huge that it breaks out of the unending successive approximations to 1/pi and takes you to 1/pi exactly. The way to get to the end of an unending process is to go through every step of the process. This whole line of reasoning is also true for an arbitrary infinite binary string. k === Subject: Re: Infinite Binary Strings: A Question > I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is > a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. In the present context > that is the arbitrary point P. If we had chosen the particular point P > such > that AB/AP = pi, then the string we produce for this ratio by applying a > fromula for pi has its anchorage in the definite position this point has > on > the line, so that geometry serves a kind of GPS for this string. (It is > not > necessary to believe that pi is a geometrical notion -- it is sufficient > that it is believed to be a precise ratio, which can then certainly be > represented as a point on a line.) In short, I don't believe that the > infinite binary string associated with an arbitrary point on the line > segment has an independent existence apart from that point. On that basis, > it cannot determine anything. If that's not complete bollocks, I'd be glad >There really is no problem with arbitrary infinite binary strings. For any >given line segment, you can use such a string to reach many points on the >line segment. In the case of a line segment like [0,1], the infinite binary >string for 1/pi is completely determined but there are 2^ALEPH_0 locations >on [0,1] whose infinite binary strings cannot be completely determined in >any finite way. Those strings still exist but they have no finite >representation and no physical instantiation. The total string for 1/pi has >no physical instantiation but the formula for 1/pi can serve as an algorithm >for computing it. Using the string for 1/pi, the location 1/pi can be >reached by going left or light, in each successive half-size segment, at >time t=1-1/k, for the kth character in 1/pi's string. At time t=1 you will >be at the location 1/pi. ALEPH_0 is so huge that it breaks out of the >unending successive approximations to 1/pi and takes you to 1/pi exactly. >The way to get to the end of an unending process is to go through every step >of the process. This whole line of reasoning is also true for an arbitrary >infinite binary string. >k > Let me try and explain why I think there is a problem with arbitrary infinite strings. This will be more of a rough sketch of an argument, might be completely potty, but if it is on the wrong track you will probably be able to tell me how. If I want to make an arbitrary choice from a set of finite strings, say binary strings of length n, there are various ways I can do this. I can write it out. I can toss a coin n times. I can pick one from a list, perhaps by picking a number between 1 and 2^n. None of these methods is available to me if I wish to choose an arbitrary infinite string. We should also note that if a finite string arbitrarily chosen happens to have some 'structure', for example if it happens to consist of alternating 1s and 0s beginning with 1, this is incidental to its arbitrary status. This would be a way of identifying the string alternative to its position as say, the k-th string in the list. Now consider the infinite binary string which consists of alternating 1s and 0s, beginning with 1. The description I have just given is no longer a dispensable way of identifying the string, but rather the string is just that string as I have described it. This is its identity. This is how such periodic strings are made, one might say. Start with 10 and repeat. When we make a list of finite strings of length n, it is natural to start with all 0s, or alternatively all 1s, and then, starting at either end, systematically build up the possible strings by incremental changes. We could of course start with an arbitrary string, and list all the possible transformations of that string. We could write a transformation, acting upon say a nine digit string, as follows: CLLCLCCCL which means: Change the first digit in the string to be transformed (from 1 to 0 or 0 to 1), Leave the second digit as it is (0 -> 0; 1 -> 1), and so on. But then the list of transformations is in effect the same binary list that we enumerated in the first place, with Cs and Ls instead of 1s and 0s. How would we go about making a list of all the systematic changes that can be made to an infinite binary string? Such a list must be a list of all the infinite periodic binary strings. All those strings which consist of an initial (possibly null) finite string followed by a repeating string of a set length, which can indeed be put into a single list. So if we were to try and make a list of all infinite binary strings and began with the pi string (the binary equivalent of pi, dropping the radix point), the strings produced would be periodic transformations of the pi string. In other words, the only strings that could be produced would be predictable variations of the pi string. But since in general an aperiodic string is quite unpredictable, is explicated by computation with no short-cut pattern in the digits, it is impossible for say pi and some quite other irrational quantity to appear in the same list. Why make such a fuss about this when there is the very fine and well known argument that the power set of an infinite set 'exceeds' the set (which since select/deselect is a binary structure is equivalent to an argument about infinite binary strings)? But the argument is too dazzling, and prevents us from seeing whats involved in the notion of a list beyond the fact that it's 1:1 with the natural numbers. All the aperiodic strings cannot be in the same list because, at bottom, the idea of just two aperodic strings being in the same list is oxymoronic. It's not about there being some mysteriously supernumerous quantity of infinite strings. Infinite strings always come with some means of production, either some rule of composition (typically producing a periodic string, but there are more subtle rules eg for Champernowne numbers and the like), or some specified computation, or in the case of the original question, a point on a line segment. Of course we can think of the string in isolation from its production method, just as we can think of a painting in isolation from the artist. For example, Constable's Haywain hangs in the National Gallery, but John Constable himself has long since fused with the earth in the churchyard of St John's at Hampstead. And just as two distinct paintings can be distinguished by their material differences, two infinite binary strings can always be distinguished by some variation at a finite place in their development. But all infinite strings are produceable, just as all paintings have to be painted. This is not a restriction on what can be an infinite string, or a real number. (Any result of an arithmetical operation on a number or numbers is automatically another produceable number. Completeness isn't a worry.) It's an analysis of what an infinite binary string is. The notion of a completely arbitrary infinite binary strings seems to me to be empty. leon === Subject: Re: Infinite Binary Strings: A Question <87irc4ps2mem2gmm8nkd5gph4bncsqcco1@4ax.com> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) [yet another argument elided] Infinite strings always come with some means of production, [...] Well, here is were you are wrong. -- m === Subject: Re: Infinite Binary Strings: A Question posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) On 15 Sep, 08:12, Mariano Su.87rez-Alvarez [yet another argument elided] Infinite strings always come with some means of production, [...] Well, here is were you are wrong. Of course, obsession with the means of production is a classic Marxist shibboleth. Victor Meldrew I don't believe it! === Subject: Re: Infinite Binary Strings: A Question [yet another argument elided] Infinite strings always come with some means of production, [...] Well, here is were you are wrong. Karl Marx says the workers should control the means of production! I can see it now, a socialist paradise of workers hammering out sequences of ones and zeros all day long, singing songs of solidarity with the oppressed masses of the world. === Subject: Re: Infinite Binary Strings: A Question >There really is no problem with arbitrary infinite binary strings. For >any >given line segment, you can use such a string to reach many points on the >line segment. In the case of a line segment like [0,1], the infinite >binary >string for 1/pi is completely determined but there are 2^ALEPH_0 locations >on [0,1] whose infinite binary strings cannot be completely determined in >any finite way. Those strings still exist but they have no finite >representation and no physical instantiation. The total string for 1/pi >has >no physical instantiation but the formula for 1/pi can serve as an >algorithm >for computing it. Using the string for 1/pi, the location 1/pi can be >reached by going left or light, in each successive half-size segment, at >time t=1-1/k, for the kth character in 1/pi's string. At time t=1 you >will >be at the location 1/pi. ALEPH_0 is so huge that it breaks out of the >unending successive approximations to 1/pi and takes you to 1/pi exactly. >The way to get to the end of an unending process is to go through every >step >of the process. This whole line of reasoning is also true for an >arbitrary >infinite binary string. >k Let me try and explain why I think there is a problem with > arbitrary infinite strings. This will be more of a rough sketch of an > argument, might be completely potty, but if it is on the wrong track you > will probably be able to tell me how. If I want to make an arbitrary choice from a set of finite strings, > say binary strings of length n, there are various ways I can do this. I > can > write it out. I can toss a coin n times. I can pick one from a list, > perhaps by picking a number between 1 and 2^n. None of these methods is > available to me if I wish to choose an arbitrary infinite string. There are methods but they cannot be physically instantiated. You can do your kth coin toss at time t=1-1/k and at time t=1 you will have an arbitrary infinite string. Again, the way to get to the end of an endless task is to do each step of the task. > also note that if a finite string arbitrarily chosen happens to have some > 'structure', for example if it happens to consist of alternating 1s and 0s > beginning with 1, this is incidental to its arbitrary status. This would > be > a way of identifying the string alternative to its position as say, the > k-th string in the list. Now consider the infinite binary string which > consists of alternating 1s and 0s, beginning with 1. The description I > have > just given is no longer a dispensable way of identifying the string, but > rather the string is just that string as I have described it. This is its > identity. This is how such periodic strings are made, one might say. Start > with 10 and repeat. When we make a list of finite strings of length n, it is natural to > start with all 0s, or alternatively all 1s, and then, starting at either > end, systematically build up the possible strings by incremental changes. > We could of course start with an arbitrary string, and list all the > possible transformations of that string. We could write a transformation, > acting upon say a nine digit string, as follows: CLLCLCCCL > which means: Change the first digit in the string to be transformed (from > 1 > to 0 or 0 to 1), Leave the second digit as it is (0 -> 0; 1 -> 1), and so > on. But then the list of transformations is in effect the same binary list > that we enumerated in the first place, with Cs and Ls instead of 1s and > 0s. > How would we go about making a list of all the systematic changes that can > be made to an infinite binary string? This is a deep question in set theory. If Godel's axiom of construction, V=L, is true then all infinite binary strings (of the kind you are describing) could be well ordered by their descriptions. > Such a list must be a list of all the > infinite periodic binary strings. All those strings which consist of an > initial (possibly null) finite string followed by a repeating string of a > set length, which can indeed be put into a single list. So if we were to > try and make a list of all infinite binary strings and began with the pi > string (the binary equivalent of pi, dropping the radix point), the > strings > produced would be periodic transformations of the pi string. In other > words, the only strings that could be produced would be predictable > variations of the pi string. But since in general an aperiodic string is > quite unpredictable, is explicated by computation with no short-cut > pattern > in the digits, it is impossible for say pi and some quite other irrational > quantity to appear in the same list. Those infinite strings that cannot be encapsulated in any finite way require, in your case, ALEPH_0 bits of information. A given string of this kind is only `unpredictable' in the sense that it cannot be characterized in any finite way. > Why make such a fuss about this when there is the very fine and > well known argument that the power set of an infinite set 'exceeds' the > set > (which since select/deselect is a binary structure is equivalent to an > argument about infinite binary strings)? But the argument is too dazzling, > and prevents us from seeing whats involved in the notion of a list beyond > the fact that it's 1:1 with the natural numbers. All the aperiodic strings > cannot be in the same list because, at bottom, the idea of just two > aperodic strings being in the same list is oxymoronic. It's not about > there > being some mysteriously supernumerous quantity of infinite strings. There are 2^ALEPH_0 infinite strings of the kind this thread has been discussing. > Infinite strings always come with some means of production, either > some rule of composition (typically producing a periodic string, but there > are more subtle rules eg for Champernowne numbers and the like), or some > specified computation, or in the case of the original question, a point on > a line segment. Many infinite strings do not have finite rules that fully characterize them, but these strings do have infinite rules that do fully characterize them. An infinite string that can only be specified by ALEPH_0 bits of information are computable via all ALEPH_0 bits of information and this computation can be completed at time t=1, where the kth character of the string is computed at time t=1-1/k.. k === Subject: Re: Infinite Binary Strings: A Question <87irc4ps2mem2gmm8nkd5gph4bncsqcco1@4ax.com> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) >There really is no problem with arbitrary infinite binary strings. æFor >any >given line segment, you can use such a string to reach many points on the >line segment. æIn the case of a line segment like [0,1], the infinite >binary >string for 1/pi is completely determined but there are 2^ALEPH 0 locations >on [0,1] whose infinite binary strings cannot be completely determined in >any finite way. æThose strings still exist but they have no finite >representation and no physical instantiation. æThe total string for 1/pi >has >no physical instantiation but the formula for 1/pi can serve as an >algorithm >for computing it. æUsing the string for 1/pi, the location 1/pi can be >reached by going left or light, in each successive half-size segment, at >time t=1-1/k, for the kth character in 1/pi's string. æAt time t=1 you >will >be at the location 1/pi. æALEPH 0 is so huge that it breaks out of the >unending successive approximations to 1/pi and takes you to 1/pi exactly. >The way to get to the end of an unending process is to go through every >step >of the process. æThis whole line of reasoning is also true for an >arbitrary >infinite binary string. >k Let me try and explain why I think there is a problem with > arbitrary infinite strings. This will be more of a rough sketch of an > argument, might be completely potty, but if it is on the wrong track you > will probably be able to tell me how. If I want to make an arbitrary choice from a set of finite strings, > say binary strings of length n, there are various ways I can do this. I > can > write it out. I can toss a coin n times. I can pick one from a list, > perhaps by picking a number between 1 and 2^n. None of these methods is > available to me if I wish to choose an arbitrary infinite string. There are methods but they cannot be physically instantiated. æYou can do > your kth coin toss at time t=1-1/k and at time t=1 you will have an > arbitrary infinite string. æAgain, the way to get to the end of an endless > task is to do each step of the task. also note that if a finite string arbitrarily chosen happens to have some > 'structure', for example if it happens to consist of alternating 1s and 0s > beginning with 1, this is incidental to its arbitrary status. This would > be > a way of identifying the string alternative to its position as say, the > k-th string in the list. Now consider the infinite binary string which > consists of alternating 1s and 0s, beginning with 1. The description I > have > just given is no longer a dispensable way of identifying the string, but > rather the string is just that string as I have described it. This is its > identity. This is how such periodic strings are made, one might say. Start > with 10 and repeat. When we make a list of finite strings of length n, it is natural to > start with all 0s, or alternatively all 1s, and then, starting at either > end, systematically build up the possible strings by incremental changes. > We could of course start with an arbitrary string, and list all the > possible transformations of that string. We could write a transformation, > acting upon say a nine digit string, as follows: CLLCLCCCL > which means: Change the first digit in the string to be transformed (from > 1 > to 0 or 0 to 1), Leave the second digit as it is (0 -> 0; 1 -> 1), and so > on. But then the list of transformations is in effect the same binary list > that we enumerated in the first place, with Cs and Ls instead of 1s and > 0s. > How would we go about making a list of all the systematic changes that can > be made to an infinite binary string? This is a deep question in set theory. æIf Godel's axiom of construction, > V=L, is true then all infinite binary strings (of the kind you are > describing) could be well ordered by their descriptions. Such a list must be a list of all the > infinite periodic binary strings. All those strings which consist of an > initial (possibly null) finite string followed by a repeating string of a > set length, which can indeed be put into a single list. So if we were to > try and make a list of all infinite binary strings and began with the pi > string (the binary equivalent of pi, dropping the radix point), the > strings > produced would be periodic transformations of the pi string. In other > words, the only strings that could be produced would be predictable > variations of the pi string. But since in general an aperiodic string is > quite unpredictable, is explicated by computation with no short-cut > pattern > in the digits, it is impossible for say pi and some quite other irrational > quantity to appear in the same list. Those infinite strings that cannot be encapsulated in any finite way > require, in your case, ALEPH 0 bits of information. æA given string of this > kind is only `unpredictable' in the sense that it cannot be characterized in > any finite way. Why make such a fuss about this when there is the very fine and > well known argument that the power set of an infinite set 'exceeds' the > set > (which since select/deselect is a binary structure is equivalent to an > argument about infinite binary strings)? But the argument is too dazzling, > and prevents us from seeing whats involved in the notion of a list beyond > the fact that it's 1:1 with the natural numbers. All the aperiodic strings > cannot be in the same list because, at bottom, the idea of just two > aperodic strings being in the same list is oxymoronic. It's not about > there > being some mysteriously supernumerous quantity of infinite strings. There are 2^ALEPH 0 infinite strings of the kind this thread has been > discussing. Infinite strings always come with some means of production, either > some rule of composition (typically producing a periodic string, but there > are more subtle rules eg for Champernowne numbers and the like), or some > specified computation, or in the case of the original question, a point on > a line segment. Many infinite strings do not have finite rules that fully characterize them, > but these strings do have infinite rules that do fully characterize them. > An infinite string that can only be specified by ALEPH 0 bits of information > are computable via all ALEPH 0 bits of information and this computation can > be completed at time t=1, where the kth character of the string is computed > at time t=1-1/k.. Well, if you are going to allow infinite rules you can simply put the actual digits in the string in the rules... === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is > a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. In the present context > that is the arbitrary point P. If we had chosen the particular point P > such > that AB/AP = pi, then the string we produce for this ratio by applying a > fromula for pi has its anchorage in the definite position this point has > on > the line, so that geometry serves a kind of GPS for this string. (It is > not > necessary to believe that pi is a geometrical notion -- it is sufficient > that it is believed to be a precise ratio, which can then certainly be > represented as a point on a line.) In short, I don't believe that the > infinite binary string associated with an arbitrary point on the line > segment has an independent existence apart from that point. On that basis, > it cannot determine anything. If that's not complete bollocks, I'd be glad There really is no problem with arbitrary infinite binary strings. æFor any > given line segment, you can use such a string to reach many points on the > line segment. æIn the case of a line segment like [0,1], the infinite binary > string for 1/pi is completely determined but there are 2^ALEPH 0 locations > on [0,1] whose infinite binary strings cannot be completely determined in > any finite way. æThose strings still exist but they have no finite > representation and no physical instantiation. Could you please elaborate a little bit? Which strings/numbers are you hinting at here? Uncomputables? Why exactly 2^ALEPH 0 of them? Shouldn't 2^ALEPH 0 rather be the length of the segment [0,1] itself? > The total string for 1/pi has > no physical instantiation but the formula for 1/pi can serve as an algorithm > for computing it. æUsing the string for 1/pi, the location 1/pi can be > reached by going left or light, in each successive half-size segment, at > time t=1-1/k, for the kth character in 1/pi's string. æAt time t=1 you will > be at the location 1/pi. You mean (by de-mapping) at time T = k = oo, that is, in *exactly* ALEPH 0 steps, right? (This too is interesting because we have an external time, and the external environment is the locus of continuity, and where the infinite strings come from at all... hmm, wandering...) -LV > ALEPH 0 is so huge that it breaks out of the > unending successive approximations to 1/pi and takes you to 1/pi exactly. > The way to get to the end of an unending process is to go through every step > of the process. æThis whole line of reasoning is also true for an arbitrary > infinite binary string. === Subject: Re: Infinite Binary Strings: A Question posting-account=Z3AipgkAAABkoMfyNwddSxsYhXHi5CDt CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > æ æ æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æ æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? To put it a bit simplistically: >At the end of the 19th century, it was Dedekind that wanted this 'to be >so', and he was thrilled when he heard about a guy called Cantor, who >had a theory providing just that: set theory. Nowadays, it's a necessary consequence of the way the real number line >is -defined- using set theory (either via Dedekind cuts or via Cauchy >sequences, see for example >http://en.wikipedia.org/wiki/Construction of real numbers). In short, the answer to your question is: the standard -definition- of >the real number set is the bit of mathematics that you're looking for. æ æ æ æ But isn't what's at stake here more general than the issue of the > definition of the real number? And I'm still struck by the feeling that the > idea of this binary sequence of lefts and rights determining a point is > unconvincing. Of course I need to say why. æ æ æ æ Part of what I have in mind is that an aperiodic string is in > general, if not always, chaotic or unpredictable. By that I suppose I > should mean something specific to the effect that there is no significantly > shorter way to determine the k-th digit (for some arbitrary, perhaps large > k) than to compute the string up to the k-th digit, or something like that. > At any rate, the last digit computed so far flips erratically between 0 and > 1 as the string is explicated. Where is the precision in such a concept? æ æ æ æ I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. In the present context > that is the arbitrary point P. If we had chosen the particular point P such > that AB/AP = pi, then the string we produce for this ratio by applying a > fromula for pi has its anchorage in the definite position this point has on > the line, so that geometry serves a kind of GPS for this string. (It is not > necessary to believe that pi is a geometrical notion -- it is sufficient > that it is believed to be a precise ratio, which can then certainly be > represented as a point on a line.) In short, I don't believe that the > infinite binary string associated with an arbitrary point on the line > segment has an independent existence apart from that point. On that basis, > it cannot determine anything. If that's not complete bollocks, I'd be glad Leon- Hide quoted text - - Show quoted text - My own intuition of the line is that it has no holes and that it is obviously possible to chop it into rational number sized bits. Had Pythagoras not made his unfortunate discovery, that would have been enough for a long time. Now, suppose we start chopping the line into halves - that is chop the original line into halves, then pick one of two created halves and iterate. My personal intuition is that there cannot be hole that we are chopping toward. In terms of the infinite descending sequence of closed intervals, it means that there must be some point in their intersection. This is, of course, assured by compactness, which you don't have until you make some formal definition of the meaning of the points on a line, but my personal intuition says that the point must be there no matter how we chop to get it. As other posters have mentioned, you can study the formal definition of the real numbers, either the Dedekind constructions (which I don't prefer) or the Cantor construction by equivalence classes of Cauchy sequences - which, with a minor twist, also gets you the p-adic numbers and is more akin to what is done by analysts. Your real problem seems to be with the logical foundations. For you, I think, sets must be built by construction; they can't just exist. My personal intution has no problem with sets just existing, and so I have essentially no insight into why many people have a problem with this. This is also known as the problem of actual infinity versus potential infinity. For me, the axiom of choice is obviously true. In fact, when I was an undergraduate, the common saying among my math major friends was, The axiom of choice is obviously true, the well- ordering theorem is obviously false, and Zorn's lemma was so weird that there was available intuition about it. I agree with the first and last statements. I have no intuition either for or against the well-ordering theorem. My own approach is practical. Zorn's lemma, equivalent to the obviously true axiom of choice, is essential in many parts of mathematics. I mostly know about the one's in algebra where its use originally surprised me a great deal. Since it is so useful, and since no contradiction has every come from using it, and also since the axiom of choice is obviously true, why not use it? The truth or falsity of axioms is, of course, in a particularly weird philosophical place. It is an axiom, so what is meant by either true or false? Here we must be guided by our intuition and by our experience. Logic is helpful but not directly helpful. Finally, I was told that the introduction to some book on foundations had an introduction or preface which told the tale of a great castle which had stood for centuries. In the basement of the castle were a large number of spiders very actively weaving webs all through the basement. One day a property manager was hired and he had the entire basement cleaned up. The spiders survived, but their webs did not. The became frantic, spinning webs as quickly as they could, because they were convinced that if they did not, the entire structure would collapse. Achava === Subject: Re: Infinite Binary Strings: A Question > æ æ æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æ æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? >To put it a bit simplistically: >At the end of the 19th century, it was Dedekind that wanted this 'to be >so', and he was thrilled when he heard about a guy called Cantor, who >had a theory providing just that: set theory. >Nowadays, it's a necessary consequence of the way the real number line >is -defined- using set theory (either via Dedekind cuts or via Cauchy >sequences, see for example >http://en.wikipedia.org/wiki/Construction_of_real_numbers). >In short, the answer to your question is: the standard -definition- of >the real number set is the bit of mathematics that you're looking for. > æ æ æ æ But isn't what's at stake here more general than the issue of the > definition of the real number? And I'm still struck by the feeling that the > idea of this binary sequence of lefts and rights determining a point is > unconvincing. Of course I need to say why. > æ æ æ æ Part of what I have in mind is that an aperiodic string is in > general, if not always, chaotic or unpredictable. By that I suppose I > should mean something specific to the effect that there is no significantly > shorter way to determine the k-th digit (for some arbitrary, perhaps large > k) than to compute the string up to the k-th digit, or something like that. > At any rate, the last digit computed so far flips erratically between 0 and > 1 as the string is explicated. Where is the precision in such a concept? > æ æ æ æ I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. In the present context > that is the arbitrary point P. If we had chosen the particular point P such > that AB/AP = pi, then the string we produce for this ratio by applying a > fromula for pi has its anchorage in the definite position this point has on > the line, so that geometry serves a kind of GPS for this string. (It is not > necessary to believe that pi is a geometrical notion -- it is sufficient > that it is believed to be a precise ratio, which can then certainly be > represented as a point on a line.) In short, I don't believe that the > infinite binary string associated with an arbitrary point on the line > segment has an independent existence apart from that point. On that basis, > it cannot determine anything. If that's not complete bollocks, I'd be glad > Leon- Hide quoted text - > - Show quoted text - My own intuition of the line is that it has no holes and that it is >obviously possible to chop it into rational number sized bits. Had >Pythagoras not made his unfortunate discovery, that would have been >enough for a long time. Now, suppose we start chopping the line into >halves - that is chop the original line into halves, then pick one of >two created halves and iterate. My personal intuition is that there >cannot be hole that we are chopping toward. In terms of the infinite >descending sequence of closed intervals, it means that there must be >some point in their intersection. This is, of course, assured by >compactness, which you don't have until you make some formal >definition of the meaning of the points on a line, but my personal >intuition says that the point must be there no matter how we chop to >get it. My intuition of the spatial line is that it has no holes. My intuition of the number line is that it has had its holes filled in. But in the light of the replies I've had, here from you and from others, particularly Ramsay, I've got a lot to think about. My original question was perhaps slightly faux naif, but not that faux! I hadn't really considered the analytic geometry aspects at all, for example. >As other posters have mentioned, you can study the formal definition >of the real numbers, either the Dedekind constructions (which I don't >prefer) or the Cantor construction by equivalence classes of Cauchy >sequences - which, with a minor twist, also gets you the p-adic >numbers and is more akin to what is done by analysts. Your real problem seems to be with the logical foundations. Perhaps fundamentals, rather than foundations. >For you, >I think, sets must be built by construction; they can't just exist. >My personal intution has no problem with sets just existing, and so I >have essentially no insight into why many people have a problem with >this. This is also known as the problem of actual infinity versus >potential infinity. For me, the axiom of choice is obviously true. >In fact, when I was an undergraduate, the common saying among my math >major friends was, The axiom of choice is obviously true, the well- >ordering theorem is obviously false, and Zorn's lemma was so weird >that there was available intuition about it. I agree with the first >and last statements. I have no intuition either for or against the >well-ordering theorem. I've come across that before. I was reading something by a mathematician about the axiom of choice recently to the effect that the point of the independence of the axiom is that it is true (or 'true', at least) in some contexts and false in others. He gave as an example of the latter case something to do with Ramsey theory, which I couldn't follow (though I've since just started to read up on Ramsey theory), in that to accept the axiom would lead to markedly artificial and counter-intuitive results, and as an example of the former something I think to do with infinite variable spaces, the point being I think that we clearly need to be able to think of an arbitrary point, one with some arbitrary value in each of the infinitely many dimensions. My own approach is practical. Zorn's lemma, equivalent to the >obviously true axiom of choice, is essential in many parts of >mathematics. I mostly know about the one's in algebra where its use >originally surprised me a great deal. Since it is so useful, and >since no contradiction has every come from using it, and also since >the axiom of choice is obviously true, why not use it? The truth or >falsity of axioms is, of course, in a particularly weird philosophical >place. It is an axiom, so what is meant by either true or false? >Here we must be guided by our intuition and by our experience. Logic >is helpful but not directly helpful. Finally, I was told that the introduction to some book on foundations >had an introduction or preface which told the tale of a great castle >which had stood for centuries. In the basement of the castle were a >large number of spiders very actively weaving webs all through the >basement. One day a property manager was hired and he had the entire >basement cleaned up. The spiders survived, but their webs did not. >The became frantic, spinning webs as quickly as they could, because >they were convinced that if they did not, the entire structure would >collapse. I can appreciate that an obsession with foundations may be more of historical interest than a pressing modern concern. >Achava leon === Subject: Re: Infinite Binary Strings: A Question > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? To put it a bit simplistically: >At the end of the 19th century, it was Dedekind that wanted this 'to be >so', and he was thrilled when he heard about a guy called Cantor, who >had a theory providing just that: set theory. Nowadays, it's a necessary consequence of the way the real number line >is -defined- using set theory (either via Dedekind cuts or via Cauchy >sequences, see for example >http://en.wikipedia.org/wiki/Construction_of_real_numbers). In short, the answer to your question is: the standard -definition- of >the real number set is the bit of mathematics that you're looking for. But isn't what's at stake here more general than the issue of the > definition of the real number? And I'm still struck by the feeling that the > idea of this binary sequence of lefts and rights determining a point is > unconvincing. Of course I need to say why. Part of what I have in mind is that an aperiodic string is in > general, if not always, chaotic or unpredictable. By that I suppose I > should mean something specific to the effect that there is no significantly > shorter way to determine the k-th digit (for some arbitrary, perhaps large > k) than to compute the string up to the k-th digit, or something like that. > At any rate, the last digit computed so far flips erratically between 0 and > 1 as the string is explicated. Where is the precision in such a concept? If there is some rule, however complex, by which, for any given positive natural n, the nth digit can be determined to be a 0 or a 1, then the expansion is computable', which is satisfactory for the existence of the the number represented by that string for all mathematics, including constructionist mathematics. It can be proved indirectly that there are infinite sequences of 0's and 1's for which no such rule can exist. Constructionists reject the existence of such numbers, but in standard mathematics they are regarded as existing but inaccessible. I also have in mind an argument which I couldn't lay out in a > couple of paragraphs but I can summarize here. (I might set it out in a > separate thread if I could get more confident about it.) I think there is a > difficulty with the notion of an arbitray infinite binary string, > understood as a point in a uniform combinatorial space of 2^infinity > possibilities. I believe an infinite binary string has to come from > somewhere, has to have something that produces it. In the present context > that is the arbitrary point P. If we had chosen the particular point P such > that AB/AP = pi, then the string we produce for this ratio by applying a > fromula for pi has its anchorage in the definite position this point has on > the line, so that geometry serves a kind of GPS for this string. (It is not > necessary to believe that pi is a geometrical notion -- it is sufficient > that it is believed to be a precise ratio, which can then certainly be > represented as a point on a line.) In short, I don't believe that the > infinite binary string associated with an arbitrary point on the line > segment has an independent existence apart from that point. On that basis, > it cannot determine anything. If that's not complete bollocks, I'd be glad > Leon === Subject: Re: Infinite Binary Strings: A Question > But isn't what's at stake here more general than the issue of the > definition of the real number? And I'm still struck by the feeling that the > idea of this binary sequence of lefts and rights determining a point is > unconvincing. Of course I need to say why. > Part of what I have in mind is that an aperiodic string is in > general, if not always, chaotic or unpredictable. By that I suppose I > should mean something specific to the effect that there is no significantly > shorter way to determine the k-th digit (for some arbitrary, perhaps large > k) than to compute the string up to the k-th digit, or something like that. > At any rate, the last digit computed so far flips erratically between 0 and > 1 as the string is explicated. Where is the precision in such a concept? If there is some rule, however complex, by which, for any given positive >natural n, the nth digit can be determined to be a 0 or a 1, then the >expansion is computable', which is satisfactory for the existence of >the the number represented by that string for all mathematics, including >constructionist mathematics. It can be proved indirectly that there are infinite sequences of 0's >and 1's for which no such rule can exist. Constructionists reject the >existence of such numbers, but in standard mathematics they are >regarded as existing but inaccessible. > remarks about arbitrary infinite binary strings, which may have a bearing on this. leon === Subject: Re: Infinite Binary Strings: A Question > If there is some rule, however complex, by which, for any given positive > natural n, the nth digit can be determined to be a 0 or a 1, then the > expansion is computable', which is satisfactory for the existence of > the the number represented by that string for all mathematics, including > constructionist mathematics. It can be proved indirectly that there are infinite sequences of 0's > and 1's for which no such rule can exist. Constructionists Do you mean constructivists? > reject the > existence of such numbers FYI, there are flavours of intuitionism that don't. > , but in standard mathematics they are > regarded as existing but inaccessible. -- Herman Jurjus === Subject: Re: Infinite Binary Strings: A Question <48cae2b3$0$27216$ba620dc5@text.nova.planet.nl> posting-account=UJeUTgkAAADYai-ULU41ORCvNnkXmdRu Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) |> If there is some rule, however complex, by which, for any given positive |> natural n, the nth digit can be determined to be a 0 or a 1, then the |> expansion is computable', which is satisfactory for the existence of |> the the number represented by that string for all mathematics, including |> constructionist mathematics. | |> It can be proved indirectly that there are infinite sequences of 0's |> and 1's for which no such rule can exist. Constructionists | |Do you mean constructivists? | |> reject the |> existence of such numbers | |FYI, there are flavours of intuitionism that don't. I think essentially anything that you could call intuitionism has a theory of free choice sequences. The notion of free choice sequences was introduced by Brouwer. A free choice sequence is given term by term without at any point committing to a rule. Intuitionism is essentially Brouwer's name for his own philosophy, and while people who continue to develop it may not always agree completely, to say that such a notion isn't valid probably means that one isn't really developing his philosophy further, but developing something else. In order to say that a real number r given by a free choice sequence is equal to 0, say, the generator of the sequence would have to commit to following a certain rule, hence it's not the case that r=0. Likewise for the number being equal to a given computable number c. This is distinct from saying that for each computable number c, there exists an positive integer n such that |r-c|>1/n, because that would mean committing to some scheme for ensuring that one is departing from the rule. In classical logic the two notions of uncomputable are of course the same thing, since the absurdity of r being equal to c is considered sufficient reason to say that there exists such an n. But constructive existence of such an n is not guaranteed. Brouwer also had a more general notion of a sequence given freely, but with the option of committing to a rule at some point, but he seems to have decided it was less worth pursuing. You might suppose that constructists who aren't intuitonists would be unhappy with all this somehow, but I'm not convinced that you would be right. Bishop has some remarks on Brouwer's real analysis in _Foundations of Constructive Analysis_. My impression is that non-intuitionist constructivists often just don't consider the study of free-choice sequences to be worth spending much effort on. In my opinion, I think we have to accept it as at least being coherent and meaningful. Constructivists in the Markov school are supposed to have explicitly assumed that each sequence is computable. Constructive mathematics done neither by intuitionist nor Markov-school constructivists usually just omits to assume anything special about how a sequence is generated. One omits the law of excluded middle or any of its equivalents, that implies the existence of uncomputable sequences. This is less a matter of denying the existence of uncomputable sequences as a matter of just remaining neutral. As long as there's nothing to be gained by making these extra assumptions, why not just leave them out? You might, for example, want to apply your reasoning to a sequence generated by some random natural process, and normally this is perfectly fine. |> , but in standard mathematics they are |> regarded as existing but inaccessible. Inaccessible only in the sense of not being computable. Keith Ramsay === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for examplehttp://en.wikipedia.org/wiki/Construction of real numbers). In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. So, to each point corresponds one, and only one, (infinite) string, and -- conversely -- to each (infinite) string corresponds one, and only one, point. Correct? -LV === Subject: Re: Infinite Binary Strings: A Question > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such > that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell > us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction_of_real_numbers). In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. Correct? Wrong! That uniqueness of representation is true for exactly those points for which there are infinitely many left intervals and infinitely many right intervals in the sequence of nested intervals. I.e., an infinite string is unique for points which are INTERIOR to every interval in its sequence of narrowings (not the endpoint of any such interval), but there are dual strings for any point which is an endpoint of any such interval. For example, using julio's own notataion, 0(1) and 1(0) are different infinite strings representing the same point. === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such > that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell > us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction of real numbers). > In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. Correct? Wrong! That uniqueness of representation is true for exactly those points for > which there are infinitely many left intervals and infinitely many right > intervals in the sequence of nested intervals. I.e., an infinite string is unique for points which are INTERIOR to > every interval in its sequence of narrowings (not the endpoint of any > such interval), but there are dual strings for any point which is an > endpoint of any such interval. For example, using julio's own notataion, 0(1) and 1(0) are different > infinite strings representing the same point. Of course that's not what I said. I am making a distinction between the *sequences* and the infinite *strings* which you have missed. That might indeed be a delicate question. Again: the two *sequences* converge to a unique *point*/*string*. IOW, there is a distinction between the sequences of digits, as -say- may be output by a TM, which are finite although unbounded, and the infinite strings corresponding to points. In fact, we pass from sequences to infinite strings through limits: they are distinct entities. Back to our example, the two *sequences* can be written, to avoid ambiguity: lower seq. := 0,1,1,1,1,1,1,1,1,1,1,1,1,... upper seq. := 1,0,0,0,0,0,0,0,0,0,0,0,0,... Still the unique midpoint here corresponds to the *string*: midpoint == 10000000000000000000000...0 (There is a reason why I am putting a final 0 there. In any case, we can always rewrite 1(0).) -LV === Subject: Re: Infinite Binary Strings: A Question > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P > lies > upon, and so on repeatedly. If we happen to have chosen a point P > such > that > AP is incommensurable with AB, the point P will never lie exactly at > the > end of any half interval. (It will never lie at the end of any > fractional > interval of the line segment.) æSo the point P produces an infinite, > and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would > tell > us > that you cannot get to a point by this repeated narrowing down -- > it's > intervals all the way down. Mathematics seems to be telling us that, > by > somehow treating the infinite sequence of narrowings down as a whole, > an > infinite binary string would indeed determine a precise point on AB. > My > question is: which bit of mathematics is it, exactly, that is telling > us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction_of_real_numbers). > In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. > So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. > Correct? Wrong! That uniqueness of representation is true for exactly those points for > which there are infinitely many left intervals and infinitely many right > intervals in the sequence of nested intervals. I.e., an infinite string is unique for points which are INTERIOR to > every interval in its sequence of narrowings (not the endpoint of any > such interval), but there are dual strings for any point which is an > endpoint of any such interval. For example, using julio's own notataion, 0(1) and 1(0) are different > infinite strings representing the same point. Of course that's not what I said. I am making a distinction between > the *sequences* and the infinite *strings* which you have missed. So what is this critical difference you make between an infinite sequence of 0's and 1's and an infinite string of 0's and 1's? That might indeed be a delicate question. Again: the two *sequences* converge to a unique *point*/*string*. IOW, > there is a distinction between the sequences of digits, as -say- may > be output by a TM, which are finite although unbounded, and the > infinite strings corresponding to points. In fact, we pass from > sequences to infinite strings through limits: they _are_ distinct > entities. To mathematicians, sequences need not be finite unless explicitely declared as such, and, as you failed to make any such declaration, they need not be finite. Back to our example, the two *sequences* can be written, to avoid > ambiguity: lower seq. := 0,1,1,1,1,1,1,1,1,1,1,1,1,... > upper seq. := 1,0,0,0,0,0,0,0,0,0,0,0,0,... Still the _unique_ midpoint here corresponds to the *string*: midpoint == 10000000000000000000000...0 (There is a reason why I am putting a final 0 there. In any case, we > can always rewrite 1(0).) ( If you are going to do that, why not simply write it as 1 (or as 1() to indicate infinitely many empty strings follow the 1) to eliminate all those unnecessary 0's. To write 'midpoint == 10000000000000000000000...0' falsely indicates that the process of bisecting intervals must have an end while the corresponding 011111.... need not have an end. === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) the *sequences* and the infinite *strings* which you have missed. So what is this critical difference you make between an infinite > sequence of 0's and 1's and an infinite string of 0's and 1's? The sequences are *not* infinite, they are finite although unbounded. That's critical. -LV === Subject: Re: Infinite Binary Strings: A Question posting-account=Y877ggoAAAB-75kOgSLDnLSoS2_jn90U Gecko/20071126 Fedora/1.5.0.12-7.fc6 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > Of course that's not what I said. I am making a distinction between > the *sequences* and the infinite *strings* which you have missed. So what is this critical difference you make between an infinite > sequence of 0's and 1's and an infinite string of 0's and 1's? The sequences are *not* infinite, they are finite although unbounded. > That's critical. -LV The term 'unbounded' is sometimes used to describe that there is no upper bound on the elements of e.g. a, possibly implicit, set, sequence or function. However, here we speak only of particular strings, like 0111... and 1000... . Could you describe what you mean by these strings being unbounded? Are you saying that we should instead consider constructs such as these to represent (the limit of?) the sequences 0, 01, 011, ..., and 1, 10, 100, ..., respectively? === Subject: Re: Infinite Binary Strings: A Question > Of course that's not what I said. I am making a distinction between > the *sequences* and the infinite *strings* which you have missed. So what is this critical difference you make between an infinite > sequence of 0's and 1's and an infinite string of 0's and 1's? The sequences are *not* infinite, they are finite although unbounded. > That's critical. No single sequence, or even finite set of sequences, can be finite but unbounded, it requires an infinite sets of sequences to be unbounded. So that what is critical for you is also impossible. === Subject: Re: Infinite Binary Strings: A Question posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for examplehttp://en.wikipedia.org/wiki/Construction of real numbers). In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. Correct? To some points there corresponds *two* infinite sequences. -- m === Subject: Re: Infinite Binary Strings: A Question <48ca7f87$0$27228$ba620dc5@text.nova.planet.nl> æ æGiven a line segment AB, and a point P arbitrarily chosen upon > æ æit, > one can ask which half of AB P lies on, left or right, then having > selected the half interval P lies on we can ask which half of that > interval P lies upon, and so on repeatedly. If we happen to have > chosen a point P such that AP is incommensurable with AB, the point > P will never lie exactly at the end of any half interval. (It will > never lie at the end of any fractional interval of the line > segment.) æSo the point P produces an infinite, and aperiodic, > infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic > æ æbinary > string pick out a precise point on AB? Comon sense, perhaps, would > tell us that you cannot get to a point by this repeated narrowing > down -- it's intervals all the way down. Mathematics seems to be > telling us that, by somehow treating the infinite sequence of > narrowings down as a whole, an infinite binary string would indeed > determine a precise point on AB. My question is: which bit of > mathematics is it, exactly, that is telling us that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to > be so', and he was thrilled when he heard about a guy called Cantor, > who had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number > line is -defined- using set theory (either via Dedekind cuts or via > Cauchy sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction_of_real_numbers). > In short, the answer to your question is: the standard -definition- > of the real number set is the bit of mathematics that you're looking > for. > So, to each point corresponds one, and only one, (infinite) string, and > -- conversely -- to each (infinite) string corresponds one, and only > one, point. > Correct? To some points there corresponds *two* infinite sequences. -- m For example, to the midpoint correspond the two strings 011111111111111111111111111111111111..... and 100000000000000000000000000000000000..... -- hendrik === Subject: Re: Infinite Binary Strings: A Question <8ab91$48caa9e4$d88ac3c2$20798@PRIMUS.CA> posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > æ æGiven a line segment AB, and a point P arbitrarily chosen upon > æ æit, > one can ask which half of AB P lies on, left or right, then having > selected the half interval P lies on we can ask which half of that > interval P lies upon, and so on repeatedly. If we happen to have > chosen a point P such that AP is incommensurable with AB, the point > P will never lie exactly at the end of any half interval. (It will > never lie at the end of any fractional interval of the line > segment.) æSo the point P produces an infinite, and aperiodic, > infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic > æ æbinary > string pick out a precise point on AB? Comon sense, perhaps, would > tell us that you cannot get to a point by this repeated narrowing > down -- it's intervals all the way down. Mathematics seems to be > telling us that, by somehow treating the infinite sequence of > narrowings down as a whole, an infinite binary string would indeed > determine a precise point on AB. My question is: which bit of > mathematics is it, exactly, that is telling us that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to > be so', and he was thrilled when he heard about a guy called Cantor, > who had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number > line is -defined- using set theory (either via Dedekind cuts or via > Cauchy sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction of real numbers). > In short, the answer to your question is: the standard -definition- > of the real number set is the bit of mathematics that you're looking > for. > So, to each point corresponds one, and only one, (infinite) string, and > -- conversely -- to each (infinite) string corresponds one, and only > one, point. > Correct? To some points there corresponds *two* infinite sequences. -- m For example, to the midpoint correspond the two strings æ æ011111111111111111111111111111111111..... > and > æ æ100000000000000000000000000000000000..... Which point in the middle of what? If those are the two bounding sequences, they still converge to a unique limit (string), namely the very 1.0000000000000000... Correct? -LV === Subject: Re: Infinite Binary Strings: A Question posting-account=yxbZkgkAAABQBvyYeebYQ-PAvi0uT3tG Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > To some points there corresponds *two* infinite sequences. > For example, to the midpoint correspond the two strings > 011111111111111111111111111111111111..... > and > 100000000000000000000000000000000000..... Which point in the middle of what? If those are the two bounding sequences, they still converge to a > unique limit (string), namely the very 1.0000000000000000... No, the two infinite strings 0.111... and 1.000... are two different digit strings representing exactly the same real. === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > To some points there corresponds *two* infinite sequences. > For example, to the midpoint correspond the two strings > 011111111111111111111111111111111111..... > and > 100000000000000000000000000000000000..... Which point in the middle of what? If those are the two bounding sequences, they still converge to a > unique limit (string), namely the very 1.0000000000000000... No, the two infinite strings 0.111... and 1.000... are two > different digit strings representing exactly the same real. You too, please note that I am stressing a distinction in meaning between the _words_: *SEQUENCES* vs. *STRINGS*/POINTS/NUMBERS. That's crucial for any progress in the discussion, even if just to show such approach invalid. -LV === Subject: Re: Infinite Binary Strings: A Question > To some points there corresponds *two* infinite sequences. > For example, to the midpoint correspond the two strings > 011111111111111111111111111111111111..... > and > 100000000000000000000000000000000000..... > Which point in the middle of what? > If those are the two bounding sequences, they still converge to a > unique limit (string), namely the very 1.0000000000000000... No, the two infinite strings 0.111... and 1.000... are two > different digit strings representing exactly the same real. You too, please note that I am stressing a distinction in meaning > between the _words_: *SEQUENCES* vs. *STRINGS* A string is a sequence of characters, and all sequences require that between any two of their members there are only finitely many other members. Since julio's *STRINGS* do not behave this way, they are garbage. === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > To some points there corresponds *two* infinite sequences. > For example, to the midpoint correspond the two strings 011111111111111111111111111111111111..... > and > 100000000000000000000000000000000000..... > Which point in the middle of what? > If those are the two bounding sequences, they still converge to a > unique limit (string), namely the very 1.0000000000000000... > No, the two infinite strings 0.111... and 1.000... are two > different digit strings representing exactly the same real. You too, please note that I am stressing a distinction in meaning >between the _words_: *SEQUENCES* vs. *STRINGS*/POINTS/NUMBERS. >That's crucial for any progress in the discussion, even if just to >show such approach invalid. What's crucial is that when we say something about strings > you understand what we mean by the term. If you want to > use the same words to mean something else it's crucial that > you use different words instead. You have no monopoly on reasoning or language. You just confirm youself a pernicious troll, and overall simply an idiot. -LV === Subject: Re: Infinite Binary Strings: A Question > To some points there corresponds *two* infinite sequences. > For example, to the midpoint correspond the two strings > 011111111111111111111111111111111111..... > and > 100000000000000000000000000000000000..... > Which point in the middle of what? > If those are the two bounding sequences, they still converge to a > unique limit (string), namely the very 1.0000000000000000... > No, the two infinite strings 0.111... and 1.000... are two > different digit strings representing exactly the same real. >You too, please note that I am stressing a distinction in meaning >between the _words_: *SEQUENCES* vs. *STRINGS*/POINTS/NUMBERS. >That's crucial for any progress in the discussion, even if just to >show such approach invalid. What's crucial is that when we say something about strings > you understand what we mean by the term. If you want to > use the same words to mean something else it's crucial that > you use different words instead. You have no monopoly on reasoning or language. He does in comparison to julio's reasoning and language. You just confirm youself a pernicious troll, and overall simply an > idiot. Returning such insults when bested, as julio has been, are the mark of a poor but habitual loser. === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > æ æGiven a line segment AB, and a point P arbitrarily chosen upon > æ æit, > one can ask which half of AB P lies on, left or right, then having > selected the half interval P lies on we can ask which half of that > interval P lies upon, and so on repeatedly. If we happen to have > chosen a point P such that AP is incommensurable with AB, the point > P will never lie exactly at the end of any half interval. (It will > never lie at the end of any fractional interval of the line > segment.) æSo the point P produces an infinite, and aperiodic, > infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic > æ æbinary > string pick out a precise point on AB? Comon sense, perhaps, would > tell us that you cannot get to a point by this repeated narrowing > down -- it's intervals all the way down. Mathematics seems to be > telling us that, by somehow treating the infinite sequence of > narrowings down as a whole, an infinite binary string would indeed > determine a precise point on AB. My question is: which bit of > mathematics is it, exactly, that is telling us that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to > be so', and he was thrilled when he heard about a guy called Cantor, > who had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number > line is -defined- using set theory (either via Dedekind cuts or via > Cauchy sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction of real numbers). > In short, the answer to your question is: the standard -definition- > of the real number set is the bit of mathematics that you're looking > for. > So, to each point corresponds one, and only one, (infinite) string, and > -- conversely -- to each (infinite) string corresponds one, and only > one, point. > Correct? > To some points there corresponds *two* infinite sequences. > -- m For example, to the midpoint correspond the two strings æ æ011111111111111111111111111111111111..... > and > æ æ100000000000000000000000000000000000..... Which point in the middle of what? If those are the two bounding sequences, they still converge to a > unique limit (string), namely the very 1.0000000000000000... Correct? Typo, should of course read: 1000000000000000000000000000000.... the (unique) middle point/string. -LV === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On 12 Sep, 16:47, Mariano Su.87rez-Alvarez > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for examplehttp://en.wikipedia.org/wiki/Construction of real numbers). > In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. Correct? To some points there corresponds *two* infinite sequences. Two *sequences* converging to a unique limit that is the above infinite *string*. And so, again: to each *point* corresponds one, and only one, (infinite) *string*; and viceversa. Correct? -LV === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On 12 Sep, 16:47, Mariano Su.87rez-Alvarez > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for examplehttp://en.wikipedia.org/wiki/Construction of real numbers). > In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. > So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. > Correct? > To some points there corresponds *two* infinite sequences. Two *sequences* converging to a unique limit that is the above >infinite *string*. And so, again: to each *point* corresponds one, and only one, >(infinite) *string*; and viceversa. Correct? No. It's curious how you misunderstand clearly > stated things. The statement some points > correspond to two infinite sequences means > exactly that some points correspond to two > infinite sequences. What you reiterate above is of course clear. I am *adding* to the discussion. You just confirm youself a pernicious troll, and overall simply an idiot. -LV === Subject: Re: Infinite Binary Strings: A Question >On 12 Sep, 16:47, Mariano Su.87rez-Alvarez > æ æGiven a line segment AB, and a point P arbitrarily chosen upon > it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval > P lies > upon, and so on repeatedly. If we happen to have chosen a point P > such that > AP is incommensurable with AB, the point P will never lie exactly > at the > end of any half interval. (It will never lie at the end of any > fractional > interval of the line segment.) æSo the point P produces an > infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic > binary > string pick out a precise point on AB? Comon sense, perhaps, would > tell us > that you cannot get to a point by this repeated narrowing down -- > it's > intervals all the way down. Mathematics seems to be telling us > that, by > somehow treating the infinite sequence of narrowings down as a > whole, an > infinite binary string would indeed determine a precise point on > AB. My > question is: which bit of mathematics is it, exactly, that is > telling us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to > be > so', and he was thrilled when he heard about a guy called Cantor, > who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number > line > is -defined- using set theory (either via Dedekind cuts or via > Cauchy > sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction_of_real_numbers). > In short, the answer to your question is: the standard -definition- > of > the real number set is the bit of mathematics that you're looking > for. > So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. > Correct? > To some points there corresponds *two* infinite sequences. >Two *sequences* converging to a _unique_ limit that is the above >infinite *string*. >And so, again: to each *point* corresponds one, and only one, >(infinite) *string*; and viceversa. >Correct? No. It's curious how you misunderstand clearly > stated things. The statement some points > correspond to two infinite sequences means > exactly that some points correspond to two > infinite sequences. What you reiterate above is of course clear. I am *adding* to the > discussion. What you are adding lessens the content. You just confirm youself a pernicious troll, and overall simply an > idiot. Whenever anyone points out some obvious foolishness or idiocy in julio's postings, he responds with insults, as above. Weighing the relevance and quality of julio's posts against David's gives a ration of infinitesimal value. === Subject: Re: Infinite Binary Strings: A Question > On 12 Sep, 16:47, Mariano Su.87rez-Alvarez > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval P > lies > upon, and so on repeatedly. If we happen to have chosen a point P > such that > AP is incommensurable with AB, the point P will never lie exactly at > the > end of any half interval. (It will never lie at the end of any > fractional > interval of the line segment.) æSo the point P produces an infinite, > and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would > tell us > that you cannot get to a point by this repeated narrowing down -- > it's > intervals all the way down. Mathematics seems to be telling us that, > by > somehow treating the infinite sequence of narrowings down as a whole, > an > infinite binary string would indeed determine a precise point on AB. > My > question is: which bit of mathematics is it, exactly, that is telling > us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction_of_real_numbers). > In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. > So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. > Correct? To some points there corresponds *two* infinite sequences. Two *sequences* converging to a _unique_ limit that is the above > infinite *string*. And so, again: to each *point* corresponds one, and only one, > (infinite) *string*; and viceversa. Correct? -LV WRONG! In julioÍs own notation, the two different infinite strings 0(1) and 1(0) represent the same point ( or number) === Subject: Re: Infinite Binary Strings: A Question posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > On 12 Sep, 16:47, Mariano Su.87rez-Alvarez > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for examplehttp://en.wikipedia.org/wiki/Construction of real numbers). > In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. > So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. > Correct? To some points there corresponds *two* infinite sequences. Two *sequences* converging to a unique limit that is the above > infinite *string*. And so, again: to each *point* corresponds one, and only one, > (infinite) *string*; and viceversa. Correct? -- m === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On 12 Sep, 19:13, Mariano Su.87rez-Alvarez On 12 Sep, 16:47, Mariano Su.87rez-Alvarez > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for examplehttp://en.wikipedia.org/wiki/Construction of real numbers). > In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. > So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. > Correct? > To some points there corresponds *two* infinite sequences. Two *sequences* converging to a unique limit that is the above > infinite *string*. And so, again: to each *point* corresponds one, and only one, > (infinite) *string*; and viceversa. Correct? > Ah, then it's correct! -LV === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On 12 Sep, 16:47, Mariano Su.87rez-Alvarez > æ æGiven a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down -- it's > intervals all the way down. Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. My > question is: which bit of mathematics is it, exactly, that is telling us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for examplehttp://en.wikipedia.org/wiki/Construction of real numbers). > In short, the answer to your question is: the standard -definition- of > the real number set is the bit of mathematics that you're looking for. > So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. > Correct? To some points there corresponds *two* infinite sequences. Two *sequences* converging to a unique limit that is the above > infinite *string*. And so, again: to each *point* corresponds one, and only one, > (infinite) *string*; and viceversa. Correct? I guess, in a strict sense, the point just IS the string. -LV === Subject: Re: Infinite Binary Strings: A Question > On 12 Sep, 16:47, Mariano Su.87rez-Alvarez > æ æGiven a line segment AB, and a point P arbitrarily chosen upon > it, > one can ask which half of AB P lies on, left or right, then having > selected > the half interval P lies on we can ask which half of that interval > P lies > upon, and so on repeatedly. If we happen to have chosen a point P > such that > AP is incommensurable with AB, the point P will never lie exactly > at the > end of any half interval. (It will never lie at the end of any > fractional > interval of the line segment.) æSo the point P produces an > infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æIs the converse true? That is, does an infinite, aperiodic > binary > string pick out a precise point on AB? Comon sense, perhaps, would > tell us > that you cannot get to a point by this repeated narrowing down -- > it's > intervals all the way down. Mathematics seems to be telling us > that, by > somehow treating the infinite sequence of narrowings down as a > whole, an > infinite binary string would indeed determine a precise point on > AB. My > question is: which bit of mathematics is it, exactly, that is > telling us > that this is so? > To put it a bit simplistically: > At the end of the 19th century, it was Dedekind that wanted this 'to > be > so', and he was thrilled when he heard about a guy called Cantor, who > had a theory providing just that: set theory. > Nowadays, it's a necessary consequence of the way the real number > line > is -defined- using set theory (either via Dedekind cuts or via Cauchy > sequences, see for > examplehttp://en.wikipedia.org/wiki/Construction_of_real_numbers). > In short, the answer to your question is: the standard -definition- > of > the real number set is the bit of mathematics that you're looking > for. > So, to each point corresponds one, and only one, (infinite) string, > and -- conversely -- to each (infinite) string corresponds one, and > only one, point. > Correct? > To some points there corresponds *two* infinite sequences. Two *sequences* converging to a _unique_ limit that is the above > infinite *string*. And so, again: to each *point* corresponds one, and only one, > (infinite) *string*; and viceversa. Correct? I guess, in a strict sense, the point just IS the string. Then some points must be like siamese twins, like the point having (in julio's own notation) both 0(1) and 1(0) as its twin infinite strings. === Subject: Re: Infinite Binary Strings: A Question posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Such a process will pick out a real number. > Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down That's not what my common sense tells me. > Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. What do you mean by a line? If you agree that line segments are parameterized by intervals in the real numbers, then it would. If you don't agree this, then you may have a different conception of line, which may or may not admit a mathematical model. (Maybe you think that Conway's surreal numbers better model your concept?) Your citation of common sense would indicate that you do have some other notion of line. But if your concept of line cannot be mathematically modelled, then it is useless asking mathematicians about it. Victor Meldrew I don't believe it! === Subject: Re: Infinite Binary Strings: A Question > Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) So the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Such a process will pick out a real number. > Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down That's not what my common sense tells me. > Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. What do you mean by a line? If you agree that line segments are >parameterized by intervals in the real numbers, then it would. If you don't agree this, then you may have a different conception of >line, which may or may not admit a mathematical model. (Maybe you >think >that Conway's surreal numbers better model your concept?) Your >citation >of common sense would indicate that you do have some other notion of >line. But if your concept of line cannot be mathematically modelled, >then it is useless asking mathematicians about it. Victor Meldrew >I don't believe it! I'm surprised that I need a particular conception of the line to frame this question. I would have thought any common or garden conception would do. And all I need to divide a line segment in two is a concept of rational number. My half-hearted appeal to common sense is meant to suggest that there is an issue here, which I believe to be an issue about infinite strings. The point of framing the question this way was to try and avoid getting bogged down with questions about real numbers. But to try and satisfy your scruples, I can put the question another way. What reason is there to believe that an infinite binary string as in, eg. 0.0110101...., has a precise value, to believe that no half plus one quarter plus one eighth plus no sixteenth etc has a precise sum, or in other words to believe that the sequence of progressive sums here (.0, .01, .011 etc) has a Least Upper Bound? There is a perfectly good reason, in that we want the real numbers to serve as a metric for continuously variable media (space, time and the like). But is it inherently plausible? I gather there are constructivist positions in which the LUB axiom would be rejected. I have also read that the LUB axiom can be proved in set theory. Do you know where I can find the proof? Is it hard? Does it involve the Axiom of Choice? These are all questions I am interested in. (Though whether I would be able to follow the answers is another matter.) But as I say, I think there is an issue here about infinite strings, in all generality, though obviously real numbers are a massive application of the concept. Leon === Subject: Re: Infinite Binary Strings: A Question <3jclc4ttuv4tmq6k7pd6c5nh6hrs6auj8l@4ax.com> posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > æ æ æ æ Given a line segment AB, and a point P arbitrarily chosen upon it, > one can ask which half of AB P lies on, left or right, then having selected > the half interval P lies on we can ask which half of that interval P lies > upon, and so on repeatedly. If we happen to have chosen a point P such that > AP is incommensurable with AB, the point P will never lie exactly at the > end of any half interval. (It will never lie at the end of any fractional > interval of the line segment.) æSo the point P produces an infinite, and > aperiodic, infinite string eg LRRLLLR...... > æ æ æ æ Is the converse true? That is, does an infinite, aperiodic binary > string pick out a precise point on AB? Such a process will pick out a real number. > Comon sense, perhaps, would tell us > that you cannot get to a point by this repeated narrowing down That's not what my common sense tells me. > Mathematics seems to be telling us that, by > somehow treating the infinite sequence of narrowings down as a whole, an > infinite binary string would indeed determine a precise point on AB. What do you mean by a line? If you agree that line segments are >parameterized by intervals in the real numbers, then it would. If you don't agree this, then you may have a different conception of >line, which may or may not admit a mathematical model. (Maybe you >think >that Conway's surreal numbers better model your concept?) Your >citation >of common sense would indicate that you do have some other notion of >line. But if your concept of line cannot be mathematically modelled, >then it is useless asking mathematicians about it. Victor Meldrew >I don't believe it! æ æ æ æ I'm surprised that I need a particular conception of the line to > frame this question. I would have thought any common or garden conception > would do. And all I need to divide a line segment in two is a concept of > rational number. æ æ æ æ My half-hearted appeal to common sense is meant to suggest that > there is an issue here, which I believe to be an issue about infinite > strings. The point of framing the question this way was to try and avoid > getting bogged down with questions about real numbers. But to try and > satisfy your scruples, I can put the question another way. What reason is > there to believe that an infinite binary string as in, eg. 0.0110101...., > has a precise value, to believe that æno half plus one quarter plus one > eighth plus no sixteenth etc has a precise sum, or in other words to > believe that the sequence of progressive sums here (.0, .01, .011 etc) has > a Least Upper Bound? There is a perfectly good reason, in that we want the > real numbers to serve as a metric for continuously variable media (space, > time and the like). But is it inherently plausible? I gather there are > constructivist positions in which the LUB axiom would be rejected. I have > also read that the LUB axiom can be proved in set theory. Do you know where > I can find the proof? Is it hard? Does it involve the Axiom of Choice? > These are all questions I am interested in. (Though whether I would be able > to follow the answers is another matter.) But as I say, I think there is an > issue here about infinite strings, in all generality, though obviously real > numbers are a massive application of the concept. I too would be very interested in more insights on this. My guess, for the sake, is that rejecting a LUB axiom amounts to rejecting an axiom of infinity, and an axiomatization of induction with it, so much more than just rejecting continuity; I'd expect only strict finitist to do such a thing. But I am more or less guessing... -LV === Subject: Re: Infinite Binary Strings: A Question > But I am more or less > guessing... -LV More! === Subject: Re: Infinite Binary Strings: A Question posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) rejecting an axiom of infinity, and an axiomatization of induction >with it, so much more than just rejecting continuity; I'd expect >only strict finitist to do such a thing. But I am more or less >guessing... At least here you admit you're just guessing, although it's > not clear why you think your guess will be of general interest. The at least is completely unwarranted, as this reiterated ad hominem. > In fact you're totally wrong. For example, if the only numbers > we admit are rationals then the LUB property fails - this > has nothing to do with the axiom of infinity or with induction. Then you are just after something dirrent than the OP. Actually, you just confirm yourself a pernicious troll, and overall simply an idiot and a waste of time. Bye David. -LV === Subject: How to check distributivity of a finite lattice posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) When we have a finite lattice expressed with a Hasse diagram, how to check whether it is distributive? Should one (or a program) to check all combinations of elements? Should I write a program which does this? Particularly I'm interested to know distributivity the following lattice which William Elliot proposed as the solution of my problem 1 | a / x y / 0 See http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice for the updated version of the problem which requires the involved lattices to be distributive. If the above William Elliot's lattice is distributive then it solves the renewed problem. If it is not distributive we should search for an other example for my conjecture. Is it distributive? === Subject: Re: How to check distributivity of a finite lattice posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > When we have a finite lattice expressed with a Hasse diagram, how to > check whether it is distributive? Use the finite form of Stone's theorem. It is distributive iff it's isomorphic to the lattice of down-sets in the lattice of its join-irreducible elements. > Particularly I'm interested to know distributivity the following > lattice which William Elliot proposed as the solution of my problem > æ 1 > æ | > æ a > æ/ > x æ y > æ / > æ 0 > If the above William Elliot's lattice is distributive then it solves > the renewed problem. If it is not distributive we should search for an > other example for my conjecture. Is it distributive? Yes. It's isomorphic to a lattice of sets via 0 -> {} x -> {1} y -> {2} a -> {1,2} 1 -> {1,2,3}. Do you still want your Abel prize? Victor Meldrew I don't believe it! === Subject: Re: How to check distributivity of a finite lattice posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/20080703 Mandriva/2.0.0.16-1.1mdv2008.1 (2008.1) Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Do you still want your Abel prize? Yes, and 15 September has not yet passed, so that you can nominate my works for Abel Prize: http://www.mathematics21.org/abel-prize.html My works (see http://www.mathematics21.org/algebraic-general-topology.html) are great even despite of some my ignorance in lattice theory such as not remembering the criterion for a lattice to be distributive: http://en.wikipedia.org/wiki/Distributive_lattice#Characteristic_properties === Subject: Re: How to check distributivity of a finite lattice posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > When we have a finite lattice expressed with a Hasse diagram, how to > check whether it is distributive? Should one (or a program) to check all combinations of elements? Should I write a program which does this? Particularly I'm interested to know distributivity the following > lattice which William Elliot proposed as the solution of my problem > æ 1 > æ | > æ a > æ/ > x æ y > æ / > æ 0 Seehttp://garden.irmacs.sfu.ca/?q=op/non separable center of a lattice > for the updated version of the problem which requires the involved > lattices to be distributive. If the above William Elliot's lattice is distributive then it solves > the renewed problem. If it is not distributive we should search for an > other example for my conjecture. Is it distributive? A lattice is distributive iff it does not contain as sublattices the two lattices you'll see pictured at In particular, every 5-element lattice which is not one of those two is distributive. -- m === Subject: solutions manual (To search click in keyboard Ctrl+F) posting-account=y7Z6OAoAAAD9FX_IL8yyi-5ioDwYBttu CLR 2.0.50727),gzip(gfe),gzip(gfe) solutions manual (To search click in keyboard Ctrl+F) Solutions Manuals in Electronic (PDF)Format! 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Martin(chapter 1 to chapter15) solution manual for Probability and Statistical Inference ( 7th edition by Hogg & Tanis) solution manual for Fundamentals of Communication Systems by John G. Proakis ,Masoud Salehi === Subject: functional notation? posting-account=QUdeoQoAAAAQEvTvNNcbdGUXxcUUVVJ5 3.2.0; .NET CLR 1.1.4322; OfficeLiveConnector.1.2),gzip(gfe),gzip(gfe) In their book _Structure and Interpretation of Classical Mechanics_, Gerald Sussman and Jack Wisdom are critical of prevailing mathematical notations used in high school and university texts. For example, in their preface, they include the following passage: In his book on mathematical pedagogy, Hans Freudenthal argues that the reliance on ambiguous, unstated notational conventions in such expressions as f(x) and df(x)/dx makes mathematics, and especially introductory calculus, extremely confusing for beginning students; and he enjoins mathematics educators to use more formal modern notation. They go on to praise Michael Spivak and his book _Calculus on Manifolds_, where he uses so-called functional notation. They argue the advantages of functional notation over classical notation as justification for using it in their own book. For anyone curious, the full quote is here: http://mitpress.mit.edu/SICM/book-Z-H-5.html#%_chap_Temp_2 (see the footnotes) I'm curious; is there general agreement in the mathematics community on this point? And can someone outline for me the major differences between classical versus functional notation as they relate to topics a student might encounter in an *introductory* (1st year undergraduate) calculus class. Sussman, Wisdom and Spivak's books were too advanced for me to appreciate the notational differences they are talking about. === Subject: Re: functional notation? > In their book _Structure and Interpretation of Classical Mechanics_, > Gerald Sussman and Jack Wisdom are critical of prevailing mathematical > notations used in high school and university texts. For example, in > their preface, they include the following passage: In his book on mathematical pedagogy, Hans Freudenthal argues that > the reliance on ambiguous, unstated notational conventions in such > expressions as f(x) and df(x)/dx makes mathematics, and especially > introductory calculus, extremely confusing for beginning students; and > he enjoins mathematics educators to use more formal modern notation. They go on to praise Michael Spivak and his book _Calculus on > Manifolds_, where he uses so-called functional notation. They argue > the advantages of functional notation over classical notation as > justification for using it in their own book. For anyone curious, the > full quote is here: http://mitpress.mit.edu/SICM/book-Z-H-5.html#%_chap_Temp_2 (see the > footnotes) I'm curious; is there general agreement in the mathematics community > on this point? And can someone outline for me the major differences > between classical versus functional notation as they relate to topics > a student might encounter in an *introductory* (1st year > undergraduate) calculus class. Sussman, Wisdom and Spivak's books were > too advanced for me to appreciate the notational differences they are > talking about. Bizarrely, there is only one example of an equation in functional notation The so-called traditional notation for Lagrange's equations seemed to me perfectly clear. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: functional notation? > In their book _Structure and Interpretation of Classical Mechanics_, > Gerald Sussman and Jack Wisdom are critical of prevailing mathematical > notations used in high school and university texts. For example, in > their preface, they include the following passage: In his book on mathematical pedagogy, Hans Freudenthal argues that the > reliance on ambiguous, unstated notational conventions in such > expressions as f(x) and df(x)/dx makes mathematics, and especially > introductory calculus, extremely confusing for beginning students; and > he enjoins mathematics educators to use more formal modern notation. They go on to praise Michael Spivak and his book _Calculus on > Manifolds_, where he uses so-called functional notation. They argue the > advantages of functional notation over classical notation as > justification for using it in their own book. For anyone curious, the > full quote is here: http://mitpress.mit.edu/SICM/book-Z-H-5.html#%_chap_Temp_2 (see the > footnotes) I'm curious; is there general agreement in the mathematics community on > this point? And can someone outline for me the major differences between > classical versus functional notation as they relate to topics a student > might encounter in an *introductory* (1st year undergraduate) calculus > class. Sussman, Wisdom and Spivak's books were too advanced for me to > appreciate the notational differences they are talking about. Notations like dy/dx are primarily useful when a number of dependent variables depend on one independent variable. The canonical example is physical variables depending on time. If that's the case, you can talk about dy/dx, dx/dt, dy/dt, and they're all well-defined. But if you have multiple independent variables, or multidimensional situations (like static geometry, or the P,V and T of thermodynamics) where it's not clear what the independent variables are, or even whether there are any, these notations break down. -- hendrik === Subject: Re: Simple squared squares posting-account=G_G-iQoAAAB08LNQidt_LsMkopmIb4ZS Gecko/20060111 Firefox/1.5.0.1 Mnenhy/0.7.3.0,gzip(gfe),gzip(gfe) > I'm looking for an agorithm that generates SSSs (don't have to be > perfect). > results are dirty as well: very often more than N 1x1 squares in an NxN > square :-( > TIA > Steven Nobody?? > :-( My guess is SSS = Square root of the Sum of the Squares! Bill J === Subject: Re: Simple squared squares > I'm looking for an agorithm that generates SSSs > (don't have to be > perfect). > but as expected the > results are dirty as well: very often more than N > 1x1 squares in an NxN > square :-( > TIA > Steven > Suppose I can guess your problem as: for given square m*n lets find its squared value p = (m*n)^(1/2) where p is the closest natural value to some correct real value q = (m*n)^(1/2) If so the simple procedure begins once comparing m and n for mn and p = cm but there p>q Your fault q-p could be taken down while taking decimal fractions ( once improving Your number n*m*10^2k and finally p = cm/10^k ) Eventually just take full procedure of finding m*n squared... Ro-Bin > Nobody?? > :-( === Subject: Re: FLT: a short amateur proof for case 2: where is the mistake? posting-account=dVG9FwoAAACJIwmFpOf2dGcPVQGlrAsl 5.1),gzip(gfe),gzip(gfe) Still another version is posted at: http://groups.msn.com/flt2008/fltproof.msnw With three different versions, one is bound to be correct. === Subject: Re: FLT: a short amateur proof for case 2: where is the mistake? rcb@powerpuff.com a .8ecrit : > Still another version is posted at: http://groups.msn.com/flt2008/fltproof.msnw With three different versions, one is bound to be correct. You forgot the smiley === Subject: extension of a continuous mapping Hi all, There is a conclusion [Royden: Real analysis, Ex. 40, Chap. 2] stating the following: If F subset R is closed, f is a real-valued function that is continuous on F, then f can be extended to a continuous function g on R. Best, Yihong === Subject: Re: extension of a continuous mapping posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Hi all, There is a conclusion [Royden: Real analysis, Ex. 40, Chap. 2] stating the following: > If F subset R is closed, f is a real-valued function that is continuous on F, then f can be extended to a continuous function g on R. > Yes, this is a special case of the Tietze extension theorem. Every continuous function from a closed subset A of the normal space X to R can be extended to a continuous function from X to R. OK, this is the n = 1 version, but replacing R by R^n follows easily. Victor Meldrew I don't believe it === Subject: Number with die-hard. Hello teacher~ In the movie Die Hard 3, our heros, John McClain (Bruce Willis) and Zeus (Samuel L. Jackson), are at the bidding of the evil Peter Krieg (Jeremy Irons). First they are sent to the pay phone, then to the subway, and finally to the park. And they find a bomb set on a five-minute timer. It is here that they must make exactly four gallons from five and three gallon jugs. They did it just in time. Now it's your time. In fact, it not easy in dire straits... === Subject: Re: Number with die-hard. >Hello teacher~ In the movie Die Hard 3, >our heros, John McClain (Bruce Willis) and Zeus (Samuel L. Jackson), >are at the bidding of the evil Peter Krieg (Jeremy Irons). First they are sent to the pay phone, then to the subway, and finally to the >park. >And they find a bomb set on a five-minute timer. >It is here that they must make exactly four gallons from five and three >gallon jugs. They did it just in time. >Now it's your time. In fact, it not easy in dire straits... > Fill the 5 then use it to fill the 3, dump the 3 and pour the remaining two from the 5 to the 3. Fill the 5, use it to top off the 3 and you have 4 left in the 5. --Lynn === Subject: Re: Number with die-hard. Distribution: world >It is here that they must make exactly four gallons from five and three >gallon jugs. >They did it just in time. >Now it's your time. >Fill the 5 then use it to fill the 3, dump the 3 and pour the >remaining two from the 5 to the 3. Fill the 5, use it to top off the 3 >and you have 4 left in the 5. Fill the three and dump it into the five. Fill the three and use it to top off the five. Dump the five, and pour the remaining gallon from the three into it. Fill the three one last time, and pour it into the five. -- Michael F. Stemper #include If this is our corporate opinion, you will be billed for it. === Subject: Re: Number with die-hard. Distribution: world posting-account=Z3AipgkAAABkoMfyNwddSxsYhXHi5CDt CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) On Sep 12, 9:59æam, mstem...@walkabout.empros.com (Michael Stemper) >It is here that they must make exactly four gallons from five and three >gallon jugs. >They did it just in time. >Now it's your time. >Fill the 5 then use it to fill the 3, dump the 3 and pour the >remaining two from the 5 to the 3. Fill the 5, use it to top off the 3 >and you have 4 left in the 5. Fill the three and dump it into the five. Fill the three and use it to top off > the five. Dump the five, and pour the remaining gallon from the three into > it. Fill the three one last time, and pour it into the five. -- > Michael F. Stemper > #include Need to pick a simple topic for a short project about 10 min. Must use > true data > from ætrue sources. > Topics may include but no limited to Elementary introduction with applications. Basic probability models. > Combinatorics. Random variables. Discrete and continuous probability > distributions. Statistical estimation and testing. Confidence > intervals. Introduction to linear regression. How about testing a cryptographically strong random number generator using the following library? http://www.iro.umontreal.ca/~simardr/testu01/tu01.html === Subject: Re: Statisitc Help posting-account=aZAW_QoAAABHaPf02XTsrle7hWr4wpvx Need to pick a simple topic for a short project about 10 min. Must use > true data > from ætrue sources. > Topics may include but no limited to Elementary introduction with applications. Basic probability models. > Combinatorics. Random variables. Discrete and continuous probability > distributions. Statistical estimation and testing. Confidence > intervals. Introduction to linear regression. How about testing a cryptographically strong random number generator > using the following library? http://www.iro.umontreal.ca/~simardr/testu01/tu01.html Sound good but not sure if I can present it in 10 min === Subject: Coordinate frame transformations (i686-redhat-linux-gnu) I have a terminological question regarding linear coordinate frame transformations. The definition of a coordinate frame transformation is the transformation that maps the basis vectors for the original frame onto the basis vectors for the destination frame. Typically, however, what one wishes to do is not to map basis vectors, but rather to map coordinates. I.e., what you typically want to do is to map the coordinates in the original frame to coordinates in the destination frame. The transformation that does this is the inverse of the coordinate frame transformation. So, my terminological question is, is there a name for this transformation that is more intuitive and less unwieldy than the inverse of the coordinate frame transformation? E.g., a term that makes it obvious, for instance, that you will be using the transformation to map coordinates, and not basis vectors. |>oug === Subject: Re: Coordinate frame transformations > I have a terminological question regarding linear coordinate frame > transformations. The definition of a coordinate frame > transformation is the transformation that maps the basis vectors for > the original frame onto the basis vectors for the destination frame. Typically, however, what one wishes to do is not to map basis vectors, > but rather to map coordinates. I.e., what you typically want to do is > to map the coordinates in the original frame to coordinates in the > destination frame. The transformation that does this is the inverse > of the coordinate frame transformation. So, my terminological question is, is there a name for this > transformation that is more intuitive and less unwieldy than the > inverse of the coordinate frame transformation? E.g., a term that > makes it obvious, for instance, that you will be using the > transformation to map coordinates, and not basis vectors. |>oug hi doug, i know of only two names [CapitalEth] but i wouldn't be surprised at more [CapitalEth], and they apply to matrices, not to the transformations they represent. if i read you correctly, the matrix for what you call the coordinate frame transformation is called the transition matrix; its transpose is called the attitude matrix. i know of no general name for the inverse transition matrix, which is the matrix which maps old components to new. (of course, if the basis is orthonormal, then the inverse transition matrix is the attitude matrix.) if you want more detail, you can look at the following blog post: http://rip94550.wordpress.com/2008/05/15/attitude-transition-matrices-etc / vale, rip -- NB eddress is r i p 1 AT c o m c a s t DOT n e t === Subject: Re: Coordinate frame transformations (i686-redhat-linux-gnu) > i know of no general name for the inverse transition matrix, which > is the matrix which maps old components to new. Doh! Clearly mathematicians don't care about the needs of us lowly software engineers! |>oug === Subject: Re: Coordinate frame transformations This is one of the advantages of using tensor notation rather than matrices. Matrices lack 'context' information so you have to make up all these different names for the matrix, its inverse, its transpose and its inverse transpose and then remember which matrix goes with which calculation. Or if you are reading some author, trace back to find which matrix he associates with which name. (Things get even worse if you are trying to use arrays for higher order objects. There can be many different ways of transposing a third or higher order array.) Using tensor notation you could define a basic second order tensor for a transistion between two coordinate systems. You could use primes (or with a computer color) on the indices for one of the coordinate systems. The tradition is to always write the tensor in an up/down index format, i.e., first index up and second index down. You then have just two basic tensors, depending on which index has the prime, and they are inverses of each other. Then when you want to calculate a transformation all you have to do is line up the indices properly. The indices carry context information and this makes everything much easier. -- David Park djmpark@comcast.net http://home.comcast.net/~djmpark/ >I have a terminological question regarding linear coordinate frame > transformations. The definition of a coordinate frame > transformation is the transformation that maps the basis vectors for > the original frame onto the basis vectors for the destination frame. Typically, however, what one wishes to do is not to map basis vectors, > but rather to map coordinates. I.e., what you typically want to do is > to map the coordinates in the original frame to coordinates in the > destination frame. The transformation that does this is the inverse > of the coordinate frame transformation. So, my terminological question is, is there a name for this > transformation that is more intuitive and less unwieldy than the > inverse of the coordinate frame transformation? E.g., a term that > makes it obvious, for instance, that you will be using the > transformation to map coordinates, and not basis vectors. |>oug === Subject: Re: Coordinate frame transformations (i686-redhat-linux-gnu) > This is one of the advantages of using tensor notation rather than > matrices. Matrices lack 'context' information so you have to make up > all these different names for the matrix, its inverse, its transpose > and its inverse transpose and then remember which matrix goes with > which calculation. I guess I should learn something about tensors, but alas I know little about them at the moment. Also, my question is for the purpose of writing intelligible software, and I have easy access to matrix libraries. I'm not so sure that I have the same for tensors, unless the latter is trivially implementable in terms of the former. > Using tensor notation you could define a basic second order tensor for > a transistion between two coordinate systems. > The definition of a coordinate frame transformation is the > transformation that maps the basis vectors for the original frame > onto the basis vectors for the destination frame. I inadvertently left out saying that a coordinate frame transformation also maps the origins of the coordinate frames, not just the basis vectors. That's the distinction between a coordinate SYSTEM In order to have affine coordinate frame transformations be linear, you have to move to a higher dimensional projective space. I.e., homogeneous coordinates. (You don't need to resort to this trick for coordinate system transformation.) Can you pull the same trick with tensors? |>oug === Subject: :: proper finite-index subgroups:: Let G be any group, and let H be a proper, finite-index subgroup. Why can't G be written as the union of all conjugates of H ? I know that this result is true when G is finite, as a simple counting argument will show. But what happens when G is arbitrary? Should we consider the action of G on the cosets of H? In that case, |G/K| is at most n! where [G : H] = n and K is the G-core of H. Also, K is a proper subgroup of G. of course, G *can* be written as the union of conjugates of H). At least we know that G can be written as a finite union of distinct cosets of K... What I'm hoping to see it a proof, in some detail (or in enough detail so as to be able to work out the rest myself). === Subject: Re: :: proper finite-index subgroups:: > Let G be any group, and let H be a proper, > finite-index subgroup. Why can't G be written as the union of all conjugates > of H ? I know that this result is true when G is finite, > as a simple counting argument will show. > But what happens when G is arbitrary? Should we consider the action of G on the cosets of > H? > In that case, |G/K| is at most n! where [G : H] = n > and K is the G-core of H. Also, K is a proper > subgroup of G. > contradiction (assuming, > of course, G *can* be written as the union of > conjugates of H). At least we know that G can be > written as a finite > union of distinct cosets of K... What I'm hoping to see it a proof, in some detail > (or in enough detail so as to be able to work out the > rest myself). > Count the cosets of K. There should be [G:K] of them, but in fact there are at most [G:K] + (1-[G:H]) of them. === Subject: Re: :: proper finite-index subgroups:: days. My association with the Department is that of an alumnus. >Let G be any group, and let H be a proper, finite-index subgroup. Why can't G be written as the union of all conjugates of H ? I know that this result is true when G is finite, >as a simple counting argument will show. Every finite index subgroup contains a finite index normal subgroup. Taking the quotient will drop you to the finite case. >But what happens when G is arbitrary? Should we consider the action of G on the cosets of H? >In that case, |G/K| is at most n! where [G : H] = n and K is the >G-core of H. Also, K is a proper subgroup of G. Exactly. And the lattice isomorphism gives you a 1-to-1, inclusion preserving correspondence between the subgroups of G/K and the subgroups of G that contain K (including H and all its conjugates). >of course, G *can* be written as the union of conjugates of H). At >least we know that G can be written as a finite >union of distinct cosets of K... But G/K cannot be written as a finite union of conjugates of H/K; but you would be able to do it if you could do it with G and H. -- magidin-at-member-ams-org === Subject: Re: :: proper finite-index subgroups:: But G/K cannot be written as a finite union of > conjugates of H/K; but > you would be able to do it if you could do it with G > and H. I see what's going on (more or less). Just a quick question about notation: if we write G = U_{g in G} H^g where H^g denotes the conjugate of H by g, it should follow that G/K = U_{g in G} (H^g)/K and I've noticed that (H/K)^{gK} is the same as (H^g)/K [since H^g indeed contains K, as K is normal in G]. But is this the same as the union over all cosets gK in G/K ? Or does it really matter, since, if g runs over the elements of G, then gK runs over all the elements of G/K, and vice versa? It would seem so, since all cosets gK of K in G are 'indexed' by the elements of G anyway? Looking forward to clarification. === Subject: Re: :: proper finite-index subgroups:: days. My association with the Department is that of an alumnus. > But G/K cannot be written as a finite union of > conjugates of H/K; but > you would be able to do it if you could do it with G > and H. I see what's going on (more or less). Just a quick question about notation: if we write G = U_{g in G} H^g where H^g denotes the conjugate of H by g, it should follow that G/K = U_{g in G} (H^g)/K In fact, it will follow that G/K = U_{ gK in G/K} (H/K)^{gK} >and I've noticed that (H/K)^{gK} is the same >as (H^g)/K [since H^g indeed contains K, as K is >normal in G]. Yes. This is just the isomorphism theorems. >But is this the same as the union over all cosets gK in G/K ? Yes. You want to show that if xK = yK, then (H/K)^{xK} = (H/K)^{yK}. But indeed, if xK=yK, then xy^{-1} is in K (since K is normal), so for every h in H, hK = (xy^{-1})h(xy^{-1})^{-1} K = x(y^{-1}hy)x^{-1}K hence (H/K)^{xy^{-1}K} = (H/K), so (H/K)^{xK} = ((H/K)^{xy^{-1}}K)^{yK} = (H/K)^{yK}. So your union is in fact taken over cosets of K rather than elements of G (of course, taking it over elements of G just repeats terms). This check is more than you need, but it may be useful for other situations where you only have certain of the conjugates under consideration. -- magidin-at-member-ams-org === Subject: Re: Limit of a simple recurrence sequence <30645104.1221119430763.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=HR1dqAkAAAA9E7mXiqvduHAelsrIxH3e Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) >If you go p(k) > 0, then, yes, it converges. But if you >allow >p(k) = 0, it may not, e.g., m = 2, p(1) = 1, p(2) = 0 >goes >x(1), x(2), x(1), x(2), etc. >-- >Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email) > For a two term series you can prove it converges and what it converges to. The standard technique for these series works. You have x(n +2)=a*x(n+1) + b*x(n) with a+b=1, both>0. Assume a solution of the form x(i)=c*r^i and substitute in r^2= a*r + b. The roots are r=1 and r=-b. The general solution is then x(i)=c1 + c2*i^(-b). Since |b|<1 (because a>0) that term will decay away leaving the limit as c1. Given a, b, x(0), and x(1) you can determine c1, which will be the limit. For larger numbers of terms, I think that your condition sum p(k)=1 assures that r=1 will be a solution of the recurrence and if all the p(k) are positive (and necessarily <1) all the other roots will be less than 1 in magnitude. I think proving that r=1 is a solution is straightforward by doing the polynomial division. I don't know how to prove that the other roots are small. === Subject: Re: Limit of a simple recurrence sequence >If you go p(k) > 0, then, yes, it converges. But if you >allow >p(k) = 0, it may not, e.g., m = 2, p(1) = 1, p(2) = 0 >goes >x(1), x(2), x(1), x(2), etc. >-- >Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) > The recurrence in question was x(n+m+1) = sum_{k=1}^m [p(k) * x(n+k)] where sum_k p(k) = 1. If X(n) is the column vector [ x(n), x(n+1),...,x(n+m) ]^T, your recurrence can be written as X(n+1) = P X(n) where P is the m x m matrix with entries P_{i,i+1}=1, P_{m,j} = p(k), all others 0. Let w be a left eigenvector of P for eigenvalue 1, i.e. w^T P = w^T, normalized so that sum_i w_i = 1. Then the limit will be w^T X(1). -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Limit of a simple recurrence sequence > for the limit of x(n)? The recurrence in question was x(n+m+1) = sum_{k=1}^m > [p(k) * x(n+k)] > where sum_k p(k) = 1. If X(n) is the column vector > [ x(n), x(n+1),...,x(n+m) ]^T, your recurrence can be > written as > X(n+1) = P X(n) > where P is the m x m matrix with entries P_{i,i+1}=1, > P_{m,j} = p(k), > all others 0. Let w be a left eigenvector of P for > eigenvalue 1, > i.e. w^T P = w^T, normalized so that sum_i w_i = 1. > Then the limit > will be w^T X(1). > -- > Robert Israel > israel@math.MyUniversitysInitials.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, > BC, Canada lim_{n->inf} x(n) = sum_{k=1}^m [sum_{i=1}^k [p(i) * x(k)]] / sum_{k=1}^m [(m-k+1)*p(k)] However, I wonder why the matrix P is diagonizable, and why the absolute values of other eigenvalues except 1 are all strictly smaller than 1, although from the Gershgorin circle theorem, they must not be larger than 1. === Subject: need hint for real analysis question Need to prove the limit of a sequence... Prove lim sqrt(n)* ( sqrt(2n+2) - sqrt (2n+1) ) = 1/(2*sqrt(2)) I am just not sure where to start. I know we need to show that |X_n - L| < .83Ì However, solving for n in this mess looks like the wrong direction... I need help getting started since it seems at first(to me!) that this isn't even a convergent sequence. Any advice? === Subject: Re: need hint for real analysis question > Need to prove the limit of a sequence... > Prove > lim sqrt(n)* ( sqrt(2n+2) - sqrt (2n+1) ) = 1/(2*sqrt(2)) > I am just not sure where to start. > I know we need to show that |X_n - L| < .91[Micro] > However, solving for n in this mess looks like the wrong direction... > I need help getting started since it seems at first(to me!) that this > isn't even a convergent sequence. > Any advice? If 0 < p < 1 and 0 < x < oo, then [(x+1)^p - x^p] = p*c^(p-1) for some c in (x, x+1), by the mean value theorem. This will give your result and a host of others without using the conjugate trick. === Subject: Re: need hint for real analysis question , > Need to prove the limit of a sequence... > Prove > lim sqrt(n)* ( sqrt(2n+2) - sqrt (2n+1) ) = 1/(2*sqrt(2)) > I am just not sure where to start. > I know we need to show that |X_n - L| < .91[Micro] > However, solving for n in this mess looks like the wrong direction... > I need help getting started since it seems at first(to me!) that this > isn't even a convergent sequence. > Any advice? If 0 < p < 1 and 0 < x < oo, then [(x+1)^p - x^p] = p*c^(p-1) for some > c in (x, x+1), by the mean value theorem. This will give your result > and a host of others without using the conjugate trick. Actually, for any real p. === Subject: Re: need hint for real analysis question posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > Need to prove the limit of a sequence... > Prove > lim sqrt(n)* ( sqrt(2n+2) - sqrt (2n+1) ) = 1/(2*sqrt(2)) > I am just not sure where to start. > I know we need to show that |X_n - L| < epsilon > However, solving for n in this mess looks like the wrong > direction... I need help getting started since it seems > at first(to me!) that this isn't even a convergent sequence. > Any advice? One thing you can try is to consider how you'd evaulate the limit in a precalculus or calculus 1 class. (It's very similar to many I've worked and given on tests in these classes over the years.) Multiply numerator and denominator by the conjugate of the binomial factor in the numerator and then multiply and divide by the highest power of n, which is sqrt(n) in this case. Try an epsilon-L proof using the rewritten expression. Dave L. Renfro === Subject: Re: need hint for real analysis question > Need to prove the limit of a sequence... > Prove > lim sqrt(n)* ( sqrt(2n+2) - sqrt (2n+1) ) = 1/(2*sqrt(2)) > I am just not sure where to start. > I know we need to show that |X_n - L| < ? > However, solving for n in this mess looks like the wrong direction... > I need help getting started since it seems at first(to me!) that this > isn't even a convergent sequence. > Any advice? The usual trick to leading with differences of square roots is multiply an divide for their sum. It says, L = Lim(sqrt(n)(sqrt(2n + 2) - sqrt(2n+1)) = Lim(sqrt(n)(sqrt(2n + 2) - sqrt(2n + 1)(sqrt(2n + 2) + sqrt(2n + 1)/(sqrt(2n + 2) + sqrt(2n + 1)) = Lim(sqrt(n)((2n + 2) - (2n + 1)/(sqrt(2n + 2) + sqrt(2n + 1)) = Lim(sqrt(n)/(sqrt(2n + 2) + sqrt(2n + 1)) = Lim(1/(sqrt(2 + 2/n) + sqrt(2 + 1/n)) = 1/(2sqrt(2)) -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: Reason for answering.... or not posting-account=UmSM6QoAAAAHQsbQSpqx55ht5J9R5UV_ CLR 1.1.4322; InfoPath.1),gzip(gfe),gzip(gfe) > If I do NOT respond to YOUR NEXT POST > about the note(s) above then it is because (1) chances are > xx % > that you have said something so utterly stupid > that it must remain there, as a tail-end turd, > for the world to see the low albedo of your > pitiful demented intellect... ahaha... AHAHA. æ------------ æ However, æ----------------- (2) although chances are < 100-xx %,.... you > actually may have said something interesting > enough that you deserve to have the last word... You be the judge for the value xx. Whatever > your judgement is, it was fun talking to you. > (1) chances are > xx % > that you have said something so utterly stupid > that it must remain there, as a tail-end turd, > for the world to see the low albedo of your > pitiful demented intellect... ahaha... AHAHA that actually made me laugh funny stuff and most of my post weather they actually work or not are designed to stimulate thought and rebuttal to further research on why they wont work or why they perform the way they do so constructive criticism is much appreciated. === Subject: Re: minimum cycle length a^k mod n I'm reposting this because I'm still interested in the question ... How does one find the minimum cycle length of 3^k mod 10^i, for arbitrary i (other than by brute force computation)? E.g., for i = 10^10? > As examples, consider the minumum cycle length of 3^k mod 10^i, for > 1 <= i <= 8; according to my computations, these are, respectively, > 4, 20, 100, 500, 5000, 50000, 500000, 5000000. I don't see anything posted so far that explains all of these values. > In the last five cases (i = 4, 5, 6, 7, 8) the minimum cycle length > is 10^i/20 -- how would one determine that fact other than by brute > force? Is the formula 10^i/20 valid for all higher moduli 10^i? Not coincidentally, this relates to some properties of power towers > mentioned in the section Decimal digits of Graham's number at > http://en.wikipedia.org/wiki/Graham's_number (and the discussion page). === Subject: Re: minimum cycle length a^k mod n > I'm reposting this because I'm still interested in the question ... How does one find the minimum cycle length of 3^k mod 10^i, > for arbitrary i (other than by brute force computation)? E.g., for i = > 10^10? > As examples, consider the minumum cycle length of 3^k mod 10^i, for > 1 <= i <= 8; according to my computations, these are, respectively, > 4, 20, 100, 500, 5000, 50000, 500000, 5000000. I don't see anything posted so far that explains all of these values. > In the last five cases (i = 4, 5, 6, 7, 8) the minimum cycle length > is 10^i/20 -- how would one determine that fact other than by brute > force? Is the formula 10^i/20 valid for all higher moduli 10^i? Not coincidentally, this relates to some properties of power towers > mentioned in the section Decimal digits of Graham's number at > http://en.wikipedia.org/wiki/Graham's_number (and the discussion page). Hint: 3^500 == 1 + 10^4 mod 10^5 Prove by induction that 3^(5*10^n) == 1 + 10^(n+2) mod 10^(n+3) for n >= 2. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: minimum cycle length a^k mod n > How does one find the minimum cycle length of 3^k mod 10^i, > for arbitrary i (other than by brute force computation)? > E.g., for i = 10^10? > Hint: > 3^500 == 1 + 10^4 mod 10^5 > Prove by induction that > 3^(5*10^n) == 1 + 10^(n+2) mod 10^(n+3) for n >= 2. cycle length of 10^i/20 for all i >= 4; I need to give more thought to why it must be minimal. === Subject: Re: Periodic Points of a Quadratic Function Yours is a very interesitng observation! I have been looking at what can be said about the number of n-cycles of polynomials (as endomorphisms of the real line), with the idea of counting them. One might do that inductively, if he has determined all cycles of order properly dividing n. Some are full, like the one described (all roots of the nth iterate which is a 2^n degree polynomial are distinct and real - therefore members of cycles of interest), and others are empty as would be Q(x)= x^2+1 (since Q(x)>x for all real x). I am curious about what intermediate cases exist (and whether results of the type sought exist for higher degree polynomials). I guess the question may be reduced to finding a pattern to the number of real distinct roots of iterates, and how it may be determinable from some structural property of the original polynomial. === Subject: Re: Periodic Points of a Quadratic Function <27101691.1221252518613.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=06BQLAoAAADoC7Y4z9FWcUwGvMa7xMG9 7.4),gzip(gfe),gzip(gfe) On 12 sep, 22:48, wheier...@corunduminium.com Yours is a very interesitng observation! > I have been looking at what can be said about the number > of n-cycles of polynomials (as endomorphisms of the real > line), with the idea of counting them. æOne might do > that inductively, if he has determined all cycles of > order properly dividing n. æSome are full, like the one > described (all roots of the nth iterate which is a 2^n degree polynomial are distinct and real - therefore members of cycles of interest), and others are empty as would be > Q(x)= x^2+1 (since Q(x)>x for all real x). æ > I am curious about what intermediate cases exist (and > whether results of the type sought exist for higher degree polynomials). æI guess the question may be reduced to finding a pattern to the number of real distinct roots of iterates, and how it may be determinable from some structural property of the original polynomial. some notices : 1) for q(x) =x^2 +1 and large x we may propose 'near by' companions such as : g(x) = {(exp(2^x) + sqrt((exp(2^(x+1) -2)}/2 Thence q^[n](x) ~= g( n +g^-1(x)) Approx. 2)towards higher power polynomials Example: g(x) =2*exp(3^x) + 1 f(2*exp(3^x) + 1) = 2*exp(3^(x +1)) + 1 f(u +1 ) = u^3/4 + 1 f(u) = (u -1)^3/4 + 1 Hope it helps you, Alain === Subject: Re: Periodic Points of a Quadratic Function posting-account=06BQLAoAAADoC7Y4z9FWcUwGvMa7xMG9 7.4),gzip(gfe),gzip(gfe) On 9 sep, 14:02, wheier...@corunduminium.com > In one of the important papers on the Sarkovskii > ordering (I have lost the reference), the function > P(x)=x^2-2 is used as an example. æAs I recall, its > fixed points (solutions of P(x)=x) and 2-periodic points > (solutions of P(P(x))=x that are not fixed points) are > algebraically computed. My question is: æis anyone aware of the publication of > all n-periodic points of this function? æ The nth iterate of P, a polynomial of degree 2^n, has a > full set of 2^n real fixed points, implying every zero > is p-periodic for some divisor p of n. All this fell out of the observation that > æ æ æ æ æ P(2 cos[t])=2 cos[2t], > implying all these zeroes are of the form 2 cost where t > is a solution of the elementary trigonometric equation > æ æ æ æ æ cos[(2^n)t]=cos t. > This allows systematic counting and explicit expression > of all n-cycles generated by iterates of P. Other polynomials, such as Q(x)=x+1, have no periodic > points of any order. æThis raises other questions about > polynomials in general, and what is known about their > (real) periodic points of given order. æI am as a matter > of curiosity exploring the situation, and would greatly Bonsoir, It is very fruitful to consider a companion function g for f such as f(g(x)) = g(x +1) , thence f^[n](x) = g(x +n). Example 1: g(x) = exp(2^x) f(exp(2^x)) = exp(2^(x+1)) = exp(2^x)^2 and f(x) = x^2 Example 2: g(x) = 2exp(2^x) + 1 f(2exp(2^x) + 1) = 2exp(2^(x+1)) + 1 Alldone f(x) = 2*((x-1)/2)^2 + 1 And f^[n](x) = 2*((x-1)/2)^(2^n) + 1 ***YOUR CASE g(x) = 2cos(2^x) f(2cos(2^x)) = 2cos(2^(x+1)) = (2cos(2^x))^2 - 2 and f(x) = x^2 - 2 We've got f^[n](2cos(2^x)) = 2cos(2^(x+n)) All done f^[n](x) = 2cos(2^n*arccos(x/2)) NOtice : unluckily companion doesn't exist for every quadratic function, Alain === Subject: Re: cubing homomorphism for finite groups > Suppose G is a finite group such that the map f from > G into G defined by f(x) = x^3 is a homomorphism; suppose further that G does not have any elements of > order 3. > How do I show that G is abelian? > Would it be enough to show that the squaring or > inversion map on G is a homomorphism? I've had a couple of ideas, but neither of them seem > to work (for me). Any ideas? > Such groups were studied by Levi. It is easy to show that cubes and squares commute in a group where cubing is a homomorphism. If G is finite of order coprime to 3, then the squares are central. However, the subgroup generated by the squares always contains the derived subgroup so such a group is two step nilpotent. The formula for powers in such groups reveals that every commutator has order 3. === Subject: Re: cubing homomorphism for finite groups days. My association with the Department is that of an alumnus. [..piggybacking..] > Suppose G is a finite group such that the map f from > G into G defined by f(x) = x^3 is a homomorphism; > suppose further that G does not have any elements of > order 3. > How do I show that G is abelian? This has been asked a number of times before. Here was my answer from 2001, which is elementary: commutes with every cube; once you have that, from the fact that G does not have any elements of order 3 you can deduce that every element can be expressed as a cube, and therefore that squares in G are central. Once you know that squares are central, you can use this and the fact that cubing is a homomorphism to obtain that (ab)^2 = a^2b^2 for all a and b in G, which in turn is well-known to imply that G is abelian. Check out the entire thread; it also contains a neat follow-up problem from Derek Holt. -- magidin-at-member-ams-org === Subject: Re: cubing homomorphism for finite groups > <20959005.1221252573735.JavaMail.jakarta@nitrogen.math > forum.org>, [..piggybacking..] > Suppose G is a finite group such that the map f from > G into G defined by f(x) = x^3 is a homomorphism; > suppose further that G does not have any elements of > order 3. .. > Check out the entire thread; it also contains a neat > follow-up problem from Derek Holt. .. > Arturo Magidin case and the splitting result for arbitrary primes are nice. I vaguely remember that Levi's characterization of the 3-groups with the property was fairly nice. He also studied groups where n'th powers were endomorphisms, for integers n, but I think n=3 was one of the nicer cases. He also had the splitting result (I think not restricted to primes, but I could be misremembering). Baer had some followup paper as well. === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=G-TjQAkAAADYg6rno3bWQPnIwKFBrf1t 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; InfoPath.2),gzip(gfe),gzip(gfe) > Anyway, I reject your main thesis that Galileo's persecution had > anything to do with proof. æThere was less proof for their Earth- > centered theory. æGalileo was persecuted because Heliocentric Theory > contracdicted the Bible as it was interpreted by the Church at that > tim As noted explicitly both by Bellarmine (who was a *moderate* on the > question in 1616) and by the Inquisition long after Bellarmine was dead. The Church condemned Galileo because he was wrong. But what they neglected to mention is that they were wrong too. Both sides were still attached to the premise of a fixed reference frame and that (in modern language) there were only 7 degrees of symmetry governing space and time, rather than 10 governing spacetime: 6 of the symmetries pertained to space alone; 1 to time alone. The remaining 3, which involve both space and time were not recognized by either Galileo (or other heliocentrists), nor by the Church. On the larger issue, the Church had a point: Galileo and his contemporaries were doing nothing more, in effect, than replacing one sacrosanct ground by another -- the sun. It was ONLY during his confinement that Galileo recognized the error of his ways -- and the error of everyone else's ways. What's not widely advertised and what's widely forgotten is that (1) Galileo was the one who first formulated the Principle of Relativity (i.e. the remaining 3 degrees of symmetry), and (2) he did this while in confinement. Very likely, being forced to admit he was wrong before a committee had the effect of planting the seeds of doubt in his own mind. It's always a healthy exercise to actually do this at some point (even if you don't believe you're wrong). It forces you to think out of the box. It was only during confinement that he finally realised what landlubbers seem to be completely blind to; and what any sailor or navigator for the past 5000 or 10000 years or more could have told you: motion is relative! At the time (and even at the present time) people were so used ot thinking in terms of the the ground as being the Cosmic Floor it didn't fully dawn on them (even the most progressive thinkers of the time) just what the consequence of the ground NOT being the ultimate ground of the cosmos really was. Only a few thought that far ahead. Kepler, for instance -- though not widely known -- was one of the first to write a Science Fiction story about going to the Moon. I haven't seen the story, but clearly he must have realised that out there in space there are no landmarks and that, like the sailor in the middle of the ocean on a downwind heading where everything seems perfectly still, no matter how fast you move in any direction you'll seem like you're standing absolutely still. The vacuum has no reference speed attached to it. If that seems obvious to you, in retrospect, that I'd argue that even YOU don't believe it! You just think you do! For, one of the consequences of that fact drives a stake deep into concepts that you and almost everyone still adhere to -- the Absoluteness of Genidentity. If motion is relative, then a consequence is that the sameness of two points at different times is relative too. You can no longer say that New York in 2008 is the same place or even the different place as New York in 2001. Therefore, it makes no more sense to have a memorial there, than it does to have it somewhere else at the same latitude on Earth (where the Earth would have been oriented in 2001 relative to some fixed axes), or out in the middle of space where the Earth used to be in 2001. The property whereby two things at different times are considered the same thing is called Genidentity. Almost everyone holds to Genidentity of points as a fundamental assumption and they act on that belief. Even if they say they don't, they believe it (even if they BELIEVE they don't believe if, they do). Actions count, not words. Nor is it totally obvious from the point of view of a typical freshman Physics course. A 200 level course generally only treats rotational and translation symmetries (and, correspondingly, the conservation of angular momentum and linear monentum); and time translation (and consequently the conservation of energy). Few that I know of explicitly account for boost-invariance (much less the relativity of genidentity), much less the corresponding conserved quantity (moving mass moment). Not too surprisingly the only place where you see this conserved quantity in prominent use is by those involved with navigation (moments are important for aircraft, ships and rockets) -- again, the landlubbers are one step behind. Boost invariance, along with its full ramifications (relativity of genidentity, explicit treatment of the moving mass moment) is the odd one out, from a typical 200 level course. Nor was boost-invariance obvious to those who were around even as late as the 19th century. Even Maxwell, who was famous for articulating the vivid description of 'space having no reference speed or landmarks nonetheless attached a reference speed to the vacuum in his equations of electromagnetism -- the G vector. So, his equation for the E field, in terms of the potentials, read E = -grad phi - dA/dt + G x B; and he = -grad phi - dA/dt, so that Maxwell's original equations (in modern form) read: D = epsilon (E + G x B). Notice there's no G vector in Maxwell's list: A, B, C (= J + dD/dt), D, E, F (= rho E + J x B), H, I (= magnetization), J? Lorentz, in effect, took it out by raising to a postulate the empirical fact that G = 0 in every frame of reference. Yet, even he tried to explain away the newly (re)found boost invariance in terms of something similar to Maxwell's boost non-invariant theory. Only Einstein (re-)raised Galileo's axiom of boost invariance back to a postulate ... albeit at the cost of replacing the (infinite) absolute speed by a (finite) absolute speed. This is why, despite the occurrence of an absolute speed in the foundation, the theory is designated Relativity. The name is meant to contrast it to the boost symmetry-breaking formalisms that immediately preceded it. Even Newton had a major issue with it and tried to argue against it -- he implicitly recognized that with boost invariance, and the corresponding loss of genidentity, there could no longer be a cohesive notion of spatial geometry -- since the very notion of point is premised on genidentity. With relativity, your only out is to delve one level deeper and replace the concept of a point by the even more basic concept of a point-at-an-instant -- what we now call an event. Thus, it's not Minkowski's geometry which forced space and time into a union, but Galileo's axiom of relativity! Minkowski merely consummated the union that was already in place; the boost symmetries already mix space and time (in one direction); the Poincare' version of boost mix space and time in the reciprocal direction, but did not innovate the mixing of space and time. Newton was not in a position to handle that, and specifically set out to formulate his physics on SPATIAL geometry. So, this forced his hand: he had no choice but to reject the relativity of motion and reject the notion of boost-invariance. His arguments about acceleration being absolute were specious (all that proves is that velocities are elements of an affine geometry, not that they too are absolute!) They were all wrong, and only the later (post-confinement) Galileo had it right. But the (pre-confiment) Galileo was just as wrong as the Church. === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=5ayZ-goAAABGZmmwx8zZEwz6gU2OuVSd CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) The Church condemned Galileo because he was wrong. But what they > neglected to mention is that they were wrong too. If Galileo had not satirized the Church they would not have condemned him. His arrogance, ultimately, was what got him into trouble. === Subject: Re: What if: the Church had NOT condemned Galileo > >The Church condemned Galileo because he was wrong. But what they >neglected to mention is that they were wrong too. > If Galileo had not satirized the Church they would not have condemned > him. His arrogance, ultimately, was what got him into trouble. Correct. Dissing Pope Urban was not a peachy keen career move for Mr. G. Mr. G. put all of Aristotle's howlers in the mouth of Simplicio who was a surrogate for Pope Urban. Bob Kolker === Subject: Re: What if: the Church had NOT condemned Galileo <2qKdnXE3B-k3eFbVnZ2dnUVZ_rPinZ2d@comcast.com> posting-account=5ayZ-goAAABGZmmwx8zZEwz6gU2OuVSd CLR 2.0.50727; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) >The Church condemned Galileo because he was wrong. But what they >neglected to mention is that they were wrong too. If Galileo had not satirized the Church they would not have condemned > him. æHis arrogance, ultimately, was what got him into trouble. Correct. Dissing Pope Urban was not a peachy keen career move for Mr. G. Mr. G. put all of Aristotle's howlers in the mouth of Simplicio who was > a surrogate for Pope Urban. Bob Kolker- Hide quoted text - - Show quoted text - And, one could argue that all the Church was trying to do was to get Galileo to do his job: provide clear proofs and an alternative, comprehensive theoretical formulation to Aristotle. They were hoping Galileo would be the next Thomas Aquinas. Instead, Isaac Newton was. The Church condemned Galileo for being arrogant, impertinent, disruptive and incompetent. They were right to do so! === Subject: Re: What if: the Church had NOT condemned Galileo >The Church condemned Galileo because he was wrong. But what they >neglected to mention is that they were wrong too. >If Galileo had not satirized the Church they would not have condemned >him. His arrogance, ultimately, was what got him into trouble. >Correct. Dissing Pope Urban was not a peachy keen career move for Mr. G. >Mr. G. put all of Aristotle's howlers in the mouth of Simplicio who was >a surrogate for Pope Urban. >Bob Kolker- Hide quoted text - >- Show quoted text - > And, one could argue that all the Church was trying to do was to get > Galileo to do his job: provide clear proofs and an alternative, > comprehensive theoretical formulation to Aristotle. They were hoping > Galileo would be the next Thomas Aquinas. Instead, Isaac Newton was. > The Church condemned Galileo for being arrogant, impertinent, > disruptive and incompetent. They were right to do so! The inquisition was not noted for asking people to continue their work and finish it in a form that the church would like better. Having your head removed is not condusive to finishing things. You cannot even begin to take this seriously can you? This is a total joke. The church was afraid that people would find out they were completely wrong about science and did not want it shown. > === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=dznm2ggAAADYkc-DGNLw7vGl4Jo4f4Ds 5.0),gzip(gfe),gzip(gfe) >The Church condemned Galileo because he was wrong. But what they >neglected to mention is that they were wrong too. > If Galileo had not satirized the Church they would not have condemned > him. æHis arrogance, ultimately, was what got him into trouble. Correct. Dissing Pope Urban was not a peachy keen career move for Mr. G. Mr. G. put all of Aristotle's howlers in the mouth of Simplicio who was > a surrogate for Pope Urban. Bob Kolker- Hide quoted text - - Show quoted text - And, one could argue that all the Church was trying to do was to get > Galileo to do his job: æprovide clear proofs and an alternative, > comprehensive theoretical formulation to Aristotle. æ They were hoping > Galileo would be the next Thomas Aquinas. HUH???? Thomas Aquinas had to flee an investigation by the Bishop of Paris. He was excommunicated after his death. Unlike Galileo, Aquinas was lucky to have lived out his life before the Chuch created the office of Grand Inquisitor. Learn something about the Catholic Church! === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=5ayZ-goAAABGZmmwx8zZEwz6gU2OuVSd CLR 2.0.50727; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) >The Church condemned Galileo because he was wrong. But what they >neglected to mention is that they were wrong too. > If Galileo had not satirized the Church they would not have condemned > him. æHis arrogance, ultimately, was what got him into trouble. > Correct. Dissing Pope Urban was not a peachy keen career move for Mr. G. > Mr. G. put all of Aristotle's howlers in the mouth of Simplicio who was > a surrogate for Pope Urban. > Bob Kolker- Hide quoted text - > - Show quoted text - And, one could argue that all the Church was trying to do was to get > Galileo to do his job: æprovide clear proofs and an alternative, > comprehensive theoretical formulation to Aristotle. æ They were hoping > Galileo would be the next Thomas Aquinas. HUH???? Thomas Aquinas had to flee an investigation by the Bishop of Paris. > He was excommunicated after his death. Unlike Galileo, Aquinas was lucky to have lived out his life before > the Chuch created the office of Grand Inquisitor. Learn something about the Catholic Church!- Hide quoted text - - Show quoted text - Ummm... Thomas Aquinas was canonized as a Saint fifty years after his death. Learn something about the Catholic Church! === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=dznm2ggAAADYkc-DGNLw7vGl4Jo4f4Ds 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >The Church condemned Galileo because he was wrong. But what they >neglected to mention is that they were wrong too. > If Galileo had not satirized the Church they would not have condemned > him. æHis arrogance, ultimately, was what got him into trouble. > Correct. Dissing Pope Urban was not a peachy keen career move for Mr. G. > Mr. G. put all of Aristotle's howlers in the mouth of Simplicio who was > a surrogate for Pope Urban. > Bob Kolker- Hide quoted text - > - Show quoted text - > And, one could argue that all the Church was trying to do was to get > Galileo to do his job: æprovide clear proofs and an alternative, > comprehensive theoretical formulation to Aristotle. æ They were hoping > Galileo would be the next Thomas Aquinas. HUH???? Thomas Aquinas had to flee an investigation by the Bishop of Paris. > He was excommunicated after his death. Unlike Galileo, Aquinas was lucky to have lived out his life before > the Chuch created the office of Grand Inquisitor. Learn something about the Catholic Church!- Hide quoted text - - Show quoted text - Ummm... Thomas Aquinas was canonized as a Saint fifty years after his > death. æLearn something about the Catholic Church!- Hide quoted text - - Show quoted text - They excommunicated Aquinas and tried to bring him up on charges of heresy, but he fled. Just like with Galileo, they have never admitted that they were wrong about the charges and the excommunication. The Catholic Church never admits that it is wrong. === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=5ayZ-goAAABGZmmwx8zZEwz6gU2OuVSd CLR 2.0.50727; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) On Sep 15, 12:54æpm, tadams...@yahoo.com The Church condemned Galileo because he was wrong. But what they >neglected to mention is that they were wrong too. > If Galileo had not satirized the Church they would not have condemned > him. æHis arrogance, ultimately, was what got him into trouble. > Correct. Dissing Pope Urban was not a peachy keen career move for Mr. G. > Mr. G. put all of Aristotle's howlers in the mouth of Simplicio who was > a surrogate for Pope Urban. > Bob Kolker- Hide quoted text - > - Show quoted text - > And, one could argue that all the Church was trying to do was to get > Galileo to do his job: æprovide clear proofs and an alternative, > comprehensive theoretical formulation to Aristotle. æ They were hoping > Galileo would be the next Thomas Aquinas. > HUH???? > Thomas Aquinas had to flee an investigation by the Bishop of Paris. > He was excommunicated after his death. > Unlike Galileo, Aquinas was lucky to have lived out his life before > the Chuch created the office of Grand Inquisitor. > Learn something about the Catholic Church!- Hide quoted text - > - Show quoted text - Ummm... Thomas Aquinas was canonized as a Saint fifty years after his > death. æLearn something about the Catholic Church!- Hide quoted text - - Show quoted text - They excommunicated Aquinas and tried to bring him up on charges of > heresy, but he fled. æJust like with Galileo, they have never admitted > that they were wrong about the charges and the excommunication. The Catholic Church never admits that it is wrong.- Hide quoted text - - Show quoted text - I think you can take the canonization as an admission of error. Just my opinion, of course. === Subject: Re: What if: the Church had NOT condemned Galileo >The Church condemned Galileo because he was wrong. But what they >neglected to mention is that they were wrong too. >If Galileo had not satirized the Church they would not have condemned >him. His arrogance, ultimately, was what got him into trouble. >Correct. Dissing Pope Urban was not a peachy keen career move for Mr. G. >Mr. G. put all of Aristotle's howlers in the mouth of Simplicio who was >a surrogate for Pope Urban. >Bob Kolker- Hide quoted text - >- Show quoted text - >And, one could argue that all the Church was trying to do was to get >Galileo to do his job: provide clear proofs and an alternative, >comprehensive theoretical formulation to Aristotle. They were hoping >Galileo would be the next Thomas Aquinas. > HUH???? Thomas Aquinas had to flee an investigation by the Bishop of Paris. > He was excommunicated after his death. Unlike Galileo, Aquinas was lucky to have lived out his life before > the Chuch created the office of Grand Inquisitor. Learn something about the Catholic Church! > He is into revisionist history which is immune to facts. All you have to do is insert your hopes for facts. He does this repeatedly. === Subject: Re: What if: the Church had NOT condemned Galileo <6iohn3Frnc3tU1@mid.individual.net> posting-account=5ayZ-goAAABGZmmwx8zZEwz6gU2OuVSd CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >Oh, certainly, I agree the Church's objections were psychological -- >or, shall we say, socio-psychological. Ö ÖThey feared social >disruption from crticism of the current conception of things without >a >comprehensive alternative being presented. ÖAs for Newton being just >a >link in the chain, Newton's Principia is a pretty big link! ÖAnd, >probably the final one. ÖA truly comprenhensive system of things to >be >put alongside Thomas Aquinas' work. ÖI doubt Galileo was capable of >work of this type, I suspect he was more of an engineer than a >scientist. ÖAnd, he got out of his depth, to his cost. >-- >Michael Press >Galileo's work with the inclined plane alone >qualifies him as a great scientist. > Galileo may have been a great scientist. ÖHe was not a great > theoretician. Ö Which may have been what was bothering the Church > authorities. > This has got to be one of the all time unbelievably silly statements > even for usenet. > Ö He was not providing a comprehensive theoretical > alternative to Aristotle's ideas, which had been integrated with > Catholic theology by Thomas Aquinas. > He was presenting an idea that the church did not like because > it threatened their control. > Ö ÖNewton's work did. ÖGalileo was > ridiculing the existing theoretical conception while providing no > alternative to it. > Presenting facts is not ridiculing. It is called science. > - Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - > Presenting facts is not ridiculing. æCalling the Pope a fool in print > is. æThis is exactly what Galileo did. æThis is what got him into > trouble. > Umm, would you be interested in replying to any of the posts in which I > have noted that this is false? And maybe support your argument, as I > supported mine? We could even have a useful debate. I? we? > You have 'asserted' that it is false - your 'noting' of æa position æis not > strictly 'falsification' - notwithatanding what yhur brother may once have > thought. You have noticed, of course, that falsifying this took place once is > different in kind from falsifying this never took place. If we were > silly enough to be utterly pure falsificationists, the burden of > falsifying my negative assertion would be on you. Negatives can be proved > in mathematics, but not in empirical matters except by exhaustion. But of course we're not as silly as that; the real issue is whether my > claims are too nonsensical to worth answering. Your position is backed by > a famous novelist; mine by historians of science. The side too weak to > bother answering isn't actually my side. Again, there is no historical evidence whatever of an intent on Galileo's > part to insult the Pope, except an assertion by the Inquisition. If you > knew of any, you might condescend to tell us of it rather than debating > epistemology. -- > Dan Drake > d...@dandrake.comhttp://www.dandrake.com/ > porlockjr.blogspot.com- Hide quoted text - - Show quoted text - Pope Urban VIII personally asked Galileo to give arguments for and against heliocentrism in the book, and to be careful not to advocate heliocentrism. He made another request, that his own views on the matter be included in Galileo's book. Only the latter of those requests was fulfilled by Galileo. Whether unknowingly or deliberate, Simplicius, the defender of the Aristotelian Geocentric view in Dialogue Concerning the Two Chief World Systems, was often caught in his own errors and sometimes came across as a fool. This fact made Dialogue Concerning the Two Chief World Systems appear as an advocacy book; an attack on Aristotelian geocentrism and defense of the Copernican theory. To add insult to injury, Galileo put the words of Pope Urban VIII into the mouth of Simplicius. Most historians agree Galileo did not act out of malice and felt blindsided by the reaction to his book.[80] However, the Pope did not take the suspected public ridicule lightly, nor the blatant bias. Galileo had alienated one of his biggest and most powerful supporters, the Pope, and was called to Rome to defend his writings. http://en.wikipedia.org/wiki/Galileo Galilei Simplicius means simpler, stupider, dumber, by the way. Whether Galileo intended to insult the Pope, or simply was tactless and foolish enough to insult the Pope, is a moot point. Clearly, Galileo insulted the Pope. === Subject: Re: What if: the Church had NOT condemned Galileo <6iohn3Frnc3tU1@mid.individual.net> posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Again, there is no historical evidence whatever of an intent on Galileo's > part to insult the Pope, except an assertion by the Inquisition. If you > knew of any, you might condescend to tell us of it rather than debating > epistemology. Pope Urban VIII personally asked Galileo to give arguments for and > against heliocentrism in the book, and to be careful not to advocate > heliocentrism. He made another request, that his own views on the > matter be included in Galileo's book. Only the latter of those > requests was fulfilled by Galileo. Whether unknowingly or deliberate, > Simplicius, the defender of the Aristotelian Geocentric view in > Dialogue Concerning the Two Chief World Systems, was often caught in > his own errors and sometimes came across as a fool. This fact made > Dialogue Concerning the Two Chief World Systems appear as an advocacy > book; an attack on Aristotelian geocentrism and defense of the > Copernican theory. To add insult to injury, Galileo put the words of > Pope Urban VIII into the mouth of Simplicius. Most historians agree > Galileo did not act out of malice and felt blindsided by the reaction > to his book.[80] However, the Pope did not take the suspected public > ridicule lightly, nor the blatant bias. Galileo had alienated one of > his biggest and most powerful supporters, the Pope, and was called to > Rome to defend his writings. http://en.wikipedia.org/wiki/Galileo Galilei That's still a nice story, maybe good for a movie, but simply no historical evidence there. > Simplicius means simpler, stupider, dumber, by the way. Simplicius means simple, not simpler. In that context, it stands for someone who looks at things with simplicity and innocence, not for someone who is stupid. > Whether Galileo intended to insult the Pope, or simply was tactless > and foolish enough to insult the Pope, is a moot point. æClearly, > Galileo insulted the Pope. Clearly you are not an aspiring historian. -LV === Subject: Re: What if: the Church had NOT condemned Galileo <6iohn3Frnc3tU1@mid.individual.net> posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) Galileo is part of the Venetian dead hand faction of science. thus quoth: And, of course, it doesn't work. The whole idea of Thorp, Black, Merton, et al., is to create risk-free betting, by coming up with mathematical formulas which will always guarantee a profit. This insane idea, which is derived from a very old gambling method known as Dutch Book, is why you hear people say that derivatives have made the financial markets safer and more stable. It's the blind leading the blind. First off, as Mr. LaRouche pointed out in his June 21 address, use of the Black-Scholes and related methods is now universal. They're all using the same formulas. Picture what would happen if every blackjack player in a Las Vegas casino was part of the MIT blackjack team, and you should understand what is wrong with that picture. Second, they demand to set their own rules. As even Black and Scholes point out in their 1973 paper, the success of their formula depends on very specific criteria, e.g., a constant flow of cash at a risk-free interest rate (provided in the real world of 2007 by the yen carry trade), no transaction costs or taxes, the possibility to always sell a stock short, etc. In other words, they have constructed an artificial game, supposedly rigged to always win. A fantasy! We have seen this before: Galileo Galilei (Concerning an Investigation on Dice, 1630), Giralamo Cardano (Book on Games of Chance, 1633), and Abraham de Moivre (Doctrine of Chances, 1718 [dedicated to Isaac Newton]), all examined the idea of using mathematical formulas to win at gambling. Their methods came into widespread use in 17th-Century Amsterdam, with the creation of speculative options trading. The result was the Tulip craze, and the South Sea and John Law bubbles. http://www.larouchepub.com/other/2007/3429it_is_gambling.html thus: Universe is about half of antimatter in the Alfven cosmology, which is based upon plasma physics from the lab; their is no light & antilight, though; antimatter looks like matter! thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch_brown.html http://wlym.com === Subject: :: n-divisible groups :: Let n be a natural number. A group G is said to be n-divisible if the map G --> G defined by x |--> x^n is surjective. In other words, every element of G is an n-th power, and we express this by writing G = G^n. Why is it that a group G is n-divisible iff it is p-divisible for each prime p dividing n? One direction is clear: if G = G^n and p|n, then write n = pk, so that G = G^n = (G^k)^p, and indeed, every element of G is a p-th power. What about the converse? Write n = p_1^{a_1}..p_r^{a_r} for distinct primes p_i and a_i >= 1. Now observe that if G is p-divisible for some prime p, then it's also p^m divisible for every m > 0, hence whenever G is p_i-divisible as above, then it must be n-divisible. Would this work? === Subject: Re: :: n-divisible groups :: days. My association with the Department is that of an alumnus. >Let n be a natural number. A group G is said to be n-divisible if the map G --> G defined by x |--> x^n is surjective. In other words, every element of G is an n-th power, >and we express this by writing G = G^n. Why is it that a group G is n-divisible >iff it is p-divisible for each prime p dividing n? One direction is clear: if G = G^n and p|n, then >write n = pk, so that G = G^n = (G^k)^p, and indeed, every element of G is a p-th power. >What about the converse? >Write n = p_1^{a_1}..p_r^{a_r} for distinct primes p_i and a_i >= 1. Now observe that if G is p-divisible for some prime p, then it's >also p^m divisible for every m > 0, hence >whenever G is p_i-divisible as above, then it must be n-divisible. Would this work? Basically, yes. You might also go this route: if every element is an n-th power and every element is an mth power, then every element is an nm-th power: for given x, we can write x = a^n, and we can write a=b^m, so x = a^n = (b^m)^n = b^{nm}. Then simply proceed by induction on a_1+...+a_r. -- magidin-at-member-ams-org === Subject: Re: :: n-divisible groups :: > <24616241.1221261986114.JavaMail.jakarta@nitrogen.math > forum.org>, >Let n be a natural number. A group G is said to be n-divisible if the map G --> G defined by x |--> x^n is surjective. In other words, every element of G is an n-th power, >and we express this by writing G = G^n. Why is it that a group G is n-divisible >iff it is p-divisible for each prime p dividing n? One direction is clear: if G = G^n and p|n, then >write n = pk, so that G = G^n = (G^k)^p, and indeed, every element of G is > a p-th power. >What about the converse? >Write n = p_1^{a_1}..p_r^{a_r} for distinct primes p_i and a_i >= 1. Now observe that if G is p-divisible for some prime > p, then it's >also p^m divisible for every m > 0, hence >whenever G is p_i-divisible as above, then it must > be n-divisible. Would this work? Basically, yes. You might also go this route: if > every element is an > n-th power and every element is an mth power, then > every element is an > nm-th power: for given x, we can write x = a^n, and > we can write > a=b^m, so x = a^n = (b^m)^n = b^{nm}. Then simply proceed by induction on a_1+...+a_r. But I'm not too sure about the last sentence. What do you mean precisely by inducting on a sum of variables, rather than just one ? === Subject: Re: :: n-divisible groups :: days. My association with the Department is that of an alumnus. [.G is n-divisible iff it is p-divisible for every prime p|n.] > Basically, yes. You might also go this route: if > every element is an > n-th power and every element is an mth power, then > every element is an > nm-th power: for given x, we can write x = a^n, and > we can write > a=b^m, so x = a^n = (b^m)^n = b^{nm}. > Then simply proceed by induction on a_1+...+a_r. But I'm not too sure about the last sentence. What do you mean >precisely by inducting on a sum of variables, >rather than just one ? Okay, fair enough. Here is how the induction would proceed in all its gory details: We want to prove that if G is p-divisible for every p|n, then G is n-divisible. We do induction on k, the number of (possibly nondistinct) prime factors of n, which in yoru notation would equal a_1+...+a_r (note that this is not, as you called it, a sum of variables, but rather a number that is associated to n). The case where k=1 (n = p is a prime) is immediate: G is n-divisible if and only if it is p-divisible. Assume the result holds for any m has strictly fewer than k>1 prime factors, and that n has exactly k>1 prime factors. Write n=p*m, where p is prime, and m>1. By assumptions, for every prime q such that q|m, G is q-divisible, so by the induction hypothesis (since m has fewer than k prime factors) G is m-divisible. We also know that G is p-divisible. Therefore, by the argument given above, G is p*m-divisible, i.e., G is n-divisible. This proves the induction, and thus the result. -- magidin-at-member-ams-org === Subject: Re: :: n-divisible groups :: > Okay, fair enough. Here is how the induction would > proceed in all its > gory details: We want to prove that if G is p-divisible for every > p|n, then G is > n-divisible. We do induction on k, the number of (possibly > nondistinct) prime > factors of n, which in yoru notation would equal > a_1+...+a_r (note > that this is not, as you called it, a sum of > variables, but rather a > number that is associated to n). The case where k=1 (n = p is a prime) is immediate: G > is n-divisible > if and only if it is p-divisible. Assume the result holds for any m has strictly fewer > than k>1 prime factors, > and that n has exactly k>1 prime factors. Write > n=p*m, where p is > prime, and m>1. > <<< I've almost got it: when you write n = p m, are you not assuming that has k - 1 prime factors, since n has exactly k prime factors, and p is prime? The rest seems to follow easily, so I have no further questions about this proof. > By assumptions, for every prime q such that q|m, G is > q-divisible, so > by the induction hypothesis (since m has fewer than k > prime factors) G > is m-divisible. We also know that G is p-divisible. > Therefore, by the > argument given above, G is p*m-divisible, i.e., G is > n-divisible. This > proves the induction, and thus the result. > === Subject: Re: :: n-divisible groups :: days. My association with the Department is that of an alumnus. > Okay, fair enough. Here is how the induction would > proceed in all its > gory details: > We want to prove that if G is p-divisible for every > p|n, then G is > n-divisible. > We do induction on k, the number of (possibly > nondistinct) prime > factors of n, which in yoru notation would equal > a_1+...+a_r (note > that this is not, as you called it, a sum of > variables, but rather a > number that is associated to n). > The case where k=1 (n = p is a prime) is immediate: G > is n-divisible > if and only if it is p-divisible. > Assume the result holds for any m has strictly fewer > than k>1 prime factors, > and that n has exactly k>1 prime factors. Write > n=p*m, where p is > prime, and m>1. > I've almost got it: when you write n = p m, are you not assuming >that has k - 1 prime factors, since n has exactly k prime factors, >and p is prime? I'm using unique factorization. The factorizatioon of n will be p*(factorization of m); uniqueness guarantees that #(factors of n) = #(factors of p) + #(factors of m). In fact, the argument could be done using strong induction on n instead, which would not rely on unique factorization. Assume the result holds for all k < n. We prove it for n. The result is clear if n is prime. If n is not prime, then n=a*b with 1 1) .. Or, are we using complete induction, in the sense that, for the induction step, we're assuming that the result is true for each t < k. Perhaps not; for then n may not have exactly k prime factors when we write n = pm. Sorry, just a bit confused at the moment! === Subject: Re: :: n-divisible groups :: days. My association with the Department is that of an alumnus. >. that m has k - 1 prime factors (k > 1) .. >Or, are we using complete induction, in the sense >that, for the induction step, we're assuming that the result is true >for each t < k. The induction is ->on the number of prime factors of n<-, not on n. So we prove the result if n has 1 prime factor, then assume the result holds for every number that has fewer than k (you can do it assuming it has exactly k-1, no problem) and that n has exactly k prime factors. Take one of them, say p, Then n = p*m, and m has exactly k-1 prime factors by unique factorization... Yes, we are using unique factorization implicitly; you can avoid that assumption by doing what I call strong induction and you seem to be calling complete induction; see my other post. >Perhaps not; for then n may not have exactly k prime factors when we >write n = pm. We are assuming the result holds for every number with exactly k-1 prime factors, and want to prove the result for numbers with exactly k prime factors. Let n be such a number, and we proceed. That's how induction works! Remember: the induction is not on the number n itself, but on the number of prime factors your given number has. That's why the basis of the induction is n is a prime, and not n is 1. -- magidin-at-member-ams-org === Subject: Re: :: n-divisible groups :: Oops, that's not quite true. But if G is p^m-divisible, then it's p-divisible, indeed. What should I do about the converse? === Subject: Re: Ways to solve second order ODE in real time? > I think this is an old problem but still I'm not sure how to do it > correctly. I need to solve a linear constant coefficient ODE on a > sample-by-sample basis. > y(t)+2*zeta*wn*y'(t)+wn*y(t) = x(t) with y'(0)=y(0) = 0; > further more, all poles are in the left half of s-plane so that the system > doesn't oscillate. > The input x(t) is sampled every T seconds, i.e. x(nT)=x[n]. For each sampled > x[n], I want to solve y[n]. > Currently, I'm doing it in a rather dumb way -- solve the impulse response > h(t) by setting x(t) to delta(t), then select a Tau such that the transient > response dies out, then sample the h[n] into a Finite Impulse Response (FIR) > filer kernel so that I can find y[n] by doing a discrete convolution for > h[n] and x[n]. The result is ok but not very satisfatory. > I'm thinking of taking bilinear transform to conver the analog H(s) to the z > domain H(z), but then I would need to do IIR filtering which is what I'm > trying to avoid. > What would be the popular way to solve y[n] in a sample-by-sample basis? Use Runge-Kutta method >transform your equation to the system of first order equations: >y_1(t)=y(t) >y_2(t)=y'(t) >youll get >y_1'(t)=y_2(t) >y_2'(t)=-2*zeta*wn*y_2(t)-wn*y_1(t) + x(t) >then assume that between the samples x(t) is constant and equal to the >last sample - zero order extrapolation - or use some kind of other >extrapolation scheme. >and you'll get a system of differential equations in the form: >Y'(t)=f(t,Y(t)) >(with Y=[y_1;y_2]) which is handled by RK methods efficiently so when >you'll use integration step T, you'll get your desired series of y[n]. but this makes no sense: if you approximate the input x of order zero, why > to use an integrator of higher order? using the system in first order form is o.k. Now, on startup, with no more information > you hardly could do more then just compute Y''(0) and let > Y(T)=T^2 *Y''(0)/2 as the first guess. in the next time step, you could use already x(0) and x(T), and a > predictor corrector scheme of order two > (Trapezoidal rule plus Euler explict as predictor) in order to recompute > Y(T). Then you could proceed to a two step method of order > three (two-step method, formally > implicit with a predictor to make it explict) in order to compute > Y(2*T) and so on: that means you use a fixed time step scheme and > a modest multistep method in PECE-form... It may not make sense from a numerical integration viewpoint but that could be the reason for so few usable advances since Merson showed how to integrate ODE more efficiently. Elsewhere though, they were busy well before infamous, nineteen dubious ways... hit the press. Here's one Krouse & Ward, Simulation 1971, Vol 17 Improved linear system simulation by matrix exponentiation with generalized order hold which someone posting from the academics should've been aware of. --- sdx - modeling, simulation. http://www.sdynamix.com === Subject: Can I write this with vector notation? I've got a very simple summation, which is Sum_i=0^i=n[a_i*b_(n-i)], which looks exactly like a dot product, except that the elements of b have been reversed. I don't know any vector operator that reverses the elements of b, although the transpose operator looks close but not correct. I really hate to throw away the simplicity and elegance of vector notation over something so simple, so I'm asking if anybody in here knows any way to write that summation with the bold-faced variables of vector notation and the dot product operator, rather than with subscripts and summation symbols? TIA === Subject: Re: Can I write this with vector notation? posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > I've got a very simple summation, which is Sum_i=0^i=n[a_i*b_(n-i)], which looks exactly like a dot product, except that the elements of b have > been reversed. I don't know any vector operator that reverses the elements > of b, although the transpose operator looks close but not correct. I really hate to throw away the simplicity and elegance of vector notation > over something so simple, so I'm asking if anybody in here knows any way to > write that summation with the bold-faced variables of vector notation and > the dot product operator, rather than with subscripts and summation > symbols? TIA As Dave has suggested, this is the convolution operation; it is often written as a*b (bold a, asterisk, bold b --- if you use bold letters for vectors). Although you cannot easily define x*y without using subscripts and summation, you can define it once and for all at the start of your document and thereafter use the simpler '*' notation, switching to summation and subscripts again only when you need to compute the final answer. The link Dave supplied shows the fundamental linearity properties of '*', the use of which greatly simplifies computations. Of course, you could also write x*y = x^T A y, where ^T = transpose and A is the n x n matrix that reverses the components of y. R.G. Vickson === Subject: Re: Can I write this with vector notation? > I've got a very simple summation, which is > Sum_i=0^i=n[a_i*b_(n-i)], > which looks exactly like a dot product, except that the elements of b have > been reversed. I don't know any vector operator that reverses the elements > of b, although the transpose operator looks close but not correct. > I really hate to throw away the simplicity and elegance of vector notation > over something so simple, so I'm asking if anybody in here knows any way to > write that summation with the bold-faced variables of vector notation and > the dot product operator, rather than with subscripts and summation > symbols? > TIA As Dave has suggested, this is the convolution operation; it is often > written as a*b (bold a, asterisk, bold b --- if you use bold letters > for vectors). Although you cannot easily define x*y without using > subscripts and summation, you can define it once and for all at the > start of your document and thereafter use the simpler '*' notation, > switching to summation and subscripts again only when you need to > compute the final answer. The link Dave supplied shows the fundamental > linearity properties of '*', the use of which greatly simplifies > computations. Of course, you could also write x*y = x^T A y, where ^T > = transpose and A is the n x n matrix that reverses the components of > y. R.G. Vickson Of course, this is not the full (discrete) convolution, since the convolution of a sequence of m terms with one of n terms results in a sequence of m+n-1 terms. Instead, it is one term of the convolution. One other remark (for the OP): that matrix A is just the n x n matrix with ones along the counter-diagonal, and zeros everywhere else. Dale === Subject: Re: Can I write this with vector notation? posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze Filter 1.2.0.72; .NET CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; WWTClient2),gzip(gfe),gzip(gfe) > I've got a very simple summation, which is Sum i=0^i=n[a i*b (n-i)], which looks exactly like a dot product, except that the elements of b have > been reversed. æI don't know any vector operator that reverses the elements > of b, although the transpose operator looks close but not correct. æ It is called a discrete convolution. See http://en.wikipedia.org/wiki/Convolution#Discrete convolution Dave === Subject: rudin === Subject: Re: rudin - contest for the most important function in mathematics > Rudin in his prologue to 'Real And Complex Analysis' says that exp[z] is 'the most important THE COMPLEX EXPONENTIAL FUNCTION (A) Its definition as a power series is in my opinion sheer beauty, especially when recorded as exp(z) = 1 + z/1 + (z/1)(z/2) + (z/1)(z/2)(z/3) + (z/1)(z/2)(z/3)(z/4) + ... (B) It offers characterisations of pi and e, in my opinion the two most important transcendentals in mathematics. (C) As far as I know it is the simplest entire complex function next to the complex polynomial functions. (D) It is in my opinion the simplest entire complex function with an essential singularity. (E) ... I guess one can extend this list to several dozens of items, all good reasons to highlight the importance of the exponential function. a kind of contest among proposals for the most important mathematical function. An independent jury would rank the submissions according to the numbers of good reasons to support the submitters' proposals; or according to some weighted sum, where purely mathematical reasons, aesthetic reasons, reasons referring to engineering applications, or whatever kinds of good reasons, are weighted differently. Speaking for myself, I would like to know which mathematical function would win the beauty contest... === Subject: Re: rudin daniel t a .8ecrit : > After all there are other candidates (or he wouldn't have bothered). Which other candidates? === Subject: Re: rudin posting-account=8wyvFgoAAAAJYWyfLHRzREe3lxFCHRTd MathPlayer 2.10b; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) I forgot who said it (about) 45 years ago - exp is the second most important function in mathematics ,the first being I(z)=z .However yes ,its hard to argue with this assertion about the primary imprtance of exp.Its tributaries include the trig functions and the logarithm ,the later of which is crucial for for example determining how a closed curve winds about a point. Much is reduced to the exponential law which gives a group homomorphism of C (complex numbers) with + to C with x .smn PS I don't know of any other candidates . === Subject: Re: binary integer non-linear programming excel solver premium posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Howdy, I am working a homework problem involving non-linear programming for > an operations research class. The problem is in three parts. The > objective is to minimize the total distance traveled between a > facility and thirty different towns each year by optimally locating > the facility amongst the towns. There are a fixed number of trips > between each facility and each town each year. The second and third > parts of the problem require locating two and three facilities. I > believe IÍve got the first part correct. ExcelÍs solver give a > consistent and reasonable answer. Of course, two and three facilities > seems to produce local minima. IÍve downloaded and IÍm trying to use > Frontline systems Premium solver add-in. Other than randomly sorting > through different starting points, how do I configure the premium > advice or help. -g Are the facilities restricted to being in the towns themselves? If so, you seem to have a classical p-median problem, with p = number of facilities. You can formulate the problem as a (binary) IP problem: let x(i,j) = 1 if town j is serviced from town i (because town i contains the nearest open facility) and x(i,j) = 0 otherwise. Let y(i) [ = x(i,i)] = 1 if a facility is at town i and y(i) = 0 otherwise. Let t(j) = number of trips per year to and from town j. The total distance travelled per year, to be minimized, is Z = sum {i} sum{j} d(i,j)*t(j) * x(i,j), where d(i,j) = distance from town i to town j (or, if you count both outbound and inbound trips, use Z = 2*(the above) instead--- but this '2' is just a constant factor, so let's drop it). Below, let c(i,j) = d(i,j)*t(j), which is easily-computed input data. There are two somewhat similar formulations, which behave differently in practice when you come to actually SOLVE the problem. The first (large) formulation is: minimize sum{c(I,j)*x(i,j), i=1..m, j = 1..n} subject to 1) sum{x(i,j),i=1..n} = 1 for all j (each town gets served from a single location) 2) sum{y(i),i=1..n} = p (open p facilities) 3) x(i,j) <= y(i) for i = 1,..., n and j = 1, 2, ..., n 4) x(i,j) = 0,1 and y(i) = 0,1 for all 1 <= i <= n and 1 <= j <= n. Constraints 3) say that we can serve j from i only if there is a facility at i. This formulation has n^2 binary variables x(i,j) for i =/= j and y(i), and 1 + n + n*(n-1) = n*n + 1 constraints (n of type 1),1 of type 2) and n(n-1)of type 3). The second (smaller) formulation is: minimize sum{c(i,j)*x(i,j), i=1..m, j = 1..n} subject to 1) sum{x(i,j),i=1..n} = 1 for all j (each town gets served from a single location) 2) sum{y(i),i=1..n} = p (open p facilities) 3) sum{x(i,j), j=1..n} <= n*y(i) for 1 <= i <= n . 4) x(i,j) = 0,1 and y(i) = 0,1 for all 1 <= i <= n and 1 <= j <= n. Here the objective and 1), 2) and 4) are the same, but 3) is smaller: it just has n relations. It says that town i can serve some towns only if there is a facility at i. Note that in both formulations, minimization of the objective guarantees that each town will be serviced by the closest open facility. Whether or not these formulations are practical for solving in EXCEL depends on the size of n (n = 30 in your case). You need to consult the manual or consult on-line references to see if your version of the SOLVER can handle problems as large as n = 30; the version of EXCEL XP that I have could not handle problems that large. Of course, you set them up just like ordinary LP problems, but in the constraints menu you choose X(i,j) = binary. You can let y(i) = x(i,i) for ease of modelling. There is also the issue of whether to go with the large (first) formulation of the small (second) formulation. The surprise is that /usually/ the larger formulation is much better, because when the software attacks the problem by branch-and-bound it produces significantly smaller search trees in the first formulation. Often, the LP relaxation of the first formulation gives optimality right away, with no need for branching at all, but, basically, this rarely if ever happens with the second formulation. So use the large formulation if it fits in your Solver's size limitations. You should also be aware that (in older versions, at least) the EXCEL Solver's performance on integer-programming problems was spotty: the software was reported to be unreliable, sometimes giving non-optimal results. I don't know if this is currently the case, but you could maybe find out more information by posting a message to the operations research newsgroup 'sci.op-research'. In the past this created problems in the operations research teaching community, because many of the textbooks have (mistakenly, in my opinion) gone over to almost exclusive use of EXCEL for modelling, while the tool itself was not always capable of delivering trustworthy answers. (However, it seems that marketing was won out over accuracy and reliability, so that people who write textbooks no longer seem to care about whether the answers are correct---they just use EXCEL anyway. I'm glad I'm retired and can leave that nonsense behind.) If you do a Google search on p-median problem you will see numerous papers that develop heuristics for the problem. The reason, of course, is that in large, industrial-sized real-world problems n is large, so solving to provable optimality an integer program with n^2 binary variables and n^2 constraints is beyond the bounds of practicality. You might, in future, get better feedback by posting to the group 'sci.op-research' instead of this one. R.G. Vickson Adjunct Professor, University of Waterloo === Subject: Re: binary integer non-linear programming excel solver premium posting-account=hrbKuAoAAABB-pktWwb5o12FCbdbJca0 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Professor Vickson, helpful. The facilities are not required to be co-located with the towns. The towns all lie within a 10x10 continuous grid. The facilities can be located anywhere with 100x100 grid that circumscribes the towns. I suspect, however, they're best placed very reassuring to find that mine was essentially the same. By trying a variety of different start points for each facility, I think I've closed in on a good solution. Of course, I may still spend some time working through the mechanics of Frontline Systme's premium solver to find tease out a definitive global optimum. I'd like to know if there's an easy way to force a brute force method out of either excel > R.G. Vickson > Adjunct Professor, University of Waterloo === Subject: Re: binary integer non-linear programming excel solver premium posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Professor Vickson, helpful. The facilities are not required to be co-located with the > towns. The towns all lie within a 10x10 continuous grid. The > facilities can be located anywhere with 100x100 grid that > circumscribes the towns. I suspect, however, they're best placed > very reassuring to find that mine was essentially the same. By trying > a variety of different start points for each facility, I think I've > closed in on a good solution. Of course, I may still spend some time > working through the mechanics of Frontline Systme's premium solver to > find tease out a definitive global optimum. I'd like to know if > there's an easy way to force a brute force method out of either excel R.G. Vickson > Adjunct Professor, University of Waterloo It sounds like you have a problem that is closer to the p-center problem than the p-median problem (although most descriptions of the p- center problem try to minimize the maximum distance, rather than the summed distance). One entry dealing with something close to your problem is the Google books entry &source=web&ots=3Rd7VD9Xis&sig=DKt7d67uDmF6cIdgXmpVOlV3Yv4&hl=en&sa=X&oi=boo k_result &resnum=4&ct=result#PPA187,M1 (where I have wrapped the long URL by putting carriage returns before two of the &s. There, Eiselt and Sandblom present a formulation much like the one I gave you, but the locations i of the facilities can be in a different set than the locations j of the customers. They also state that the allocation variables x(i,j) can be relaxed to continuous variables with 0 <= x(i,j) <= 1 (at least for some of the LP solutions), leaving only the location variables y(i) as binary. This is a tremendous saving. Note that if you allow the facility locations i to be in a much larger (discrete) set than the towns, you will have a much larger problem than the one I outlined before. Eiselt and Sandblom present a binary search algorithm for the problem that would be much, much simpler and more practical than the full-blown IP formulation. It requires solving set covering problems as subproblems, but these are many times easier than the original p-center problem itself, and are well-doable within the bounds of the EXCEL Solver. Note that the freely-viewable exerpt of the Eiselt-Sandblom book is incomplete (some pages are omitted) but there is enough there to give you about all you need. Details on set covering problems are available on line through a Google search. The full-blown version of your problem, in which the facility locations can be anywhere in a continuum, is much harder; restricting to a large, discrete set of possibilities is what Eiselt and Sandblom are talking about. Another Google book (Handbook of Transportation Science, by Randolph W. Hall) &source=web&ots=atK3FVrfnS&sig=NYwKQhHLsa3kOw-SxZ-etZ85sT8&hl=en&sa=X &oi=book_result&resnum=1&ct=result#PPA332,M1 presents (on page 332) the following important result: you can restrict the facility locations to nodal locations without loss of optimality; that is, there exists an optimal solution in which the facilities ARE located in the towns! (Again, this is for a formulation in which the facility locations sets I and town locations sets J can be different, but (I guess) with J being a subset of I.) Actually, Hall shows only that any facility *on* a link joining two towns can be moved to one of the towns without loss of optimality; it seems to me that he still leaves open the issue of whether facilities that are not on any link between towns can be moved to a link (and then to a town) without loss of optimality. For the classical p-center problem (minimizing the maximum distance) the result above is false: some optimal solutions are strictly between towns. However, your version is more like the p-median problem, which behaves somewhat differently. Good luck. R.G. Vickson === Subject: Re: combinatorics question posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > I've been trying to figure out what's happening in the Clay > Institute's P v. NP Problem description. It talks about the > difficulty of generating the answers to the following scenario: you > have 400 students, and you want to figure out how many ways you can > put them into a dormitory that only holds 100 students. Also, there > are certain pairs of students who can't be in the dorm together. I > think that the number of combinations of students overall would be > 400!/(300!)*(100!). If this is correct, a question: what difference > would it make to the problem if instead of just considering the 100 > students as an undifferentiated mass, you had to generate the number > of *pairs* of students that would fit in the dorm at once? Does > that even make sense? I haven't the slightest idea how you would go > about eliminating prohibited pairs of students without generating > the whole list and striking off those pairs. > Isn't this just the difference between the marriage problem (which has a polynomial algorithm) and the homosexual marriage problem (which is NP-hard)? R.G. Vickson === Subject: Windows6.0 PPC system/Wifi/ standard USB interface/ hardware hot boot/ Touch FLO dynamic operation (paypal posting-account=2OV1VAoAAAC_BqC54DITSFDesCsctnNJ CIBA),gzip(gfe),gzip(gfe) (paypal payment)(www.goodsaler.com ) Operating Frequency:GSM;Network Frequency:900/1800/1900MHz;More information:MP3/MP4/Handsfree/SMS group sending/Voice recorder/WAP/Handwritten input/Bluetooth/GPRS download/Infrared interface/MMS/Smart phone/Memory extended/E-book/ Coming call firewall/E-dictionary/Windows6.0 PPC system/Wifi/ standard USB interface/ hardware hot boot/ Touch FLO dynamic operation (paypal payment)(www.goodsaler.com ) === Subject: Problem with an integral posting-account=13KbngoAAABfTatbRX9gGDZXi3iESPtC AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.22,gzip(gfe),gzip(gfe) Hi there! Can you please help me this integral? integral of dV/[(V^2)*(V-nb)^(1-g)] where nb ang g are constants === Subject: Re: Problem with an integral > Hi there! > Can you please help me this integral? > integral of dV/[(V^2)*(V-nb)^(1-g)] where nb ang g > are constants > Try the symbolic integrator at http://integrals.wolfram.com The result involves hypergeometric functions. Best wishes Torsten. === Subject: Re: Problem with an integral <965146.1221303669977.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=13KbngoAAABfTatbRX9gGDZXi3iESPtC AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.22,gzip(gfe),gzip(gfe) The result involves hypergeometric functions. Best wishes > Torsten. hypergeometric funtions is my problem... I don't know what it is.... if 1-g=0.33 the function is F(1.33, 2, 2.33, 1, -1.x/a) is there a way to evaluate it? otherwise... is there a way to eliminate the hypergeometrical function? === Subject: Re: Problem with an integral posting-account=13KbngoAAABfTatbRX9gGDZXi3iESPtC AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.22,gzip(gfe),gzip(gfe) I need this integral to solve this ODE: T'+{R/[c(V-nb)]}*T=(na)/(c*V^2) === Subject: Dual group: name of theorem? Is there a name attached to the theorem that the dual G* of a locally compact abelian group G is itself locally compact? Could one call it Pontryagin's Theorem, although as far as I can see he only proved it when G is compact or discrete? -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Dual group: name of theorem? posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Is there a name attached to the theorem > that the dual G* of a locally compact abelian group G > is itself locally compact? Could one call it Pontryagin's Theorem, > although as far as I can see he only proved it > when G is compact or discrete? It is usually called like that. Or, rather, Pontryagin's Duality Theorem, but it was proved by van Kampen and Weyl in the case of locally compact groups (There are also extensions to `bigger' abelian groups...) -- m === Subject: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) I'm an amateur in this topic. When my friend told me about Cantor's cardinalities, the first thing that seemed odd to me was the 1-to-1 mapping used for comparing the sizes of two infinite sets. I'm not sure if it is proved that 1-to-1 correspondence is valid for comparing two sets of infinite size. Not because of the practical difficulties in comparing or the time it takes for the comparing process to complete. I think, atleast it should be proved logically that such mapping is valid between two infinite sets. For example, the mappings such as y=2x showing that there are as many even numbers as integers seem dubious to me because there is no proof that such function is meaningful when values of x and y reach infinite. I'm sure these questions are not new, but I would like to see some proofs. Any references or proofs in this regard? Venkat === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? Distribution: world >two sets of infinite size. Not because of the practical difficulties >in comparing or the time it takes for the comparing process to >complete. I think, atleast it should be proved logically that such >mapping is valid between two infinite sets. Please define what you mean by the term valid. >For example, the mappings such as y=2x showing that there are as many >even numbers as integers seem dubious to me because there is no proof >that such function is meaningful when values of x and y reach >infinite. If you take x and y from the integers, as your remark implies, neither x nor y will ever be infinite. Every integer is finite. The set of integers is infinite, but the set of integers is not an integer (any more than the set of bicycles is a bicycle), so it's not ever going to be x or y, which are integers. -- Michael F. Stemper #include This sentence no verb. === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. Please define what you mean by the term valid. My attempt to rephrase venkat's question: Is there a convincing argument (an informal one will do) why cardinality is an accurate formalization of the intuitive notion of 'size', for infinite sets? -- Herman Jurjus === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? <48ce9b90$0$27228$ba620dc5@text.nova.planet.nl two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. > Please define what you mean by the term valid. > My attempt to rephrase venkat's question: > Is there a convincing argument (an informal one will do) why > cardinality is an accurate formalization of the intuitive notion of > 'size', for infinite sets? Changing the name of every element of a set does not change the size of the set. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. It is not proved, it cannot be, you need to define size. However, 1-1 mapping seems an intuitvely nice notion. If 1-1 mapping is not valid then there must be a set whose size we can change by changing the names of its elements. Since we would like a size that does not have this property we are stuck with 1-1 mapping and the counterintuitive results that go along with it. Yes, you can get rid of these results by saying that there is no ordered entity (changing the name, e.g. from set to potential set, does not help, you have to get rid of the entity) without last element, but this leads to results (e.g. there is a largest natural) that are even more counterintuitive. - William Hughes === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. It is not proved, it cannot be, you need to define size. > However, 1-1 mapping seems an intuitvely nice notion. > If 1-1 mapping is not valid then there must be a set whose size > we can change by changing the names of its elements. æSince we would > like a size that does not have this property we are stuck with > 1-1 mapping and the counterintuitive results that go along with > it. æ Yes, you can get rid of these results by saying that > there is no ordered entity (changing the name, e.g. from set > to potential set, does not help, you have to get rid of > the entity) without last element, but this leads > to results (e.g. there is a largest natural) that are even > more counterintuitive. May be we are accepting 1-1 because we don't any better thing, while we are forced to compare the size of infinite sets, and brushing off the questions as related to intuition? I would say, actually believing that 1-1 can be used for sizing the infinite sets is based on a poor intuition which extends this discreteness or distinctness of things for ever. Just like Euclidean spaces extends straightness infinitely. æ æ æ æ æ æ æ æ æ æ æ æ æ - William Hughes === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. It is not proved, it cannot be, you need to define size. > However, 1-1 mapping seems an intuitvely nice notion. > If 1-1 mapping is not valid then there must be a set whose size > we can change by changing the names of its elements. Since we would > like a size that does not have this property we are stuck with > 1-1 mapping and the counterintuitive results that go along with > it. Yes, you can get rid of these results by saying that > there is no ordered entity (changing the name, e.g. from set > to potential set, does not help, you have to get rid of > the entity) without last element, but this leads > to results (e.g. there is a largest natural) that are even > more counterintuitive. May be we are accepting 1-1 because we don't any better thing, while > we are forced to compare the size of infinite sets, and brushing off > the questions as related to intuition? I would say, actually believing > that 1-1 can be used for sizing the infinite sets is based on a poor > intuition which extends this discreteness or distinctness of things > for ever. Just like Euclidean spaces extends straightness infinitely. It is clear that you do not like the results of defining the size of infinite sets by 1-1 mapping. Your claim is that this defintion is based on poor intuition. However, you have entirely ignored the question of whether or not a size of a set must have the property that changing the name of every element in the set does not change the size of the set. You are left with two choices for your better thing: infinite sets do not have a size, or they have a size but there is an infinite set whose size changes when you change element names. - William Hughes === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. > It is not proved, it cannot be, you need to define size. > However, 1-1 mapping seems an intuitvely nice notion. > If 1-1 mapping is not valid then there must be a set whose size > we can change by changing the names of its elements. æSince we would > like a size that does not have this property we are stuck with > 1-1 mapping and the counterintuitive results that go along with > it. æ Yes, you can get rid of these results by saying that > there is no ordered entity (changing the name, e.g. from set > to potential set, does not help, you have to get rid of > the entity) without last element, but this leads > to results (e.g. there is a largest natural) that are even > more counterintuitive. May be we are accepting 1-1 because we don't any better thing, while > we are forced to compare the size of infinite sets, and brushing off > the questions as related to intuition? I would say, actually believing > that 1-1 can be used for sizing the infinite sets is based on a poor > intuition which extends this discreteness or distinctness of things > for ever. Just like Euclidean spaces extends straightness infinitely. It is clear that æyou do not like the results of defining > the size of infinite sets by 1-1 mapping. > Your claim is that this defintion is based on poor intuition. > However, you have entirely > ignored the question of whether or not a size of a set must have the > property > that changing the name of every element in the set does not change > the size of the set. æYou are left with two choices for your better > thing: > infinite sets do not have a size, or they have a size but there is > an infinite set whose size changes when you change element names. Sorry, I didn't give enough thought to changing names changes size. Let me think about it right now. So, you seem to say that names are some labels attached to entities which exist by themselves. But I think names *are* the entities. There is no entity separate from its name (if there is, how do you prove its existence). If I go by my view, changing a name would mean an atomic process of removing an existing entity from the set and adding a new entity. However, if the mappings are dependent on the old name, that is old entity, and if the mappings handled by the old entity can't be moved to new entity, and if the size depends on these mappings, yes, size is affected. But I'm not sure how this relates to defending that 1-1 mapping can be used for sizing the infinite sets. æ æ æ æ æ æ æ æ æ æ æ - William Hughes === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. > It is not proved, it cannot be, you need to define size. > However, 1-1 mapping seems an intuitvely nice notion. > If 1-1 mapping is not valid then there must be a set whose size > we can change by changing the names of its elements. Since we would > like a size that does not have this property we are stuck with > 1-1 mapping and the counterintuitive results that go along with > it. Yes, you can get rid of these results by saying that > there is no ordered entity (changing the name, e.g. from set > to potential set, does not help, you have to get rid of > the entity) without last element, but this leads > to results (e.g. there is a largest natural) that are even > more counterintuitive. > May be we are accepting 1-1 because we don't any better thing, while > we are forced to compare the size of infinite sets, and brushing off > the questions as related to intuition? I would say, actually believing > that 1-1 can be used for sizing the infinite sets is based on a poor > intuition which extends this discreteness or distinctness of things > for ever. Just like Euclidean spaces extends straightness infinitely. It is clear that you do not like the results of defining > the size of infinite sets by 1-1 mapping. > Your claim is that this defintion is based on poor intuition. > However, you have entirely > ignored the question of whether or not a size of a set must have the > property > that changing the name of every element in the set does not change > the size of the set. You are left with two choices for your better > thing: > infinite sets do not have a size, or they have a size but there is > an infinite set whose size changes when you change element names. Sorry, I didn't give enough thought to changing names changes size. > Let me think about it right now. So, you seem to say that names are > some labels attached to entities which exist by themselves. But I > think names *are* the entities. There is no entity separate from its > name (if there is, how do you prove its existence). I do not see any real difference. Instead of a set with a number of entites with names name_1, name_2, name_3 ... you have a set with entities, name_1, name_2, name_3 ... We need to specify a size of a set. Intutively, changing a name (whether we think of this as changing the name of an entity, or replacing an entity with another entity) should not change the size of the set. - William Hughes === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? > May be we are accepting 1-1 because we don't any better thing, while > we are forced to compare the size of infinite sets, and brushing off > the questions as related to intuition? I would say, actually believing > that 1-1 can be used for sizing the infinite sets is based on a poor > intuition The only other standard way to compare set sizes for infinite sets as well as finite ones is by the the subset relation, and, one notes, that whenever Set A is smaller than set B by the subset relation, it is also smaller by injection, so injection works perfectly wherever one has any alternate way of testing for size. The only other alternative is to refuse to make any comparisons between sizes of infinite sets other than by the subset test. === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) May be we are accepting 1-1 because we don't any better thing, while > we are forced to compare the size of infinite sets, and brushing off > the questions as related to intuition? I would say, actually believing > that 1-1 can be used for sizing the infinite sets is based on a poor > intuition The only other standard way to compare set sizes for infinite sets as > well as finite ones is by the the subset relation, and, one notes, that > whenever Set A is smaller than set B by the subset relation, it is also > smaller by injection, so injection works perfectly wherever one has any > alternate way of testing for size. Can you please provide any web page links to this technique of subset The only other alternative is to refuse to make any comparisons between > sizes of æinfinite sets other than by the subset test. === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? Supersedes: <1tcayezrg62fg.vkfegf86tobi.dlg@40tude.net > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. > It's not clear to me what valid might mean in this context. At least it's _sensible_ to use 1-to-1 correspondence as a means to compare two sets concerning their size. Actually, we define the notion /equipollent/ (or /equipotent/) or /of the same power/ or /equivalent/, to mean just this: there exists a 1-to-1 correspondence between the two sets considered. (For finite sets, this obviously coincides with our intuitive notion of the same size. For infinite sets this may be considered a _generalization_ of that (intuitive) notion. Of course this leads to the somewhat puzzling result that {1, 2, 3, 4, ...} is of the same size as {2, 4, 6, 8, ...}.) Not because of the practical difficulties in comparing or the time it > takes for the comparing process to complete. > There is no such process. Actually, 1-to-1 correspondence here means that a bijective _function_ (which is a static object, not a dynamic one) exists between the two sets (i.e. maps one set onto the other). For example the function f defined with f(n) = 2*n is a bijective function from {1, 2, 3, 4, ...} onto {2, 4, 6, 8, ...}. So there _is_ (exists) such a function. Hence we have established that {1, 2, 3, 4, ...} ~ {2, 4, 6, 8, ...}. I think, at least it should be proved logically that such > mapping is valid between two infinite sets. > Whatever you mean with valid. Actually, in the context of _axiomatic set theory_ we can PROVE that such a function between {1, 2, 3, 4, ...} (or the set of natural numbers) and {2, 4, 6, 8, ...} (or the set of even numbers) exists. For example, the mappings such as y = 2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. > Neither x nor y _ever_ reach infinite. (!) The function f defined with f: {1, 2, 3, 4, ...} --> {2, 4, 6, 8, ...} n |-> 2*n just maps ANY (each and every) element in {1, 2, 3, 4, ...} (i.e. any natural number) to an element in {2, 4, 6, 8, ...}. With other words, it maps the (an arbitrary) element n e {1, 2, 3, 4, ...} to an element 2*n e {2, 4, 6, 8, ...}. Since every element in {1, 2, 3, 4, ...} is FINITE there's no problem with the mapping n |-> 2*n. (Neither n nor 2*n ever reach infinite; each and any n [and hence 2*n] is finite.) I'm sure these questions are not new, but I would like to see some > proofs. > Proofs for what? That all n e IN = {1, 2, 3, 4, ...} are finite? :-) Here's a rather simply one, working with mathematical induction: 1 e IN is (obviously) finite. If n e IN is finite, (obviously) n+1 e IN is finite too. Hence all n e IN are finite (by induction). [] B. === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. But x and y never do reach infinite. The issue is that in comparing N, the set of naturals and , say E, for the set of evens, is there a correspondence which does not use twice nor omit any member of either set. That is certainly a valid comparison for finite sets. Can you think of a better way of comparing sets for relative sizes that works for finite sets? If not why object to it for infinite sets? I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? Since one DEFINES relative size of sets by injection, surjection and bijection, what is there to be proven? === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. But x and y never do reach infinite. The issue is that in comparing N, the set of naturals and , say E, for > the set of evens, is there a correspondence which does not use twice nor > omit any member of either set. That is certainly a valid comparison for finite sets. Can you think of a better way of comparing sets for relative sizes that > works for finite sets? If not why object to it for infinite sets? The 1-to-1 correspondence works well as long as n is different from n +1, which is true for a natural number since it is defined to be finite. May be the roots of my perceived problem is in the definition of natural numbers as an infinite set while claiming that none of its elements is infinite. I read that the natural number n is defined as {0,1,2,...,n-1} where n can't be called infinite and n is also the count of all numbers in the finite set. Then we extended this to set containing all natural numbers but we now find none of its members can represent its size? First of all, there doesn't seem to be any such thing as all natural numbers. The standard mathematical language saying For all x in the set I, x+1 also belongs to I, and zero belongs to I does seem to do justice in the usage of the word all. What do you mean by all (or every)? By using this word one is assuming that all is already understood and defined in this context, which, in fact, is what we are trying to define. I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? Since one DEFINES relative size of sets by injection, surjection and > bijection, what is there to be proven? === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? venkat.6123@gmail.com a .8ecrit : > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. > For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. > But x and y never do reach infinite. > The issue is that in comparing N, the set of naturals and , say E, for > the set of evens, is there a correspondence which does not use twice nor > omit any member of either set. > That is certainly a valid comparison for finite sets. > Can you think of a better way of comparing sets for relative sizes that > works for finite sets? If not why object to it for infinite sets? The 1-to-1 correspondence works well as long as n is different from n > +1, which is true for a natural number since it is defined to be > finite. May be the roots of my perceived problem is in the definition > of natural numbers as an infinite set while claiming that none of its > elements is infinite. I read that the natural number n is defined as {0,1,2,...,n-1} where n > can't be called infinite and n is also the count of all numbers in the > finite set. Then we extended this to set containing all natural > numbers but we now find none of its members can represent its size? > First of all, there doesn't seem to be any such thing as all natural > numbers. The standard mathematical language saying For all x in the > set I, x+1 also belongs to I, and zero belongs to I does seem to do > justice in the usage of the word all. What do you mean by all (or > every)? By using this word one is assuming that all is already > understood and defined in this context, which, in fact, is what we are > trying to define. > At this stage, you are obviously trolling. I take the opportunity to note you use the same devices that known trolls. Is there a troll school somewhere ? [plonk] I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? > Since one DEFINES relative size of sets by injection, surjection and > bijection, what is there to be proven? === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? <48cce93c$0$7098$7a628cd7@news.club-internet.fr> posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 14, 3:36æpm, Denis Feldmann I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. > For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. > But x and y never do reach infinite. > The issue is that in comparing N, the set of naturals and , say E, for > the set of evens, is there a correspondence which does not use twice nor > omit any member of either set. > That is certainly a valid comparison for finite sets. > Can you think of a better way of comparing sets for relative sizes that > works for finite sets? If not why object to it for infinite sets? The 1-to-1 correspondence works well as long as n is different from n > +1, which is true for a natural number since it is defined to be > finite. May be the roots of my perceived problem is in the definition > of natural numbers as an infinite set while claiming that none of its > elements is infinite. I read that the natural number n is defined as {0,1,2,...,n-1} where n > can't be called infinite and n is also the count of all numbers in the > finite set. Then we extended this to set containing all natural > numbers but we now find none of its members can represent its size? > First of all, there doesn't seem to be any such thing as all natural > numbers. The standard mathematical language saying For all x in the > set I, x+1 also belongs to I, and zero belongs to I does seem to do > justice in the usage of the word all. What do you mean by all (or > every)? By using this word one is assuming that all is already > understood and defined in this context, which, in fact, is what we are > trying to define. At this stage, you are obviously trolling. I take the opportunity to > note you use the same devices that known trolls. Is there a troll school > somewhere ? æ[plonk] I don't mind being called troll, because I don't care about whatever it means. I'm just trying to ask a few questions that bother me. It seems I was not precise enough in my questions. Let me make one more attempt at making my questions much clearer. 1) If none of the natural numbers is infinite, how can the size of the set of natural numbers be called infinite? Is there some finite n where n+1 becomes infinite? 2) The statement zero belongs to I, and if x belongs to I, x+1 also belongs I simply seeding off the set with zero and attempts to build it with a chain relationship. But it is cleverly evading the task of identifying all elements of the set. This cleverness is useful in situations like summing up some diminishing infinite series, but not in this case. How can this definition be allowed to define a complete set? I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? > Since one DEFINES relative size of sets by injection, surjection and > bijection, what is there to be proven? === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? > On Sep 14, 3:36æpm, Denis Feldmann I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. > For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. > But x and y never do reach infinite. > The issue is that in comparing N, the set of naturals and , say E, for > the set of evens, is there a correspondence which does not use twice nor > omit any member of either set. > That is certainly a valid comparison for finite sets. > Can you think of a better way of comparing sets for relative sizes that > works for finite sets? If not why object to it for infinite sets? > The 1-to-1 correspondence works well as long as n is different from n > +1, which is true for a natural number since it is defined to be > finite. May be the roots of my perceived problem is in the definition > of natural numbers as an infinite set while claiming that none of its > elements is infinite. > I read that the natural number n is defined as {0,1,2,...,n-1} where n > can't be called infinite and n is also the count of all numbers in the > finite set. Then we extended this to set containing all natural > numbers but we now find none of its members can represent its size? > First of all, there doesn't seem to be any such thing as all natural > numbers. The standard mathematical language saying For all x in the > set I, x+1 also belongs to I, and zero belongs to I does seem to do > justice in the usage of the word all. What do you mean by all (or > every)? By using this word one is assuming that all is already > understood and defined in this context, which, in fact, is what we are > trying to define. At this stage, you are obviously trolling. I take the opportunity to > note you use the same devices that known trolls. Is there a troll school > somewhere ? æ[plonk] I don't mind being called troll, because I don't care about whatever > it means. I'm just trying to ask a few questions that bother me. It seems I was not precise enough in my questions. Let me make one > more attempt at making my questions much clearer. 1) If none of the natural numbers is infinite, how can the size of the > set of natural numbers be called infinite? Is there some finite n > where n+1 becomes infinite? The size of a set may bear little resemblance to the size of its members. The set of rational fractions strictly between 0 and 1 is infinite even though none of its members is even as large as 1. One reasonably useful definition of a set being infinite is that it is, or at least can be, ordered so as not to have any last member. Do you wish to assert to the contrary that, with the usual ordering, the set of naturals HAS a last member? 2) The statement zero belongs to I, and if x belongs to I, x+1 also > belongs I simply seeding off the set with zero and attempts to build > it with a chain relationship. But it is cleverly evading the task of > identifying all elements of the set. There are a lot of sets of numbers having that *inductive* property. The uniqueness of the naturals comes in being a subset of any set with that inductive property, i.e., being the smallest inductive set. > This cleverness is useful in > situations like summing up some diminishing infinite series, but not > in this case. How can this definition be allowed to define a complete > set? If by complete you mean a set all of whose members have been explicitly listed, there are enough finite sets which are equally incomplete to make the question irrelevant. === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 14, 3:36æpm, Denis Feldmann venkat.6...@gmail.com a .8ecrit : > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. > For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. > But x and y never do reach infinite. > The issue is that in comparing N, the set of naturals and , say E, for > the set of evens, is there a correspondence which does not use twice nor > omit any member of either set. > That is certainly a valid comparison for finite sets. > Can you think of a better way of comparing sets for relative sizes that > works for finite sets? If not why object to it for infinite sets? > The 1-to-1 correspondence works well as long as n is different from n > +1, which is true for a natural number since it is defined to be > finite. May be the roots of my perceived problem is in the definition > of natural numbers as an infinite set while claiming that none of its > elements is infinite. > I read that the natural number n is defined as {0,1,2,...,n-1} where n > can't be called infinite and n is also the count of all numbers in the > finite set. Then we extended this to set containing all natural > numbers but we now find none of its members can represent its size? > First of all, there doesn't seem to be any such thing as all natural > numbers. The standard mathematical language saying For all x in the > set I, x+1 also belongs to I, and zero belongs to I does seem to do > justice in the usage of the word all. What do you mean by all (or > every)? By using this word one is assuming that all is already > understood and defined in this context, which, in fact, is what we are > trying to define. > At this stage, you are obviously trolling. I take the opportunity to > note you use the same devices that known trolls. Is there a troll school > somewhere ? æ[plonk] I don't mind being called troll, because I don't care about whatever > it means. I'm just trying to ask a few questions that bother me. It seems I was not precise enough in my questions. Let me make one > more attempt at making my questions much clearer. 1) If none of the natural numbers is infinite, how can the size of the > set of natural numbers be called infinite? Is there some finite n > where n+1 becomes infinite? The size of a set may bear little resemblance to the size of its members. > The set of rational fractions strictly between 0 and 1 is infinite even > though none of its members is even as large as 1. But in case of naturals, the resemblance is already established with the usual ordering. Isn't that how we defined n as {0, 1, 2, ..., n-1} implying that the size of this finite set is 1 more than the size of its largest member? This resemblance is dropped when it comes to the set of naturals, for some reason. I feel, with my limited intuition and knowledge, may be we should not define an infinite set or its size. The set and its size is unknown or indeterministic at best. You can produce paradoxical results by extending on these indeterministic concepts and still get away with them. Were there any issues solved in physical world by defining such set and its size? One reasonably useful definition of a set being infinite is that it is, > or at least can be, ordered so as not to have any last member. Do you wish to assert to the contrary that, with the usual ordering, the > set of naturals HAS a last member? No, I don't. I wish to refuse the concept that naturals can be wrapped into a set and establish its size as a number. 2) The statement zero belongs to I, and if x belongs to I, x+1 also > belongs I simply seeding off the set with zero and attempts to build > it with a chain relationship. But it is cleverly evading the task of > identifying all elements of the set. There are a lot of sets of numbers having that *inductive* property. > The uniqueness of the naturals comes in being a subset of any set with > that inductive property, i.e., being the smallest inductive set. This cleverness is useful in > situations like summing up some diminishing infinite series, but not > in this case. How can this definition be allowed to define a complete > set? If by complete you mean a set all of whose members have been > explicitly listed, there are enough finite sets which are equally > incomplete to make the question irrelevant. May be I can't define completeness enough. But an infinite set lacks the important property of a finite set where the members are distinct and size can represented using a natural number. An infinite set is not guaranteed to have distinct members and its size can't be expressed using a natural number. May be I should talk more about distinctness. We need to be able tell the members apart. To do this, first we need to be able to identify or express the existence of a member. Recursive expressions such as if an x is a member, x+1 is a member are based on the ability to express x. If the x can't be expressed the statement is not worth it. How can you prove that all the members in an infinite set can be expressed uniquely? === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. > For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. But x and y never do reach infinite. The issue is that in comparing N, the set of naturals and , say E, for > the set of evens, is there a correspondence which does not use twice nor > omit any member of either set. That is certainly a valid comparison for finite sets. Can you think of a better way of comparing sets for relative sizes that > works for finite sets? If not why object to it for infinite sets? The 1-to-1 correspondence works well as long as n is different from n > +1, which is true for a natural number since it is defined to be > finite. May be the roots of my perceived problem is in the definition > of natural numbers as an infinite set while claiming that none of its > elements is infinite. I read that the natural number n is defined as {0,1,2,...,n-1} where n > can't be called infinite and n is also the count of all numbers in the > finite set. Then we extended this to set containing all natural > numbers but we now find none of its members can represent its size? > First of all, there doesn't seem to be any such thing as all natural > numbers. The standard mathematical language saying For all x in the > set I, x+1 also belongs to I, and zero belongs to I does seem to do > justice in the usage of the word all. What do you mean by all (or > every)? By using this word one is assuming that all is already > understood and defined in this context, which, in fact, is what we are > trying to define. correction: ... doesn't seem to do justice in the usage of the word all ... > I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? Since one DEFINES relative size of sets by injection, surjection and > bijection, what is there to be proven? === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? Venkat To begin with a historical note: the idea of comparing infinite sets by 1-1 correspondences was around long before Cantor. I think it was known to Galileo, and I think he too found it puzzling and counter-intuitive that there are as many even numbers as there are numbers. Is it valid - ? Most mathematicians are persuaded that this question cannot arise because, for infinite sets, the statement S has at least as many elements as T is *defined* to mean there is a one-to-one map from T into S. This settles the question, unless -- (a) one already has another notion of what as many elements means; or (b) one doubts that there is a one-to-one map really makes sense. As to (a), I do not think that we really do have another notion of as many as. Indeed, in everyday life, when we count things we are are placing them in one-to-one correspondence with number words. Or, in even simpler cases, with fingers. Someone who had pearls and diamonds, but no number words, could decide whether he had more pearls than diamonds by lining them up, one pearl by one diamond. As to (b), questions can arise in general, or in particular. The general question: what is a so-called one-to-one map anyway? - is best answered by displaying a few examples. A particular question would be whether someone's alleged one-to-one map is well-defined and in fact one-to-one. Venkat has[1] raised this question for the usual map from integers to even integers. There are two ways to describe a particular map m. One is to list the pairs of objects (x, m(x)), where x is something in the domain of the map and m(x) is the value which the map associates to x. In the present case, the list might be { (0, 0), (1, 2), {2, 4), ..., (137, 274), ... } The other description of m consists in specifying the domain (that is, what x does m apply to), and giving a rule that determines m(x) for a given x. In the present case, the domain is the natural numbers, that is whole numbers, non-negative, and finite. The rule is, m(x) = 2x. At long last I am ready, I hope, to address Venkat's objection, that the function is not meaningful when x is not finite. Indeed it isn't, but then it doesn't need to be. Each natural number is finite. (Wasn't it subtle of me to sneak that word into the previous paragraph?) The set of natural numbers has a property which no member of it has, namely the property of being infinite. [1] I mention Venkat rather than addressing him directly because a newsgroup is a public place and any posting is being read by lots of people. This formality tends to keep the discussion cool, I hope. Note that in deliberative bodies such as Congress or Parliament, one addresses the chairman or Speaker. -- Christopher J. Henrich chenrich@monmouth.com htp://www.mathinteract.com === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. 1-to-1 correspondence is one way to compare two sets of infinite size. There may be other ways, too. > Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. What do you mean by valid? For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. When we compare the integers to the even integers, we don't have to worry about this, since none of them reach infinite. I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? Most texts on set theory discusss this, perhaps in a section on cardinality. Venkat === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? <130920081034221497%anniel@nym.alias.net.invalid> posting-account=8eIz8woAAADrOkmI_WKvJlIYV3yGdH_d Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. 1-to-1 correspondence is one way to compare two sets of infinite size. > There may be other ways, too. Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. What do you mean by valid? May be valid is not the right word. I thought that the very possibility (or imagination) of getting hold of an element in a set for mapping with some other element, makes its ordinality as finite and determinate. I think this problem in observability (mathematically) of infinite as a number makes it unsuitable for mapping. And if we don't extend this mapping to infinite, then we can't say anything about equi-numerosity of the infinite-sized sets. Am I making sense? For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. When we compare the integers to the even integers, we don't have > to worry about this, since none of them reach infinite. I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? Most texts on set theory discusss this, perhaps in a section on > cardinality. Venkat === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? What do you mean by valid? May be valid is not the right word. I thought that the very > possibility (or imagination) of getting hold of an element in a set > for mapping with some other element, makes its ordinality as finite > and determinate. I don't know what you mean by ordinality or determinate ... > I think this problem in observability > (mathematically) of infinite as a number makes it unsuitable for > mapping. And if we don't extend this mapping to infinite, then we > can't say anything about equi-numerosity of the infinite-sized sets. Am I making sense? no === Subject: Re: Is one-to-one mapping valid for comparing infinite-sized sets? > I'm an amateur in this topic. When my friend told me about Cantor's > cardinalities, the first thing that seemed odd to me was the 1-to-1 > mapping used for comparing the sizes of two infinite sets. I'm not > sure if it is proved that 1-to-1 correspondence is valid for comparing > two sets of infinite size. > 1-to-1 correspondence is one way to compare two sets of infinite size. > There may be other ways, too. > Not because of the practical difficulties > in comparing or the time it takes for the comparing process to > complete. I think, atleast it should be proved logically that such > mapping is valid between two infinite sets. > What do you mean by valid? > May be valid is not the right word. I thought that the very > possibility (or imagination) of getting hold of an element in a set > for mapping with some other element, makes its ordinality as finite > and determinate. How so? Suppose A = { N, R } and B = { Q, Z }. Each member of A and each member of B is infinite. Yet there is a bijection (a 1-1 correspondence) between A and B. Each has two elements. Nothing much changes if the sets A and B happen to be infinite. The members of A and B may be finite or infinite. It makes no difference. If there is a bijection between A and B, we say they have the same cardinality. For example, N and Q are infinite sets having the same cardinality, but each member of N and each member of Q is finite. >I think this problem in observability > (mathematically) of infinite as a number makes it unsuitable for > mapping. And if we don't extend this mapping to infinite, then we > can't say anything about equi-numerosity of the infinite-sized sets. Unsuitable in what way? > Am I making sense? You haven't explained what you mean. > For example, the mappings such as y=2x showing that there are as many > even numbers as integers seem dubious to me because there is no proof > that such function is meaningful when values of x and y reach > infinite. > When we compare the integers to the even integers, we don't have > to worry about this, since none of them reach infinite. > I'm sure these questions are not new, but I would like to see some > proofs. Any references or proofs in this regard? Proofs of what? That each natural number is finite? What do you mean by finite? The definition I know is that a set is called finite if it can be mapped bijectively to some natural number, viewed as a set. With that definition, your question reduces to a triviality. You must have some different meaning of finite in mind, and that's why I ask the question. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Have I understood the axiom of choice? posting-account=_F3ICAoAAAD89283jXG9Iyx11EtgOOuB Gecko/2008071616 CentOS/3.0.1-1.el5.centos Firefox/3.0.1,gzip(gfe),gzip(gfe) I need to verify my current understanding of (the need for) the axiom of choice. Let us work in ZF. Let S be a function on I, where I can be any infinite set. It is easy to prove the existence of the set F of all functions f on I such that f(i) in S(i) for all i in I. My understanding of the axiom of choice is that it is equivalent to the statement that F is non-empty. Informally, although we know that the set of all sequences on an indexed family of sets exists, we cannot, in general, prove that this set is not empty without the axiom of choice. I also remember reading somewhere (perhaps in a newsgroup like this one) that its possible to trace the whole issue to the distributivity of 'and' over 'or', or something to that effect. I can't seem to find that reference anywhere. Does this ring a bell with anyone? /ALiX === Subject: Re: Have I understood the axiom of choice? posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > I also remember reading somewhere (perhaps in a newsgroup > like this one) that its possible to trace the whole issue > to the distributivity of 'and' over 'or', or something to > that effect. I can't seem to find that reference anywhere. > Does this ring a bell with anyone? I mentioned this in two posts a little over two years ago: sci.logic & sci.math (21 July 2006) sci.logic & sci.math (22 July 2006) This post might also be of interest: sci.math (30 April 2007) Dave L. Renfro === Subject: --- --- --- Consistency of an equation Cc: deepkdeb@yahoo.com posting-account=iJfBPwgAAACaBAH7DreA6VfAOYvL5VDz 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; IEMB3; IEMB3),gzip(gfe),gzip(gfe) Consider the following equation under the given conditions. U = k^T[(a^2 - b^2)^t)/(2ab)^t] (1) Conditions: T = (mk-1)/k , t = 1/k, (a, b) = 1, 2k|b integer m, a, b, t > 0, k is a prime > 3, U is rational > 0. Assertion: (1) cannot be consistent. Any comments about the correctness of the assertion will be appreciated. === Subject: Solutions manua to Numerical methods for engineers 5th by Chapra posting-account=y7Z6OAoAAAD9FX_IL8yyi-5ioDwYBttu CLR 2.0.50727),gzip(gfe),gzip(gfe) solutions manual (To search click in keyboard Ctrl+F) Solutions Manuals in Electronic (PDF)Format! 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There was some mild outrage at the question, which was considered by some to be outside the subject-matter envisaged in the competition. [I think the questions are supposed to involve only first-year university math, whatever that means.] I wonder what people think? Is this problem very difficult? -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: A problem on finite groups posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > The following problem was asked in the IMC > (International Competition in Mathematics) > in Bulgaria this summer. > (See I didn't find this years problems on-line.) Find a finite group G and subgroup H such that |Aut(H)| > |Aut(G)|. I wonder what people think? > Is this problem very difficult? Not for anyone with good experience of group theory. Probably for schoolkids though. Elementary Abelian groups have lots of automorphisms, so why not embed one in a group with not so many? For instance the elementary Abelian group of 32 elements embeds very obviously in S_10. The 32-element group has 31 x 30 x 28 x 24 x 16 = 9999360 automorphisms. S_10 has 10! = 3628800 automorphisms. This solution involves knowing the automorphism group of elementary Abelian groups (essentially linear algebra) and knowing that for n > 6, S_n is its own automorphism group (probably not first-year maths in many places!). Victor Meldrew I don't believe it! === Subject: Re: A problem on finite groups > I wonder what people think? > Is this problem very difficult? Not for anyone with good experience of group theory. Sorry, I missed out a vital word: > Find a finite group G and NORMAL subgroup H such that |Aut(H)| > |Aut(G)|. > Probably for schoolkids though. The competition is for university students. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: A problem on finite groups posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > I wonder what people think? > Is this problem very difficult? Not for anyone with good experience of group theory. Sorry, I missed out a vital word: Find a finite group G and NORMAL subgroup H such that |Aut(H)| > |Aut(G)|. > Probably for schoolkids though. The competition is for university students. The first idea I had was for a normal subgroup, namely G = AGL(1,q), the group of maps x |--> ax + b on GF(q) with a nonzero. There's a bit of a hassle with this, as calculating Aut(G) is a bit fiddly, but one can get an easy upper bound on its order. Let's take q = 32. The subgroup H is the group of all translations x |--> x + b and as in my previous posting Aut(H) has 9999360 elements. Now G is generated by the element x |--> x + 1 or order 2 and an element x |--> ax of order 31. So Aut(G) is bounded by the product of the number of order 2 elements and the number of order 31 elements, which is 31 x 32 x 30 --- much less than a million. (Actually it's even smaller than this). I on't think this is a good competition problem. It's bnot clear how much detail in calculating automorphism groups is required. There are lots of automorphism groups that are well-known, e.g. that of A_n, but proving them is rather intricate. Victor Meldrew I don't believe it! === Subject: Re: A problem on finite groups > Sorry, I missed out a vital word: > Find a finite group G and NORMAL subgroup H such that |Aut(H)| > |Aut(G)|. Oddly this made it easier for me to solve. The idea is to choose H highly symmetric, but have G break the symmetry. So let H be elementary abelian, but let it not be a G-isotypic module. For instance, the semidirect product of C_2 by the normal subgroup C_3 x C_3, where the element of order 2 acts as the diagonal matrix diag(1,-1) works. The automorphism group of the semidirect product is somewhat small as it has to respect the module structure. In particular, it only has 12 elements (and when they are restricted to just C_3 x C_3, there are only 4). The automorphism group of C_3 x C_3 has no such restriction, and so has 48 elements. I don't understand contests, so I don't really have any idea whether the question is reasonable. It seems like a reasonable thing to include as an example in lectures. If the students were actually familiar with group theory, then the example of A5 x A5 in S5 x A5 is probably better. Aut(A5^n) = Aut(A5) wr Sym(n), but including the S5 breaks the symmetry, so its automorphism group is only half as big. === Subject: Re: A problem on finite groups Reply-to: weu_rznvy-hfrarg@lnubb.pbz.invalid On 13-Sep-2008, Jack Schmidt <12486464.1221355131722.JavaMail.jakarta@nitrogen.mathforum.org>: > Sorry, I missed out a vital word: > Find a finite group G and NORMAL subgroup H such that |Aut(H)| > |Aut(G)|. Oddly this made it easier for me to solve. The > idea is to choose H highly symmetric, but have G > break the symmetry. So let H be elementary abelian, > but let it not be a G-isotypic module. For instance, the semidirect product of C_2 by the > normal subgroup C_3 x C_3, where the element of order > 2 acts as the diagonal matrix diag(1,-1) works. The automorphism group of the semidirect product is > somewhat small as it has to respect the module structure. > In particular, it only has 12 elements (and when they > are restricted to just C_3 x C_3, there are only 4). Right. This is arguably the 'easiest' group to see that satisfies the criterion, although it's not the smallest. In fact, G is the direct product of two characteristic subgroups, G = S_3 x C_3, so Aut(G) = Aut(S_3) x Aut(C_3) = S_3 x C_2. > The automorphism group of C_3 x C_3 has no such > restriction, and so has 48 elements. Exactly. Aut(C_3 x C_3) = GL(2,3) of course, of order (3^2 - 3^0) * (3^2 - 3^1). It turns out that the smallest groups satisfying the criterion are three groups of order 16, two of which have elementary abelian subgroups H of order 8, so Aut(H) = GL(3,2), of order (2^3 - 2^0) * (2^3 - 2^1) * (2^3 - 2^2) = 168. The two groups are the direct product of the dihedral group of order 8 and the cyclic group of order 2, G = D_8 x C_2, with |Aut(G)| = 64, and the unique non-abelian semidirect product G = (C_2 x C2) x| C_4, with |Aut(G)| = 32. The third relevant group of order 16 is the semidihedral group, with presentation G = , |Aut(G)| = 16, and H = , the quaternion group or order 8, so |Aut(H)| = 24. [...] -- Jim Heckman === Subject: Re: A problem on finite groups > Find a finite group G and NORMAL subgroup H such that |Aut(H)| > |Aut(G)|. > For instance, the semidirect product of C_2 by the > normal subgroup C_3 x C_3, where the element of order > 2 acts as the diagonal matrix diag(1,-1) works. All the solutions offered (by the examiners) were along the above lines, using semi-direct products. As I said, what surprised me was that none of the 300 or so competitors - most of whom would probably have been final year university students - managed to solve this, although they solved what seem to me much more difficult problems, eg Let H be an infinite-dimensional real Hilbert space, let d > 0, and suppose that S is a set of points in H such that the distance between any two points in S is equal to d. Show that there is a point y in H such that {sqrt{2}(x-y)/d: x in S} is an orthonormal system of vectors in H. This was also objected to on the grounds that its subject matter lay outside the specified area of first-year university mathematics; but it was solved by several of the competitors, including the 3 joint winners, from Poland, Hungary and Moscow, IIRC. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: A problem on finite groups Reply-to: weu_rznvy-hfrarg@lnubb.pbz.invalid On 13-Sep-2008, Timothy Murphy I wonder what people think? > Is this problem very difficult? Not for anyone with good experience of group theory. Sorry, I missed out a vital word: Find a finite group G and NORMAL subgroup H such that |Aut(H)| > |Aut(G)|. I'm pressed for time at the moment and haven't verified this, but I seem to recall that if S is (finite?) non-abelian simple and S <= G <= Aut(S) = A, then Aut(G) = N_A(G). If this is correct, then one need only find an S with Out(S) having a non-normal subgroup, of which there are of course lots and lots among the groups of Lie type; browsing the _ATLAS of Finite Groups_, it looks like the smallest might be S = PSU(3,5) with Out(S) = S_3, so choose G = S.2, the upward extension of S by its outer automorphism of order 2. > Probably for schoolkids though. The competition is for university students. -- Jim Heckman === Subject: Re: General Topology by John L. Kelley posting-account=h0BplggAAACbakJwttbpVF72VZ8jVCAq Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Let us try to make this real simple, stupid. The book which has now been reprinted was published in 1955. The author is dead. No copyright by the author has ever been registered on the original book. http://www.amazon.com/dp/0923891552 What part of this do you not understand? Sam Sloan === Subject: Re: General Topology by John L. Kelley posting-account=F3xv6QoAAAAg6DXPK6QdNB3Zr2WSoRAd rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Let us try to make this real simple, stupid. Sloan, you understand nothing of US copyright law. The fact that the 1955 Van Nostrand edition is out of print does not mean that the text is now in the public domain. The fact that Professor Kelley is dead does not mean that the text is now in the public domain. You need to get out of publishing because you don't understand how this stuff works at all. And you have the temerity to call me stupid? I am dying laughing. J. === Subject: Re: General Topology by John L. Kelley posting-account=h0BplggAAACbakJwttbpVF72VZ8jVCAq Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Copyrights by publishers are no longer valid (except in the case of works for hire). The copyrights belong to the authors, not to the publishers. Sam Sloan === Subject: Re: General Topology by John L. Kelley posting-account=F3xv6QoAAAAg6DXPK6QdNB3Zr2WSoRAd rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > Copyrights by publishers are no longer valid (except in the case of > works for hire). The copyrights belong to the authors, not to the publishers. Sam Sloan That's not true at all. Authors can assign copyright to publishers in their contract. Where are you getting the idea that publishers' copyrights were somehow magically invalidated? Do either you or Bozulich have a lawyer? Because you'll need one. J. === Subject: Re: General Topology by John L. Kelley Angus Rodgers a .8ecrit : > Copyrights by publishers are no longer valid (except in the case of > works for hire). > The copyrights belong to the authors, not to the publishers. Oops, guess I was the only one here who didn't know that! Sorry. > But death of the author doesn't stop the copyright (if he has heirs, and even then) for 70 years , and no registration is necessary. I wonder how much he will have to pay if someone wakes up... (not to mention the usual total immorality of the guy) === Subject: Re: General Topology by John L. Kelley .... Interestingly, the Ishi Press website > is down. Coincidence? Or lawyers? Always think incompetence rather than conspiracy. -- He is not here; but far away The noise of life begins again And ghastly thro' the drizzling rain On the bald street breaks the blank day. === Subject: Re: General Topology by John L. Kelley posting-account=F3xv6QoAAAAg6DXPK6QdNB3Zr2WSoRAd rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Sorry, but it is you who do not know how this copyright thing works. HA HA HA Oh my sides. The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at Berkeley in > 1962-1964, when Kelley was a professor there. Yes. That is the edition published by the Van Nostrand Press of Princeton. IS WHAT IT SAYS ON THAT BOOK'S COPYRIGHT NOTICE PAGE. And in any case, a book published in 1955 is not out of copyright today in the United States. http://www.amazon.com/dp/0923891552 Springer was not the publisher of that book. No. Van Nostrand was. Then Springer acquired it for the Graduate Texts in Mathematics series. The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) had > nothing to do with the 1955 book that I am reprinting. They were the editors of the Graduate Texts in Mathematics series, and were responsible for acquiring and editing the previously- published (by Van Nostrand) book. It is the same book. AND A BOOK PUBLISHED IN 1955 IN THE US WOULD NOT NECESSARILY BE OUT OF COPYRIGHT TODAY, EVEN IF IT HAD NOT BEEN REISSUED IN 1975. What part of US copyright law is so difficult for you and the other people at Ishi Press to understand? Before 1978, copyright was awarded on the date of publication or registration (so 1955 in this case) for a term of 28 years. So the book would have been protected by its first grant of copyright until 1983, at which time it was renewable for a second term--which This is not rocket science. You guys need to do basic research to make sure a book is not still in print, and not still protected by copyright. J. === Subject: Re: General Topology by John L. Kelley > It has long been out of print, Bollocks. http://www.springer.com/math/geometry/book/978-0-387-90125-1 for a mere forty four quid. -- He is not here; but far away The noise of life begins again And ghastly thro' the drizzling rain On the bald street breaks the blank day. === Subject: Re: General Topology by John L. Kelley posting-account=llxexwoAAADqntGDNkT4ytZG1pkEWVd9 SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; Zune 2.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) >GeneralTopologyis not only a textbook, it is also an invaluable > reference work for all mathematicians working the field of analysis. > It has long been out of print, > In what sense is it out of print? It is often part of Springer's > yearly Yellow Sale. I bought a copy last year from amazon.com and > have just checked that plenty of copies are still available there. > Are you saying you have acquired the printing rights from Springer? > No, he's not saying that at all. Mr. Sloan seems not to understand > how the whole copyright thing works. He appears to have announced a > reprint of a book that is still in print from the world's largest > publisher of mathematics texts. Interestingly, the Ishi Press website > is down. Coincidence? Or lawyers? > J. > Sorry, but it is you who do not know how this copyright thing works. > The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at Berkeley in > 1962-1964, when Kelley was a professor there. > æhttp://www.amazon.com/dp/0923891552 > Springer was not the publisher of that book. > The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) > S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) had > nothing to do with the 1955 book that I am reprinting. > Sam Sloan Sam, it was a later edition of the same book. The copyright notice on > the Springer edition (which you can view on line if you have three > functioning brain cells) says Copyright 1955 by J. L. Kelley. æNot > that I care particularly if you get sued into the ground, but you > would be well advised to talk to a lawyer before making a fool of > yourself again. Once you get into copyright law, you will find that many copyright > notices are false notices. Here is an example: Dover has reprinted a book called Sloan's Victorian Buildings by Sam > Sloan. Inside there is a copyright notice saying Copyright © 1980 by Dover > Publishing. However, this book was first published in 1851 by my distant relative > who was also named Sam Sloan. Obviously, the book is in public domain. Nevertheless, Dover has put a > copyright notice on it. The Real Sam Sloan- Hide quoted text - - Show quoted text - That book may be in the public domain, but I have doubts as to whether General Topology by Kelley is similarly situated. published before 1923 have expired copyrights and are in the public domain. In addition, works published before 1964 that did not have their copyrights renewed 28 years after first publication year also are in the public domain.... Maybe this gives you a loophole, provided that nobody else renewed the copyright. But since Springer is publishing an edition, it is likely that the copyright was renewed. I am not a legal expert, but it would seem to me that investing a few hundred in a legal opinion might prevent many thousands of dollars from fleeing your bank account. === Subject: Sam Sloan Re: General Topology by John L. Kelley Mail-To-News-Contact: abuse@dizum.com The poster formerly known as Colleyville Alan The Real Sam Sloan . . . > I am not a legal expert, but it would seem to me that investing a few > hundred in a legal opinion might prevent many thousands of dollars > from fleeing your bank account. There never was any possibility of many thousands of dollars fleeing the bank account of Sam Sloan. Sam's never had thousands of dollars. Sam has been judgement-proof for all his life. But for this fact, this P.O.S. would not merely have lost all 30 of his kooksuits (as he has), but would have been successfully sued by scores of people that he has libeled, lied about, scammed, stolen from or molested. Figure out why he has ed a series of imported retards to produce DAUGHTERS? You need to broaden your mind and research more. === Subject: Re: Sam Sloan Re: General Topology by John L. Kelley posting-account=h0BplggAAACbakJwttbpVF72VZ8jVCAq Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > The poster formerly known as Colleyville Alan The Real Sam Sloan > æ. . . > I am not a legal expert, but it would seem to me that investing a few > hundred in a legal opinion might prevent many thousands of dollars > from fleeing your bank account. There never was any possibility of many thousands of dollars fleeing > the bank account of Sam Sloan. Sam's never had thousands of dollars. Sam has been judgement-proof for all his life. But for this fact, this > P.O.S. would not merely have lost all 30 of his kooksuits (as he has), > but would have been successfully sued by scores of people that he has > libeled, lied about, scammed, stolen from or molested. Figure out why he has ed a series of imported retards to produce > DAUGHTERS? You need to broaden your mind and research more. === Subject: Re: Sam Sloan Re: General Topology by John L. Kelley Mail-To-News-Contact: abuse@dizum.com > Figure out why he has ed a series of imported retards to produce > DAUGHTERS? You need to broaden your mind and research more. > Why should you thank Mr. Truong for comments which he did not make? Are you confusing him with Creighton again? You were warned that if you did not get your mental illnesses treated they would progress, Sam. If you delay any longer it can become untreatable. So you believe falsities (eg, Mr. Truong was the non-satirical FSS) and disbelieve truths (eg, Kayo did not get your children aborted). Why do the bats keep following you, Sam? === Subject: Re: General Topology by John L. Kelley posting-account=fk5d_woAAAAWJdzhm0GQEtVEiuLmnAo8 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) >GeneralTopologyis not only a textbook, it is also an invaluable > reference work for all mathematicians working the field of analysis. > It has long been out of print, > In what sense is it out of print? It is often part of Springer's > yearly Yellow Sale. I bought a copy last year from amazon.com and > have just checked that plenty of copies are still available there. > Are you saying you have acquired the printing rights from Springer? > No, he's not saying that at all. Mr. Sloan seems not to understand > how the whole copyright thing works. He appears to have announced a > reprint of a book that is still in print from the world's largest > publisher of mathematics texts. Interestingly, the Ishi Press website > is down. Coincidence? Or lawyers? > J. > Sorry, but it is you who do not know how this copyright thing works. > The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at Berkeley in > 1962-1964, when Kelley was a professor there. > ?http://www.amazon.com/dp/0923891552 > Springer was not the publisher of that book. > The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) > S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) had > nothing to do with the 1955 book that I am reprinting. > Sam Sloan > Sam, it was a later edition of the same book. The copyright notice on > the Springer edition (which you can view on line if you have three > functioning brain cells) says Copyright 1955 by J. L. Kelley. ?Not > that I care particularly if you get sued into the ground, but you > would be well advised to talk to a lawyer before making a fool of > yourself again. Once you get into copyright law, you will find that many copyright > notices are false notices. Here is an example: Dover has reprinted a book called Sloan's Victorian Buildings by Sam > Sloan. Inside there is a copyright notice saying Copyright ? 1980 by Dover > Publishing. However, this book was first published in 1851 by my distant relative > who was also named Sam Sloan. Obviously, the book is in public domain. Nevertheless, Dover has put a > copyright notice on it. The Real Sam Sloan- Hide quoted text - - Show quoted text - That book may be in the public domain, but I have doubts as to whether > General Topology by Kelley is similarly situated. published before 1923 have expired copyrights and are in the public > domain. In addition, works published before 1964 that did not have > their copyrights renewed 28 years after first publication year also > are in the public domain.... Maybe this gives you a loophole, > provided that nobody else renewed the copyright. But since Springer > is publishing an edition, it is likely that the copyright was renewed. I am not a legal expert, but it would seem to me that investing a few > hundred in a legal opinion might prevent many thousands of dollars > from fleeing your bank account. That Wikipedia paragraph is poorly phrased, though literally true. The law was changed to eliminate renewal requirements in 1978. Anything that crossed the line in either first- or second-term copyright got a one-shot extension such it expired 75 years after the initial copyright. (There's a bit more to it, but it's not relevant here.) The point is that, since Kelley's book seems to have been first published in 1955, it could not have fallen out of copyright before the 1978 changeover. There are other ways that the book could have fallen into public domain, but they require the consent of the copyright owner. === Subject: Re: General Topology by John L. Kelley posting-account=llxexwoAAADqntGDNkT4ytZG1pkEWVd9 SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; Zune 2.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) >GeneralTopologyis not only a textbook, it is also an invaluable > reference work for all mathematicians working the field of analysis. > It has long been out of print, > In what sense is it out of print? It is often part of Springer's > yearly Yellow Sale. I bought a copy last year from amazon.com and > have just checked that plenty of copies are still available there. > Are you saying you have acquired the printing rights from Springer? > No, he's not saying that at all. Mr. Sloan seems not to understand > how the whole copyright thing works. He appears to have announced a > reprint of a book that is still in print from the world's largest > publisher of mathematics texts. Interestingly, the Ishi Press website > is down. Coincidence? Or lawyers? > J. > Sorry, but it is you who do not know how this copyright thing works. > The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at Berkeley in > 1962-1964, when Kelley was a professor there. >http://www.amazon.com/dp/0923891552 > Springer was not the publisher of that book. > The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) > S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) had > nothing to do with the 1955 book that I am reprinting. > Sam Sloan > Sam, it was a later edition of the same book. The copyright notice on > the Springer edition (which you can view on line if you have three > functioning brain cells) says Copyright 1955 by J. L. Kelley. Not > that I care particularly if you get sued into the ground, but you > would be well advised to talk to a lawyer before making a fool of > yourself again. > Once you get into copyright law, you will find that many copyright > notices are false notices. > Here is an example: > Dover has reprinted a book called Sloan's Victorian Buildings by Sam > Sloan. > Inside there is a copyright notice saying Copyright 1980 by Dover > Publishing. > However, this book was first published in 1851 by my distant relative > who was also named Sam Sloan. > Obviously, the book is in public domain. Nevertheless, Dover has put a > copyright notice on it. > The Real Sam Sloan- Hide quoted text - > - Show quoted text - That book may be in the public domain, but I have doubts as to whether > General Topology by Kelley is similarly situated. published before 1923 have expired copyrights and are in the public > domain. In addition, works published before 1964 that did not have > their copyrights renewed 28 years after first publication year also > are in the public domain.... æMaybe this gives you a loophole, > provided that nobody else renewed the copyright. æBut since Springer > is publishing an edition, it is likely that the copyright was renewed. I am not a legal expert, but it would seem to me that investing a few > hundred in a legal opinion might prevent many thousands of dollars > from fleeing your bank account. That Wikipedia paragraph is poorly phrased, though literally true. The > law was changed to eliminate renewal requirements in 1978. Anything > that crossed the line in either first- or second-term copyright got a > one-shot extension such it expired 75 years after the initial > copyright. (There's a bit more to it, but it's not relevant here.) The > point is that, since Kelley's book seems to have been first published > in 1955, it could not have fallen out of copyright before the 1978 > changeover. There are other ways that the book could have fallen into > public domain, but they require the consent of the copyright owner.- Hide quoted text - - Show quoted text - was for 75 years but that many, many years earlier (like in the early 1900's) it had been for a shorter timeframe. Anyhow, it does confirm my gut suspicion that with Springer publishing it, there was very little likelihood of it being in public domain. Now I can change that to zero likelihood. I used to read a little bit in one of the chess Usenet groups and I knew that many there did not like Sam Sloan. So, too, it appears that a few in Sci.Math do not like him very much either. I did not hang around the chess groups long enough to understand the history of who-said-what-to-whom. In any event, I do not have any animosity towards him. I hope he comes to his senses before he ends up on the wrong side of a lawsuit with Springer. Probably by this point, someone has sent an email to Springer notifying them of Sloan's intentions and they are politely informing him of their intention to break his legs (figuratively and in a court of law). === Subject: Re: General Topology by John L. Kelley posting-account=h0BplggAAACbakJwttbpVF72VZ8jVCAq Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 12, 9:15æpm, The poster formerly known as Colleyville Alan >GeneralTopologyis not only a textbook, it is also an invaluable > reference work for all mathematicians working the field of analysis. > It has long been out of print, > In what sense is it out of print? It is often part of Springer's > yearly Yellow Sale. I bought a copy last year from amazon.com and > have just checked that plenty of copies are still available there. > Are you saying you have acquired the printing rights from Springer? > No, he's not saying that at all. Mr. Sloan seems not to understand > how the whole copyright thing works. He appears to have announced a > reprint of a book that is still in print from the world's largest > publisher of mathematics texts. Interestingly, the Ishi Press website > is down. Coincidence? Or lawyers? > J. > Sorry, but it is you who do not know how this copyright thing works. > The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at Berkeley in > 1962-1964, when Kelley was a professor there. >http://www.amazon.com/dp/0923891552 > Springer was not the publisher of that book. > The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) > S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) had > nothing to do with the 1955 book that I am reprinting. > Sam Sloan > Sam, it was a later edition of the same book. The copyright notice on > the Springer edition (which you can view on line if you have three > functioning brain cells) says Copyright 1955 by J. L. Kelley. Not > that I care particularly if you get sued into the ground, but you > would be well advised to talk to a lawyer before making a fool of > yourself again. > Once you get into copyright law, you will find that many copyright > notices are false notices. > Here is an example: > Dover has reprinted a book called Sloan's Victorian Buildings by Sam > Sloan. > Inside there is a copyright notice saying Copyright 1980 by Dover > Publishing. > However, this book was first published in 1851 by my distant relative > who was also named Sam Sloan. > Obviously, the book is in public domain. Nevertheless, Dover has put a > copyright notice on it. > The Real Sam Sloan- Hide quoted text - > - Show quoted text - > That book may be in the public domain, but I have doubts as to whether > General Topology by Kelley is similarly situated. > published before 1923 have expired copyrights and are in the public > domain. In addition, works published before 1964 that did not have > their copyrights renewed 28 years after first publication year also > are in the public domain.... æMaybe this gives you a loophole, > provided that nobody else renewed the copyright. æBut since Springer > is publishing an edition, it is likely that the copyright was renewed. > I am not a legal expert, but it would seem to me that investing a few > hundred in a legal opinion might prevent many thousands of dollars > from fleeing your bank account. That Wikipedia paragraph is poorly phrased, though literally true. The > law was changed to eliminate renewal requirements in 1978. Anything > that crossed the line in either first- or second-term copyright got a > one-shot extension such it expired 75 years after the initial > copyright. (There's a bit more to it, but it's not relevant here.) The > point is that, since Kelley's book seems to have been first published > in 1955, it could not have fallen out of copyright before the 1978 > changeover. There are other ways that the book could have fallen into > public domain, but they require the consent of the copyright owner.- Hide quoted text - - Show quoted text - was for 75 years but that many, many years earlier (like in the early > 1900's) it had been for a shorter timeframe. æAnyhow, it does confirm > my gut suspicion that with Springer publishing it, there was very > little likelihood of it being in public domain. æNow I can change that > to zero likelihood. I used to read a little bit in one of the chess Usenet groups and I > knew that many there did not like Sam Sloan. æSo, too, it appears that > a few in Sci.Math do not like him very much either. I did not hang around the chess groups long enough to understand the > history of who-said-what-to-whom. æIn any event, I do not have any > animosity towards him. æI hope he comes to his senses before he ends > up on the wrong side of a lawsuit with Springer. æProbably by this > point, someone has sent an email to Springer notifying them of Sloan's > intentions and they are politely informing him of their intention to > break his legs (figuratively and in a court of law). You might consider the possibility, unlikely though it may seem, that I have already checked with the copyright records office and have found that there is no copyright on this 1955 book. Sam Sloan === Subject: Re: General Topology by John L. Kelley > You might consider the possibility, unlikely though it may seem, that > I have already checked with the copyright records office and have > found that there is no copyright on this 1955 book. Careful. Under the 1909 Copyright Act, copyright came into existence if > the work was published with copyright notice. The author was required > to promptly register the work and deposit a copy with the Library of > Congress. Subsequent court cases have construed promptly rather broadly (there's > one case I know about where the registration was 27 years after > publication and that was found to be prompt!). The net outcome of the court cases is that failing to register or > deposit generally does NOT forfeit your copyright right, unless the > Copyright Office has specifically requested that you register and > deposit. What this means for your situation is that you should not conclude from > lack of registration that there is no copyright. At best, you can > conclude that the copyright status of the book might have been unclear > in 1955 and the following years. Then the law changed, adding more > complications. How did that affect things that were unclear? I have no > idea. Proceeding in your endeavor without consulting a copyright attorney > would not be wide. Just to add (and IANAL): in the European Community the copyright is almost automatic (i.e. without explicit and detailed permission you are not allowed to make use) and always bound to the author. === Subject: Re: General Topology by John L. Kelley posting-account=fk5d_woAAAAWJdzhm0GQEtVEiuLmnAo8 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > On Sep 12, 9:15?pm, The poster formerly known as Colleyville Alan >GeneralTopologyis not only a textbook, it is also an invaluable > reference work for all mathematicians working the field of analysis. > It has long been out of print, > In what sense is it out of print? It is often part of Springer's > yearly Yellow Sale. I bought a copy last year from amazon.com and > have just checked that plenty of copies are still available there. > Are you saying you have acquired the printing rights from Springer? > No, he's not saying that at all. Mr. Sloan seems not to understand > how the whole copyright thing works. He appears to have announced a > reprint of a book that is still in print from the world's largest > publisher of mathematics texts. Interestingly, the Ishi Press website > is down. Coincidence? Or lawyers? > J. > Sorry, but it is you who do not know how this copyright thing works. > The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at Berkeley in > 1962-1964, when Kelley was a professor there. >http://www.amazon.com/dp/0923891552 > Springer was not the publisher of that book. > The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) > S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) had > nothing to do with the 1955 book that I am reprinting. > Sam Sloan > Sam, it was a later edition of the same book. The copyright notice on > the Springer edition (which you can view on line if you have three > functioning brain cells) says Copyright 1955 by J. L. Kelley. Not > that I care particularly if you get sued into the ground, but you > would be well advised to talk to a lawyer before making a fool of > yourself again. > Once you get into copyright law, you will find that many copyright > notices are false notices. > Here is an example: > Dover has reprinted a book called Sloan's Victorian Buildings by Sam > Sloan. > Inside there is a copyright notice saying Copyright 1980 by Dover > Publishing. > However, this book was first published in 1851 by my distant relative > who was also named Sam Sloan. > Obviously, the book is in public domain. Nevertheless, Dover has put a > copyright notice on it. > The Real Sam Sloan- Hide quoted text - > - Show quoted text - > That book may be in the public domain, but I have doubts as to whether > General Topology by Kelley is similarly situated. > published before 1923 have expired copyrights and are in the public > domain. In addition, works published before 1964 that did not have > their copyrights renewed 28 years after first publication year also > are in the public domain.... ?Maybe this gives you a loophole, > provided that nobody else renewed the copyright. ?But since Springer > is publishing an edition, it is likely that the copyright was renewed. > I am not a legal expert, but it would seem to me that investing a few > hundred in a legal opinion might prevent many thousands of dollars > from fleeing your bank account. > That Wikipedia paragraph is poorly phrased, though literally true. The > law was changed to eliminate renewal requirements in 1978. Anything > that crossed the line in either first- or second-term copyright got a > one-shot extension such it expired 75 years after the initial > copyright. (There's a bit more to it, but it's not relevant here.) The > point is that, since Kelley's book seems to have been first published > in 1955, it could not have fallen out of copyright before the 1978 > changeover. There are other ways that the book could have fallen into > public domain, but they require the consent of the copyright owner.- Hide quoted text - > - Show quoted text - was for 75 years but that many, many years earlier (like in the early > 1900's) it had been for a shorter timeframe. ?Anyhow, it does confirm > my gut suspicion that with Springer publishing it, there was very > little likelihood of it being in public domain. ?Now I can change that > to zero likelihood. I used to read a little bit in one of the chess Usenet groups and I > knew that many there did not like Sam Sloan. ?So, too, it appears that > a few in Sci.Math do not like him very much either. I did not hang around the chess groups long enough to understand the > history of who-said-what-to-whom. ?In any event, I do not have any > animosity towards him. ?I hope he comes to his senses before he ends > up on the wrong side of a lawsuit with Springer. ?Probably by this > point, someone has sent an email to Springer notifying them of Sloan's > intentions and they are politely informing him of their intention to > break his legs (figuratively and in a court of law). You might consider the possibility, unlikely though it may seem, that > I have already checked with the copyright records office and have > found that there is no copyright on this 1955 book. Sam Sloan And how exactly did you check, Sam? If you tried to do it online, you found nothing because the records prior to 1978 are not available on line. Did you go to DC and look it up yourself? Did you hire someone who actually knew where to look? Are you a cretin? (Oh, well, we know the answer to that one.) === Subject: Re: General Topology by John L. Kelley posting-account=llxexwoAAADqntGDNkT4ytZG1pkEWVd9 SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; Zune 2.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > On Sep 12, 9:15æpm, The poster formerly known as Colleyville Alan >GeneralTopologyis not only a textbook, it is also an invaluable > reference work for all mathematicians working the field of analysis. > It has long been out of print, > In what sense is it out of print? It is often part of Springer's > yearly Yellow Sale. I bought a copy last year from amazon.com and > have just checked that plenty of copies are still available there. > Are you saying you have acquired the printing rights from Springer? > No, he's not saying that at all. Mr. Sloan seems not to understand > how the whole copyright thing works. He appears to have announced a > reprint of a book that is still in print from the world's largest > publisher of mathematics texts. Interestingly, the Ishi Press website > is down. Coincidence? Or lawyers? > J. > Sorry, but it is you who do not know how this copyright thing works. > The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at Berkeley in > 1962-1964, when Kelley was a professor there. >http://www.amazon.com/dp/0923891552 > Springer was not the publisher of that book. > The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) > S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) had > nothing to do with the 1955 book that I am reprinting. > Sam Sloan > Sam, it was a later edition of the same book. The copyright notice on > the Springer edition (which you can view on line if you have three > functioning brain cells) says Copyright 1955 by J. L. Kelley. Not > that I care particularly if you get sued into the ground, but you > would be well advised to talk to a lawyer before making a fool of > yourself again. > Once you get into copyright law, you will find that many copyright > notices are false notices. > Here is an example: > Dover has reprinted a book called Sloan's Victorian Buildings by Sam > Sloan. > Inside there is a copyright notice saying Copyright 1980 by Dover > Publishing. > However, this book was first published in 1851 by my distant relative > who was also named Sam Sloan. > Obviously, the book is in public domain. Nevertheless, Dover has put a > copyright notice on it. > The Real Sam Sloan- Hide quoted text - > - Show quoted text - > That book may be in the public domain, but I have doubts as to whether > General Topology by Kelley is similarly situated. > published before 1923 have expired copyrights and are in the public > domain. In addition, works published before 1964 that did not have > their copyrights renewed 28 years after first publication year also > are in the public domain.... æMaybe this gives you a loophole, > provided that nobody else renewed the copyright. æBut since Springer > is publishing an edition, it is likely that the copyright was renewed. > I am not a legal expert, but it would seem to me that investing a few > hundred in a legal opinion might prevent many thousands of dollars > from fleeing your bank account. > That Wikipedia paragraph is poorly phrased, though literally true. The > law was changed to eliminate renewal requirements in 1978. Anything > that crossed the line in either first- or second-term copyright got a > one-shot extension such it expired 75 years after the initial > copyright. (There's a bit more to it, but it's not relevant here.) The > point is that, since Kelley's book seems to have been first published > in 1955, it could not have fallen out of copyright before the 1978 > changeover. There are other ways that the book could have fallen into > public domain, but they require the consent of the copyright owner.- Hide quoted text - > - Show quoted text - was for 75 years but that many, many years earlier (like in the early > 1900's) it had been for a shorter timeframe. æAnyhow, it does confirm > my gut suspicion that with Springer publishing it, there was very > little likelihood of it being in public domain. æNow I can change that > to zero likelihood. I used to read a little bit in one of the chess Usenet groups and I > knew that many there did not like Sam Sloan. æSo, too, it appears that > a few in Sci.Math do not like him very much either. I did not hang around the chess groups long enough to understand the > history of who-said-what-to-whom. æIn any event, I do not have any > animosity towards him. æI hope he comes to his senses before he ends > up on the wrong side of a lawsuit with Springer. æProbably by this > point, someone has sent an email to Springer notifying them of Sloan's > intentions and they are politely informing him of their intention to > break his legs (figuratively and in a court of law). You might consider the possibility, unlikely though it may seem, that > I have already checked with the copyright records office and have > found that there is no copyright on this 1955 book. Sam Sloan- Hide quoted text - - Show quoted text - As I stated, I have no animosity towards you, so the comments that I make are not intended to be derogatory. You may have checked with the copyright records office, but since Springer publishes Kelley's book and claims a 1955 copyright, you may have incomplete knowledge of the situation. I think that it would not hurt for you to contact Springer - you may have missed something in your understanding of copyright law, notwithstanding the fact that you checked with the copyright records office. If you are correct, such an inquiry will do you no harm; if you are mistaken, it could save you a great deal of anguish. Just a thought. Oh, and here is the legalese written inside the cover of the book published by Springer. Looks to me like they are claiming copyright. It clearly states that it is a reprint of the 1955 book. This book was originally published in 1955 by Van Nostrand in the University Series in Higher Mathematics (Editorial Board: M.A. Stone, L. Nirenberg, S.S. Chern) Library of Congress Cataloging in Publication Data Kelley, John L. General Topology (Graduate texts in mathematics; 27) Reprint of the 1955 ed. Published by Van Nostrand. New York in The University series in higher mathematics. M. A. Stone, L. Nirenberg, and S. S. Chern, eds. Bibliography: p. 282 Includes index. I. Topology. I. Title II. Series QA611.K4 1975 514.3 75-14364 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1955 by J. L. Kelley === Subject: Re: General Topology by John L. Kelley posting-account=h0BplggAAACbakJwttbpVF72VZ8jVCAq Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 12, 10:30æpm, The poster formerly known as Colleyville Alan > You may have checked with the copyright records office, but since > Springer publishes Kelley's book and claims a 1955 copyright, Do you have the original 1955 book? Have you noticed that the original 1955 book does not say that? Sam Sloan === Subject: Re: General Topology by John L. Kelley posting-account=llxexwoAAADqntGDNkT4ytZG1pkEWVd9 SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; Zune 2.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > On Sep 12, 10:30æpm, The poster formerly known as Colleyville Alan You may have checked with the copyright records office, but since > Springer publishes Kelley's book and claims a 1955 copyright, Do you have the original 1955 book? Have you noticed that the original 1955 book does not say that? Sam Sloan No I do not have the original 1955 book. But Springer is showing that it was copyrighted by John Kelley. The book by Springer clearly shows © 1955 by J. L. Kelley. Moreover, per the UC system: http://www.universityofcalifornia.edu/copyright/ownership.html Copyrights can be bought, sold, willed to others, or given away. A transfer of the copyright or an exclusive grant or license to use the work is a transaction that must be conveyed in writing. Publishers commonly require an author to transfer his/her copyright to the publisher as a condition of publication. You say that copyrights by publishers are no longer valid, but the information above seems to contradict this. If, as seems likely to me, John Kelley sold his copyright to Van Nostrand who later sold it to Springer, then it is not in the public domain. I looked at the copyright search at the US Copyright Office website: http://cocatalog.loc.gov/cgi-bin/Pwebrecon.cgi?DB=local&PAGE=First It says that Works registered prior to 1978 may be found only in the Copyright Public Records Reading Room. So unless you went there to check this, you IMO are mistaken about this being a public domain book. Several people have asked about whether you checked with a lawyer knowledgeable in copyright law, how you determined that it was public domain, etc. You have been inexact in your replies. As far as I can tell, you are basing everything on what you see or fail to see in the 1955 textbook. That, in my opinion, is a good way to get yourself into legal hot water. === Subject: AW: General Topology by John L. Kelley > GeneralTopologyis not only a textbook, it is also an invaluable > reference work for all mathematicians working the field of > analysis. It has long been out of print, In what sense is it out of print? It is often part of Springer's > yearly Yellow Sale. I bought a copy last year from amazon.com > and have just checked that plenty of copies are still available > there. Are you saying you have acquired the printing rights from > Springer? > No, he's not saying that at all. Mr. Sloan seems not to > understand how the whole copyright thing works. He appears to > have announced a reprint of a book that is still in print from > the world's largest publisher of mathematics texts. > Interestingly, the Ishi Press website is down. Coincidence? Or > lawyers? > J. Sorry, but it is you who do not know how this copyright thing > works. The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at > Berkeley in 1962-1964, when Kelley was a professor there. ?http://www.amazon.com/dp/0923891552 Springer was not the publisher of that book. The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) > had nothing to do with the 1955 book that I am reprinting. Sam Sloan > Sam, it was a later edition of the same book. The copyright notice > on the Springer edition (which you can view on line if you have > three functioning brain cells) says Copyright 1955 by J. L. > Kelley. ?Not that I care particularly if you get sued into the > ground, but you would be well advised to talk to a lawyer before > making a fool of yourself again. Once you get into copyright law, you will find that many copyright > notices are false notices. Here is an example: Dover has reprinted a book called Sloan's Victorian Buildings by > Sam Sloan. Inside there is a copyright notice saying Copyright ? 1980 by Dover > Publishing. However, this book was first published in 1851 by my distant > relative who was also named Sam Sloan. Obviously, the book is in public domain. Nevertheless, Dover has > put a copyright notice on it. The Real Sam Sloan- Hide quoted text - - Show quoted text - > That book may be in the public domain, but I have doubts as to > whether General Topology by Kelley is similarly situated. > published before 1923 have expired copyrights and are in the public > domain. In addition, works published before 1964 that did not have > their copyrights renewed 28 years after first publication year also > are in the public domain.... Maybe this gives you a loophole, > provided that nobody else renewed the copyright. But since Springer > is publishing an edition, it is likely that the copyright was > renewed. > I am not a legal expert, but it would seem to me that investing a few > hundred in a legal opinion might prevent many thousands of dollars > from fleeing your bank account. > That Wikipedia paragraph is poorly phrased, though literally true. The > law was changed to eliminate renewal requirements in 1978. Anything > that crossed the line in either first- or second-term copyright got a > one-shot extension such it expired 75 years after the initial > copyright. (There's a bit more to it, but it's not relevant here.) The > point is that, since Kelley's book seems to have been first published > in 1955, it could not have fallen out of copyright before the 1978 > changeover. There are other ways that the book could have fallen into > public domain, but they require the consent of the copyright owner. There is another aspect to fishi-type publishing. The copyright we are talking about here applies to content, e.g. the text of a book. There are also originator's rights that apply to editing and presentation. Even fonts are protected. When Sloan is xeroxing he is infringing in more than one way. He has no qualm's about xeroxing dust jackets, for example. To be clear of these problems you would have to do the type- setting and the art-work yourself. The link http://www.amazon.com/dp/0923891552 which he gave for the Kelley book is dead, so it might be that Amazon woke up. Incidentally, among the many entries for this book on Amazon there is a previous on-demand 'publisher', but that is now listed as 'currently not available'. === Subject: Re: General Topology by John L. Kelley posting-account=fk5d_woAAAAWJdzhm0GQEtVEiuLmnAo8 Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) >GeneralTopologyis not only a textbook, it is also an invaluable > reference work for all mathematicians working the field of analysis. > It has long been out of print, > In what sense is it out of print? It is often part of Springer's > yearly Yellow Sale. I bought a copy last year from amazon.com and > have just checked that plenty of copies are still available there. > Are you saying you have acquired the printing rights from Springer? > No, he's not saying that at all. Mr. Sloan seems not to understand > how the whole copyright thing works. He appears to have announced a > reprint of a book that is still in print from the world's largest > publisher of mathematics texts. Interestingly, the Ishi Press website > is down. Coincidence? Or lawyers? > J. > Sorry, but it is you who do not know how this copyright thing works. > The book I am reprinting was published in 1955, the same book I > studied as a math major at the University of California at Berkeley in > 1962-1964, when Kelley was a professor there. > ?http://www.amazon.com/dp/0923891552 > Springer was not the publisher of that book. > The Springer book was first published in 1975. It was edited by S. > Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) > S. Axler (Editor), F. W. Gehring (Editor), P. R. Halmos (Editor) had > nothing to do with the 1955 book that I am reprinting. > Sam Sloan Sam, it was a later edition of the same book. The copyright notice on > the Springer edition (which you can view on line if you have three > functioning brain cells) says Copyright 1955 by J. L. Kelley. ?Not > that I care particularly if you get sued into the ground, but you > would be well advised to talk to a lawyer before making a fool of > yourself again. Once you get into copyright law, you will find that many copyright > notices are false notices. Here is an example: Dover has reprinted a book called Sloan's Victorian Buildings by Sam > Sloan. Inside there is a copyright notice saying Copyright ? 1980 by Dover > Publishing. However, this book was first published in 1851 by my distant relative > who was also named Sam Sloan. Obviously, the book is in public domain. Nevertheless, Dover has put a > copyright notice on it. The Real Sam Sloan You're baying at the moon, Sam. And, as pointed out above, you could now get five years in jail for pirating the Kelley book. We can only hope. (BTW, all copyright registrations are filed with the U.S. Copyright Office. Real publishers do their research. Have you? Can pigs fly?) === Subject: Re: Solution manual to a First Course in Differential Equations with Modeling Applications (7th ed.) posting-account=D5vvugoAAADp7CyENrbRir2UeJOjEDxp AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.22,gzip(gfe),gzip(gfe) I am a solutions manual collector, I offer solutions manual and ebook > services > Note: > all solutions manual in soft copy > that mean in Adobe Acrobat Reader (PDF ) format. if you want any book > not just solutions just contact with us.81B to get the solution manual > you want .81Cplease send message to > sharesolut...@msn.com .81Csharesolution(at)msn.com.81C replace (at) to > @ ,please email to me . > This is my part of solutions manual list ,If you want any other > solutions manual which is not in my solutions list, don't give > up .please email to sharesolution(at)msn.com Solutions manual to Vector Mechanics Dynamics Beer 8th Edition > Solutions manual to Fundamentals of Applied Electromagnetics 5th > edition by Fawwaz T. 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John Finnemore, Joseph B Franzin > Solutions manual to Derivatives Markets, 2nd , by Robert L. > McDonald (Solutions by Yufeng Guo) > Solutions manual to Options, Futures and Other Derivatives, 4th, > Solutions manual to a First Course in Differential Equations with > Modeling Applications (7th ed.) and Zill & Cullen's Diferential > Equations with Boundary-Value Problems (5th ed.) By Peter Tye hello, I am looking for , solution manual, for the book.A first course > in differential equations, by zill, 7th, 8th 0r 9th edition. hello, did you by any chance find the solutions manual for a first course in differential equations, by zill, 8th edition ? I would gladly pay for it. === Subject: Re: Solution manual to a First Course in Differential Equations with Modeling Applications (7th ed.) posting-account=5ngaNAoAAABlQRjFahg9gbUgPk9gGvED Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Hey I would really appreciate it if I could get your most recent volume of A First Course in Differential Equations with Modeling Applications. I'm taking the class now and I'm behind.. it'd really help me just understand the concepts faster. Let me know if I can get === === Subject: Re: Very basic mistakes > Rational numbers are as countable as are natural numbers. > Cantor's illusion was that real numbers include > irrational numbers and can nonetheless be counted too. The set of real number is uncountable. Well, Cantor was correct when he found out that there is no bijection between reals and rationals. I meant something else. Cantor imagined the reals like discrete numbers, each single of which could be separated and calculated with. > Genuine reals would be uncountable. The set of real numbers is uncountable. I know the dogma and I would nonetheless like to state: Every single (genuine) real number has the property to be only complete as p/q with p, q infinite. In other words, it describes a point on the line with infinite acuity. > Mathematics has been mutilated since as to > understand real numbers as if they were rational numbers at a time. This is nonsense: we understand the real number sqrt(2) is > not rational. There is no alternative as to either calculate with irrational numbers as if they were rational, i.e. using an approximation or letting the command sqrt(2) unresolved. > A set cannot be uncountable if it has been set element by element. has been set element by element is word salad: each interval > in R remains an uncountable set. Each interval of IR represents a sauce of fictitious uncountables. I object to the idea that this interval is a set because a genuine continuum cannot be resolved into single elements (points). The other way round, you can set together as many single points as you like and will never get a genuine continuum. > You cannot eschew religious matters without loosing G. Cantor loosing? To loose means to set free? I can't see what you mean > here. (A frequent event on reading your postings.) Georg Cantor's mathematical thinking cannot be separated from his religious belief. He believed that the infinitum aeternum increatum sive absolutum referred to god. He tried in vain to convince kardinal franzelin that his infinitum creatum sive transfinitum was not just a finitum ordinatum. > My main topic is IR+. However you ignored what I am claiming. > You have neither defined IR+ (that's not a standard notation) > Mathematicians understand IR+ like IR but restricted to positive values. Again, what is IR? That is not standard notation. IR stands for the blackboard letter R. If you were using a blackboard, then you could easily write letters with bold vertical lines in order to emphase and distinguish them. Dedekind still used R for the body of rationals, not just the reals. > The entity of ALL rational numbers is something that cannot be > quantified. Whether or not the set Q of rational numbers can be quantified > it can be put into bijection with N and so is countable. The attribute countable is a bit misleading. It would be more appropriate to call rational numbers with finite p, q discrete but rational numbers with infinite p, q irrational. > Therefore it is something quite different from any rational number. Only an eejit would suppose that Q was itself a rational number. Sometimes we agree. However, I meant: The same originally rational number gets an irrational (fictitious) one if and so long as it is thougt as a constituent part of a genuine continuum. Victor Meldrew > I don't believe it! Mathematics should be free of belief. > Salviati: > ... in ultima conclusione, gli attributi di eguale > maggiore e minore non aver luogo ne gl'infiniti, > ma solo nelle quantit.88 terminate. > IR>|>IR+=|=IR === Subject: Breadline schmuck's basic mistakes <48cbefda$0$3550$9b4e6d93@newsspool3.arcor-online.net> posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > The set of real number is uncountable. Well, Cantor was correct when he found out that there is no bijection > between reals and rationals. Yes. > I meant something else. Cantor imagined the > reals like discrete numbers, each single of which could be separated and > calculated with. separated? > I know the dogma and I would nonetheless like to state: > Every single (genuine) real number has the property > to be only complete as p/q with p, q infinite. So, Breadline Schmuck's counterdogma is that every irrational number is a quotient of infinite thingies. So what could those infinite thingies be when x = sqrt(2) or x = pi? > This is nonsense: we understand the real number sqrt(2) is > not rational. There is no alternative as to either calculate with irrational numbers > as if they were rational, I don't think any mathematicians suppose that irrational numbers are rational when they calculate with them. > Each interval of IR represents a sauce of fictitious uncountables. IR? Do you mean R? What sort of sauce? Bechamel sauce? tomato sauce? Worcestershire sauce? > I object to the idea that this interval is a set because a genuine > continuum cannot be resolved into single elements (points). Private language again --- genuine continuum --- whassat? > Georg Cantor's mathematical thinking cannot be separated from his > religious belief. One can do set theory with or without religious belief. What Cantor thought about gods is of no relevance. > Mathematicians understand IR+ like IR but restricted to positive values. Again, what is IR? That is not standard notation. IR stands for the blackboard letter R. If you were using a blackboard, > then you could easily write letters with bold vertical lines > in order to emphase and distinguish them. Dedekind still used R for > the body of rationals, not just the reals. Aha. But you are an imbecile, Breadline Schmuck, this is not a blackboard. > The attribute countable is a bit misleading. It would be more appropriate to > call rational numbers with finite p, q discrete but rational numbers with > infinite p, q irrational. So Breadline Schumck believes in *infinite* numbers! How queer. > Sometimes we agree. However, I meant: The same originally > rational number gets an irrational (fictitious) one if and so long as > it is thougt as a constituent part of a genuine continuum. Again, more jabbering Herr Schmuck: what is a genuine continuum? Victor Meldrew I don't believe it! === Subject: Markov bound? Hi all, I do not understand one step in a lemma and it would be nice if someone of you can help me. Let A be a random variable over {0,1}^n and let 0 <= t_a <= 1. Let further: sum_{a in {0,1}^n}Pr[a=A]*t_a > d. Let now be B subset {0,1}^n such that for b in B we have t_b >= d/2. Then in the Lemma it says that by Markov it follows: Pr[A in B] >= d/2 Do you know which Markov bound is used here and why this acutally should be the case? Bernd === Subject: min-entropy I have an additional question: Let A be a random variable uniform over {0,1}^n. Let A_m be a random variable with min-entropy n-m. Now my question is: Why is it possible to define an event E (depending on A) such that Pr[E] >= 2^{m} and conditioned on E, A has the same distribution as A_m? I mean, obviously it is possible to define such an event, but why does A conditioned on E have the same distribution as A_m? And how does the min- entropy helps here? Bernd === Subject: Re: min-entropy does nobody have an idea? I have an additional question: Let A be a random variable uniform over {0,1}^n. Let A_m be a random > variable with min-entropy n-m. Now my question is: Why is it possible to define an event E (depending > on A) such that Pr[E] >= 2^{m} and conditioned on E, A has the same > distribution as A_m? I mean, obviously it is possible to define such an event, but why does > A conditioned on E have the same distribution as A_m? And how does the > min- entropy helps here? Bernd > === Subject: Absent-mindedness and focus among great mathematicians posting-account=fVOpuAkAAAB0gOUkQMH0DG_KdwTVgKXP CLR 2.0.50727),gzip(gfe),gzip(gfe) I have met and even got to know many very well-known mathematicians. Some of them are very absent-minded and unfocused with regard to the non-mathematical aspects of life. They think about mathematics while washing the dishes and while driving, (and probably while making love), and don't listen if anyone tries to talk to them about anything non-mathematical. Consequently, if they go shopping, and kindly ask me some soup? I really feel like bread and soup today, they will give you fifty bars of soap (the root cause of the miscommunication being that they didn't listen to your second sentence.) However, some are extremely focused on all tasks whether mathematical or not, apparently subscribing to a philosophy that the most productive way to live is always to focus on the task at hand. Despite the fact that some present-minded individuals have achieved great success in mathematics, my intuition is that the absent-minded approach has a mathematical pay-off in that the time spent on mathematical thought is thereby much greater. For example, suppose two mathematicians wash up. One pays attention to the task, and clean dishes result. The other goes through the motions while thinking about mathematics, resulting in knives and plates with bits of chicken grease stuck to them. Has the extra mathematical thought-time of the absent-minded mathematician paid off, or are there hidden advantages to living life in such a way that you actually pay attention to what you do? Paul Epstein === Subject: Re: Absent-mindedness and focus among great mathematicians > I have met and even got to know many very well-known mathematicians. > Some of them are very absent-minded and unfocused with regard to the > non-mathematical aspects of life. They think about mathematics while > washing the dishes and while driving, (and probably while making > love), and don't listen if anyone tries to talk to them about anything > non-mathematical. Consequently, if they go shopping, and kindly ask > me some soup? I really feel like bread and soup today, they will > give you fifty bars of soap (the root cause of the miscommunication > being that they didn't listen to your second sentence.) However, some are extremely focused on all tasks whether mathematical > or not, apparently subscribing to a philosophy that the most > productive way to live is always to focus on the task at hand. Despite the fact that some present-minded individuals have achieved > great success in mathematics, my intuition is that the absent-minded > approach has a mathematical pay-off in that the time spent on > mathematical thought is thereby much greater. For example, suppose > two mathematicians wash up. One pays attention to the task, and clean > dishes result. The other goes through the motions while thinking > about mathematics, resulting in knives and plates with bits of chicken > grease stuck to them. Has the extra mathematical thought-time of the > absent-minded mathematician paid off, or are there hidden advantages > to living life in such a way that you actually pay attention to what > you do? When I was an undergrad at UIC, A.O.L. Atkin was my NT 360 professor. http://en.wikipedia.org/wiki/A._O._L._Atkin It was really amusing seeing him enter the SEO elevator on the ground floor with me, with the elevator stopping on the 2nd floor and him getting out mumbling, only to run back in quickly, mumbling again, before the elevator door closed, because his office was really on the 3rd floor ;o) He was also friends with my piano teacher at the music department where one could see him occasionally walking in the basement halls of the music dept., seemingly randomly, holding a plastic bag ;o) In class, his implies steps, required at least half an hour of thought from the students and he could calculate Jacobi symbols in seconds. I think (but I am not sure) that he had something to do with factoring F_9, or some other F_n, as well. > Paul Epstein -- I.N. Galidakis === Subject: Absent-mindedness and focus among great mathematicians posting-account=fVOpuAkAAAB0gOUkQMH0DG_KdwTVgKXP CLR 2.0.50727),gzip(gfe),gzip(gfe) I have met and even got to know many very well-known mathematicians. Some of them are very absent-minded and unfocused with regard to the non-mathematical aspects of life. They think about mathematics while washing the dishes and while driving, (and probably while making love), and don't listen if anyone tries to talk to them about anything non-mathematical. Consequently, if they go shopping, and kindly ask me some soup? I really feel like bread and soup today, they will give you fifty bars of soap (the root cause of the miscommunication being that they didn't listen to your second sentence.) However, some are extremely focused on all tasks whether mathematical or not, apparently subscribing to a philosophy that the most productive way to live is always to focus on the task at hand. Despite the fact that some present-minded individuals have achieved great success in mathematics, my intuition is that the absent-minded approach has a mathematical pay-off in that the time spent on mathematical thought is thereby much greater. For example, suppose two mathematicians wash up. One pays attention to the task, and clean dishes result. The other goes through the motions while thinking about mathematics, resulting in knives and plates with bits of chicken grease stuck to them. Has the extra mathematical thought-time of the absent-minded mathematician paid off, or are there hidden advantages to living life in such a way that you actually pay attention to what you do? Paul Epstein === Subject: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) I have completed the basic theory for 2 variable Diophantine equations. That is, the mathematical theory covering equations of the form c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y. That theory gives the equation for determining existence of integer solutions, as well as a method using what I call Diophantine chains to actually find solutions when they exist. The paper does include some basic Pell's Equation results as well, like the result that for every solution to Pell's Equation of the form x^2 - 2y^2 = 1 you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = -1. I am publishing through Google Docs: James Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > I have completed the basic theory for 2 variable Diophantine > equations. æThat is, the mathematical theory covering equations of the > form c 1*x^2 + c 2*xy + c 3*y^2 = c 4 + c 5*x + c 6*y. That theory gives the equation for determining existence of integer > solutions, as well as a method using what I call Diophantine chains to > actually find solutions when they exist. The paper does include some basic Pell's Equation results as well, > like the result that for every solution to Pell's Equation of the form x^2 - 2y^2 = 1 you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = > -1. I am publishing through Google Docs: > I did want to put the new mathematical tools forward but also I was curious to find out things like, can people actually access the document? And more importantly, will anyone reply with issues with the details of the theory itself? After all I found a proof of Fermat's Last Theorem over 6 years ago. I no longer believe in the academic system. I read news reports of supposed findings, or listen to babbling about String theory or supposedly how old the universe is, and now kind of yawn, if I bother at all. It is weird though. I discretized conic sections using my tautological spaces. And added to my math yet another remarkable result, which took me 8 days to do, with mathematical tools so powerful they can do that with little to no effort. 2000 years of mathematical research in that area. 8 days for me to cover with a more powerful and succinct mathematical theory. So I had this one idea I call tautological spaces and it can do so much, but it doesn't seem like that big of a deal. But I remember being a kid playing with parabolas, graphing them over and over again, and being excited about their properties. Reading other people's math. Now I read my math. And to my math that I read I now have a full discrete theory for conics in general, figured out within a few days, using a mathematical technique, I invented. And the world calls me a crackpot. I no longer believe in people, with good reasons. Over six years of good reasons with the knowledge of my accomplishments and the world's reaction to them. But at least I still like people, mostly. I just kind of see most as, primitive. I believe in mathematics. Long after all of you are dead. Long after the sun has cooled and the Milky Way has drifted apart, what I've discovered will still be true. And I defined mathematical proof. So I know of what I speak. James Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) > I have completed the basic theory for 2 variable Diophantine > equations. æThat is, the mathematical theory covering equations > of the form c 1*x^2 + c 2*xy + c 3*y^2 = c 4 + c 5*x + c 6*y. Just so we know what to look out for, which parts of the theory > did you provide that were missed by Euler, Gauss, Lagrange, > H J S Smith, Hurwitz, Minkowski, Kneser, Watson, Hasse, > and a few dozen others? I've simplified. Just the number of names shows that all of their research couldn't go into 6 pages. I give a complete theory in only 6 pages. Mathematicians might not care about such a simplification, though I hope they do, but for physicists simpler mathematical tools are gold. They're easier to use, can be used more widely, and can point to underlying physical rules. Like here, there could be greater indications of discrete behavior beyond known quantum behavior, which could advance discrete physics theory. Discrete mathematics is such a difficult area that physics theories that assume more discrete behavior in our world--like discrete space-- are hard to mathematicize. One of my long time goals has been opening the door to checking such ideas, which requires advancing the mathematical tools. Taking an area of number theory with 2000 years of prior history and simplifying with a 6 page paper which encompasses everything previously done indicates that the mathematical tool advancements are achievable. It could mean a total transformation of our understanding of behavior in certain areas, and might allow the merging of Einstein's gravitational theories with quantum mechanics. The two theories kind of don't seem to like each other now. James Harris. === Subject: JSH: Complete theory for 2 variable Diophantine equations, paper now available posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Discrete mathematics is such a difficult area that physics theories > that assume more discrete behavior in our world--like discrete space-- > are hard to mathematicize. Wrong. In fact one of the main reasons for introducing lattice field theory - a huge area of research in current physics - is that lattice theories are much /easier/ to mathematicize than their continuum counterparts. === Subject: Re: JSH: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Discrete mathematics is such a difficult area that physics theories > that assume more discrete behavior in our world--like discrete space-- > are hard to mathematicize. Wrong. In fact one of the main reasons for introducing lattice field > theory - a huge area of research in current physics - is that lattice > theories are much /easier/ to mathematicize than their continuum > counterparts. Ok then, quantize gravitational theory and report back. Or simply give an overview of current efforts with quantum gravity. James Harris === Subject: Re: JSH: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Discrete mathematics is such a difficult area that physics theories > that assume more discrete behavior in our world--like discrete space-- > are hard to mathematicize. Wrong. In fact one of the main reasons for introducing lattice field > theory - a huge area of research in current physics - is that lattice > theories are much /easier/ to mathematicize than their continuum > counterparts. Ok then, quantize gravitational theory and report back. Or simply give an overview of current efforts with quantum gravity. James Harris === Subject: Re: JSH: Complete theory for 2 variable Diophantine equations, paper now available posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Discrete mathematics is such a difficult area that physics theories > that assume more discrete behavior in our world--like discrete space-- > are hard to mathematicize. Wrong. In fact one of the main reasons for introducing lattice field > theory - a huge area of research in current physics - is that lattice > theories are much /easier/ to mathematicize than their continuum > counterparts. Ok then, quantize gravitational theory and report back. Huh? What on Earth do you think this question has to do with what I That doesn't change the fact that lattice theories are much easier to mathematicize than their continuum counterparts. In fact the main difficulty with quantising gravity is the fact that the theory is not perturbatively renormalizable, and last I heard it was not known whether there its RG flow has a UV fixed point - these are problems which arise precisely because the theory treats spacetime as continuous, rather than discrete. So your question is utterly irrelevant. On the other hand you claim that physical theories with discrete space are harder to mathematicize, so it only seems fair to ask you to rigorously define quantised Yang-Mills theory on Minkowski space. After all, the discrete-space version of the theory has been defined for decades, so if it's true that discrete-space theories are harder to mathematicize than their continuum counterparts then defining Yang- Mills theory on continuous spacetime should be a doddle. Also there's potentially a million dollars in it for you, since it's one of the Clay institute problems. So get cracking, and report back. >æOr simply give an overview of current efforts with quantum gravity. === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available > Taking an area of number theory with 2000 years of prior history and > simplifying with a 6 page paper which encompasses everything > previously done indicates that the mathematical tool advancements are > achievable. > It could mean a total transformation of our understanding of behavior > in certain areas, and might allow the merging of Einstein's > gravitational theories with quantum mechanics. All by rearranging the terms of a quadratic equation? > That's ... literally incredible! > Yes, but don't forget the tautological space. That makes a difference. === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) > Taking an area of number theory with 2000 years of prior history and > simplifying with a 6 page paper which encompasses everything > previously done indicates that the mathematical tool advancements are > achievable. > It could mean a total transformation of our understanding of behavior > in certain areas, and might allow the merging of Einstein's > gravitational theories with quantum mechanics. All by rearranging the terms of a quadratic equation? > That's ... literally incredible! Yes, but don't forget the tautological space. æThat makes a difference. For the physicists what I do is artificially add extra degrees of freedom. It's a kind of wacky technique that came to me back in December 1999, where you take an expression like x+y+vz = x+y+vz and express it with mod as x+y+vz = 0(mod x+y+vz) and then do some basic algebraic manipulations, and subtract out an equation to be analyzed, and consider the residue which is how I got my Quadratic Diophantine Theorem, which allowed me to encompass 2000 years of mathematical research on 2 variable quadratic Diophantines, in 6 pages. It is an incredibly powerful analysis technique which is unfortunately facing a vicious political war with current number theorists--fighting over what they see as their turf against an outsider--doing their best from what I've seen to suppress the technique by suffocation-- literally sucking the air away from knowledge of it by not acknowledging it, or by some criticizing results from it. But for physicists it is really just adding degrees of freedom. The v variable is totally free and carefully setting it to various values is what allows you to get results. If physicists do not help me with the political war, I have no doubt number theorists will continue to work to suppress the technology. So far they have done so since December 1999--to give you perspective. publishing a paper of mine with analysis done using this technique. So you see, it IS a war of information suppression and a dead mathematical journal is one of the casualties. James Harris === Subject: Re: JSH Complete theory for 2 variable Diophantine equations, paper now available Alert Blocking level 7 all Cisco Routers JSH Block imposed Further attempts by JSH to post Math JSH will be truncated with extreme prejudice === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) >Taking an area of number theory with 2000 years of prior history and >simplifying with a 6 page paper which encompasses everything >previously done indicates that the mathematical tool advancements are >achievable. It could mean a total transformation of our understanding of behavior >in certain areas, and might allow the merging of Einstein's >gravitational theories with quantum mechanics. All by rearranging the terms of a quadratic equation? > That's ... literally incredible! My position is that the difficulties in reconciling the general theory of relativity with quantum mechanics have to do with the discrete. There are rules that are emergent with discrete results that are not there with continuous functions, which can even be seen with my current paper where there is a mod p result that exists based on how quadratic residues behave. It disappears in the field of reals or complex numbers. The issue is a huge one in physics. And the question really is about describing our world. You see, in physics, what works is what works in the real world. Better mathematical tools are needed to explore this area. My research may be a foot in the door... James Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available >Taking an area of number theory with 2000 years of prior history and >simplifying with a 6 page paper which encompasses everything >previously done indicates that the mathematical tool advancements are >achievable. It could mean a total transformation of our understanding of behavior >in certain areas, and might allow the merging of Einstein's >gravitational theories with quantum mechanics. All by rearranging the terms of a quadratic equation? > That's ... literally incredible! >My position is that the difficulties in reconciling the general theory >of relativity with quantum mechanics have to do with the discrete. piss-off, moron. >There are rules that are emergent with discrete results that are not >there with continuous functions, which can even be seen with my >current paper where there is a mod p result that exists based on how >quadratic residues behave. new buzzwords will not help you. >It disappears in the field of reals or complex numbers. So is your intelect, disappeared into complex spoofarinies. >The issue is a huge one in physics. And the question really is about >describing our world. Wrong. You need attention to exist, post your crap in alt.highschool.morons.with.NPD. >You see, in physics, what works is what works in the real world. You have no idea of what you are talking about. >Better mathematical tools are needed to explore this area. My >research may be a foot in the door... Rather a turd on the Lawn. >James Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available reply-type=response >Taking an area of number theory with 2000 years of prior history and >simplifying with a 6 page paper which encompasses everything >previously done indicates that the mathematical tool advancements are >achievable. >It could mean a total transformation of our understanding of behavior >in certain areas, and might allow the merging of Einstein's >gravitational theories with quantum mechanics. > All by rearranging the terms of a quadratic equation? > That's ... literally incredible! >My position is that the difficulties in reconciling the general theory >of relativity with quantum mechanics have to do with the discrete. piss-off, moron. Yeah, why don't you off, stooopid emotional ignorant cunt? === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available I have completed the basic theory for 2 variable Diophantine > equations. That is, the mathematical theory covering equations of the > form c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y. That theory gives the equation for determining existence of integer > solutions, as well as a method using what I call Diophantine chains to > actually find solutions when they exist. The paper does include some basic Pell's Equation results as well, > like the result that for every solution to Pell's Equation of the form x^2 - 2y^2 = 1 you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = > -1. I am publishing through Google Docs: > James Harris What are you telling us for. If it's new publich it in a joural or conference. === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) I have completed the basic theory for 2 variable Diophantine > equations. æThat is, the mathematical theory covering equations of the > form c 1*x^2 + c 2*xy + c 3*y^2 = c 4 + c 5*x + c 6*y. That theory gives the equation for determining existence of integer > solutions, as well as a method using what I call Diophantine chains to > actually find solutions when they exist. The paper does include some basic Pell's Equation results as well, > like the result that for every solution to Pell's Equation of the form x^2 - 2y^2 = 1 you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = > -1. I am publishing through Google Docs: > James Harris What are you telling us for. If it's new publich it in a joural or > conference. It's at a journal, but that takes time. And they might decide not to publish anyway. But I can provide the mathematical tools which DO have relevance to current physics research, as well as research in other sciences, today. So hey, don't worry. Paper is at a journal--a major one. But in the meantime, I can also publish directly and physicists can use the mathematical tools now--without waiting on some editors and publication cycles of a major journal. Oh, also, I'm not a big fan of journals though I do play the game, and with some revolutionary mathematics in it--relying on similar techniques to those used for this latest research--and some freaking sci.math'ers not only managed to get the editors of that journal to pull my paper after publication with some concerted emails, wouldn't you know the damn journal DIED a few months later. Yeah. Died. The entire freaking journal keeled over and quietly died. It was a 10 year old journal. Yeah. It's like that. The journal system is weak. But I know, yeah, it's the main system, so I at least DID send this latest paper to one--and then published myself. James Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available > Oh, also, I'm not a big fan of journals though I do play the game, and > with some revolutionary mathematics in it--relying on similar > techniques to those used for this latest research--and some freaking > sci.math'ers not only managed to get the editors of that journal to > pull my paper after publication with some concerted emails, wouldn't > you know the damn journal DIED a few months later. /tried/ submitting the paper to another journal? > But I know, yeah, it's the main system, so I at least DID send this > latest paper to one--and then published myself. Pray tell, could you post the reviewers' comments when they come back? === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) > Oh, also, I'm not a big fan of journals though I do play the game, and > with some revolutionary mathematics in it--relying on similar > techniques to those used for this latest research--and some freaking > sci.math'ers not only managed to get the editors of that journal to > pull my paper after publication with some concerted emails, wouldn't > you know the damn journal DIED a few months later. /tried/ submitting the paper to another journal? Yes. > But I know, yeah, it's the main system, so I at least DID send this > latest paper to one--and then published myself. Pray tell, could you post the reviewers' comments when they come back? They never send them to me. Never. Guess they realize I'd just post them and rip on them. The journals do not follow the rules with my research. They can't. As if they followed the rules, they'd publish. Like with this latest result, it IS a general theory for 2 variable Diophantine equations which is simpler than anything else previously known. I've boiled down 2000 years of math history to a 6 page paper that represents research I started 8 days ago. I'm shutting down just one number theory area, for now, by completing the research. I can take them all away. Of course, number theorists can pretend I did not, but then, they're more than just pathetic, now aren't they? And physicists will have to not use tools readily available to give in to mathematicians who are, well, weak. It took me one week to take away 2 variable Diophantine equations. One week. James Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available > Pray tell, could you post the reviewers' comments when they come back? They never send them to me. Never. Guess they realize I'd just post them and rip on them. Editors have no obligation to send papers to reviewers before rejecting them. If an editor thinks that it is plainly obvious the paper does not meet the journal standards, then he'll reject it himself and no reviewer will see it. So, with that in mind, there are two possibilities: (1) You don't get reviewer comments because no reviewer has seen it. (2) You don't get reviewer comments because editors are scared of adverse publicity from the Google-Master. Yeah, probably (2). -- And I wish some of you would grow past thinking that you've discovered some extraordinary thing [...] as if you found the Holy Grail or something, when I acknowledge a mistake. After all, I've had to do it quite a few times. It's not like it's news. --James S. Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available > But I know, yeah, it's the main system, so I at least DID send this > latest paper to one--and then published myself. > Pray tell, could you post the reviewers' comments when they come back? They never send them to me. Never. Guess they realize I'd just post them and rip on them. Let me see if I got this one. You are claiming that when your paper was accepted the editor did not send you the reviewers' comments because he realized that you would just post them and rip on them. Did I get that right? Jose Carlos Santos === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available <6j40d9F1c46bU1@mid.individual.net> posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) But I know, yeah, it's the main system, so I at least DID send this > latest paper to one--and then published myself. > Pray tell, could you post the reviewers' comments when they come back? They never send them to me. æNever. Guess they realize I'd just post them and rip on them. Let me see if I got this one. You are claiming that when your paper was > accepted the editor did not send you the reviewers' comments because he > realized that you would just post them and rip on them. Did I get that > right? > all the arguing on sci.math so I told the editors upfront that I was an amateur researcher, and even had to correct them when they were sending emails to me with Dr. Harris. And they stopped doing that and switched to Mr. Harris. If I'd realized that I should get reviewers comments I would have requested them, but the editors may have been hedging their bets by not sending them. One of the more telling emails I think was when I was effusively thanking them for publishing my paper--I had been in contact regularly with the journal editors for 9 months keeping up with how things were going--one of the editors emailed me back that it didn't matter who found a result, but only its correctness mattered, even if that person were a janitor. I thought that was kind of odd thing to say, but maybe a reference to the movie Good Will Hunting and an indication that the editors realized how big my paper was. And later after the debacle I was in contact with other editors on the journal after chief editor refused to reply back to me after sending an email that he was taking a sabbatical, so no, I don't accept the explanations that sci.math posters give claiming that it was all just a mistake. And posters claim I knowingly sent an erroneous paper to the journal but Barry Mazur himself commented on an early draft!!! I forwarded his email to Ralph McKenzie when he was claiming problems with the paper when I'd sent it to him for publication in a journal for which he was editor, and after Dr. Mazur's paper he offered I could explain to him in person. And I DID explain to him in person driving over 4 hours each way from the Atlanta metro area, so I put in the legwork for that paper and had feedback from top mathematicians--none of whom gave the sci.math'er complaints. NONE OF THEM VALIDATED ANY OF YOUR COMPLAINTS ABOUT THE PAPER OR ITS METHOD. The editors knew what they were doing when they published my paper. The journal used two reviewers. I never got their reports. And the sci.math email campaign worked against 9 months of effort, where I was upfront with those editors. So it was the sci.math'ers who deliberately with their emails broke the journal system, and even that journal itself died. The sci.math'ers showed the weakness of the system, and maybe the editors did just kind of give up, as in one of their first editions they'd published a paper claiming to prove P=NP, nearly 10 years before. And don't blather on about what the author of that paper says today, as after 10 years why would he hold onto his claims if his society doesn't play by its own rules, if he wants to keep working as a mathematician? Maybe a lot of mathematicians themselves have given up on the math system. James Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available > But I know, yeah, it's the main system, so I at least DID send this > latest paper to one--and then published myself. > Pray tell, could you post the reviewers' comments when they come back? > They never send them to me. Never. > Guess they realize I'd just post them and rip on them. > Let me see if I got this one. You are claiming that when your paper was > accepted the editor did not send you the reviewers' comments because he > realized that you would just post them and rip on them. Did I get that > right? all the arguing on sci.math so I told the editors upfront that I was > an amateur researcher, and even had to correct them when they were > sending emails to me with Dr. Harris. And they stopped doing that > and switched to Mr. Harris. If I'd realized that I should get reviewers comments I would have > requested them, but the editors may have been hedging their bets by > not sending them. One of the more telling emails I think was when I was effusively > thanking them for publishing my paper--I had been in contact regularly > with the journal editors for 9 months keeping up with how things were > going--one of the editors emailed me back that it didn't matter who > found a result, but only its correctness mattered, even if that person > were a janitor. I fully agree with the editor on this. > I thought that was kind of odd thing to say, but maybe a reference to > the movie Good Will Hunting and an indication that the editors > realized how big my paper was. How did you jump from correct to big paper? > And later after the debacle I was in contact with other editors on the > journal after chief editor refused to reply back to me after sending > an email that he was taking a sabbatical, so no, I don't accept the > explanations that sci.math posters give claiming that it was all just > a mistake. And posters claim I knowingly sent an erroneous paper to the journal > but Barry Mazur himself commented on an early draft!!! I forwarded his email to Ralph McKenzie when he was claiming problems > with the paper when I'd sent it to him for publication in a journal > for which he was editor, and after Dr. Mazur's paper he offered I > could explain to him in person. And I DID explain to him in person driving over 4 hours each way from > the Atlanta metro area, so I put in the legwork for that paper and had > feedback from top mathematicians--none of whom gave the sci.math'er > complaints. NONE OF THEM VALIDATED ANY OF YOUR COMPLAINTS ABOUT THE PAPER OR ITS > METHOD. The editors knew what they were doing when they published my paper. > The journal used two reviewers. How do you know that? > I never got their reports. You never even *saw* them. How do you know that they ever existed? > And the sci.math email campaign worked against 9 months of effort, > where I was upfront with those editors. Effort? You sent the paper and, nine months later they published. What effort did you do during those months? > So it was the sci.math'ers who deliberately with their emails broke > the journal system, and even that journal itself died. Which proofs do you have concerning the connection between your paper being withdrawn and the end of the journal? Jose Carlos Santos === Subject: Re: JSH Complete theory for 2 variable Diophantine equations, paper now available > But I know, yeah, it's the main system, so I at least DID send this > latest paper to one--and then published myself. > Pray tell, could you post the reviewers' comments when they come back? > They never send them to me. Never. > Guess they realize I'd just post them and rip on them. > Let me see if I got this one. You are claiming that when your paper was > accepted the editor did not send you the reviewers' comments because he > realized that you would just post them and rip on them. Did I get that > right? > all the arguing on sci.math so I told the editors upfront that I was > an amateur researcher, and even had to correct them when they were > sending emails to me with Dr. Harris. And they stopped doing that > and switched to Mr. Harris. > If I'd realized that I should get reviewers comments I would have > requested them, but the editors may have been hedging their bets by > not sending them. > One of the more telling emails I think was when I was effusively > thanking them for publishing my paper--I had been in contact regularly > with the journal editors for 9 months keeping up with how things were > going--one of the editors emailed me back that it didn't matter who > found a result, but only its correctness mattered, even if that person > were a janitor. I fully agree with the editor on this. Agree here too, it has always been that way. > I thought that was kind of odd thing to say, but maybe a reference to > the movie Good Will Hunting and an indication that the editors > realized how big my paper was. How did you jump from correct to big paper? Wrong conclusion. (JSH unfamiliar with process now conjectures failing to realize that most all papers are small.) > And later after the debacle I was in contact with other editors on the > journal after chief editor refused to reply back to me after sending > an email that he was taking a sabbatical, so no, I don't accept the > explanations that sci.math posters give claiming that it was all just > a mistake. So? he was taking a sabbatical. What has the rest of above to do with that? > And posters claim I knowingly sent an erroneous paper to the journal > but Barry Mazur himself commented on an early draft!!! > I forwarded his email to Ralph McKenzie when he was claiming problems > with the paper when I'd sent it to him for publication in a journal > for which he was editor, and after Dr. Mazur's paper he offered I > could explain to him in person. > And I DID explain to him in person driving over 4 hours each way from > the Atlanta metro area, so I put in the legwork for that paper and had > feedback from top mathematicians--none of whom gave the sci.math'er > complaints. didn't you threaten him at the chalkboard? Did you know it is very easy to talk with most all professors at universities? Did you listen to him or try to ram your math down his throat? > NONE OF THEM VALIDATED ANY OF YOUR COMPLAINTS ABOUT THE PAPER OR ITS > METHOD. You must show it is correct. Did any of them validate it was correct ?? > The editors knew what they were doing when they published my paper. > The journal used two reviewers. How do you know that? How? > I never got their reports. You never even *saw* them. How do you know that they ever existed? Obviously, JSH confrontational personality was known by them and for reasons of personal safety instead of telling him to go back to high school, they dropped it. Or nobody really reviewed it at all, then the paper was withdrawn, JSH got caught with his spoof. > And the sci.math email campaign worked against 9 months of effort, > where I was upfront with those editors. Effort? You sent the paper and, nine months later they published. What > effort did you do during those months? > So it was the sci.math'ers who deliberately with their emails broke > the journal system, and even that journal itself died. Wrong again. Accept the fact that your paper is junk, written by a troll. Which proofs do you have concerning the connection between your paper > being withdrawn and the end of the journal? > Jose Carlos Santos === Subject: Re: JSH Complete theory for 2 variable Diophantine equations, paper now available > One of the more telling emails I think was when I was effusively > thanking them for publishing my paper--I had been in contact regularly > with the journal editors for 9 months keeping up with how things were > going--one of the editors emailed me back that it didn't matter who > found a result, but only its correctness mattered, even if that person > were a janitor. > I fully agree with the editor on this. > Agree here too, it has always been that way. OK, now I'm curious. Some of the great mathematicians of the past were what we today would > call amateurs. But has any amateur actually found anything significant > in modern times? Marjorie Rice: http://www.ivanrival.com/docs/picturepuzzling_2.pdf Jose Carlos Santos === Subject: Re: JSH Complete theory for 2 variable Diophantine equations, paper now available Tim Smith a .8ecrit : > One of the more telling emails I think was when I was effusively > thanking them for publishing my paper--I had been in contact regularly > with the journal editors for 9 months keeping up with how things were > going--one of the editors emailed me back that it didn't matter who > found a result, but only its correctness mattered, even if that person > were a janitor. > I fully agree with the editor on this. > Agree here too, it has always been that way. OK, now I'm curious. Some of the great mathematicians of the past were what we today would > call amateurs. But has any amateur actually found anything significant > in modern times? Let's define amateur as someone without a PhD in mathematics (or a > related field). Strange def ; usually amateurs are non-professionals, i.e. not doing this as a way of earning their life... What is the most significant amateur contribution in, > say, the last 20 years? Last 50 years? Last 100 years? > === Subject: Re: JSH Complete theory for 2 variable Diophantine equations, paper now available >One of the more telling emails I think was when I was effusively >thanking them for publishing my paper--I had been in contact regularly >with the journal editors for 9 months keeping up with how things were >going--one of the editors emailed me back that it didn't matter who >found a result, but only its correctness mattered, even if that person >were a janitor. I fully agree with the editor on this. >Agree here too, it has always been that way. OK, now I'm curious. Some of the great mathematicians of the past were what we today would > call amateurs. But has any amateur actually found anything significant > in modern times? Let's define amateur as someone without a PhD in mathematics (or a > related field). What is the most significant amateur contribution in, > say, the last 20 years? Last 50 years? Last 100 years? Ramanujan's work, perhaps ? http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ramanujan.html Han de Bruijn === Subject: Re: JSH Complete theory for 2 variable Diophantine equations, paper now available One of the more telling emails I think was when I was effusively >thanking them for publishing my paper--I had been in contact regularly >with the journal editors for 9 months keeping up with how things were >going--one of the editors emailed me back that it didn't matter who >found a result, but only its correctness mattered, even if that person >were a janitor. >I fully agree with the editor on this. Agree here too, it has always been that way. > OK, now I'm curious. > Some of the great mathematicians of the past were what we today would > call amateurs. But has any amateur actually found anything significant > in modern times? > Let's define amateur as someone without a PhD in mathematics (or a > related field). What is the most significant amateur contribution in, > say, the last 20 years? Last 50 years? Last 100 years? Ramanujan's work, perhaps ? http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ramanujan.html Han de Bruijn Ramanujan was amature, He was a real genius. After reading his works, one concludes that this JSH stuff is complete 100% crap. === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available > Oh, also, I'm not a big fan of journals though I do play the game, and > with some revolutionary mathematics in it--relying on similar > techniques to those used for this latest research--and some freaking > sci.math'ers not only managed to get the editors of that journal to > pull my paper after publication with some concerted emails, wouldn't > you know the damn journal DIED a few months later. > /tried/ submitting the paper to another journal? Yes. > But I know, yeah, it's the main system, so I at least DID send this > latest paper to one--and then published myself. > Pray tell, could you post the reviewers' comments when they come back? They never send them to me. Never. Guess they realize I'd just post them and rip on them. That's one possible explanation. === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 2.0.50727),gzip(gfe),gzip(gfe) > I have completed the basic theory for 2 variable Diophantine > equations. æThat is, the mathematical theory covering equations of the > form > c 1*x^2 + c 2*xy + c 3*y^2 = c 4 + c 5*x + c 6*y. > That theory gives the equation for determining existence of integer > solutions, as well as a method using what I call Diophantine chains to > actually find solutions when they exist. > The paper does include some basic Pell's Equation results as well, > like the result that for every solution to Pell's Equation of the form > x^2 - 2y^2 = 1 > you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = > -1. > I am publishing through Google Docs: > James Harris What are you telling us for. If it's new publich it in a joural or > conference. It's at a journal, but that takes time. æAnd they might decide not to > publish anyway. Ya think? But I can provide the mathematical tools which DO have relevance to > current physics research, as well as research in other sciences, > today. Yeah, sure. So hey, don't worry. æPaper is at a journal--a major one. æ Be sure and let us know what they say when they reject it. > But in the > meantime, I can also publish directly and physicists can use the > mathematical tools now--without waiting on some editors and > publication cycles of a major journal. Or waiting for a peer review. Oh, also, I'm not a big fan of journals though I do play the game, and > with some revolutionary mathematics in it--relying on similar > techniques to those used for this latest research--and some freaking > sci.math'ers not only managed to get the editors of that journal to > pull my paper after publication with some concerted emails, wouldn't > you know the damn journal DIED a few months later. You've still never said. Who put you up to sending it to Yeah. æDied. æThe entire freaking journal keeled over and quietly > died. æIt was a 10 year old journal. æYeah. æIt's like that. The journal system is weak. So are your papers. But I know, yeah, it's the main system, so I at least DID send this > latest paper to one--and then published myself. Anybody can post anything. James Harris === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available > I have completed the basic theory for 2 variable Diophantine > equations. That is, the mathematical theory covering equations of the > form > c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y. > That theory gives the equation for determining existence of integer > solutions, as well as a method using what I call Diophantine chains to > actually find solutions when they exist. > The paper does include some basic Pell's Equation results as well, > like the result that for every solution to Pell's Equation of the form > x^2 - 2y^2 = 1 > you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = > -1. > I am publishing through Google Docs: > James Harris What are you telling us for. If it's new publich it in a joural or > conference. You just answered your own question. === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) well, seeing that he seems to imply the definition of a tuatological space (est-ce que un espace tautologique/ QuestionMark), it sort-of is a demonstration of modular arithmetic, at least the additive part of it, that I read up to/UnScrolled). that was my question, iff I had asked. yeah, what you call some thing is what you call something; the map is supposed to be like the terrain, preferably & suitably earthlike-ish. > You just answered your own question. thus: the additive inverse of X is -2X?... I mean, not summorial of -2X iff defined. sorry, about the outburst from Descartes CIRCA Fermat; didn't realize that it may be thought, ambivalent! === Subject: Re: Complete theory for 2 variable Diophantine equations, paper now available > I have completed the basic theory for 2 variable Diophantine > equations. That hardly belongs in a physics newsgroup, blazo :-) [followup set appropriately] Dirk Vdm === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation >Now that I have a general theory for all 2 variable quadratic >Diophantine equations I'm not even sure you know what that means. >it's worth coming back to note again the weird >connection I found between certain Pythagorean Triplets and Pell's >Equation in the form x^2 - Dy^2 = 1 when D-1 is a perfect square. For instance for D=2, I have that for >every solution of Pell's Equation you have a Pythagorean Triplet! I'm hardly an expert here, but let me make the following comments: 1. It's an interesting result 2. People should stop bashing you because it is, as I said, an interesting result. Seriously guys. He's come up with something correct for a change. Give the man some credit. 3. It's cute. Not groundbreaking. Not weird. The relationship between Pell's Equation and Pythagorean Triples has been known for a long time. A new one wouldn't be a big surprise. 4. If you'd present the results without so much ego you'd get a better reception (yes, I know your history) Alan -- Defendit numerus === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Now that I have a general theory for all 2 variable quadratic > Diophantine equations it's worth coming back to note again the weird > connection I found between certain Pythagorean Triplets and Pell's > Equation in the form x^2 - Dy^2 = 1 when D-1 is a perfect square. æFor instance for D=2, I have that for > every solution of Pell's Equation you have a Pythagorean Triplet! But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. > The connection is that w is x+y from Pell's Equation. The more general result is that u = sqrt(D-1)j, and v = j+1, while w > still equals x+y. Intriguingly that means that proof that there are an infinite number > of solutions for certain Pell's Equations is proof that there are an > infinity of Pythagorean Triplets of a certain form! An easy example with D=2, is x=17, y=12, where notice you are paired > with the triplet 20, 21, 29. That is just some low-hanging fruit that I thought I'd mention. æKind > of been a whirlwind of results flowing from playing with my > Diophantine Quadratic Theorem. New argument now I'm starting to see is that I've found nothing new, though I will add that for me the Pell's Equation result is just a fun tidbit which is nothing compared to the main result of generally solving the 2 variable Diophantine equation. A succinct example of the tidbit result claimed to not be new is the easy to show case that for EVERY solution to x^2 - 2y^2 = 1 you have a solution to the negative Pell's Equation: z^2 - 2(x+y)^2 = -1. For instance, x=17, y = 12 is a solution to the first as 17^2 - 2(12)^2 = 1 and with x+y=29, you get z=41 for the second, as 41^2 - 2(29)^2 = -1. To me that it's easy to explain so I have to wonder why no one it seems has said it in that way in human history before... 2000 years of mathematical history with Pell's Equation. The will to lie about a subject that old is a powerful and demonic one, and for those of you who have wondered how I could be right, if so many people are arguing with me, here you can see. They argue with me because these battles are supposed to be hard. If it were easy then there wouldn't be a choice, now would there? I'm set. It's you who has a fate in the balance. It's your life that is being decided now. Not mine. What are you made of? Who are you really? In a sense, me and the others here are just agents to test your mettle. God's way of testing your worth as human beings. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) >[...] A succinct example of the tidbit result claimed to not be new is the >easy to show case that for EVERY solution to x^2 - 2y^2 = 1 you have a solution to the negative Pell's Equation: z^2 - 2(x+y)^2 = -1. For instance, x=17, y = 12 is a solution to the first as 17^2 - 2(12)^2 = 1 and with x+y=29, you get z=41 for the second, as 41^2 - 2(29)^2 = -1. To me that it's easy to explain so I have to wonder why no one it >seems has said it in that way in human history before... 2000 years of mathematical history with Pell's Equation. Oh, gawd, is this a joke? Charlie Brown rushes up to kick the football again: æ æ(x + 2y)^2 - 2(x + y)^2 æ= -(x^2 - 2y^2) æ= -1 Yup. It's EASY. Therefore the past mathematicians who didn't discover it, were they ignorant? Or maybe just, um, not so great? After all, I discovered it in a couple of days. Now, go, cite the result from any other source that Pell's Equation x^2 - 2y^2 = 1 is directly connected to the negative Pell's Equation z^2 - 2(x+y)^2 = -1. Just try to rescue the blind belief that brilliance has truly defined past mathematical efforts in this area. I suggest instead, you are making the argument that others simply failed to see the simple. > Even by your standards (not to mention mine), this is silly! AAUUGGGHH! Come on James, admit it: you wanted to see if anyone was daft > enough to believe that you could be daft enough to mean this > seriously! Um, I went from not to my knowledge even knowing about Pell's Equation last Friday, to giving some remarkably simple results that I can't find anywhere else despite their obvious simplicity, in a few days. 2000 years of mathematical history traversed by me completely within 4 days. It seems to me you make a brilliant argument that past mathematicians were not, after all, all that brilliant, unless you wish to cite someone, anyone, before me, noting that EVERY case of x^2 - 2y^2 = 1 connected to z^2 - 2(x+y)^2 = -1 as that's a kind of beautiful symmetry. Now I know how hard mathematical discovery is, so I want to say that I DO think those past mathematicians were quite brilliant, but mathematics is an infinite subject, though many of you seem to forget it. How many results would it take then? I can wipe out entire areas of number theory in a single week, as I just demonstrated. The psychological wars you people fight are just signs of weakness to me, and I know human psychology better than you do. I know how your brains are wired and why you fight. It is a primitive male urge to solidify control over women, whether you have women or not or are even gay. You are naked apes, doing what apes do. I long since tired of contacting mathematicians directly with my research as they'd always go sort of oddly silent. I'll never forget the colleague of someone I knew who was a professor who he'd contacted about my research who left the country on a 6 month sabbatical, and when he came back, claimed he didn't remember ever hearing about it. paper, went on a sabbatical as well, immediately thereafter. I wonder what happened to him. Did you people destroy his career? But blame him? I can take mathematicians out one by one at will just with an email of one of my papers. It's not the single one of you that has power. It is the bulk of all of you together. So as this proceeds, you force me to find a result that handles every mathematicians around the world at the same time. Is that challenge worthy of me? Do you think? (Oh, and their students, as well. LOL.) James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation > Um, I went from not to my knowledge even knowing about Pell's Equation > last Friday, to giving some remarkably simple results that I can't > find anywhere else despite their obvious simplicity, in a few days. > 2000 years of mathematical history traversed by me completely within 4 > days. > It seems to me you make a brilliant argument that past mathematicians > were not, after all, all that brilliant, unless you wish to cite > someone, anyone, before me, noting that EVERY case of > x^2 - 2y^2 = 1 > connected to > z^2 - 2(x+y)^2 = -1 > as that's a kind of beautiful symmetry. I earnestly recommend that you read John Stillwell, /Mathematics and > Its History/. I'm sorry to say this, Angus, because I know you mean well, but you are really being a bit stupid. It should be completely obvious to you by now that James has not the slightest interest in reading any mathematical texts. Giving him advice like this is equivalent to pissing into the wind. === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation Angus Rodgers a .8ecrit : > I'm sorry to say this, Angus, because I know you mean well, but you are > really being a bit stupid. It should be completely obvious to you by > now that James has not the slightest interest in reading any > mathematical texts. Giving him advice like this is equivalent to > pissing into the wind. But James isn't going to go away, is he? I might as well get used > to him. And in a weird kind of way, I'm enjoying this, while fully > realising that it might be an ancient sci.math rite of passage, and > it might not be fruitful at any level whatsoever. The way in which > the conversation keeps going even when it appears to have become re- > duced to utter nothingness seems almost like Zen (whatever that is). > I have to agree the ribbet, ribbet part sounded soewhat Zen (not to mention the Moo (or Mu) part, of course) . But what I like best in your discussions with James is what is the sound of one person having a dialogue? === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation >Angus Rodgers a .8ecrit : > I'm sorry to say this, Angus, because I know you mean well, but you are > really being a bit stupid. It should be completely obvious to you by > now that James has not the slightest interest in reading any > mathematical texts. Giving him advice like this is equivalent to > pissing into the wind. > But James isn't going to go away, is he? I might as well get used > to him. And in a weird kind of way, I'm enjoying this, while fully > realising that it might be an ancient sci.math rite of passage, and > it might not be fruitful at any level whatsoever. The way in which > the conversation keeps going even when it appears to have become re- > duced to utter nothingness seems almost like Zen (whatever that is). >I have to agree the ribbet, ribbet part sounded soewhat Zen (not to >mention the Moo (or Mu) part, of course) . But what I like best in your >discussions with James is what is the sound of one person having a >dialogue? Remember that mu is the Japanese equivalent of a Chinese original, which sounded much more like wu: Does a dog have Buddha-nature? Wu! Woof! is more appropriate than moo. rossum === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation > Angus Rodgers a .8ecrit : > I'm sorry to say this, Angus, because I know you mean well, but you are > really being a bit stupid. It should be completely obvious to you by > now that James has not the slightest interest in reading any > mathematical texts. Giving him advice like this is equivalent to > pissing into the wind. > But James isn't going to go away, is he? I might as well get used > to him. And in a weird kind of way, I'm enjoying this, while fully > realising that it might be an ancient sci.math rite of passage, and > it might not be fruitful at any level whatsoever. The way in which > the conversation keeps going even when it appears to have become re- > duced to utter nothingness seems almost like Zen (whatever that is). > I have to agree the ribbet, ribbet part sounded soewhat Zen [...] > I object. I'm partial to Theravada and have no love for them new-fangled variants on Buddhism. They're just too darned easy. Bodhisattvas are for amateurs. -- Just because you're ... in a Ph.d program it does not mean that you're up to the challenge of being a real mathematician. Only those who have a purity of mind and dedication to the truth as the highest ideal have a chance. --James Harris, as Sir Galahad the Pure. === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation > I'm sorry to say this, Angus, because I know you mean well, but you are > really being a bit stupid. It should be completely obvious to you by > now that James has not the slightest interest in reading any > mathematical texts. Giving him advice like this is equivalent to > pissing into the wind. But James isn't going to go away, is he? I might as well get used > to him. And in a weird kind of way, I'm enjoying this, while fully > realising that it might be an ancient sci.math rite of passage, and > it might not be fruitful at any level whatsoever. The way in which > the conversation keeps going even when it appears to have become re- > duced to utter nothingness seems almost like Zen (whatever that is). > I understand the perverse appeal of interacting with him, and he does read all the posts (he probably re-reads his own many times over). My point was simply that advising him to read anything else is a waste of keystrokes. === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) I earnestly recommend that you read John Stillwell, /Mathematics and > Its History/. æ I'm sorry to say this, Angus, because I know you mean well, but you are > really being a bit stupid. æIt should be completely obvious to you by > now that James has not the slightest interest in reading any > mathematical texts. æGiving him advice like this is equivalent to > pissing into the wind. Now give Angus his due. Angus plays Socrates to James's Meno in a fashion that even Plato would applaud. But you are right that both are stupid in the sense that neither can see the endgame. Angus will eventually snap and James will win, just as surely as James will never solve anything of any real significance. Problem is that telling them does no good. ;>) M === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation > I'm sorry to say this, Angus, because I know you mean well, > but you are really being a bit stupid. It should be completely > obvious to you by now that James has not the slightest interest > in reading any mathematical texts. Giving him advice like this > is equivalent to pissing into the wind. But James isn't going to go away, is he? I might as well get used > to him. And in a weird kind of way, I'm enjoying this, while fully > realising that it might be an ancient sci.math rite of passage, and > it might not be fruitful at any level whatsoever. The way in which > the conversation keeps going even when it appears to have > become reduced to utter nothingness seems almost like Zen > (whatever that is). You are romanticizing the situation. James does go away from time to time, and the newsgroup is the better for it. He leaves when the chorus of boos becomes loud enough. === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation > It now seems > worth showing a bit of persistence myself, and doing a bit of > work to try to understand this ... well, this person. William Baerg, an arachnologist at the University of Arkansas purposely allowed himself to be bitten by a black widow spider in 1922: three days of pain and delirium in a hospital. But that was not enough. In 1933 William Blair, MD, of the University of Alabama took up the torch and allowed a black widow spider to bite him. The pain lasted a week. His skin ithced, burned, and peeled for a further two weeks -- Michael Press === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) > I'm sorry to say this, Angus, because I know you mean well, > but you are really being a bit stupid. æIt should be completely > obvious to you by now that James has not the slightest interest > in reading any mathematical texts. æGiving him advice like this > is equivalent to pissing into the wind. But James isn't going to go away, is he? æI might as well get used > to him. æAnd in a weird kind of way, I'm enjoying this, while fully > realising that it might be an ancient sci.math rite of passage, and > it might not be fruitful at any level whatsoever. æThe way in which > the conversation keeps going even when it appears to have > become reduced to utter nothingness seems almost like Zen > (whatever that is). You are romanticizing the situation. > James does go away from time to time, > and the newsgroup is the better for it. > He leaves when the chorus of boos > becomes loud enough. Cite then old man. Give ANY math text that has the result that with x^2 - 2y^2 = 1 you automatically have another answer with z^2 - 2(x+y)^2 = -1. You are just an old blow-hard trying to protect your turf, and willing to piss on mathematics for it because you were NEVER a real mathematician, now were you? Were you old man? Were you? James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) Cite then old man. Give ANY math text that has the result that with x^2 - 2y^2 = 1 you automatically have another answer with z^2 - 2(x+y)^2 = -1. You are just an old blow-hard trying to protect your turf, and willing > to piss on mathematics for it because you were NEVER a real > mathematician, now were you? Were you old man? Were you? Did you look at the references Angus Rodgers gave? æOn page 17 of Scanned them quickly... > Number Theory: An Approach Through History from Hammurapi to Legendre, > by Andre Weil, there is this: æ æ(x+2y)^2 - 2(x+y)^2 = -(x^2-2y^2) That's a pure superset of your result. æIt shows that if æ æx^2-2y^2 = N then you automatically have: æ æz^2-2(x+y)^2 = -N and, unlike yours, it actually gives you z = x+2y. I wouldn't doubt next there will be a barrage of requests for me to apologize to the other posters, but my question is, why couldn't they do what Tim Smith did? Succinctly and clearly state what is given versus sending me fishing for it. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation > and, unlike yours, it actually gives you z = x+2y. > I wouldn't doubt next there will be a barrage of requests for me to > apologize to the other posters, but my question is, why couldn't they > do what Tim Smith did? My question is: why couldn't you do what Tim Smith did? === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) > and, unlike yours, it actually gives you z = x+2y. > I wouldn't doubt next there will be a barrage of requests for me to > apologize to the other posters, but my question is, why couldn't they > do what Tim Smith did? My question is: æwhy couldn't you do what Tim Smith did? I looked over both links. The second link is a Google scan of a book by Weil and it just so happens that the required page for some reason scanned in differently, and hurrying along I brushed past it. Tim Smith may simply have looked more carefully than I did. But regardless the simpler thing would have been for the poster to give the one link and note on what page the result was shown. Given his preening behavior afterwards it seems possible to me he preferred a continuing argument where I looked stupid. Hey, I look stupid all the time. That's not news. But people who make it harder to get information and think they're gaining points by making me waste time learn soon enough how quickly they can be flushed down the newsgroup toilet. This guy may think he's joining the sci.math elite or something. So he's testing my will to back away from considering the implications of the latest research to talking about idiot behavior from wannabes coming into the newsgroup thinking they have a quick route to popularity by making me look stupid. Moron. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation >I looked over both links. The second link is a Google scan of a book >by Weil and it just so happens that the required page for some reason >scanned in differently, and hurrying along I brushed past it. This is mathematics James, you cannot just brush past things. You need to read and understand things. It is strange of you to ask for references and then just brush past the references you are given. Having asked for them it would be polite of you to read them carefully. Tim Smith may simply have looked more carefully than I did. Which may be part of the reason Tim Smith gets treated with more respect here than you generally do. Actions have consequences James. If you do not liek the consequences then do not do the actions. But regardless the simpler thing would have been for the poster to >give the one link and note on what page the result was shown. The posted merely assumed that you were able to read a web reference. In this the posted made a mistake, but that is more down to you brushing past rather than reading with care and attention. rossum === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 FunWebProducts; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) spider-mtc-tf02.proxy.aol.com[400C70A2] (Prism/1.2.1), HTTP/1.1 cache-mtc-af14.proxy.aol.com[400C754E] (Traffic-Server/6.1.5 [uScM]) > and, unlike yours, it actually gives you z = x+2y. > I wouldn't doubt next there will be a barrage of requests for me to > apologize to the other posters, but my question is, why couldn't they > do what Tim Smith did? My question is: ?why couldn't you do what Tim Smith did? I looked over both links. ?The second link is a Google scan of a book > by Weil and it just so happens that the required page for some reason > scanned in differently, and hurrying along I brushed past it. Tim Smith may simply have looked more carefully than I did. But regardless the simpler thing would have been for the poster to > give the one link and note on what page the result was shown. Given his preening behavior afterwards it seems possible to me he > preferred a continuing argument where I looked stupid. Hey, I look stupid all the time. ?That's not news. But people who make it harder to get information and think they're > gaining points by making me waste time learn soon enough how quickly > they can be flushed down the newsgroup toilet. This guy may think he's joining the sci.math elite or something. So he's testing my will to back away from considering the implications > of the latest research to talking about idiot behavior from wannabes > coming into the newsgroup thinking they have a quick route to > popularity by making me look stupid. What journal did you send your paper to? Moron. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation Tim Smith a .8ecrit : > Cite then old man. > Give ANY math text that has the result that with > x^2 - 2y^2 = 1 > you automatically have another answer with > z^2 - 2(x+y)^2 = -1. > You are just an old blow-hard trying to protect your turf, and willing > to piss on mathematics for it because you were NEVER a real > mathematician, now were you? > Were you old man? > Were you? Did you look at the references Angus Rodgers gave? It's a conspiracy, insnt'it? You *know* maths books make him ill ! It's a little bit like Necronomicon ; full of powerfull spells, but yo.8dur mental health drop down if you try to read them... === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation Angus Rodgers a .8ecrit : > Cite then old man. > Give ANY math text that has the result that with > x^2 - 2y^2 = 1 > you automatically have another answer with > z^2 - 2(x+y)^2 = -1. > You are just an old blow-hard trying to protect your turf, and willing > to piss on mathematics for it because you were NEVER a real > mathematician, now were you? > Were you old man? > Were you? This must be more Zen, I suppose. Whatever it is, you are clearly > the master of it. I agree : the classic koan is a unanswerable question and the real answer being even more absurd (like Mu,or putting one's shoe on one's head). James ask a stupid question with lots of innuendos (have you stop beating your wife becomes did you succeed in proving sqrt (42) is rational and if not is it because you are weak or because you are lying ?), then his counter to painstakingly devised answers is to ask *the same question*, with a twist (on the innuendos, of course) I am lost in amazement that you can keep asking > the same question, which has already been answered as plainly as it > could possibly be! Anyway, for the third time: The first is a Google Books scan of some pages from the Stillwell > book I mentioned. The second is a Google Books scan of some pages > from Andre Weil, /Number Theory: An approach through history from > Hammurapi to Legendre/ (Birkh.8auser 1987, 4th pr. in paperback 2007). The Weil extract gives EXACTLY what you are asking for. (Stillwell > merely implies it, so that any rational person would consider it > established, but you are obviously already able to wriggle out of > that kind of implication.) This is already clear on page 16; and > if you squint hard at the deliberately rather blurry scan of page > 17 (something to do with copyright, I presume), you can just see > the EXACT same identity - that very, very easy identity! - that I > typed out a couple of days ago. (No ... good gawd! ... it was only > a little over one day and two nights ago ... this is all obviously > starting to get to me.) I repeat, the result is approximately 2600 years old. What are you trying to do ? Make him read some math books ? You *know* he is allergic to math books ! You want to drive him crazy, or what ? (borrowed from Peanuts, when Charlie Brown tries to teach Sally fractions...) > === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation >Do you *really* believe saying the same thing 3 times will magically >open James' eyes ? Well no more that the fact *he* says the same tning >30000 times dont make it truer... Just the place for a Snark! the Bellman cried, As he landed his crew with care; Supporting each man on the top of the tide By a finger entwined in his hair. Just the place for a Snark! I have said it twice: That alone should encourage the crew. Just the place for a Snark! I have said it thrice: What I tell you three times is true. The Hunting of the Snark - Lewis Carroll rossum === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation > I'm sorry to say this, Angus, because I know you mean well, > but you are really being a bit stupid. It should be completely > obvious to you by now that James has not the slightest interest > in reading any mathematical texts. Giving him advice like this > is equivalent to pissing into the wind. > But James isn't going to go away, is he? I might as well get used > to him. And in a weird kind of way, I'm enjoying this, while fully > realising that it might be an ancient sci.math rite of passage, and > it might not be fruitful at any level whatsoever. The way in which > the conversation keeps going even when it appears to have > become reduced to utter nothingness seems almost like Zen > (whatever that is). You are romanticizing the situation. > James does go away from time to time, > and the newsgroup is the better for it. A matter of opinion! > He leaves when the chorus of boos > becomes loud enough. Oh, please. He's thick-skinned. He simply goes through cycles and I doubt the chorus of boos has a damned thing to do with it. -- I have to break the code of how [mere humans] work, and I have made a lot of progress over years of effort, and I feel like I am close to figuring out all the inner details of human wiring. -- James S. Harris on the extra problems of conveying his research === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=Z3AipgkAAABkoMfyNwddSxsYhXHi5CDt CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) I'm sorry to say this, Angus, because I know you mean well, but you are >really being a bit stupid. æIt should be completely obvious to you by >now that James has not the slightest interest in reading any >mathematical texts. æGiving him advice like this is equivalent to >pissing into the wind. But James isn't going to go away, is he? æI might as well get used > to him. æAnd in a weird kind of way, I'm enjoying this, while fully > realising that it might be an ancient sci.math rite of passage, and > it might not be fruitful at any level whatsoever. æThe way in which > the conversation keeps going even when it appears to have become re- > duced to utter nothingness seems almost like Zen (whatever that is). -- > Angus Rodgers > Contains mild peril Angus, James is demanidng a citation in which the author explicitly says that given a solution to x^2 - 2y^2 = 1, you also get a solution to x^2 - 2y^2 = -1. I don't see that you have done that, although I didn't read through all of the citations that you posted. Therefore James will not stop. It doesn't matter that this is screamingly obvious from some of what you posted. He did, in fact, notice this point and seems to think that this is a world-shaking mathematical discovery. It is actually an interesting point, since it gives you a unit of norm -1 from a unit of norm 1,l and so looking at general Pell equations, it can only be done for some of them. Which ones? Possibly interesting thing to have a casual look at. Now the mathematics is so easy that is difficult to believe that it wasn't published or sent in a letter or something some long while ago. The only reasons it wouldn't have been is that everybody thought it was such a minor point that it wasn't worth mentioning, or else that they thought anybody reading what the did write would simply see Has anybody checked L.E. Dickson's history of number theory? Achava === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) James is demanidng a citation in which the author explicitly says that >given a solution to x^2 - 2y^2 = 1, you also get a solution to x^2 - >2y^2 = -1. æI don't see that you have done that, although I didn't >read through all of the citations that you posted. I posted these two an hour ago, in response to his insistent requests: (reference to the Stillwell book I mentioned) > (reference to a book by Andre Weil) Therefore James will not stop. I don't mind if he doesn't stop. æWhile very deluded, he is obviously* > sincere. æAlso, his creativity in keeping up these conversations (for > how many years now?) is amusing. æI've mostly avoided them in the past, Hey, I made a mistake before and didn't see the equations with the link to the book by Weil. But you COULD have, like Tim Smith posted where the equation was versus just giving links for me to look through and you gave 2 when the second would have sufficed. Instead you're grand-standing. Hey, I make lots of mistakes. I admittedly knew next to nothing about this area of number theory before a bit over a week ago, and am feeling my way along. Assholes like you may think it fun to come in and play stupid games that waste people's time but I, unlike you, am actually trying to find answers versus trying to look pretty for the crowd. Now you can see the discussion moving more towards physics where I'm making the case that math society is deliberately trying to keep advanced mathematical tools from the science community so you can begin to see that it's not just about kissing ass with math people. And you may find your place in history here, and the consequences that follow. In the physics world you don't gain points by deliberately obscuring information to try and make someone look stupid. But physicists, you see, are actually in the pursuit of knowledge. Having such a purpose can give you ethics, decency, and common sense. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) James is demanidng a citation in which the author explicitly says that >given a solution to x^2 - 2y^2 = 1, you also get a solution to x^2 - >2y^2 = -1. æI don't see that you have done that, although I didn't >read through all of the citations that you posted. I posted these two an hour ago, in response to his insistent requests: (reference to the Stillwell book I mentioned) > (reference to a book by Andre Weil) I checked the first time, but had trouble with the second page. After Tim Smith's post I looked again more closely and saw the general result, so yes, you are correct. >Therefore James will not stop. Hey, if you'd put all the information upfront like Tim Smith then scanned in clearer I probably would have noticed the first time. I really am curious about finding out what is correct here as I learn about an area I hadn't been very interested in, a little more than a week ago. I don't even remember hearing about Pell's Equation before then. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation >Perhaps I do nurse the >glorious fantasy of being The Genius Who Solved The Harris Problem! :-) I have started my own small attempt at this: JSH - an Axiomatic Approach Axiom 1: JSH is the world's greatest living mathematician. Being an axion of the system, this is unchallengable from within the system. We are at liberty to speculate whether or not JSH is the greatest mathematician ever, but we cannot challenge Axiom 1. This axiomatic system is also consistent - there is no inconsistency between the axiom and itself. The greatness of JSH is already apparent. Theorem 1: There are parts of mathematics that only JSH understands. If someone else understood all the mathematics that JSH does, then that person would be as great a mathematician as JSH, and that is not allowed by Axiom 1. Theorem 2: All mathematical results produced by JSH are new, exciting, ground breaking, revolutionary and very important. This follows directly from Axiom 1; since JSH is the world's greatest living mathematician, therefore all his results are the worlds greatest mathematical results. JSH has a complete and rigorous proof of this, but unfortunately it falls into the area of mathematics covered by Theorem 1, so we cannot hope to understand it. This theorem applies to all of JSH's results. If JSH rederives the Chinese Remainder Theorem, then that result is also new, exciting, ground breaking, revolutionary and very important. Whoever first discovered the CRT thousands of years ago was not aware of things like complex numbers, transcendental numbers and so forth that JSH is, hence JSH's result cannot be viewed in the same light as the original proof, which was made in a far less complex environment. Borges' Pierre Menard ... (http://www.coldbacon.com/writing/borges-quixote.html) is relevant here, particularly the passage discussing truth, whose mother is history, rival of time .... Corollary 2.1: JSH's factoring methods are new, exciting, ground breaking, revolutionary and very important. This follows directly from Theorem 2. Lemma 2.2: RSA factoring is in danger. By Corollary 2.1 we know the importance etc. of James' factoring ideas. This requires that these methods will be able to factor RSA numbers quickly; if they were not able to factor such numbers quickly then the methods would not be revolutionary etc. Since we know that these results are important they must have a great impact on the Factoring Problem. Once we have understood the full impact of these factoring ideas we will be able to factor very large numbers very quickly. However, due to our lack of understanding, as per Theorem 1, James has not yet been able to assign a timescale to how long it will take us to fully comprehend the depth and importance of his factoring methods. Corollary 2.3: JSH's Diophantine methods are new, exciting, ground breaking, revolutionary and very important. This follows directly from Theorem 2. Merely because we cannot see the importance of James' results does not mean that they are not important. Theorem 1 may well be in play again here. rossum === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation ;Í ;