mm-1339 === Subject: Re: How to solve logarithm simulataneous equations Excellent analysis! I grow more and more impressed with Lambert (or at least, with his W) :-) --Ron Bruck >I dunno. I think it might be easier to solve > log(m)/ log(m+1) = log(a)/log(b) >approximately, using Newton's method; if there's to be an integral >solution, this will have to be close to an integer. Purify m by >setting it to be this close integer. (The iteration is fast.) > For large m, log(m)/log(m+1) = 1 - 1/(m log(m)) + o(1/m^2), so the > initial guess could be m_0 = 1/((1-t) W(1/(1-t))) > where t = log(a)/log(b) and W is the Lambert W function (ProductLog in > Mathematica). This satisfies 1 - 1/(m_0 log(m_0)) = t. > According to Maple, m - m_0 = -1/2 + O(1/log(m)), so round(m_0-1/2) > should be good for large m. In fact, it appears that m = round(m_0-1/2) > for all positive integers m. > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2 === Subject: Topology/metric space question...hints, please Let M be the set of all bounded sequences of real numbers. Define a metric d on M by d(x_n, y_n) = sup |x_n - y_n| as n ranges through the natural numbers. The questions that I'm stuck on are: 1. Is M complete? 2. Is M or any subset of M countably compact? advance. === Subject: Re: Topology/metric space question...hints, please Originator: grubb@lola >Let M be the set of all bounded sequences of real numbers. Define a >metric d on M by d(x_n, y_n) = sup |x_n - y_n| as n ranges through the >natural numbers. The questions that I'm stuck on are: > 1. Is M complete? Hint: Think uniform convergence. > 2. Is M or any subset of M countably compact? Certainly finite subsets are :). Can you show that there is a sequence in some closed ball with no convergent subsequence? Think Dirac deltas. >advance. --Dan Grubb === Subject: Re: Topology/metric space question...hints, please > Let M be the set of all bounded sequences of real numbers. Define a > metric d on M by d(x_n, y_n) = sup |x_n - y_n| as n ranges through the > natural numbers. The questions that I'm stuck on are: This is the (real) l^infinity (that's a lower case L, it's read ell-infinity) space. It's a standard example of an important category of spaces. > 1. Is M complete? So you wish to determine whether every Cauchy sequence in M converges. Let {X1, X2, X3, ..., Xn, ... } be such a sequence. That means that for any epsilon > 0, there is an N for which m,n > N ==> |Xm - Xn| < epsilon Now, this norm is the supremum norm, right? What does that say about the individual components (terms of the sequences Xm and Xn)? The limiting sequence should be easy to write down. The trick will be to determine whether it forms a bounded sequence. It's not terribly difficult. > 2. Is M or any subset of M countably compact? For a metric space, countable (or sequential) compactness is equivalent to ordinary compactness. Do you have any idea as to the sort of space this thing is? For instance, do you see that it's a vector space over R? Do you see that it contains R as a closed subspace? Prove that, and then use some facts about closed subspaces of compact spaces, and you'll be over halfway to your answer to the first question. Second, are you familiar with any compact spaces? Does any one of them fit inside M? > advance. Hints provided. I had written out answers, until I saw how you wanted hints. Dale === Subject: Re: Topology/metric space question...hints, please > For a metric space, countable (or sequential) compactness is equivalent > to ordinary compactness. Eh? I thought countable compactness was when every open cover reduces to a countable covering... === Subject: Re: Topology/metric space question...hints, please >> For a metric space, countable (or sequential) compactness is equivalent >> to ordinary compactness. >Eh? I thought countable compactness was when every open cover reduces >to a countable covering... No, that is Lindelof. -- 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: Topology/metric space question...hints, please Originator: grubb@lola >> For a metric space, countable (or sequential) compactness is equivalent >> to ordinary compactness. >Eh? I thought countable compactness was when every open cover reduces >to a countable covering... No. That's Lindelof. Countable compactness is when every countable cover has a finite subcover. --Dan Grubb What Maxwell is famously celebrated for, unifying the representation of light and electromagnetic phenomena using a wave conception, was precisely what GaussÖand Amp.8fre before him, had rejected as an oversimplification. Amp.8fre had been so close to the development of the modern wave theory of light, that its founder, his good friend Augustin Fresnel, lived in his Paris apartment at the same time that Amp.8fre was carrying out his electrical researches. To suppose that Amp.8fre, and later Gauss, did not consider a wave representation for electromagnetic propagation is absurd. In order to establish his theory, Maxwell had to disregard the most crucial questions and anomalies that had arisen in the decades-long study of these phenomena by the greatest minds before him. Foremost among these were the angular (or relative velocity) dependency of the electrodynamic force, and the little problem of where gravitation should fit in. Gravitation The possibility of subsuming the phenomenon of gravitation under electrodynamics, came up for serious discussion early in this history. One of the more widely discussed contributions was a memoir of about 1830 by O.F. Mossotti, a French physics teacher at the University of Buenos Aires.10 Mossotti proposed to account for gravitation in the following way: If matter is assumed to be constituted of equal > distinguish between GR and some competing theories, or distinguish --ils duce d'Enron! > What Maxwell is famously celebrated for, unifying the representation > of light and electromagnetic phenomena using a wave conception, That wasn't what Maxwell did. > was > precisely what Gauss-and Amp.8fre before him, had rejected as an > oversimplification. Amp.8fre had been so close to the development of the > modern wave theory of light, that its founder, his good friend > Augustin Fresnel, lived in his Paris apartment at the same time that > Amp.8fre was carrying out his electrical researches. To suppose that > Amp.8fre, and later Gauss, did not consider a wave representation for > electromagnetic propagation is absurd. Yes. It is. > In order to establish his > theory, Maxwell had to disregard the most crucial questions and > anomalies that had arisen in the decades-long study of these phenomena > by the greatest minds before him. Foremost among these were the > angular (or relative velocity) dependency of the electrodynamic force, > and the little problem of where gravitation should fit in. Balderdash. Maxwell was the first 'real' physicist to acknowledge Faraday's work (the velocity relation between electric and magnetic forces). Others of his day shunned Faraday (the outsider). Maxwell explicitly included Faraday's work in his magnum opus, On Physical Lines of Force 1861-2. The work wherein Maxwell's equations were first derived (as opposed to empirical equations); wherein light was first identified as transverse electric and magnetic waves; wherein superconductivity was first derived; wherein superfluidity was first utilized. > Gravitation > The possibility of subsuming the phenomenon of gravitation under > electrodynamics, came up for serious discussion early in this history. Maxwell included discussion of gravitation in 1862, but only as an aside. (Whaddya want in one paper?) > One of the more widely discussed contributions was a memoir of about > 1830 by O.F. Mossotti, a French physics teacher at the University of > Buenos Aires.10 Mossotti proposed to account for gravitation in the > following way: If matter is assumed to be constituted of equal l {continuing} If matter is assumed to be constituted of equal amounts of positive and negative electricity, then, by the usual interpretation, there would be a universal tendency for attraction would result. This has nothing to do with wave representations. -- greywolf42 ubi dubium ibi libertas {remove planet for return e-mail} === Subject: Sequences and Applications? Hi all, shows how numerical sequences are applied? I have heard, and maybe seen, them used in continued fractions and the like, but what else is there? TIA, Lurch === Subject: Re: Sequences and Applications? Charlie Johnson skrev i melding > Hi all, > shows how numerical sequences are applied? I have heard, and maybe seen, > them used in continued fractions and the like, but what else is there? > TIA, > Lurch I would look to D. E. Knut's books on The Art of Computer Programming === Subject: Re: Sequences and Applications? > Hi all, > shows how numerical sequences are applied? I have heard, and maybe seen, > them used in continued fractions and the like, but what else is there? > TIA, > Lurch I have seen that others recommended some websites, and that there are tens of thousands of those. The reason is that the uses of sequences are so many. Trivialities: What are the uses of the hammer in the construction business? What are the uses of nouns in a language? What are the uses of water in earthly life? What are the uses of paper in the arts? That's how basic sequences are in mathematics. Try to be concrete, concise and exhaustive in answering the four questions above. A specialized study of sequences is called Difference Calculus, developed by Newton, Gregory, Boole, and perhaps many others before them. An excellent classic is Gelfond A., The calculus of finite differences, M., Nauka, 1967; English transl. Hindustan Publ. (M. means Moscow) === Subject: Re: Sequences and Applications? Zdislav, The only websites I was offered, as I said earlier, consisted of Analysis course descriptions. I am not interested in Infinite series, Power series, Taylor series, or anything of that ilk. I would like to see applications of infinite numerical SEQUENCES. I know that they are basic, but I can't seem to find anything on the web which demonstrates this. I have seen examples like x_n = {1, 1.4, 1.41, 1.414,.......}, whereby the lim x_n = sqrt(2). n-->oo Other than that example, I haven't seen any uses of numerical sequences. If they are so basic, why can't I find any examples? Lurch > Hi all, > shows how numerical sequences are applied? I have heard, and maybe seen, > them used in continued fractions and the like, but what else is there? > TIA, > Lurch > I have seen that others recommended some websites, and that > there are tens of thousands of those. The reason is that > the uses of sequences are so many. > Trivialities: > What are the uses of the hammer in the construction business? > What are the uses of nouns in a language? > What are the uses of water in earthly life? > What are the uses of paper in the arts? > That's how basic sequences are in mathematics. Try to be > concrete, concise and exhaustive in answering the four questions > above. > A specialized study of sequences is called Difference Calculus, > developed by Newton, Gregory, Boole, and perhaps many others > before them. An excellent classic is > Gelfond A., The calculus of finite differences, M., Nauka, > 1967; English transl. Hindustan Publ. > (M. means Moscow) === Subject: Re: Sequences and Applications? > Zdislav, > The only websites I was offered, as I said earlier, consisted of Analysis > course descriptions. I am not interested in Infinite series, Power series, > Taylor series, or anything of that ilk. I would like to see applications of > infinite numerical SEQUENCES. I don't understand your objection. Every infinite series *is* an application of an infinite numerical sequence. By definition an infinite series is the limit of the sequence of its partial sums. I guess what you are asking for is an example of a useful infinite sequence that is not associated with any series. But that seems rather silly -- if you were to come up with such a sequence, and it had a limit, then would it not always be possible to express that limit as the sum of the first term plus the successive differences between each term and the next -- in other words, a series? Or, if it didn't have a limit, how would it be useful? === Subject: Re: Sequences and Applications? > Hi all, > shows how numerical sequences are applied? I have heard, and maybe seen, > them used in continued fractions and the like, but what else is there? > TIA, > Lurch Dirk Vdm === Subject: Re: Sequences and Applications? Ya, I already tried that Dirk. All I got was descriptions of Analysis sequences to known sequences. I want to see applications to number theory or something. For example, an application where you are given a sequence and then have to find its limit. And, I guess I don't mean things like the Fibonacci sequence where its lim = +oo. I mean something like continued fractions, or something. Sorry, I can't be more specific, but I just don't know what they could be used for, other than proving that the sqrt(2) is exists. Lurch === Subject: Re: Sequences and Applications? > Ya, I already tried that Dirk. All I got was descriptions of Analysis > sequences to known sequences. I want to see applications to number theory > or something. For example, an application where you are given a sequence > and then have to find its limit. And, I guess I don't mean things like the > Fibonacci sequence where its lim = +oo. I mean something like continued > fractions, or something. Sorry, I can't be more specific, but I just don't > know what they could be used for, other than proving that the sqrt(2) is > exists. > Lurch Maybe this one then: More courses, but like you asked, there are some books as well. Dirk Vdm === Subject: Re: Sequences and Applications? Lurch > Ya, I already tried that Dirk. All I got was descriptions of Analysis > sequences to known sequences. I want to see applications to number theory > or something. For example, an application where you are given a sequence > and then have to find its limit. And, I guess I don't mean things like the > Fibonacci sequence where its lim = +oo. I mean something like continued > fractions, or something. Sorry, I can't be more specific, but I just don't > know what they could be used for, other than proving that the sqrt(2) is > exists. > Lurch > Maybe this one then: > More courses, but like you asked, there are some books as > well. > Dirk Vdm === Subject: Re: JSH: Consider Chebyshev > Tee-hee. http://www.shop4egifts.com/target.asp?item=/jpg/25112.jpg&width=314&height=4 00# http://www.fetchfido.co.uk/sound_files/giggle.wav > Guffaw. / _, _, /)| /) _'-`/(_/```,, / < ``--._ |. _ ) ` * ', ^ ^/_, * (* (#_/ `) /' * - ))_| U * _,//_ * ) >_* ((|_/ | | # ( # # # ( )/ _)> _))/(_ .-,, , //////),, ,,)/ ,. .- -'-,)/.')) http://www.georgetown.edu/faculty/ballc/animals/hyena.html === Subject: Re: JSH: Consider Chebyshev just say, What ever! > / > _, _, > /)| /) > _'-`/(_/```,, > / < ``--._ > |. _ ) ` * ', > ^ ^/_, * (* > (#_/ `) /' * - ))_| > U * _,//_ * --ils duces d'Enron! === Subject: Re: Coming soon -- new proof checking software > Within a week or so, I will be releasing a free beta version download for my > new proof checking software, DC Proof 1.0. > > In the mean time, here is a sampler from the User Guide: > > http://members.allstream.net/~dchris/DCProofT.chm > > It contains a tutorial that illustrates many of the main features of DC > Proof. Readers may be interested in both theoretical and a pedagogical > aspects of this application. Example 3, is a resolution of Russell's Paradox > without the usual prohibition on self-reference. > > Enjoy. > > Dan Christensen > Toronto, Canada > Your system appears to be a great aid to logicians writing proofs, and > will also hopefully lend more insight into the exact nature of proofs. > I will be happy to obtain a copy. > While a great bookkeeping aid, your system doesn't seem to do anything > that isn't being done by hand already. Agreed. Indeed, the user explicitly > enters the proof itself. This is a helpful tool, but I don't see how > you have resolved the Russell Paradox. You have only computerized > the same proof that is written out by hand. You correctly (IMHO) > conclude that there is no Russell Set (the set of sets that don't > contain themselves), which is the common conclusion one reaches from > seeing the contradiction. > However, the question remains, what do we do about it? How do we > define sets to include the sets that mathematicians use, but exclude > the Russell Set? Are you simply saying don't allow it? But what do > we allow? Everything else? No. But if postulating the existence of a particular set leads to a contradiction, then that set is said to not exist. To play it safe, I don't actually postulate the existence of any set in my theory of sets -- not even the empty set. That way, if I get a contradiction from, say, a premise giving Peano's Axioms for the set of natural numbers (previous posting), I would have to conclude that the set of natural numbers, as defined by PA, cannot exist. > ZF and various other axiomitizations of set theory attempt to define > sets in a way that that meets these two needs (completeness without > contradiction.) Do you really have a solution to that problem? I'm not sure. Maybe. Dan === Subject: Re: Coming soon -- new proof checking software > ZF and various other axiomitizations of set theory attempt to define > sets in a way that that meets these two needs (completeness without > contradiction.) Do you really have a solution to that problem? > I'm not sure. Maybe. > Dan Then how do you define which sets exist and how do you avoid contradiction? Charlie Volkstorf === Subject: Re: Coming soon -- new proof checking software > ZF and various other axiomitizations of set theory attempt to define > sets in a way that that meets these two needs (completeness without > contradiction.) Do you really have a solution to that problem? > I'm not sure. Maybe. > Dan > Then how do you define which sets exist and how do you avoid contradiction? I don't define which sets exist in my set theory. And I don't know if I can avoid contradictions altogether -- Frege gave me a good scare this morning! -- but I think my set theory deals nicely with some of the troublesome contradictions of naive set theory (like Russell's Paradox). Dan === Subject: Re: I WILL GET MY MONEY I did it, after dinner; he is Duly Authorized. now, can someone, please, authorize *me* ?? > |> Hi James. I checked your math - it's wrong. I will > |> so that no one will have to read them. > | > |what, exactly, gives you the authority to do such thing? > i said he could. > > and, who or what exactly, gives you the authority to do such thing? > > I said he could. > and who made you the authority??? --ils duces d'Enron! === Subject: Re: I WILL GET MY MONEY > > |> Hi James. I checked your math - it's wrong. I will > |> so that no one will have to read them. > | > |what, exactly, gives you the authority to do such thing? > i said he could. > and, who or what exactly, gives you the authority to do such thing? > > Oh, me, that was me. I did. > Ooops. Sorry, Nathan. I mis-remembered. It was you who I told > posts. > You can see how I would get a teensy bit confused, yes? That's OK. After all, I was the one who told you that you could tell me that. Nathan === Subject: Re: Mathematical Induction in Discrete Math.... (I assume that mathematical induction in discrete math is just standard mathematical induction). ical+induction&btnG=Google+Search The second hit (of the 92,700) has a clear and simple explanation: http://www.math.csusb.edu/notes/proofs/pfnot/node10.html This link has lots of examples: http://www.cs.odu.edu/~toida/nerzic/content/induction/examples.html That wasn't too hard now, was it? > can some one please break this down for me, im not all together clear > on this con cept as a whole === Subject: Re: Comment welcome -- martian mounds anomaly first of all, there is no reason for the geographical features of Mars to be random, if we can assume some sort of analog of plate tectonics (see Euler poles). after that, there will always be some minimum of patterns, which can become quite involved, as witness [place in France that's associated with Xian spooks], but it hardly means that they were built by folks on Mars. on the other hand, you can take a picture of any thing -- clouds, stucco, what ever -- and you can always manage to find close approximations to pi, phi, gamma, the dihedral of the tetrahedron etc., by drawing and measuring enough lines & angles. it is of no consequence, no matter what Art Bell's guests say about it! >> Nothing. What do the mounds have to do with math? > Well, take a look at the polygon BAGED. It displays the maximum number > of DIFFERENT parallel and perpendicular directions of lines drawn > among their vertex. The only other polygon that display this property > is a square with a fifth point being (as D in BAGED) in the > intersection of the diagonals of the square (parallelogram for BAGED) > drawn from the adjacent vertex. > It gets very much complex to say than to draw it. > I am not talking of little green martians. I am saying (and that is > objectable, but I will just consider scientific objections) that the > mounds distribution is not random. Nothing else. > And this pattern of maximality of parallel/perpendicular may suggest > some unknown geological mechanism. Of course, in order to discover it, > maths have to be applied (maths is applied in astrophysical alignments > of galaxies, for example, in a way similar to the Kolmogorov-Smirnov > test that was applied to the mounds. Seeing that the pattern may be > non-random, you go further in your analysis). --ils duces d'Enron! === Subject: newbie's question. Hi.. The terms like 'complete', 'compact' are very commonly used. What is the exact definition of those terms? or usual meaning? And is there any good reference web site to search a that kind of things? -- HJ === Subject: Re: newbie's question. > Hi.. > The terms like 'complete', 'compact' are very commonly used. > What is the exact definition of those terms? or usual meaning? > And is there any good reference web site to search a that kind of things? A terrific online reference is Interactive Real Analysis, at http://www.shu.edu/projects/reals/ A complete metric space is one in which every Cauchy sequence converges. A compact topological space is one in which every open covering of the space has a finite subcovering. === Subject: Re: newbie's question. >Hi.. >The terms like 'complete', 'compact' are very commonly used. >What is the exact definition of those terms? or usual meaning? >And is there any good reference web site to search a that kind of things? These are very basic terms in general topology (also known as point-set topology) and analysis. You really want to understand the basics of these subjects, rather than these specific definitions. If you pursue studies in mathematics, you will. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: newbie's question. Actually, I studied mathematical analysis and topology, a long time ago. I can't remember exact meaing of those words, I just understand those words very loosely. ;-( So, I need a main concept of those words. -- HJ >Hi.. >The terms like 'complete', 'compact' are very commonly used. >What is the exact definition of those terms? or usual meaning? >And is there any good reference web site to search a that kind of things? >-- >HJ > These are very basic terms in general topology (also known as point-set > topology) and analysis. You really want to understand the basics of > these subjects, rather than these specific definitions. If you pursue > studies in mathematics, you will. > -- > Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: newbie's question. > Actually, I studied mathematical analysis and topology, a long time ago. > I can't remember exact meaing of those words, I just understand those words > very loosely. ;-( > So, I need a main concept of those words. Just use Google! Or get a copy of a book like Munkres: Topology - A first course. Rene. -- Ren.8e Meyer Student of Physics & Mathematics Zhejiang University, Hangzhou, China === Subject: Re: Fourier transform of switched cos wave > 'on' over -tau/2 to tau/2, and 0 at all other points in time. > In a book i have it explains the process as convolution which is ok. > However i believe i should also be able to work out the fourier > transform using the following: > /tau/2 > | > | cos(w0.t).exp(-j.w.t) dt > | > / -tau/2 > and using the exponential form of cos to give > /tau/2 > | - - > | |exp(j.w0.t)+exp(-j.w0.t)|exp(-j.w.t) dt > | |----------------------- | > | | 2 | > | - - > / -tau/2 > and simplyfying to two integrals > /tau/2 / tau/2 > | | > | exp(j(w0-w)t) dt + | exp(-j(w0+w)t) dt > | ------------- | -------------- > | 2 | 2 > | | > / -tau/2 / -tau/2 > basically i am ending up with: > sin((w0-w)tau/2) + sin((w0+w)tau/2) > ---------------- ---------------- > w0-w w0+w > no matter how much i manipulate the equation i cannot arrive at the > same result as in my book. in particular the term w0-w in the above > equation should be the other way around as w-w0 (im sure im making a > mistake somewhere). the correct result is: > tau sin((w-w0)tau/2) + tau sin((w+w0)tau/2) > --- ---------------- --- ---------------- > 2 (w-w0)tau/2 2 (w+w0)tau/2 > could anyone kindly suggest where i am going wrong, or point me to any > sources where this problem is derived in full (with intermidiate steps > - maths isnt my strongest point!). Chris, As Glen suggests, use the Fourier Series if what you meant to say is that the waveform is really a switched cosine wave and is periodic with only the positive half cycles not zero. You will find that the sine terms go to zero because the waveform is even around t=0. So, only the cosine terms yield nonzero results. If it's not periodic, then it's been multiplied by a gate which is the same as convolving in the frequency domain by a sinc - which appears to be the form that you have. In that case, I'd not call it a switched cosine, rather I'd call it a single positive half cycle of a cosine. So, resolve (or re-state) the exponential into sines and cosines and see what you get. Fred === Subject: Re: mode question >> > >>my daughter had the following multiple choice question on her maths >>test. I >>dont agree with any of the answers. >> >>This is the question in full >> >>Q Julie kept a record of the number of pairs of shoes she had. >> >>year no of pairs of shoes >> >>1996 11 >>1997 9 >>1998 12 >>1999 10 >>2000 13 >> >>The mode is >>a. 1996 >>b 1997 >>c 1998 >>d 1999 >>e 2000 >> >>the correct answer given was e 2000. >> >>I say there is no mode and if there was a mode , then the answer >>would not >>be a year but a number of pairs of shoes. What do the statistics >>experts >>think? >> >> >> >The mode is 13, not the year. You are correct that the correct answer >was not offered. > >>Oops. Typed without thinking. 10 sqrt(7) pi / 3 lashes with a wet noodle. >>As others have pointed out, 13 is not the mode. Or rather, I would >>agree with someone else, that each of the values 9 through 13 is a mode. >>In any case, the teacher is wrong. S/he may have had something else >>(i.e., 1996 appearing eleven times, etc.) in mind, but that would be a >>rather unusual (even nonsensical) data set for this situation. >> >Minor change in question produces a plausible situation: >} Q Julie has marked each pair of shoes with the year in which she >} purchased them, and now wants to know which mark occurs most >} frequently. >} year no of pairs of shoes marked with that year >} 1996 11 >} 1997 9 >} 1998 12 >} 1999 10 >} 2000 13 >} The mode is >} a. 1996 >} b 1997 >} c 1998 >} d 1999 >} e 2000 >I think we can agree that (e) is the right answer to this question. Bestimmt. But here we are twisting the wording of the question to fit the intended answer. As far as I can tell (from this and other examples elsewhere in this thread), the teacher has the misconception that the mode is defined as the argmax of a natural-number-valued function. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: mode question >Q Julie kept a record of the number of pairs of shoes she had. > >year no of pairs of shoes > >1996 11 >1997 9 >1998 12 >1999 10 >2000 13 > >The mode is >a. 1996 >b 1997 >c 1998 >d 1999 >e 2000 > >the correct answer given was e 2000. > ... >In any case, the teacher is wrong. S/he may have had something else >(i.e., 1996 appearing eleven times, etc.) in mind, but that would be a >rather unusual (even nonsensical) data set for this situation. > >>Minor change in question produces a plausible situation: >>} Q Julie has marked each pair of shoes with the year in which she >>} purchased them, and now wants to know which mark occurs most >>} frequently. >>} year no of pairs of shoes marked with that year >>} 1996 11 >>} 1997 9 >>} 1998 12 >>} 1999 10 >>} 2000 13 >>} The mode is >>} a. 1996 >>} b 1997 >>} c 1998 >>} d 1999 >>} e 2000 >>I think we can agree that (e) is the right answer to this question. > Bestimmt. But here we are twisting the wording of the question to fit > the intended answer. As far as I can tell (from this and other examples > elsewhere in this thread), the teacher has the misconception that the > mode is defined as the argmax of a natural-number-valued function. One strategy when responding to a multiple choice question is to choose a problem interpretation that fits the available answers, even when this interpretation is literally incorrect. I've been trying to come up with a good understanding of the misunderstandings/misinterpretations/ambiguities here. And I have a few thoughts. One problem area is the presentation format for data sets. If you have a set of 55 shoes and a year of purchase for each shoe, you have two choices. You can present that as a list of 55 year numbers. Or you can present it as a frequency table of 5 ordered pairs (year, number of shoes purchased in that year) On a data set for which it is useful to calculate a mode, it is likely that the frequency table presentation format is going to be far more compact than the list format. If we assume that the question is being asked about the 55 element data set (year of purchase for each of 55 shoes) and that this data set is being presented in frequency table format then everything makes sense. In a perfect world, we'd be able to look at a data set, see that it presented as a list of ordered pairs and say ahah! -- it's a data set in frequency table format or we could look at a list of values and say ahah! -- it's a data set in straight list format. But in the real world, we'll get data presented in many ways and with lots of relevant and irrelevant data tacked on. Even with a single variable data set, we may find an index value presented. e.g 1 11 2 9 3 12 4 10 5 13 We are perfectly willing to ignore the line number when computing the mean, median or mode of the second column. And sometimes the index column is semi-meaningful. 1996 11 1997 9 1998 12 1999 10 2000 13 Again, we are perfectly correct to ignore the first column when computing the mean, median or mode of the second column. But this last format is _indistinguishable_ from the frequency table presentation format for the data set: 1996 1996 1996 1996 1996 1996 1996 1996 1996 1996 1996 1997 1997 1997 1997 1997 1997 1997 1997 1997 1998 1998 1998 1998 1998 1998 1998 1998 1998 1998 1998 1998 1999 1999 1999 1999 1999 1999 1999 1999 1999 1999 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 I know which presentation format I'd rather see when I'm asked to compute a mode. John Briggs === Subject: Re: mode question > > >The mode is 13, not the year. You are correct that the correct answer >was not offered. > > The mode is *not* 13. The mode is 2000. > > Doug > The most common value obtained in a set of observations. > from: http://mathworld.wolfram.com/Mode.html > The set of observations is here: {11,9,12,10,13}. In other words > the mode is not unique, and all five of them are the modes. > So, you are totally correct: > I say there is no mode > Yes. (or all five of them are the mode, it is nowhere states > in the definition it must be unique.) Now, I'm a little confused. According to some sources, like Dr. Math for instance, if no number occurs more than once, there is no mode. Other sources, like Mathworld, make no explicit mention of this case, but say the mode is the most frequent value -- if two or more tie, there are two or more modes. Then there are sources, like Introduction to the Theory of Statistics, by Mood, Graybill, and Boes, that also make no mention of this case, but claim the mode is the maximum value of f_X (the density function of a random variable X), which will necessarily exist for a finite range of values. Is there any kind of agreement among statisticians as to whether the mode exists for a list of numbers each appearing only once? I feel it should. According to Dr. Math, the list 2,2,3,3,4,4 is trimodal, with modes 2, 3 and 4. But, the list 2,3,4 has no mode. I don't see what is to be gained by declaring the second list to have no mode. The relative frequencies are the same for the two lists. Moreover, both lists have the same mean and the same median. Can anyone clarify this? > and if there was a mode , then the answer would not > be a year but a number of pairs of shoes. > Also correct. > Wilbert === Subject: Re: mode question Let U be a set and S be an element of U^n. Call S a data set with values in U. Define m_S: U -> N by m_S(x) = #{i <= n: x_i = x} Then x in U is a mode of S iff (m_S)(x) = max {(m_S)(y): y in U}. Thus, if S = {11,9,12,10,13}, each of the values 9 thorugh 13 are modes. Let p be a discrete distribution on some probability space S where all singletons are events. Then x in S is a mode of p iff p{x} = max { p{y}: y in S}. Let f be the density of some absolutely continuous distribution on R^n. Then x in R^n is a mode of the distribution iff f(x) = max {f(y): y in R^n} In none of these definitions is a mode necessarily unique. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: relations transitive > Soit E l'ensemble {1,2,3}. Trouvez des relations R et S sur E qui > sont transitives mais telles que R o S n'est pas transitive > Est ce possible? > Yes, for example: > R = { (1,2), (2,3), (1,3) } > S = { (1,1), (2,2) } > RoS = { (1,2), (2,3) } Oops, that was a wrong example :-( RoS = { (1,2), (2,3), (1,3) } Arturo gave a good one. Sorry. Dirk Vdm === Subject: [Fwd: Re: newbie's question.] I have a high probability that there is at least one error in the sequel. If so, I am sure someone will correct me. I meant to add some standard references: For topology, Munkres JR. Topology. For analysis, Rudin W. Principles of Mathematical Analysis. -------- Original Message -------- === Subject: Re: newbie's question. <3FCC3CE0.7090209@rutcor.rutgers.edu> <01bd01c3b8a7$b9434a70$b85c78d2@your993grr4ffc> Definitions: * A partially ordered set is complete iff every subset that has a lower bound has a greatest lower bound. * A metric space is complete iff every Cauchy sequence converges. * A topological space is compact iff every open cover has a finite subcover. * A topological space is countably compact iff every infinite subset has a limit point. * A metric space M is totally bounded iff for any e > 0, there exists a finite collection a of subsets of M such that Ua = M and diam A < e for all A in a. Facts (theorems): * In a complete set in the former sense, any subset with an upper bound has a least upper bound. * The real numbers are complete in both senses. * Any complete (in the former sense) ordered field is isomorphic to the reals. * R^n is complete (in the latter sense). * The L^p spaces are complete for 1 <= p <= infinity. * A compact subset of a Hausdorff space is closed. * A closed subset of a compact topological space is compact. * The image of continuous map on a compact space is compact. * A compact space is countably compact. * A countably compact metric space is compact * A metric space is compact iff it is complete and totally bounded. o This fact is remarkable since the latter two properties are metric, whereas the first is topological. o Corollaries: + Heine-Borel Theorem: A subset of R^n is compact iff it is closed and bounded. + Extreme Value Theorem: Any continuous real-valued function on a closed, bounded subset of R^n attains its extrema. + Bolzano-Weierstrass Theorem: A subset of R^n is countably compact iff it is closed and bounded. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Aleph0 - 1 conjecture Please can you show a proof that contradicts my conjecture, saying: By base value expansion method we can represent an aleph0-1 list of rational numbers (repetitions over scales) where the missing rational number is the diagonal rational number used as an input for Contor's function, where the result depends on some arbitrary order of the Aleph0-1 list and some rule, which is used by Cantor's function. The rule is: a) Every 0 in the original diagonal number is turned to 1 in Cantor's new number. b) Every non-zero in the original diagonal number is turned to 0 in Cantor's new number. When we have the Aleph0 complete list of rational numbers, represented by base value expansion method, then and only then we can find only some irrational diagonal number (no repetitions over scales) as an input for Cantor's function. === Subject: Re: Aleph0 - 1 conjecture > Please can you show a proof that contradicts my conjecture, saying: I read a few lines of your work at the link you provided yersterday... it's crawling with mistakes !! === Subject: Re: Aleph0 - 1 conjecture > Please can you show a proof that contradicts my conjecture, saying: > By base value expansion method we can represent an aleph0-1 list of > rational numbers (repetitions over scales) where the missing rational > number is the diagonal rational number used as an input for Contor's > function, where the result depends on some arbitrary order of the > Aleph0-1 list and some rule, which is used by Cantor's function. The set you are talking about is Q{q}, the complement in Q of some singleton set. The cardinality of the set is aleph_0, not aleph_0 - 1. It has the same cardinality as Q itself. > The rule is: > a) Every 0 in the original diagonal number is turned to 1 in Cantor's > new number. > b) Every non-zero in the original diagonal number is turned to 0 in > Cantor's new number. First, it should be noted that your rule does not work in the context of Cantor's theorem, because it fails to take account of the fact that some rational numbers have a dual representation (one ending in all 0's, and one ending in all 9's). If we apply your rule to a given list of rationals, it's entirely possible that the number produced might be something like 0.01101001110000000000000000000000000... with just a finite number of 1's and the rest 0's. By the way the number was constructed, it's clear that this particular decimal representation does not appear anywhere in the original list. However, it is possible that the original list might contain 0.01101001109999999999999999999999999... which is the same number. Hence your attempt at producing a number not in the original list fails. Cantor knew about this difficulty and avoided it by specifying the new number in a way that the problem does not arise. For example, we can change the rule to: a) Every 2 in the original diagonal number is turned to 3 in Cantor's new number. b) Every non-2 in the original diagonal number is turned to 2 in Cantor's new number. Now we can talk about your original conjecture. The answer is no. Suppose we have a list defined by a bijection f: N -> Q{q}, where q is some rational number. If the decimal representation of q contains any digits other than the two that are mentioned in the rule (2 and 3 in the version of the rule that I described), then it is obvious that q cannot be produced by the diagonal rule. Let's consider the case where q = 2/9 = 0.2222222222222222.... According to the rule, the only way this number can be produced is if every digit appearing on the diagonal is a non-2. That means that the numbers 0.02222222222222222222222... 0.12222222222222222222222... 0.32222222222222222222222... 0.42222222222222222222222... 0.52222222222222222222222... 0.62222222222222222222222... 0.72222222222222222222222... 0.82222222222222222222222... 0.92222222222222222222222... can not appear anywhere else except at the first position in the list, since any other position would result in a 2 on the diagonal. But only one of these nine numbers can possibly be located at the first position in the list, and therefore the other eight must be missing altogether. A similar argument applies to any other rational number q whose decimal representation consists entirely of 2's and 3's. We just have to vary the argument by noticing what digits are forbidden on the diagonal, and how many numbers consist entirely of forbidden digits except in one position. > When we have the Aleph0 complete list of rational numbers, represented > by base value expansion method, then and only then we can find only > some irrational diagonal number (no repetitions over scales) as an > input for Cantor's function. I'm not exactly sure what you mean by that, but I don't see any way of interpreting your statement that would make it correct. For example, there is a proper subset of the rationals that, no matter how arranged, will always produce an irrational when the diagonal rule is applied. We can achieve that by enumerating all the rationals whose decimal representations consist entirely of 2's and 3's. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Aleph0 - 1 conjecture > Please can you show a proof that contradicts my conjecture, saying: > By base value expansion method we can represent an aleph0-1 list of > rational numbers (repetitions over scales) where the missing rational > number is the diagonal rational number used as an input for Contor's > function, where the result depends on some arbitrary order of the > Aleph0-1 list and some rule, which is used by Cantor's function. infinity minus one eh? > The rule is: > a) Every 0 in the original diagonal number is turned to 1 in Cantor's > new number. > b) Every non-zero in the original diagonal number is turned to 0 in > Cantor's new number. > When we have the Aleph0 complete list of rational numbers, represented > by base value expansion method, then and only then we can find only > some irrational diagonal number (no repetitions over scales) as an > input for Cantor's function. but we already have a bijection between Q and N. i don't see the point of your conjecture. obviously the missing rational number can only have ones and zeroes in it, so unless you're doing base 2 stuff you're not going to add in all of them. this way, cos presumably you wanted to add to an aleph0 - 2 list before that... but where would that process start? certainly not with a finite list. what's this conjecture for? purely interests sake? given any rational with zeroes and ones in its expansion, then it is possible to construct an countably infinite long list of rationals for which the cantor argument produces your chosen rational. but the construction certainly doesn't contain 'many' rationals: choose them to be pretty much zero everywhere. except up to two places. === Subject: Parity of a permutation Hey, i'm preparing a test, and got stuck with the following problem Any ideas/proofs would be greatly appreciated. If mm where sigma is a permutation of S_n. Prove that the parity of the permutation in S_m stays constant when the permutation is considered in this way (as an element of S_n). === Subject: Re: Parity of a permutation > Hey, i'm preparing a test, and got > stuck with the following problem > Any ideas/proofs would be greatly appreciated. > If m assuming j (as an element of S_n) acts over 1,2,...,m,...n > as it did over 1,2,...,m and suppose sigma fixes j>m where > sigma is a permutation of S_n. Prove that the parity > of the permutation in S_m stays constant when the permutation > is considered in this way (as an element of S_n). You can write this permutation as a product of transposition in S_m. But this is also a product of transposition in S_n... -- Maxi === Subject: Re: a nice e^n like integer sequence >So D(n) grows too fast. Maybe there are other similar recurrences with >poly coeffs? Like: > a(n) = a(n-1) + (n-1)a(n-2) ... >>The sequence a(n) = a(n-1) + (n-1)a(n-2) -- how fast does it grow? >>I don't know at all how to analyze such a sequence. Any ideas? >>Based on some empirical checking, the sequence appears to actually >>grow _faster_ than e^n. > Note that a(n) grows faster than b(n) = (n-1)*b(n-2), > which grows like const*(n/e)^(n/2), of course much faster than e^n. er...yeah. hmmm...note to self, think before posting. OK, here's another admittedly half-baked idea. e is somewhere between 2 and 3. So sometimes multiply by 2, other times by 3, in a ratio that approaches e. that is e^n ~ 2^k 3^{n-k} After thinking, the values will bounce around (as opposed to behaving uniformly) but be a bit more than e^n (never more than 3/2 e^n). So what are a and b? or similarly how about (2^(lg e))^n? (which is like multiply by 4 every so often). But maybe lg e has a nice continued fraction (which by inspection it doesn't seem to). Mitch === Subject: Re: a nice e^n like integer sequence > >So D(n) grows too fast. Maybe there are other similar recurrences with >poly coeffs? Like: a(n) = a(n-1) + (n-1)a(n-2) ... >> >>The sequence a(n) = a(n-1) + (n-1)a(n-2) -- how fast does it grow? >>I don't know at all how to analyze such a sequence. Any ideas? >>Based on some empirical checking, the sequence appears to actually >>grow _faster_ than e^n. > > Note that a(n) grows faster than b(n) = (n-1)*b(n-2), > which grows like const*(n/e)^(n/2), of course much faster than e^n. > er...yeah. hmmm...note to self, think before posting. OK, here's another > admittedly half-baked idea. e is somewhere between 2 and 3. So > sometimes multiply by 2, other times by 3, in a ratio that approaches e. > that is e^n ~ 2^k 3^{n-k} > After thinking, the values will bounce around (as opposed to behaving > uniformly) but be a bit more than e^n (never more than 3/2 e^n). > So what are a and b? > or similarly how about (2^(lg e))^n? (which is like multiply by 4 every > so often). But maybe lg e has a nice continued fraction (which by > inspection it doesn't seem to). The product of the primes up to n is asymptotic to e^n, so you could use 2, 2, 2*3, 2*3, 2*3*5, 2*3*5, 2*3*5*7, 2*3*5*7, 2*3*5*7, 2*3*5*7, 2*3*5*7*11, .... -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: What are the constants in the computing sciences? What are the constants in the computing sciences? By computing sciences I mean Set Theory (ZF), the Theory of Computation (Turing), Recursion Theory (Kleene), Proof Theory (Godel), Foundations of Mathematics (Peano), Paradoxes (Russell), etc. Here are some examples: 1. Set Theory a. {} b. {{}} 2. Theory of Computation a. the smallest Turing Machine that never halts b. the smallest Universal Turing Machine (simulates any given Turing Machine plus its input) 3. Recursion Theory a. the smallest program that outputs only itself b. the function that maps a 1-input program and literal to the smallest 0-input program that uses the literal in place of the input c. the function that maps any 1-input program that computes function f(x) into the smallest program that computes f(f(x)) 4. Proof Theory a. the smallest wff in PA that is undecidable (neither provable nor refutable) 5. Foundations of Mathematics a. 0 6. Paradoxes a. the smallest English sentence that is neither true nor false In each case, we are taking a system that can be used to model objects in the real world, and instead are talking only about objects created within the theory itself. Each constant is created completely within the system. What other constants can people think of? Charlie Volkstorf Cambridge, MA http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/20021008.1/1 http://www.arxiv.org/html/cs.lo/0003071 === Subject: Re: An interesting integral >>An interesting integral just appeared in alt.math, and there should be >>a neater solution than I found. The problem is to evaluate >> >> int_0^1 int_0^{1-x} 1/(1-axy) dy dx, >> >>where |a| < 4. >> Rotate axes with x = u+v and y = u-v. Then >> |1 |1-x dy dx >> | | ------- >> | 0 | 0 1 - axy >> |1/2 | u 2 dv du >> = | | --------------- [1] >> | 0 | -u 1 - au^2 + av^2 >> Then with a simple u -> u/sqrt(a) and v -> v/sqrt(a) substitution >> [1] >> |sqrt(a)/2 | u 2/a dv du >> = | | ------------- >> | 0 | -u 1 - u^2 + v^2 >> |sqrt(a)/2 4 u >> = | ------------- atan( ----------- ) du [2] >> | 0 a sqrt(1-u^2) sqrt(1-u^2) >> Here it is pretty easy to see the proper substitution is u = sin(w) >> [2] >> |asin(sqrt(a)/2) 4 >> = | - w dw >> | 0 a >> 2 2 >> = - asin(sqrt(a)/2) [3] >> a >Marvelous! I think this one goes into my kit of >hard-problems-to-ask-students. (With the change of variables as a >hint, of course.) This only handles the case 0 <= a <= 4. For the case of negative a, we need to do more work. Rather than carry around a negative a, let us rewrite the integral, still using x = u+v and y = u-v: |1 |1-x dy dx | | ------- | 0 | 0 1 + axy |1/2 | u 2 dv du = | | --------------- [1] | 0 | -u 1 + au^2 - av^2 Again with a simple u -> u/sqrt(a) and v -> v/sqrt(a) substitution [1] |sqrt(a)/2 | u 2/a dv du = | | ------------- | 0 | -u 1 + u^2 - v^2 |sqrt(a)/2 4 u = | ------------- atanh( ----------- ) du [2] | 0 a sqrt(1+u^2) sqrt(1+u^2) If we let u = sinh(w), then sqrt(1+u^2) = cosh(w) and [2] |asinh(sqrt(a)/2) 4 = | - w d w | 0 a 2 2 = - asinh(sqrt(a)/2) [3] a Since asin(ix) = i asinh(x), plugging -a into the old result gives the new result. Rob Johnson take out the trash before replying === Subject: Re: Help / ideas with statistical prediction problem Toby. === Subject: Special Kauffman polynome values I wonder whether there are interesting parameter values for the Kauffman polynome K(A) (with the writhe variable set to 1 as the most basic invariants should be achiral). E.g. A=golden mean All knots/links have the value +-sqrt(5)^n A=-2 All knots/links are integer squares Surely this has been investigated? -- Hauke Reddmann <:-EX8 Private email:fc3a501@math.uni-hamburg.de For our chemistry workgroup,remove math from the address === Subject: is the sci.math board moderated ? how do I de-post ? Jack === Subject: Re: is the sci.math board moderated ? how do I de-post ? Hello Jack! I assume that you are using Google Groups, as in your headers it says that you are posting from Google. Google does have a way that you can delete your own posts. There is a set of instructions for doing this, and these instructions can be found at: I hope that this helps! Anthony > Jack === Subject: Re: is the sci.math board moderated ? how do I de-post ? No, sci.math is not moderated. What does de-post mean? >Jack ************************ David C. Ullrich === Subject: Re: is the sci.math board moderated ? how do I de-post ? Adjunct Assistant Professor at the University of Montana. >No, sci.math is not moderated. What does de-post mean? option, provided you are logged into the same account as the one which sent the message in the first place; from trn, simply press cntrl-c while reading the message. Google allows you to delete a message by depend on his newsreader. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: is the sci.math board moderated ? how do I de-post ? >option, provided you are logged into the same account as the one which >sent the message in the first place Of course this is not a feature of the newsreader, but the NNTP === Subject: Re: is the sci.math board moderated ? how do I de-post ? Discussion, linux) >>No, sci.math is not moderated. What does de-post mean? > option, provided you are logged into the same account as the one which > sent the message in the first place; from trn, simply press cntrl-c > while reading the message. Google allows you to delete a message by > depend on his newsreader. His newsreader is evidently Google. I know nothing about posting -- Jesse Hughes She moaned, in pain and pleasure, as, in a confused whirlwind, she glimpsed an image of Saint Sebastian riddled with arrows, crucified and impaled. --Mario Vargas Llosa on category theory === Subject: Re: is the sci.math board moderated ? how do I de-post ? Adjunct Assistant Professor at the University of Montana. >> option, provided you are logged into the same account as the one which >> sent the message in the first place; from trn, simply press cntrl-c >> while reading the message. Google allows you to delete a message by >> depend on his newsreader. >His newsreader is evidently Google. I know nothing about posting other servers, but trn; I usually use the message-ID number. If you are reading through the post, and click on Original Format, which will display the complete headers, including the message-id number). It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: is the sci.math board moderated ? how do I de-post ? ... >His newsreader is evidently Google. I know nothing about posting > other servers, but that, the probability that the message is still around at a number of pronouncing your error. Tough luck if what you did was illegal. (I once posted something to comp.sources.mac, I think, but received a message seconds later by somebody who accused me of plagiarism. I immediately I do not think the accuser had something real, I posted a completely different version ten minutes later... Finding all archives can be a pain. And that was back in 1989 or something like that.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Euclidian lattice? I have a question. I know what a lattice is but when is it called Euclidian? thank in advance, David === Subject: Re: Suggestions for Applied Maths [...] > So, I'm looking for suggestions for the following: > Areas of physics I should pursue. > Any other interesting applications of mathematics. > Good books for the above. [...] For Information Theory and Coding Theory, I'd sugest Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay in the Physics Department at Cambridge. It was published by Cambridge University Press in Information Theory, e.g. Shannon's work, Hamming codes, Turbo codes but also Machine Learning and Neural Networks (I know essentially nothing about the latter two). The whole book is available for download (5 to 9 Megabytes) from: http://www.inference.phy.cam.ac.uk/mackay/itila/p0.html It's also possible to download small chunks of the book, for example Chapter 1 from: http://www.inference.phy.cam.ac.uk/mackay/itprnn/ps/ You might be interested in learning about the recent Digital Fountain (TM) technology; for example, c.f.: http://www.digitalfountain.com/products/tf/metaContent.cfm David Bernier === Subject: Re: Suggestions for Applied Maths > [...] >>So, I'm looking for suggestions for the following: >>Areas of physics I should pursue. >>Any other interesting applications of mathematics. >>Good books for the above. > [...] > For Information Theory and Coding Theory, I'd sugest > Information Theory, Inference, and Learning Algorithms, > by David J.C. MacKay in the Physics Department at Cambridge. > It was published by Cambridge University Press in > Information Theory, e.g. Shannon's work, Hamming codes, > Turbo codes but also Machine Learning and Neural Networks > (I know essentially nothing about the latter two). > The whole book is available for download (5 to 9 Megabytes) > from: > http://www.inference.phy.cam.ac.uk/mackay/itila/p0.html > It's also possible to download small chunks of the book, > for example Chapter 1 from: > http://www.inference.phy.cam.ac.uk/mackay/itprnn/ps/ > You might be interested in learning about the recent > Digital Fountain (TM) technology; for example, c.f.: > http://www.digitalfountain.com/products/tf/metaContent.cfm > David Bernier planning to read up on the notes for our information theory course this term over the holidays (it clashed with something else I was interested in, so I wasn't able to attend both), and it looks like a very interesting subject. I'm going to have to give the digital fountain technology a miss for now, as my knowledge of computer science in general doesn't seem quite up to it yet, but will certainly file it for later reading. David === Subject: Re: Suggestions for Applied Maths > You may do worse than looking into the area of fluid mechanics. There > are interesting problems people are working on to do with the > singularities that form when droplets form and leave the bulk of the > fluid. There are also interesting problems in the area of > viscoelasticity. You may have to get into numerical analysis and > computer programming to follow this up. Problems in fluid mechanics > may also have industrial relevance. The rheology of grease, paint, > suspensions, polymers, etc flowing through pipes etc may not be > particularly exciting but is quite challenging mathematically and of > relevance to many industries. You forgot to add Fluid Dynamics is less fashionable than HEP/GR/String Theory at the moment so there will be fewer problems finding PhD places or funding. -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. === Subject: Is a retract an identification map? well, I don't know if the term identification map is the right one in English, but it means the following: It is a continuous surjective map between topological spaces X and Y, namely f:(X,Tx)->(Y,Ty) where Ty is the final topology w.r.t. f. I am now trying to prove that a retract, namely a continuous map r: X -> A, A subset X, r|_A = I_A: A -> A is such an identification map, but I don't get it. Above, A is a subspace of X, i.e. A has the subspace topology of X. In particular, I already tried the following approaches: 1) Show that the subspace topology on A and the final topology on A w.r.t. r are identical, but after some tries I gave up. 2) Use: If a map f is a continuous open surjection, then it is an identification map. Continuous surjection is given, but I can't show that r is necessarily open, especially for open subsets of X that lie partly in A and partly in X-A. 3) The same with: f:X->Y is an identification map iff ( U is open in Y iff f^{-1}(U) is open in X). The direction => is no problem as r is continuous, but with <= I got the same problem as above in showing that r is open. Can anyone give me a hint? I didn't used the property that r restricted to A is the identity map and I am really not shure where to use it. I think the problem is the case when an open subset of X lies partly in A and partly outside. Is the image of this subset an open subset of A? Rene. -- Ren.8e Meyer Student of Physics & Mathematics Zhejiang University, Hangzhou, China === Subject: Please help!! I'm puzzled!! Last week I asked for help with one of the homework's problems in a Probability & Random Signals course. The assignment is already submitted, and the teacher did indicate that we could use the almost solution from the book, as long as we justify *why* that solution is applicable. But someone indicated in my previous thread that there is an easier solution, using the fact that any linear combination of Gaussians is Gaussian, and the property about uncorrelated <==> independence for Gaussians. I can't seem to find that easier solution!!! And I normally get obsessed when I can't beat a puzzle! The original problem was: if x(t) is WSS Gaussian with zero-mean, and z(t) = x^2(t), show that Czz(tau) = 2Rxx^2(tau). The almost solution from the book is the property that if x1 and x2 are Gaussian with zero-mean, then: E{x1^2 x2^2} = E(x1^2} E{x2^2} + 2R^2{x1 x2} This directly leads to the solution if we make x1 = x(t) and x2 = x(t + tau) -- given that the process x is WSS, and thus E{x1^2} and E{x2^2} are the same, the mean of z(t). But, how can it be solved directly? More easily?? I tried things like using E{x1 + x2}, E{(x1+x2)^2}, but none of those attempts got me anywhere. I mean, I need to get x1^2 x2^2 somewhere, and as soon as I introduce a square operation, the thing stops being Gaussian and the nice properties no longer hold; with the above property (which in the book they obtain using moment generating functions), seems the only formula I know of that introduces the squares in the mix. Please help! === Subject: Re: Donald Duck's Physics Class Nothing of importance. I've assembled here the briefest summary I can of my a > priori perspective on gravity... > JS: No physicist is interested in your apriori perspective on gravity. > Apriori is not a good word to use with physicists. Jack, a priori is Latin. Apriori is not a word. David A. Smith === Subject: Integrability of over an interval I found the following exercise in a book: If f is bounded on [a, b] and integrable over [c,b] for every c in (a,b), then f is integrable over [a,d] and lim c->a Int (from c to b) f(x) dx = Int (from a to b) f(x) dx . For the first part it was sugested we show that, for every eps>0, there's a partition P of [a,b] such that U(f,p) - L(f,P) a the second member gets as small as desired. OK? Amanda === Subject: Re: Integrability of over an interval > I found the following exercise in a book: > If f is bounded on [a, b] and integrable over [c,b] for every c in > (a,b), then f is integrable over [a,d] and lim c->a Int (from c to b) > f(x) dx = Int (from a to b) f(x) dx . For the first part it was > sugested we show that, for every eps>0, there's a partition P of [a,b] > such that U(f,p) - L(f,P) Lebesgue's criterion. According to it, for every c in (a,b), the set > Dc of discontinuites of f in [c,b] is null (has measure 0). Therefore, > for every c we can cover Dc with a countable collection of intervals > whose total length is < eps. If 0< eps +eps/3, I_0 = (c -eps/2 , c+eps/2) and choose a countable collection > I ={I1, I2...} of open intervals that covers Dc and has total length < > eps. Since I_0 contains [a,c], it contains all the discontinuities of > f on [a,c]. If D is the set of discontinuities of f on [a,b], then > adjoining I_0 to I gives a countable collection J = {I_0, I-1, I-2...} > of open intervals that covers D and has total length < eps + > Length(I_0) = eps + eps = 2eps. Since eps is arbitrary, we conclude D > is null and Lebesgue's criterion ensures f is integrable over [c,d]. > Is this proof correct? It seems to be correct (I must admit I didn't check all the details of of the proof), but I do have two problems with it. Firstly, you're rather reinventing the wheel as you prove it. All you need to show that the set of discontinuities of f on [a, d] is null is countable subadditivity of the measure. Secondly it's a bit of a 'hammer to crack an eggshell' proof. Lebesgue's criterion requires a fair bit of work to prove, and there are much much more elementary proofs of this result which would give you a better understanding of what's going on. For example: Fix e > 0. Pick y > a 'quite close' to a (where how close will depend on e and an upper bound for |f|). Then by choosing an appropriately fine partition D of [y, d] (which will exist by integrability on [a, d]), you can make the difference in the upper and lower sums on D union{a} less than e. But D union {a} is a partition of [a, d]. Hence you've found such a partition, so the function is integrable on [a, d]. (You do of course need to fill in the details there). > Once we've proved the integrability of f, the second part is kinda > easy, all we have to do is observe that |Integral (a to c) f(x) dx| <= > (c-a) W(f), W(f) = oscillation of on [a,b]. Therefore, when c ->a the > second member gets as small as desired. OK? With a little bit more work the proof I sketched also gives you that the integral over [a, d] is the limit of the integral over [c, d] as c->a, but it's probably easier to do that second part like you did it anyway. (Or in a similar manner at any rate) David === Subject: relative topology in R^2 Can someone help for this; Let B is the class of {(x,y)| a <= x < b, c <= y < d} in R^2, then B is the base for a topology T on R^2. 1) Show that the relative topology T_A on the line A={(x,y)| x + y = 0 } (ie. x = -y ) is a discrete topology. 2) Show that the relative topology T_B on the line B={(x,y)| x = y } is not discrete. === Subject: Re: relative topology in R^2 Originator: grubb@lola >Can someone help for this; >Let B is the class of >{(x,y)| a <= x < b, c <= y < d} >in R^2, then B is the base for a topology T on R^2. >1) Show that the relative topology T_A on the line > A={(x,y)| x + y = 0 } (ie. x = -y ) > is a discrete topology. >2) Show that the relative topology T_B on the line > B={(x,y)| x = y } > is not discrete. Sure. First of all, what does the phrase 'relative topology' mean? What will be a base for the relative topologies on those sets? After considering that question, you may want to look at intersections of your sets A and B with elements of your original base. Drawing a picture may help. What does it mean to be the discrete topology? What are the smallest sets you can get to be open in A? What type of sets are basis sets for B? --Dan Grubb === Subject: Re: A Summary of Cubic Approximations of Sine on [-Pi/2, Pi/2] Originator: jgamble@ripco.com (John M. Gamble) >The recent sci.math.num-analysis thread sinus approximation became rather >unwieldy, and so I thought it might be helpful to summarize results which >are optimal according to various measures of error. (I have also >crossposted to sci.math for others who might be interested.) >The objective was to determine the third-degree polynomial which best >approximates the sine function on [-Pi/2, Pi/2]. Naturally, many people >think first of the Maclaurin polynomial x - x^3/6, but it does not minimize >any standard measure of error on [-Pi/2, Pi/2]. Couple of questions: Why third-degree? Was someone attempting a cubic spline variant? This problem almost seems tailored for solution by genetic algorithm. Has anyone tried that? [table of results snipped] >I must suppose that all of these approximations, none of which I had seen >before, have been previously obtained by others. Information about a source >listing these approximations would certainly be appreciated. Is there >nothing like a pseudo-comprehensive _Encyclopedia of Approximations_? Actually, wood-workers (and probably other artisan types) often use very good approximation techniques. If you're looking for that sort of thing (granted, this problem wouldn't be there) this might be a good area to start. >Thoughtful comments are invited. >David Cantrell -- -john February 28 1997: Last day libraries could order catalogue cards from the Library of Congress. === Subject: Re: A Summary of Cubic Approximations of Sine on [-Pi/2, Pi/2] >The recent sci.math.num-analysis thread sinus approximation became >rather unwieldy, and so I thought it might be helpful to summarize >results which are optimal according to various measures of error. (I >have also crossposted to sci.math for others who might be interested.) >The objective was to determine the third-degree polynomial which best >approximates the sine function on [-Pi/2, Pi/2]. Naturally, many people >think first of the Maclaurin polynomial x - x^3/6, but it does not >minimize any standard measure of error on [-Pi/2, Pi/2]. > Couple of questions: Why third-degree? The OP in sinus approximation never said why. And there are so many possible reasons that I won't even bother trying to conjecture why he wanted degree 3 in particular. > Was someone attempting a cubic spline variant? I don't think so. > This problem almost seems tailored for solution by genetic algorithm. > Has anyone tried that? > [table of results snipped] >I must suppose that all of these approximations, none of which I had >seen before, have been previously obtained by others. Information about >a source listing these approximations would certainly be appreciated. Is >there nothing like a pseudo-comprehensive _Encyclopedia of >Approximations_? > Actually, wood-workers (and probably other artisan types) often use > very good approximation techniques. If you're looking for that sort > of thing (granted, this problem wouldn't be there) this might be a > good area to start. David === Subject: Re: Group character...silly question > |Am I missing something? > i don't know. do you agree that the traces of all the ordinary powers > of a linear transformation gives a lot more information than just the > trace of the linear transformation itself? that's the only real point > i was trying to make. I agree. That's almost what I want to know. > (in the present context i suspect the traces of the ordinary powers > completely determine the linear transformation up to similarity, but > offhand i forget some of the details of how that should work.) And this is what I want to know (exactly)...do they? === Subject: Re: Group character...silly question |> |Am I missing something? |> i don't know. do you agree that the traces of all the ordinary powers |> of a linear transformation gives a lot more information than just the |> trace of the linear transformation itself? that's the only real point |> i was trying to make. | |I agree. That's almost what I want to know. | |> (in the present context i suspect the traces of the ordinary powers |> completely determine the linear transformation up to similarity, but |> offhand i forget some of the details of how that should work.) | |And this is what I want to know (exactly)...do they? i was and still am trying to weasel out of giving a complete answer to that question (in particular by using the imprecise weasel words in the present context above without spelling out what i actually mean by the present context) because it's such an obviously trivial question and i have a feeling that there's some well-known complete and correct answer that i've forgotten or never heard of. are you sure dave rusin didn't give a complete answer in his post in this first there's the question: can you reconstruct the distribution of eigenvalues from its moments (= the traces of the ordinary powers of the transformations, more or less)? the answer is yes or almost yes, probably, and it might be easier in the present context than in the general case. second there's the question: can you reconstruct the similarity class of the transformation from its distribution of eigenvalues? the answer is almost yes, probably, and it might be easier in the present context than in the general case. maybe someone can (or maybe dave rusin already did) clean up these almost yes's to finish giving a complete explanation of how under the appropriate assumptions a group representation is completely determined by its character. -- [e-mail address jdolan@math.ucr.edu] === Subject: Re: Group character...silly question |second there's the question: can you reconstruct the similarity class |of the transformation from its distribution of eigenvalues? the |answer is almost yes, probably, and it might be easier in the |present context than in the general case. maybe one of the magic words i neglected to exploit here is unitary. maybe there's a good reason why the linear transformations you're interested in here are unitary, and maybe that makes the problem of determining the similarity class from the distribution of eigenvalues rather easier. -- [e-mail address jdolan@math.ucr.edu] === Subject: Re: Group character...silly question > spelling it out. If I got it right, this corresponding means it > respects the exterior product like f(y ^ v) = f(y) ^ f(v)? Yes. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: An Easy Complex Quesition This should be an easy question, but I can't seem to get it. Q: Solve the following equations in polar form and locate the roots in the complex plane: c) z^4 = -1 + sqrt(-3) This is taken from Bak, Newman, Complex Analysis. Pg 18. They don't give the answer for the roots in the book, but clearly, (z^4)/2 = (-1 + sqrt(-3))/2 is the third root of unity. Let w = (-1 + sqrt(-3))/2 Then when z = (2^{1/4}) w, z^4 = 2w. Done. However, I'm stumped as to how to find the polar coordinates. Can I just say z = 2^{1/4} e^{ (2 (PI) i )/ 3} Then r = 2^{1/4} and arg z = 2 (PI) i / 3, i = 1, 2 However, this isn't the answer in the book. So, any help is appreciated. GREG === Subject: Re: An Easy Complex Quesition >This should be an easy question, but I can't seem to get it. >Q: Solve the following equations in polar form and locate the roots in the >complex plane: >c) z^4 = -1 + sqrt(-3) De Moivre's theorem. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: An Easy Complex Quesition > This should be an easy question, but I can't seem to get it. > Q: Solve the following equations in polar form and locate the roots in the > complex plane: > c) z^4 = -1 + sqrt(-3) Write z = r*(cos(Q) + i*sin(Q)) Then r^4 = |-1 + i*sqrt(3)| = 4 => r = sqrt(2). and tan(Q) = Im[z]/Re[z] = -1/sqrt(3) => Q = - pi/6 so z = sqrt(2) * (cos(pi/6) - i*sin(pi/6)) = sqrt(2) * (sqrt(3)/2 - i*(1/2)) But for any w such that w^4 = 1, (z*w)^4 = w^4 * z^4 = z^4 is a solution. The fourth roots of 1 are {1, -1, i, -i}; thus the four solutions of the equation are sqrt(2) * (sqrt(3)/2 - i/2) sqrt(2) * (-sqrt(3)/2 + i/2) sqrt(2) * (1/2 + i*sqrt(3)/2) sqrt(2) * (1/2 - i*sqrt(3)/2) -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. === Subject: Re: An Easy Complex Quesition Originator: grubb@lola >Q: Solve the following equations in polar form and locate the roots in the >complex plane: >c) z^4 = -1 + sqrt(-3) >This is taken from Bak, Newman, Complex Analysis. Pg 18. >They don't give the answer for the roots in the book, but clearly, >(z^4)/2 = (-1 + sqrt(-3))/2 is the third root of unity. >Let w = (-1 + sqrt(-3))/2 >Then when z = (2^{1/4}) w, z^4 = 2w. Done. ^^^^^^^^^^^^^^^^ You forgot to find the 4th roots of w. --Dan Grubb === Subject: Re: Complex Number > I don't know the notation for a subscript character without the proper > character set (e.g. 2^2 = two squared (superscript), but there is no down > arrow to represent subscript characters). So I'm going to use |x for this > example. An underscore is common convention here for subscripts. z_1, z_2 Both the underscore _ and the carat ^ come from TeX. - Randy === Subject: Re: Complex Number >>I don't know the notation for a subscript character without the proper >>character set (e.g. 2^2 = two squared (superscript), but there is no down >>arrow to represent subscript characters). So I'm going to use |x for this >>example. > An underscore is common convention here for subscripts. > z_1, z_2 > Both the underscore _ and the carat ^ come from TeX. > - Randy However, it's pretty certain that Knuth took them from common typewriter notation for those effects (sub- and super- script, that is). Dale. === Subject: Quadratic Forms Let q be a quadratic form on R^n with the associated matrix Q = (q_{ik})_{i,k=1}^n and q_m a quadratic form with restriction to Q_m = (q_{ik}_{i,k=1}^m. Show that q is positive definite if and only if det(Q_m) > 0 for m = 1, ..., n. Anyone has a solution for this problem? It's a part of a text I'm reading. === Subject: Re: Quadratic Forms Lukas Horosiewicz a .8ecrit dans le message de > Let q be a quadratic form on R^n with the associated matrix Q = > (q_{ik})_{i,k=1}^n and q_m a quadratic form with restriction to Q_m = > (q_{ik}_{i,k=1}^m. Show that q is positive definite if and only if > det(Q_m) > 0 for m = 1, ..., n. > Anyone has a solution for this problem? It's a part of a text I'm > reading. I don't detail but I remember it's easy by induction. === Subject: Re: problem 1 Ignacio Larrosa Ca.96estro let curve x(t)=(3t , 3t^2 , 2t^3) > find angle between all unit tangent vector on x(t) and plane x + z > = 0 > ------------------- > I am anxious about answer... > thank you very much.... >> Find the cartesian equation of the tangent plane, by evaluating the >> determinant: >> | x-x0 y-y0 z-z0 | >> | x0' y0' z0' | = 0 >> | x0'' y0'' z0'' | >> where x0=3t, y0=3t^2, z0=2t^3, x0' is the first derivative relative of >> x0 to t, x0'' the second derivative, etc. >> This will give an equation of the form Ax+By+Cz+D=0, with A,B,C,D >> depending on t because the tangent plane is not constant, it depends >> on where you are on the curve. >> A vector normal to this plane is v1=(A,B,C). >> The other plane is x+0y+z=0 so the normal vector is v2=(1,0,1). >> The angle is calculated with the dot product >> angle=arccos[ (v1 dot v2) / (||v1|| ||v2||) ] >> which also depends on t. >More directly, find the angle alfa between the tangent vector (3, 6t, 6t^2) >and the vector normal to given plane (1, 0, 1), and take the complementary >angle, pi/2 - alfa. When I calculate the tangent plane equation with the determinant I get (2t^2)x + (-2t)y + z - 2t^3 = 0 so the normal vector is (2t^2, -2t, 1). And the tangent vector is (1, 2t, 2t^2) after simplification. They are orthogonal to each other as expected. But when I calc the angle with (1,0,1) I get the same angle for both vectors, because (2t^2, -2t, 1) dot (1,0,1)=(1, 2t, 2t^2) dot (1,0,1) and I don't understand how that can be. === Subject: Re: Conjecture regarding the number 12 >Has this conjecture been made before? Proven? A similar conjecture? > Tau (X) / X < Tau (12) / 12, for any natural number X > 12. If x and y are relatively prime, then Tau(xy) = Tau(x)Tau(y); thus, Tau(xy)/(xy) = (Tau(x)/x)(Tau(y)/y). For a prime p, Tau(p^k)/p^k is (k+1)/p^k. The following table contains all cases where Tau(p^k)/p^k is at least 1/2: kp 2 3 5 +--------------- 1 | 1/1 2/3 2/5 2 | 3/4 1/3 3/25 3 | 1/2 4/27 4/125 Tau(p^k)/p^k So the only numbers with Tau(x)/x >= 1/2 are x Tau(x) 1 1 2 1 3 2/3 4 3/4 6 2/3 8 1/2 12 1/2 So for x > 6, Tau(x)/x <= 1/2. Rob Johnson take out the trash before replying === Subject: Re: Conjecture regarding the number 12 >>Has this conjecture been made before? Proven? A similar conjecture? >> Tau (X) / X < Tau (12) / 12, for any natural number X > 12. >If x and y are relatively prime, then Tau(xy) = Tau(x)Tau(y); thus, >Tau(xy)/(xy) = (Tau(x)/x)(Tau(y)/y). For a prime p, Tau(p^k)/p^k is >(k+1)/p^k. The following table contains all cases where Tau(p^k)/p^k >is at least 1/2: > kp 2 3 5 > +--------------- > 1 | 1/1 2/3 2/5 > 2 | 3/4 1/3 3/25 > 3 | 1/2 4/27 4/125 > Tau(p^k)/p^k >So the only numbers with Tau(x)/x >= 1/2 are Of course, the title of the second column below should be Tau(x)/x. >x Tau(x) >1 1 >2 1 >3 2/3 >4 3/4 >6 2/3 >8 1/2 >12 1/2 >So for x > 6, Tau(x)/x <= 1/2. Rob Johnson take out the trash before replying === Subject: Re: Conjecture regarding the number 12 > M?rio Amado Alves a .8ecrit dans le message de > Has this conjecture been made before? Proven? A similar conjecture? > Tau (X) / X < Tau (12) / 12, for any natural number X > 12. > I dunno if that's precisely true, but somehow it's trivial: a number n has > at most -say- 2*sqrt(n) divisors, and 2*sqrt(n)/n -> 0 as n grows. one of my 'real' intuition which is that the number 12 is the most 'relatively' divisible, where this 'relation' is with respect to the divided number--but obviously not in the manner of the simple ratio above. (My personal interest relates to the importance of the number 12 in music. I want to support this importance with a formal--but simple--mathematical theorem, for some music classes I am preparing. In music 12 is important because it has divisors 2, 3, 4 which are important rytmic divisions, and 12 is also the number of chromatic tones.) === Subject: Re: Conjecture regarding the number 12 > one of my 'real' intuition which is that the number 12 is the most > 'relatively' divisible, where this 'relation' is with respect to the > divided number--but obviously not in the manner of the simple ratio > above. So let's first lay out the asymptotic behavior of tau() -- are there probably be clear what the natural way of expressing this number has a lot of divisors is. Dale === Subject: [Set Theory] Class of Ordinals well-ordered ? I am assuming ZF (no choice, but regularity), I call a class any well-formed statement R(X) with one-free variable (I am not able to be more formal than that). I define the class of ordinals Ord(X): Ord(X): [ (i) for all a in X , a Ord(X)] which is not empty [i.e. there is X with R(X)], then R(X) has a smallest element, i.e. There is A [R(A) and for all B (R(B) => A B not in A) which one can prove from regularity axiom (if not, there is an infinite descent ... in An in An-1 in... in A2 in A1 etc..) and without choice. I hope I haven't said anything silly. So is it true that Ord(X) is well-ordered by inclusion and that you don't require choice to show it? Noel. === Subject: Re: [Set Theory] Class of Ordinals well-ordered ? > I am assuming ZF (no choice, but regularity), I call a class any > well-formed statement R(X) with one-free variable (I am not able to be > more formal than that). I define the class of ordinals Ord(X): Strictly speaking, ZF does not have proper classes. Everything is a set, and Ord does not exist. You need a more powerful set theory such as NBG. > Ord(X): > (i) for all a in X , a and > (ii) for all a,b in X, (a=b) or (a in b) or (b in a) > I can work out from my textbook that Ord(X) is linearly ordered by set > inclusion (denoted <). I would like to confirm that it is in fact > well-ordered, i.e. that if R(X) is a subclass of Ord(X) [i.e. R(X) =Ord(X)] which is not empty [i.e. there is X with R(X)], then R(X) has > a smallest element, i.e. > There is A [R(A) and for all B (R(B) => A It seems to me that this is a consequence of: > Lemma: If R(X) is a non-empty class (there is X with R(X)), then there > is A: > R(A) and for all B (R(B) => B not in A) > which one can prove from regularity axiom (if not, there is an > infinite descent ... in An in An-1 in... in A2 in A1 etc..) and > without choice. > I hope I haven't said anything silly. So is it true that Ord(X) is > well-ordered by inclusion and that you don't require choice to show > it? I don't see where either regularity or choice comes into play here. Ordinals are transitive sets with the property of being well-ordered by the set membership relation. This is a matter of definition, not of regularity. There may be other sets (non-ordinals) that violate the regularity axiom, but ordinals are ordinals whether you have regularity or not. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Printing Math Symbols I just learned I will need to turn in math homework and so I thought I may as well type it. What software would simplify this chore? Tom Adams === Subject: Re: Printing Math Symbols > I just learned I will need to turn in math homework and so I thought I > may as well type it. What software would simplify this chore? If you're using a PC with Microsoft Windows try TeXnicCenter for writing in TeX/LaTexX. It may be downloaded for free. Google will find it. -- G.C. === Subject: Re: Printing Math Symbols > I just learned I will need to turn in math homework and so I thought I > may as well type it. What software would simplify this chore? (1) If it's not coursework which counts toward your final grade then it's not worth typing. (2) Use LaTeX. (http://www.latex-project.org) -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. === Subject: Re: electrons cling more tightly as universe expands? >> I like Richards approach but disagree with >> his conclusion. I do agree with Richard, it >> is a fact that >> INFO = frequency * duration = invariant. >> where Richard calls INFO,. A cycle is an >> absolute event and the information... >> But the duration is expressed by (N = cycles) >> duration = N * wavelength, then >Wrong >time interval source = N / frequency >frequency = c/wavelength >Thus >time interval source = N * wavelength/ c >> INFO = N*c, (c=freq*wavelength). >INFO = N Ok, thats fine, INFO and N are invariant. >> Because the wavelength increases there is >> no loss of information carried by the photon. >True, but how are you going to shift the wavelength? Excatly the same way a photon emitted at the surface of the sun is red-shifted by gravitation as it moves up. >Each end of the >wave represents the information of the beginning of two separate >objective events, e.g. cycles, both of these information 'bits' is Richard seems to be adopting a photon *model* that carries 2 bits of information. I thinks it's best to trust the invariants we've agreed upon, c, N = INFO, and of course c =wavelength*frequency, then these must always remain reciprical analogies in any model. >propagating at the same speed wrt the observer, and thus their >instantaneous displacement in space wrt each other is fixed wrt the >observer from the time of its emission to its absorption, i.e. >throughout its entire trip. The only way to change a wavelength in >flight is to change the speed of the wave, OTOH, this will not change >the frequency. This is simple Jr. High level optics. (Simple for you maybe :-) This question remains : No information is lost when a photon is reflected from a receding mirror. A receding mirror will red-shift the photon on reflection (doppler effect) but no INFO is lost. N and c remain invariant but frequency and wavelength are altered. >> As a photon propagates across the empty voids >> of inter-galatic space, it is being deflected by all >> the matter that gravitationally influences that >> location, and gravitational deflection sucks photon >> momentum, and frequency, but INFO remains >> invariant. >The frequency never changes wrt a given inertial frame. That's a very good point, and produces a tired light analogy. Suppose a Spalding golf ball is struck very hard in some distant galaxy, and is moving near c. As it moves through intergalatic space it deaccelerates imparts acceleration to, such as galaxies. It slowly gives up momentum but the information (SPALDING) remains invariant., (written on the ball). Rather like a bullet moving threw water. >Richard Perry Ken S. Tucker === Subject: Re: Real Analysis - Book recommendations for 3 months self study > I have taken calculus, up to multivariable calculus and vector calculus, > linear algebra, discrete math and probability theorey. All undergrad > classes. Now iÇm planning to do self study in real analysis during 3 months. > The following are my criteras for a suitable textbook on real analysis: > * Not assume any previous knowledge of real analysis(or complex analysis for > that matter) > * Clear concise examples, must not leave any doubts of what is going on > * Rigorous AND reasonably easy to follow.(not skip 6-7 steps and assume the > reader still follow) > * No formulations like, it is obvious that... or intuitively we see... > * Problem sets and students solution manual available > Any input appreciated on good books appreciated I do not have a direct answer to your question. But I went to www.alltheweb.com and entered the compound search term real analysis lecture notes on the theory that a professor's lecture notes would specify a textbook to go with his instructional materials. If you don't like, don't buy. I found this: http://www.dmat.ufpe.br/~peterj/Analysis.Renfro.html and I think some of these materials would probably interest you. David Ames === Subject: can anyone solve this little problem can any one reading this slve the following geometry problem: In a triangle ABC straight lines AD, BE, CE are drawn through a common point P to meet the sides BC, CA, AB at D, E, F respectively. Prove that (PD/AD)+(PE/BE)+(PF/CF)=1 === Subject: Re: can anyone solve this little problem > .... > In a triangle ABC straight lines AD, BE, CE .... ^^^^ You probably meant CF. > .... are drawn through a common point P to meet the > sides BC, CA, AB at D, E, F respectively. Prove that > (PD/AD)+(PE/BE)+(PF/CF)=1 Think about areas of triangles. Can you find a pair of triangles whose areas are in the ratio PD/AD ? There's more than one such pair. Ken Pledger. === Subject: Re: Outer Measure in R2 >I'm wondering if it s true that the outer measure of a bdd region in R2 can >be expressed as the integral of the outer measure of its cross-section? >(I know that it is true if the set if measurable in R2, and we replaces >outer measure with measure). I think it is true, but just don't know how to >go about proving it. It is trivially false. Let A and B be disjoint subsets of R of measure 1 each, and let C be a subset of the unit interval of inner measure 0 and outer measure 1, and D be its complement in the unit interval. The set is AxC U BxD; its outer measure is 2, but the (outer) measure of the X-component for given Y is 1. Integrating the other way will get the correct value of 2. But if A U B is disjoint from the unit interval, AxC U BxD U CxA U DXB will have outer measure 4, but the outer measure of each non-trivial cross section is 1, and will integrate to 3. -- 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: Peano in Plain English? >> Is there a nice website listing Peano's assumptions in Plain English? >> And also then going on to showing them being used in nice useful >> examples. >> to his assumptions, but don't actually list them! IE, there is 100 >> pages about discussing Peano's assumptions, and the philosophical >> ramifications of that, and maybe 1 tucked in somewhere that lists >> them, and explains them. >The first thing to realize about Peano's axioms is that they are one >of the biggest examples of how the research community has failed to >understand and formalize computing. Peano did not have the benefit of >computers in the 1800s, and so described the properties of the natural >numbers in term of logical expressions written in English: >1. Zero is a number. >2. Every number has a successor. >3. Zero is not the successor of any number. >4. Different numbers have different successors. >5. The set containing zero and the successor of all elements in the >set is the set of all natural numbers. This IS plain English, and even the generalizations not identifying the initial number can be stated in plain English so that a first grader can understand them. In fact, I prefer that, with the first theorem being that there is only one non-successor. -- 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: Short Fermat Proof permission for an emailed response. > I would suppose that by now you see that the natural numbers are a minimal > model of PA has no bearing on whether any substantial theorem about them is > provable in PA. Yes, this is correct. (I knew that before, actually, but had [significantly] misplaced the nature of the boundary in question, between true of the natural numbers and provable in PA.) Thomas === Subject: Re: Short Fermat Proof permission for an emailed response. > You have misconstrued my statement. I gave the explanation of my > statement: that the natural numbers are the smallest PA model. > If by the natural numbers are *defined* by PA you meant the > natural numbers are the smallest PA model, you should have > immediately realized the irrelevance of your statement to the > question of the provability of FLT in PA, since the natural > numbers are also the smallest model of the theory obtained > from PA by dropping the induction axioms. I never said that this particular statement was directly relevant to the provability of FLT in PA. === Subject: Re: Short Fermat Proof > I never said that this particular statement was directly relevant to > the provability of FLT in PA. You might as well reply that your various comments are not supposed to be relevant to anything in particular. === Subject: Re: Short Fermat Proof permission for an emailed response. > I never said that this particular statement was directly relevant to > the provability of FLT in PA. > You might as well reply that your various comments are not supposed > to be relevant to anything in particular. You had asked me what I meant by the natural numbers (a reasonable question, given my confusion on the main question), and I gave a *correct* answer to that question. Now you are complaining that when I answer your question, the answer must be relevant to everything else I say? Thomas === Subject: Re: Axiomatic Set Theory and Foundation permission for an emailed response. SANTA CLAUS comes down a FIRE ESCAPE wearing bright blue LEG WARMERS.. He scrubs the POPE with a mild soap or detergent for 15 minutes, starring JANE FONDA!! > at 01:46 PM, tb+usenet@becket.net (Thomas Bushnell, BSG) said: >For example, you can't prove that > If G = {G} and H = {H}, then G=H. > Yes you can. That comes from Extesionality - you don't need > Foundation. Huh? Can you give me the proof please? === Subject: Re: Axiomatic Set Theory and Foundation >>For example, you can't prove that >> If G = {G} and H = {H}, then G=H. > Yes you can. That comes from Extesionality - you don't need > Foundation. > No, it doesn't. Actually, the axiom being referred to is Hyper-Extensionality; which is axiom that states that if: for all x: (x in A <-> P(x,A)) and for all x: (x in B <-> P(x,B)) then A = B. === Subject: Re: Axiomatic Set Theory and Foundation permission for an emailed response. authoritatively >>For example, you can't prove that >> If G = {G} and H = {H}, then G=H. > Yes you can. That comes from Extesionality - you don't need > Foundation. > No, it doesn't. > Actually, the axiom being referred to is Hyper-Extensionality; > which is axiom that states that if: > for all x: (x in A <-> P(x,A)) > and > for all x: (x in B <-> P(x,B)) > then A = B. Such an axiom is what I had in mind for extending Extensionality with something that observes the shape of the sets. I'm not sure I'm happy with Hyper-Extensionality though as an axiom. :) Thomas === Subject: Re: Axiomatic Set Theory and Foundation <3fcbbc14$17$fuzhry+tra$mr2ice@news.patriot.net> <874qwki1r4.fsf@phiwumbda.org> <87iskzhy6m.fsf@becket.becket.net> Discussion, linux) >>>For example, you can't prove that >>> If G = {G} and H = {H}, then G=H. >> Yes you can. That comes from Extesionality - you don't need >> Foundation. >> No, it doesn't. >> Actually, the axiom being referred to is Hyper-Extensionality; >> which is axiom that states that if: >> for all x: (x in A <-> P(x,A)) >> and >> for all x: (x in B <-> P(x,B)) >> then A = B. > Such an axiom is what I had in mind for extending Extensionality with > something that observes the shape of the sets. > I'm not sure I'm happy with Hyper-Extensionality though as an > axiom. :) Sounds like what you have in mind is the coinduction proof principle. This says that two bisimilar sets are identical. Bisimilar is similar to the shape argument you're interested in, but it's a bit different, too. For instance, the set G above is the same set as the set K = {K,{K}}, but I'm not sure if it's obvious that G and K have the same shape. (Of course, since they're identical, they must have the same shape, but that's backwards from the reasoning you want to use). -- It's an exercise in game theory[...] I've been brutally logical in my analysis on this point.[...] My analysis indicates that the optimal strategy for mathematicians is to acknowledge the result today. --JSH gives practical reasons to accept his FLT proof === Subject: Re: Axiomatic Set Theory and Foundation permission for an emailed response. > This says that two bisimilar sets are identical. Bisimilar is > similar to the shape argument you're interested in, but it's a bit > different, too. For instance, the set G above is the same set as > the set K = {K,{K}}, but I'm not sure if it's obvious that G and K > have the same shape. (Of course, since they're identical, they must > have the same shape, but that's backwards from the reasoning you want > to use). Ah, interesting. My notion of shape is of course pretty vague, and it's quite possible that there are different axioms which would capture it in different (and non-equivalent) ways. I am not at all committed to any single way of understanding it. Thomas === Subject: Re: Axiomatic Set Theory and Foundation <3fcbbc14$17$fuzhry+tra$mr2ice@news.patriot.net> <874qwki1r4.fsf@phiwumbda.org> Discussion, linux) >For example, you can't prove that > If G = {G} and H = {H}, then G=H. >> Yes you can. That comes from Extesionality - you don't need >> Foundation. >> No, it doesn't. > Actually, the axiom being referred to is Hyper-Extensionality; > which is axiom that states that if: > for all x: (x in A <-> P(x,A)) > and > for all x: (x in B <-> P(x,B)) > then A = B. I don't get it. What's P? In any case, I'm certainly not disputing that G = H is a theorem of Aczel's non-well-founded set theory. But I've never heard of Hyper-Extensionality. -- I've been thinking about my problems with getting any kind of admission that my math arguments showing the core error in mathematics are correct, so I've gone to marketing books. -- James S. Harris, on when mathematics isn't enough === Subject: Re: Axiomatic Set Theory and Foundation >>For example, you can't prove that >> If G = {G} and H = {H}, then G=H. > Yes you can. That comes from Extesionality - you don't need > Foundation. > No, it doesn't. >> Actually, the axiom being referred to is Hyper-Extensionality; >> which is axiom that states that if: >> for all x: (x in A <-> P(x,A)) >> and >> for all x: (x in B <-> P(x,B)) >> then A = B. > I don't get it. What's P? > In any case, I'm certainly not disputing that G = H is a theorem of > Aczel's non-well-founded set theory. But I've never heard of > Hyper-Extensionality. Neither have I. But concerning the question G=H: In fact it depends. I don't have the details at hand, but there are models in which there is only one single set obeying x={x} and there are models where there are many. Models are constructed via decorated graphs and bisimulations on them to tell which decorated graphs represent the same hyperset. The particular bisimulation you take determines this equality and therewith determines also if G=H. (I remember there being at least 3 different bisimulation models, called VAFA, SAFA and FAFA, at least one of which has G=/=H) -- Jasper === Subject: Re: Axiomatic Set Theory and Foundation <3fcbbc14$17$fuzhry+tra$mr2ice@news.patriot.net> <874qwki1r4.fsf@phiwumbda.org> <87ptf7b8jk.fsf@phiwumbda.org> Discussion, linux) >> In any case, I'm certainly not disputing that G = H is a theorem of >> Aczel's non-well-founded set theory. But I've never heard of >> Hyper-Extensionality. > Neither have I. But concerning the question G=H: > In fact it depends. I don't have the details at hand, but there are models > in which there is only one single set obeying x={x} and there are models > where there are many. Models are constructed via decorated graphs and > bisimulations on them to tell which decorated graphs represent the same > hyperset. The particular bisimulation you take determines this equality and > therewith determines also if G=H. (I remember there being at least 3 > different bisimulation models, called VAFA, SAFA and FAFA, at least one of > which has G=/=H) It depends on what anti-foundation axiom one takes, of course, but I specifically said that it is a theorem of Aczel's theory. This is true. -- Jesse Hughes LOL. How arrogant you are. Now when you realize that I DID prove Goldbach's conjecture and that I proved Fermat's Last Theorem as well, how are you going to feel then? -- James Harris === Subject: Re: Axiomatic Set Theory and Foundation > In any case, I'm certainly not disputing that G = H is a theorem of > Aczel's non-well-founded set theory. But I've never heard of > Hyper-Extensionality. > >> Neither have I. But concerning the question G=H: >> In fact it depends. I don't have the details at hand, but there are >> models in which there is only one single set obeying x={x} and there are >> models where there are many. Models are constructed via decorated graphs >> and bisimulations on them to tell which decorated graphs represent the >> same hyperset. The particular bisimulation you take determines this >> equality and therewith determines also if G=H. (I remember there being at >> least 3 different bisimulation models, called VAFA, SAFA and FAFA, at >> least one of which has G=/=H) > It depends on what anti-foundation axiom one takes, of course, but I > specifically said that it is a theorem of Aczel's theory. This is > true. Indeed. I believe Aczel's theory is exactly one of these VAFA, SAFA or FAFA. You were right that according to Aczel, G=H. But there are alternative formulations of ZF- possible, in which G=/=H ('sall I wanted to say). -- Jasper === Subject: Re: Vedic Mathematics --- Myth and Reality > By the by, we understood all modern sceince discoveries > (like special theory of relativity) are explained in Vedas > in detail. > No, that is wrong. The Indian philosophical thought - Sanatana > dharma, or the way of life beyond the scope of time - is completely > different from the modern and dominant Jewish [...] all this time, the other shoe has finally dropped. > But could you enlight us discoveries yet to be happened > (Like design of perpetual machine.) from Vedas? > Vimans - which should work upon that principle - are mentioned in the > ancient Indian epics. People seeking enlightenment should first get > their grammar and spelling correct. Knowledge is wasted upon > low-minded and deliberate fools. Well, 5000 years without flight in the presence of all that knowledge: clearly it must have been wasted on the whole subcontinent all those millennia. Meanwhile, the first airplanes used by Deccan Airways were made and purchased from whom? === Subject: Re: Vedic Mathematics --- Myth > I can also do it in one line: > 12 * > 34 > -------- > 48 36 408 > But seriously, this vedic method is no different as our method. > Our method is basically: > 2*4 + 10*4 + 2*30 + 10*30 = 48 + 360 = 408 > See, exactly the same as your method. > Not at all! At all! 12*34 involves 4 multiplications and 3 additions either way. For an M and N digit number, both methods involve MN multiplications and MN-1 additions. The method described before of simply enumerating the first 10 multiples and adding up appropriate results involves 10M or fewer count-by-x's (x < 10); 10M or fewer count-by-1's, (M+1)N additions, and 0 multiplications anywhere. The complexity in the first two cases is MNm + (MN-1)a, and in the last case (M+1)Na + 20Mc with M <= N chosen (without loss of generality), where 0 < c < a < m are the complexities, respectively, of counting by, adding and multiplying single-digit numbers. Taking m = 2a, a = 2c, for instance, leads to a comparison of (6MN - 2)c vs. (2MN + 2N + 20M)c. But (6MN - 2)c < (2MN + 2N + 20M)c only if 2MN < N + 10M + 1 which is only true if (noting that M <= N was chosen for the last method): M = 1 & N < 11 or M < 4 & N < 7 or M, N < 6. For M = 1, the first two methods are identical and trivial; and for the last two cases are all easily handled by even the cheapest solar-powered calculator that in many cases comes attached free to many devices that, themselves, are often given away for free. === Subject: Re: Vedic Mathematics --- Myth > I can also do it in one line: > 12 * > 34 > -------- > 48 36 408 > But seriously, this vedic method is no different as our method. > Our method is basically: > 2*4 + 10*4 + 2*30 + 10*30 = 48 + 360 = 408 > See, exactly the same as your method. > Not at all! > At all! > 12*34 involves 4 multiplications and 3 additions either way. > For an M and N digit number, both methods involve MN > multiplications and MN-1 additions. > The method described before of simply enumerating the first > 10 multiples and adding up appropriate results involves > 10M or fewer count-by-x's (x < 10); 10M or fewer count-by-1's, > (M+1)N additions, and 0 multiplications anywhere. > The complexity in the first two cases is MNm + (MN-1)a, > and in the last case (M+1)Na + 20Mc with M <= N chosen (without > loss of generality), where 0 < c < a < m are the complexities, > respectively, of counting by, adding and multiplying single-digit > numbers. > Taking m = 2a, a = 2c, for instance, leads to a comparison of > (6MN - 2)c vs. (2MN + 2N + 20M)c. > But > (6MN - 2)c < (2MN + 2N + 20M)c > only if > 2MN < N + 10M + 1 > which is only true if (noting that M <= N was chosen for the last > method): > M = 1 & N < 11 or M < 4 & N < 7 or M, N < 6. > For M = 1, the first two methods are identical and trivial; > and for the last two cases are all easily handled by even the cheapest > solar-powered calculator that in many cases comes attached free > to many devices that, themselves, are often given away for free. You have no clue about the term simplicity of genius. You have not understood *anything* of what I have been saying. Try to read my definition of Vedic multiplication, and see how you can multiply 10,000 digit numbers quite easily with elementary programming on any PC. Really, you know nothing about representing numbers! Enough from me, on this thread. Arindam Banerjee. === Subject: Re: Vedic Mathematics --- Myth <1c_pb.9702$9M3.5220@newsread2.news.atl.earthlink.net> <3faec8d7.2339223@news.clara.net> <3FAFA35E.7036073E@tue.nl> Discussion, linux) >> I can also do it in one line: >> 12 * >> 34 >> -------- >> 48 36 408 >> But seriously, this vedic method is no different as our method. >> Our method is basically: >> 2*4 + 10*4 + 2*30 + 10*30 = 48 + 360 = 408 >> See, exactly the same as your method. >> Not at all! > At all! > 12*34 involves 4 multiplications and 3 additions either way. But not with *my* way. I use base 35. 12 * 34 takes only one multiplication and no additions. Admittedly, it takes somewhat more effort to memorize the multiplication tables. -- And yes, for those who think that just maybe I did find a short proof of Fermat's Last Theorem, and THE prime counting function, if I succeed at what I'm working on now world economy as you know it will be gone. -- James Harris branches out. === Subject: Re: Vedic Mathematics --- Myth > But not with *my* way. > I use base 35. 12 * 34 takes only one multiplication and no > additions. > Admittedly, it takes somewhat more effort to memorize the > multiplication tables. I've read some stuff about Vedic Maths ... it's the most useless and supid thing i've ever heard of.... like... Voodoo Mathematics ! === Subject: Re: Vedic Mathematics --- Myth You might want to reason your comments. Reading some stuff does not suffice in this regard. -Samik > I've read some stuff about Vedic Maths ... it's the most useless and supid > thing i've ever heard of.... like... Voodoo Mathematics ! -- === Subject: Re: Vedic Mathematics --- Myth > You might want to reason your comments. Reading some stuff does not suffice in this regard. > -Samik I cannot possibly associate Vedic Nonsenses with mathematics; a mathematical creation has nothing to do with a show. I mean, we're not in a circus. So... maybe it could be impressive to see some old monkey find out the result of 1232334 ^2 in less than 10 secs, but most people agree to say it's useless, and it depends rather on how much you trained your memory to perform the trick, than how creative you are. I've heard chinese students are trained to do that (mental calculus), but till now chinese mathematicians are far from being the most prestigious ones, aren't they? Really, man, leave that job to clowns !! -- Julien Santini === Subject: Re: Vedic Mathematics --- Myth and Reality > I thought you said you did not want to continue the discussion. > I had no choice continuing, since you put me words in my > mouth that i hadn't written. > Hmm, I am trying to put words in your mouth, eh? This is something > new, and should have come up in your earlier post! > What words? These words: clarifying that Tractenberg got his stuff from 16 books on Vedic maths... Herman Jurjus === Subject: Re: Vedic Mathematics --- Myth and Reality > I thought you said you did not want to continue the discussion. > I had no choice continuing, since you put me words in my > mouth that i hadn't written. > Hmm, I am trying to put words in your mouth, eh? This is something > new, and should have come up in your earlier post! > What words? > These words: > clarifying that Tractenberg got his stuff from 16 books on Vedic > maths... point, of course. But you did say that Vedic arithmetic was so verbose that it had to be translated into 16 books. I put and two and two together, and probably got four. If Tractenberg did not get his stuff from Vedic arithmetic, which were published around the time he did his work, then where did he get it from? Did he do it independently? If he did, why was his work not accepted, given its superior ease and simplicity? Nothing but the prevailing racist and religious bigotry could have prevented their acceptance, as they hailed from an unacceptable heathen culture. Exactly as any new idea from modern heathen sources are not acceptable today in the Western world; they are all inferior by definition. No one earlier had said that Vedic arithmetic was published in 16 books around that time (1910, which was a time when new ideas from the East were coming to the West), that is your one great contribution to this thread, for which you are to be sincerely thanked. Bye, Arindam Banerjee. > Herman Jurjus === Subject: Re: Jordan curves and n holes > Let P and Q be differentiable functions continuous over a open set R > which is path connected with n-holes, if d(P)/dy= d(Q)/dx for all _R. > The number of different values for the line integrals P dx + Q dy over > regular Jordan curves is 2^(n+1) - 1. Anyone know a proof of this > using combinatorics? I read one > using mathematical induction. Any other way to prove this? Feel free > to mail. The winding number of a Jordan curve about all points in its interior is either -1 or +1 (if it's clockwise or anticlockwise). An anticockwise Jordan curve can wind positively about 2^n - 1 nonempty sets of holes, and these will determine this path integral etc. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: Riemann Surfaces in Analysis What's a good book for this sort of thing? I've got a book by Kreyzig on differential geometry. I'm slowly reading it. Will this do? -Grisha === Subject: Re: Use of Apple Key Caps for Math Symbols > .... > The last few years' versions of AppleTalk and MS Word all include .... ^^^^^^^^^ That should have been AppleWorks. Sorry. Ken Pledger. === Subject: elliptic integrals perimeter is it possible to actually compute the perimeter of ellipse? can anyone give additional info on where to refer for rectifiable curves? === Subject: Re: elliptic integrals perimeter Vikram > is it possible to actually compute the perimeter of ellipse? Not in closed form, but http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html LH === Subject: Re: elliptic integrals perimeter > Vikram > is it possible to actually compute the perimeter of ellipse? > Not in closed form, but > http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html Not in closed form in terms of elementary functions, that is. 4*a*EllipticE[1-(b/a)^2] (using Mathematica notation) is just as much in closed form as is Sin[x], the latter being elementary, the former not. So, Vikram, it is possible to actually compute the perimeter of ellipse just as much as it is possible to actually compute, say, sin(2/7). It's just that the precise expression for the perimeter isn't elementary. David Cantrell === Subject: Incompleteness Theorems of Kurt G.9adel Good Evening! I would like to know if this is the right newsgroup for asking questions about the Incompleteness Theorems of Kurt G.9adel. I am very interested in exchanging thoughts about his findings and especially the implications in all kinds of domains, perhaps not only mathematical. Can somebody give me some guidance on this subject? J. Bours jfhm-bours AT home.nl === Subject: Re: Incompleteness Theorems of Kurt =?ISO-8859-1?Q?G=F6del?= > .... > I would like to know if this is the right newsgroup for asking questions > about the Incompleteness Theorems of Kurt G.9adel.... Yes, although the news group sci.logic also is worth trying. Ken Pledger. === Subject: [OT] Fields Medal VS Abel Prize I'm an italian boy and I speak a bit english. My question is this: I know that not exist a Nobel prize for math then was instituted the Field Medal. Now, there is also the Abel Prize. What's the criterion to assign Field Medal or Abel Prize? What's the difference? -- Noixe === Subject: Re: [OT] Fields Medal VS Abel Prize > I know that not exist a Nobel prize for math then was instituted the Field > Medal. Now, there is also the Abel Prize. What's the criterion to assign > Field Medal or Abel Prize? What's the difference? Field Medal is awarded for one brilliant work (in particular), while Abel Prizes is awarded for one brilliant career. === Subject: Re: [OT] Fields Medal VS Abel Prize Julien Santini ha scritto: > Field Medal is awarded for one brilliant work (in particular), while Abel > Prizes is awarded for one brilliant career. -- Noixe === Subject: Re: [OT] Fields Medal VS Abel Prize Check out: http://www.abelprisen.no/index_english.html http://www.fields.utoronto.ca/aboutus/jcfields/fields_medal.html I think that will answer your questions. Ciao!! Lurch > I'm an italian boy and I speak a bit english. My question is this: > I know that not exist a Nobel prize for math then was instituted the Field > Medal. Now, there is also the Abel Prize. What's the criterion to assign > Field Medal or Abel Prize? What's the difference? > -- > Noixe === Subject: Re: [OT] Fields Medal VS Abel Prize > Check out: > http://www.abelprisen.no/index_english.html > http://www.fields.utoronto.ca/aboutus/jcfields/fields_medal.html > I think that will answer your questions. > Ciao!! Ciao? Do you speak italian? :) I know these difference: Fields Medal: It's assigned every 4 years, there is a limit of age (40 years old) and the amount of money is less than Abel Prize. Abel Prize: Was insituted this years, there is not a limit of age and the amount of money is 760.000 Euro. Both Fields Medal that Abel Prize are the Nobel prize for the math. But, what's the criterion to assign one or other? One of they is more prestigious? -- Noixe === Subject: Re: [OT] Fields Medal VS Abel Prize I would say that the Fields medal is more prestigious. As for the criteria, you will have to ask the board members. It is ultimately up them what is considered important. Ciao! Lurch > Check out: > http://www.abelprisen.no/index_english.html > http://www.fields.utoronto.ca/aboutus/jcfields/fields_medal.html > I think that will answer your questions. > Ciao!! > Ciao? Do you speak italian? :) > I know these difference: > Fields Medal: > It's assigned every 4 years, there is a limit of age (40 years old) and the > amount of money is less than Abel Prize. > Abel Prize: > Was insituted this years, there is not a limit of age and the amount of > money is 760.000 Euro. > Both Fields Medal that Abel Prize are the Nobel prize for the math. But, > what's the criterion to assign one or other? One of they is more > prestigious? > -- > Noixe === Subject: Re: real world problem > paulgoodhew@bigpond.com says... >hi guys, I've a real life math/physics problem that I would love help with, >and someone here might enjoy a real world problem... >I have a fish tank that is 4'x2'x2' along the horizontal axis. >the stand that the tank is on is on a slight lean, so that the water level >in the highest corner is 12mm lower than the water level at the opposite >diagonal corner. >can anyone tell me the difference in pressure exerted outwards at the >different parts of the tank? >the tank is filled with water obviously!! > 1 gallon of water occupies 231 cubic inches and weighs 8 pounds, > so a column of water 231 high (or 19' 3) weighs 8 lbs/sq_in. > The pressure at a depth of 2' (24) would be > 8 * (24/231) = 0.831169 psi > at 24 - (12/25.4) = 23.527559 it would be > 8 * (23.527559/231) = 0.814807 psi aren't you forgetting about the 15 psi added by the atmosphere that still pushes on the water? Achava === Subject: Re: Reconsidering Halton Arp >>Archimedes was a remarkable engineer and mathematician. Pretty much >>nothing is known about whether he was a very good scientist. By at >>least one canon of science, he was miserable: he kept his discoveries >>a secret. >> Ptui. How do you know that? I was in the computer biz. We >> did not document the obvious. Now that obvious is gone and >> today's developers are rediscovering the knowledge all over again. >He was famous for keeping them secret. His engineering triumphs were >military secrets. Well, hi buyoancy law wasn't secret and it was good science. Same for the lever rule. That's alone qutie enough to be considered great scientist. How many other scientists of that time you know whose results are still routinely used today. As for engineering, this was usually secret throughout history. Especially engineering possessing military applications. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same === Subject: Re: Reconsidering Halton Arp >We can't measure distances directly beyond parallax range (about >300 parsecs). The 'measurements' that we use beyond the nearest >few dozen galaxies (based mostly on cepheids) are all based >squarely on assuming the big bang. > > Describe those methods and tell me where this assumption comes into > the method, or stop repeating this ridiculous bit of nonsense. >> >>You can easily show me my error by describing (i.e. not just naming) >>one method that does NOT use the big-bang assumption -- either >>directly or indirectly (i.e. for calibration). >> >> Noted, for the record, that you made a blanket statement without any >> knowledge. >I have knowledge of several different methods of estimating distance. >But they are all based directly or indirectly on assuming the hubble >constant. Obviously, you think you know one. All it takes is one, >to show me (and the rest of the newsgroup) my error. Go on! Don't >you want to provide education? I guess not. >> Now you want me to provide the actual information so you >> can try to find a place to claim your silliness applies. >I figured you couldn't provide any. > You're wrong. I found more than enough detail in online sources to > provide a technical explanation. Bookmarked a bunch in preparation to > composing my reply. But I won't address the use the Big Bang > indirectly question till I understand what the question is. Then you should have replied. For I have now provided the definition for you. It still exists, below. Yet you still refuse to identify a single method beyond cepheid range..... >>I simpy won't bother answering an open-ended question, from someone >>who snips and ignores all prior evidence in the thread. No matter how >>many methods I describe that do use the BB directly or for >>calibration, you can always complain that I missed one. >> >> What does use the Big Bang for calibration even mean? >Every standard candle distance method requires a calibration step. >There are methods used for distance estimation that are used beyond >the range of cepheids. Look up the section entitled secondary distance >indicators in the book The Cosmological Distance Ladder. > Don't have it. I'll be replying from online sources, which include > NASA pages, published papers, Hubble pages, and various sites attached > to various astronomy and physics departments in the US and UK. You are needlessly limiting your education. Web pages are notoriously unreliable. If you don't want to look up the reference I gave you, find any method you like, that claims to measure the distance to galaxies (and/or galaxy clusters) at around 10 Mpc or beyond. > But just one point of logic: If you are trying to plot a bunch of > points to measure H0 (slope of the redshift vs. distance best-fit > line), how can you possibly use a value of H0 to get your x-axis > distance values? You don't. We're not talking about determining H0. > Won't you find kind of a perfect fit, whose slope is > exactly the value of H0 you used? Think about that in connection with > papers that have new values of H0, with error bars. No one said or implied that you used H0 to determine H0. >These are defined as ... indicators which depend for their >calibration on our knowing the distance to some representative >nearby galaxies through primary distance indicators. >Taking the first method in that section -- for no other reason than it's >first in the book -- we have the HII regions method. It is based on >the assumption that one can estimate the dimensions of core and halo >diameters within the HII regions of a galaxy (and that these surround >new O and B stars). A 'correlation' was found between the HII region >diameter and the galaxy luminosity class. There are several problems >with this method (including the fact that the relationship was not the >one that was first identied, but was 'forced' as a secondary method >when the first was found to be nearly useless), which are listed in the >book. The primary one being that the method of core and halo >diameters on the plates are subjective and liable to systematic errors. >But this method was created by first ensuring (via extinction and >selection of the 'proper' definitions for intensity) that the correlation >(which has no theoretical justification) would be both linear and >consistent with the hubble law. > OK, stop right there. Since you're claiming scientific fraud, I'm > going to ask for details. Excuse me? Who mentioned scientific fraud? There's nothing fraudulent about anything in the method I described, above. For more details, read the (several pages) of the section in the book that describes this method and it's history. I know you don't want to be bothered to learn, but that's your problem. > What is meant by extinction and selection of the proper definitions for > intensity. The determination of the amount of extinction is subjective. As is the definition of the brightness cutoff point for caluculating the diameter of the core and halo regions. (See the reference for details.) > What method did they use, > and did the authors say we want to ensure compliance with the Hubble > law so we fudged the data? Or what did they say? Surely they had a > justification for whatever this procedure is you are alluding to. What > is it? The procedure is called calibration. >> Tell me that >> and I'll describe distance measurement via Cepheid variables. Found >> several good links. >I'm not interested in links. > I described what the links are. A number of them are papers. All of > them are written by astronomers. Saying you aren't interested in > links is saying you're going to dismiss out of hand all research I > lay my hands on. Isn't that a little prejudicial? Why should I bother? > I bookmarked at least 10 pages in the last 24 hours with plans to read > the calibration procedures in detail and summarize them here. Are you > saying that by virtue of being on a web page you're going to dismiss > anything I write sight unseen? I simply noted that I am not interested in reading links, because of the following sentences. Perhaps you should pause to read an entire paragraph, before launching into a silly attack at phantoms. > And I don't need to know about estimating >WITH cepheid variables. I'm talking about distance measurments *beyond* > the distance where cepheid variables are resolvable. > You are? Yes, and always have been. > What distance do you think that is? Approximately 10 Mpc. > Cepheids are the farthest thing we've got distance measurements on. You are apparently ignorant. And apparently incapable of reading (or comprehending) your own post from 2 weeks ago (noted below). > The Hubble paper involved Cepheids out to pretty extreme ranges. Well, 2 megaparsecs may have been considered extreme in 1929, but not these days. Your own post of Nov 23rd 1996 data to 500 Mpc (after first deleting the 1996 data beyond 500 Mpc that disproves the hubble constant on the order of billions of parsecs). Now, I've defined everything for you, and yet you still refuse to identify a method. I'm done with you in the thread, unless you care to proffer a single counterexample. -- greywolf42 ubi dubium ibi libertas {remove planet for return e-mail} === Subject: Re: Reconsidering Halton Arp >> Meanwhile, I'll just note that Hubble's red shift data was published >> in 1929 (with distances measured by parallax), but the calibration >> curve for Cepheid variables was published by Henrietta Leavitt in >> 1912. >Did you have a point to make? > Cepheid calibration 1912. > Big Bang Theory post-1940s. 1927. See below, from prior post. > Cepheid calibration can't be based on Big Bang Theory. I did not claim that cepheid period-luminosity calibration was based on the big bang theory -- exactly as I said below, in the prior post. > Only understand short sentences? You apparently have a reading comprehension problem. >> Big Bang theory in its present form is mostly credited to Gamow >> in the 1940s with a successful prediction of the 3-degree background, >Which was a false claim, as the lowest temperature predicted by Gamow, >prior to Penzias and Wilson was 50 degrees (a factor of 10,000 error in >energy density -- which was the basis for his estimate). >> though Lemaitre in 1927 did propose an explosive-origin theory. >Yes. Carl Wirtz first published an empirical redshift-distance relation >in 1924 (pre Cepheid variable identification). Lemaitre's publication of >the 'expanding universe' theory came in 1927, and was based partly on >Wirtz' empirical work. Hubble's version of the redshift relation was not >published until 1929 (after Cephied variable identification made Wirtz' >relationship more certain). >There have been at least five major revisions of the explosive origin >theory that is now called the 'big bang.' Which one are you defending? Well? Which version of the big bang are you defending? >> Explain to me how Leavitt managed to use Big Bang theory for her >> calibration in 1912, >I never claimed that the cepheid period-luminosity relationship was based >on the big bang theory. What I noted was that Hubble's law was based on >the cepheid curve. >> and what use Big Bang for calibration means, >> and I'll explain both Leavitt's calibration and the Cepheid variable >> method. >Not necessary. All I've (repeatedly) asked you to do was simply describe >one, modern distance estimatation method -- applicable beyond the range >of cepheid variable resolution > As I asked in your other post, what would beyond the range of cepheid > variable resolution be, since those are as far as I know the most > distant sources used for Hubble Law tests? As I pointed out in my posts of yesterday, you could read the section entitled Secondary Distance Indicators in the book The Cosmological Distance Ladder for more information. Or simply have read the excerpt from the book that I provided. The supernovae data, for example, exceed the range of the cepheids (although these are primary data). They also disprove the hubble assumption, being non-linear. Which requires the new ad hoc postulation of 'dark energy' to save the theory. Now, I've defined everything for you, and yet you still refuse to identify a method. I'm done with you in the thread, unless you care to proffer a single counterexample. -- greywolf42 ubi dubium ibi libertas {remove planet for return e-mail} === Subject: Re: Reconsidering Halton Arp permission for an emailed response. >Archimedes was a remarkable engineer and mathematician. Pretty much >nothing is known about whether he was a very good scientist. By at >least one canon of science, he was miserable: he kept his discoveries >a secret. > Ptui. How do you know that? I was in the computer biz. We > did not document the obvious. Now that obvious is gone and > today's developers are rediscovering the knowledge all over again. He was famous for keeping them secret. His engineering triumphs were military secrets. === Subject: Re: Reconsidering Halton Arp > He was famous for keeping them secret. His engineering triumphs were > military secrets. Archimedes military applications are a matter of public record. See -Plutarch's Lives-. His greatest stunt was rigging a pulley so that King Hieron could pull a fully loaded freight ship onto shore just by turning a crank. He also made a name for himself by inventing a screw type water lifter, whose design is still in use. Bob Kolker === Subject: Re: Reconsidering Halton Arp permission for an emailed response. > He was famous for keeping them secret. His engineering triumphs were > military secrets. > Archimedes military applications are a matter of public record. See > -Plutarch's Lives-. Are you familiar with the difference between engineering and science? Thomas === Subject: Re: Reconsidering Halton Arp > Are you familiar with the difference between engineering and science? One is theoretical the other applied. They are on a continuum. When Archimedes designed his parabolic mirrors he was doing both geometry and weapons building. Engineers who design transistors are doing science of an applied sort. When Bardeen (twice winner of the Nobel in physics)working along with Schockley and Brattain were producing the first effective transistor, Bardeen was doing engineering as much as physics. Intellectual snots and snobs tend to exaggerate the distinction. Bob Kolker === Subject: Re: Reconsidering Halton Arp >But assume he is right. Then take a 100 pound bag of feathers. The >Greeks had sacks and fibre bags. Take a ten pound metal object. The ten >pound solid metal object is several times more dense then the 100 pound >bag of feather. Drop both from a height at the same time. Is the time of >the fall through an equal distance for the objects in the inverse ratio >of their densities? It is not. Was he really as specific as saying the inverse ratio of their densities? >This is an experiment which could have been done with the technology >available to the Greeks (Athens in the 4-th century b.c.e.). With error bars small enough to distinguish the truth or falsity of the inverse ratio claim? I once measured the acceleration of gravity by dropping things from as high as a few dozen feet with a stopwatch in hand. Once something starts falling, things move pretty fast. The data showed the right trend, but would not have supported a constant acceleration very well if you'd had other hypotheses in mind. The Greeks might have gotten higher, but I'm pretty sure they didn't have stopwatches. Nor did they know how to relate a pendulum's length and period. For that matter, linear regression analysis hadn't been invented yet. How should the experiment have been performed and analyzed? -- A good plan executed right now is far better than a perfect plan executed next week. -Gen. George S. Patton === Subject: Re: Reconsidering Halton Arp couldn't they use an arbitrary pendulum, a particular one, to measure a few experiments?... of course, if you look at the fractal of a feather, its density is not just its weight, divided by what it dysplaces in water; eh? > Was he really as specific as saying the inverse ratio of their densities? > With error bars small enough to distinguish the truth or falsity of the > inverse ratio claim? I once measured the acceleration of gravity by > dropping things from as high as a few dozen feet with a stopwatch in > hand. Once something starts falling, things move pretty fast. The data > showed the right trend, but would not have supported a constant > acceleration very well if you'd had other hypotheses in mind. The Greeks > might have gotten higher, but I'm pretty sure they didn't have > stopwatches. Nor did they know how to relate a pendulum's length and > period. For that matter, linear regression analysis hadn't been invented --ils duces d'Enron! === Subject: Re: Reconsidering Halton Arp <87y8tzoule.fsf@becket Discussion, linux) >>But assume he is right. Then take a 100 pound bag of feathers. The >>Greeks had sacks and fibre bags. Take a ten pound metal object. The ten >>pound solid metal object is several times more dense then the 100 pound >>bag of feather. Drop both from a height at the same time. Is the time of >>the fall through an equal distance for the objects in the inverse ratio >>of their densities? It is not. > Was he really as specific as saying the inverse ratio of their > densities? I don't know much about Aristotelian physics, but I was under the impression that Aristotle was not an advocate of applying mathematics to the physical realm. I would guess that when he spoke of proportional dependencies and the like, it was relatively loosely. If Bob can really show us where Aristotle explicitly said something about the inverse ratio of the densities, I'd be surprised (but not terribly -- I'm not a scholar on the subject, of course(. > How should the experiment have been performed and analyzed? A fine question, methinks. -- We want a single platform. We're trying to get there using the carrot, or blackmail, or rewards, or whatever you call it. -- Madison, WI, superintendent Rainwater grasps subtlety in the operating system wars. === Subject: Re: Reconsidering Halton Arp permission for an emailed response. > I don't know much about Aristotelian physics, but I was under the > impression that Aristotle was not an advocate of applying mathematics > to the physical realm. More likely just that he didn't have a sophisticated enough theory to apply math to it. He did say that the planets traveled in circles around the earth, and he did mean exact mathematical circles. Ptolemy--an experimentalist Greek, mind you--looked up at the sky with more careful recordkeeping than Aristotle, and said nope, not circles. > I would guess that when he spoke of > proportional dependencies and the like, it was relatively loosely. This is probably true. Aristotle did of course know his Euclid, and was certainly familiar with strict proportionality. But the word proportional could also mean just a direct (monotonic) relation. Thomas === Subject: Re: Reconsidering Halton Arp >>I think that Thomas answered this already more than adequately. >Wrong! Given the medium Aristotle said explicitly that heavier bodies >would move faster in it, in proportion to their weight. >> Thomas explained that the term heavy as used by Aristotle refers to >> what we call nowadays dense. >He is wrong. The context rather clearly implies that he's right. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same === Subject: Re: Reconsidering Halton Arp >> >> > > > >>I think that Thomas answered this already more than adequately. > >Wrong! Given the medium Aristotle said explicitly that heavier bodies >would move faster in it, in proportion to their weight. >> >> >>Thomas explained that the term heavy as used by Aristotle refers to >>what we call nowadays dense. >He is wrong. >> The context rather clearly implies that he's right. >It is NOT clear. I have quite the opposite impression from two distinct >translations of -The Physics-. gold is qudratzver than wood then it is quite clear that his qudratzver refers to density. ***etc***. What's not clear here. >But assume he is right. Then take a 100 pound bag of feathers. The >Greeks had sacks and fibre bags. Take a ten pound metal object. The ten >pound solid metal object is several times more dense then the 100 pound >bag of feather. Drop both from a height at the same time. Is the time of >the fall through an equal distance for the objects in the inverse ratio >of their densities? It is not. It is not clear at all whether Aristotle meant strict proportionality (which is wrong) or the plain heavier (aka denser) objects fall faster which is quite right. Again, the context is rather supportive of the latter. You're getting hang up on translations of translations of translation, not terribly reliable. Heck, I can point out to you arrors in the translation of the Bible to English (and not in some secondary passage but in the ten commandments). So? ... snip verbiage ... Sorry, by now I've sufficiently lost interest in what you've to say, to bother with reading more than few lines. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same === Subject: Re: Reconsidering Halton Arp >> >> >> >I think that Thomas answered this already more than adequately. >> >>Wrong! Given the medium Aristotle said explicitly that heavier bodies >>would move faster in it, in proportion to their weight. >Thomas explained that the term heavy as used by Aristotle refers to >what we call nowadays dense. >>He is wrong. > The context rather clearly implies that he's right. It is NOT clear. I have quite the opposite impression from two distinct translations of -The Physics-. But assume he is right. Then take a 100 pound bag of feathers. The Greeks had sacks and fibre bags. Take a ten pound metal object. The ten pound solid metal object is several times more dense then the 100 pound bag of feather. Drop both from a height at the same time. Is the time of the fall through an equal distance for the objects in the inverse ratio of their densities? It is not. This is an experiment which could have been done with the technology available to the Greeks (Athens in the 4-th century b.c.e.). Clearly the experiment was not done, for it it was the -Physics- of Aristotle would not have said what it said. So we ask why wasn't the experiment or something like it done? Possible reasons: 1. Aristotle was stupid. But I reject this. Aristotles work on logic and categories suggest he is a sophisticated and intelligent man 2. It never occurred to Aristotle to do the experiment. This is in line with my assumption that the Greeks were too easily swayed by the logic of their own arguments. 3. He did the experiment but he rejected it as a refutation of his general principle. This is possible for folks who put principles before facts. Today we call them lawyers. 4. Perhaps a later (corrected) version of -The Physics- was made but it did not survive. This is the most charitable assumption. However there is no evidence for it. In any case a simple experiment that a child of 10 could think of should have disuaded Aristotle from his conclusion. Apparently this did not happen. For all of Aristotle's skill at examining and describing animals (this is what he was best at), he apparently exhibits no crtical emprical tendencies. He simply does not put testing in a foremost priority. Like I said, he didn't check. And that is all I said. Aristotle did not always check his work in a critical manner with the intent of seeing if it was wrong. Now I am going back to the original greek to see if Aristotle intended density or weight in his analysis, in particular the paragraph I quoted. That is the only way to find out what Aristotle intended to say. I will report my findings. If weight is what he meant (as I believe it is) he was flat out wrong and could have found out easily (no great technology required). If he meant density he is still wrong an a similar easy to do experiment would have revealed it. Greek philosophy along either the Platonic or Aristotelean axis did not conduce to quantitative careful experimentation. Even John Philiponus (490 - 580 c.e.) was able to see just how wrong Aristotle was and the technology available to John was little different than available to the Greeks (no gunpowder, no telescopes). John, or someone he knew of, did the experiment concerning falling bodies. The scholars of the middle ages tried to compensate for Aristotle's blunders by inventing the concept of impetus for moving bodies. This was something akin to momentum using our technology. And contrary to popular believe these folks from the middle ages did simple experimentation (nothing as elaborate as did Galileo however). Now if sixth century Alexandrians could do simple experiments, why not Athenian Greek philosophers of the fourth century b.c.e.? Bob Kolker === Subject: Re: Reconsidering Halton Arp permission for an emailed response. > So we ask why wasn't the experiment or something like it done? Something like it *was* done---in water. Thomas === Subject: Re: Reconsidering Halton Arp >>So we ask why wasn't the experiment or something like it done? > Something like it *was* done---in water. Considering most things fall in air, not a good experiment, assuming it was done at all. Bob Kolker === Subject: Re: Reconsidering Halton Arp permission for an emailed response. Mary Tyler Moore's SEVENTH HUSBAND is wearing my DACRON TANK TOP in a cheap hotel in HONOLULU! > Considering most things fall in air, not a good experiment, assuming > it was done at all. Why? If Aristotle is taken to be referring to terminal velocity, it's much better to check in water, since terminal velocity is more easily reached. Thomas === Subject: Re: Reconsidering Halton Arp Originator: grubb@lola >Clearly the experiment was not done, for it it was the -Physics- of >Aristotle would not have said what it said. >So we ask why wasn't the experiment or something like it done? >Possible reasons: >1. Aristotle was stupid. But I reject this. Aristotles work on logic and >categories suggest he is a sophisticated and intelligent man Definitely >2. It never occurred to Aristotle to do the experiment. This is in line >with my assumption that the Greeks were too easily swayed by the logic >of their own arguments. This is quite possibly true. The Greeks were very sceptical of sensory evidence since they knew of such things as optical illusions, etc. Because of this, the philosophers were biased toward abstract argument and away from demonstrations. THey did do demonstrations, however. >3. He did the experiment but he rejected it as a refutation of his >general principle. This is possible for folks who put principles before >facts. Today we call them lawyers. Doubtful for Aristotle. >4. Perhaps a later (corrected) version of -The Physics- was made but it >did not survive. This is the most charitable assumption. However there >is no evidence for it. I have heard that he *did* do some experiments, but he did them by letting things fall in water to slow the motion. If you consider terminal velicities, this leads to his version of how objects fall. The difficulty is that terminal velocities are not reached very quickly in the air. --Dan Grubb === Subject: Re: Reconsidering Halton Arp permission for an emailed response. I hope something GOOD came in the mail today so I have a REASON to live!! >2. It never occurred to Aristotle to do the experiment. This is in line >with my assumption that the Greeks were too easily swayed by the logic >of their own arguments. > This is quite possibly true. The Greeks were very sceptical of sensory > evidence since they knew of such things as optical illusions, etc. > Because of this, the philosophers were biased toward abstract argument > and away from demonstrations. THey did do demonstrations, however. Who are the Greeks? This makes about as much sense as talking about the Italians. (And thus lumping both Galileo and his ecclesiastical abusers in one category!) Thomas === Subject: Re: Reconsidering Halton Arp > Who are the Greeks? This makes about as much sense as talking about > the Italians. (And thus lumping both Galileo and his ecclesiastical > abusers in one category!) Lisping pansies from the Acedemy, the Lyceum and the Stoa. And their students. Bob Kolker === Subject: Re: Reconsidering Halton Arp permission for an emailed response. > Who are the Greeks? This makes about as much sense as talking about > the Italians. (And thus lumping both Galileo and his ecclesiastical > abusers in one category!) > Lisping pansies from the Acedemy, the Lyceum and the Stoa. And their > students. Wonderful! So your ignorance is matched by your homophobia. Wonderful! === Subject: Re: Reconsidering Halton Arp da noive, Randy Poe; maybe you should *both* stick to the ('pure') math. I mean, just because most people use the Encyclopedia Brit. as their source for US History (the SM/MUSD is a case in point, taht I'm familiar with), doesn't mean that it's validatable -- oy heil! I leave it to the vast (and/or silent) readership to discern the parallel that I'm trying to make, herein >> 2) Measure distance to object. >We can't measure distances directly beyond parallax range (about 300 >parsecs). The 'measurements' that we use beyond the nearest few dozen > Yeah, it's odd, but that's Randy Poe. He creatively deletes to create > false implication, and then keeps posting repeatedly until he drives > away the person he's arguing with by replying, and replying, and > replying. --ils duces d'Enron! === Subject: Re: Reconsidering Halton Arp >I think that Thomas answered this already more than adequately. >>Wrong! Given the medium Aristotle said explicitly that heavier bodies >>would move faster in it, in proportion to their weight. > Thomas explained that the term heavy as used by Aristotle refers to > what we call nowadays dense. He is wrong. Bob Kolker === Subject: Re: Reconsidering Halton Arp permission for an emailed response. >> >> >> >I think that Thomas answered this already more than adequately. >> >> Wrong! Given the medium Aristotle said explicitly that heavier >> bodies would move faster in it, in proportion to their weight. > Thomas explained that the term heavy as used by Aristotle refers > to what we call nowadays dense. > He is wrong. So what do you think Aristotle means when he says that gold is heavier than wood, if he is not speaking of the relative density of the two substances? Are you telling me that Aristotle thought that a lump of gold (however small) was heavier than a lump of wood (however large)? Thomas === Subject: Re: Reconsidering Halton Arp > So what do you think Aristotle means when he says that gold is heavier > than wood, if he is not speaking of the relative density of the two > substances? Are you telling me that Aristotle thought that a lump of > gold (however small) was heavier than a lump of wood (however large)? It doesn't matter. Even if you assume weight means density, I have already show that an experiment doable by a boy scout could falsify Aristotle's conclusions. He didn't check. That is why he got motion all wrong. If he had oiled a large wooden table and gave a smooth puck (a cylindrical piece of marble polished smooth, say) a shove and it kept on going that would have blown up his hypothesis that movement required continuous force. Too bad Aristotle did not play shuffle board. He might have learned a thing or three. His ueber error, his category blunder was regarding the kosmos as alive. It isn't. The universe is mostly dead. Live things are just temporary arrangements of dead things. His nonsense on final cause and telos mark him as an inferior thinker in matters scientific. If there is any cause at all (Hume doubted this), it is efficient cause. The bat hits the ball and the ball goes flying. That is cause. None of this entelechia nonsense. Aristotle share the fault of many of the Greek thinkers. They loved their logic and their arguments so much they did not bother to check. Even John Philliponos in the sixth century c.e., a christian scholar in Alexandria knew enough to check. Fortunately for science, most of Aristotle's horse has been purged from physics. Only unschooled children buy into Aristotelan notions of movement. And best of all most of our scientist know enough to check on their conclusions. One way another, experiment has the last say. We are all empiricists now. Bob Kolker All there is, are atoms and the void. === Subject: Re: Reconsidering Halton Arp permission for an emailed response. > It doesn't matter. Even if you assume weight means density, I have > already show that an experiment doable by a boy scout could falsify > Aristotle's conclusions. He didn't check. That is why he got motion > all wrong. But he did check: in water. Think terminal velocity here. > Aristotle share the fault of many of the Greek thinkers. They loved > their logic and their arguments so much they did not bother to > check. Even John Philliponos in the sixth century c.e., a christian > scholar in Alexandria knew enough to check. What is a Greek thinker? I've asked you over and over what that broad phrase means, and you can't bother to say. > We are all empiricists now. Which, actually, we learned from Aristotle, the first great emiricist. Thomas === Subject: Re: Reconsidering Halton Arp > Which, actually, we learned from Aristotle, the first great emiricist. Real empiricists check their work carefully. Aristotle did not. He gets a D- in physics. Bob Kolker === Subject: Re: New idea in my prime counting In sci.math, James Harris understand how that partial difference equation I keep talking about > works. > Basically, for dS(x,y), where x and y are positive integers, if y is a > prime number p, dS(x,p) is the count of composites up to and including > x that have p as a factor but do NOT have primes less than p as a > factor. For instance, dS(10,3) = 1 because 9 is the only composite up > to 10 with 3 as a factor that doesn't have 2 as a factor. [rest snipped] Give it up, James. Your algorithm was plastered in my contest. :-P :-) Even if one throws out all of the memoized variants. http://home.earthlink.net/~ewill3/math/primecounters/index.html And then there's Bau. Christian Bau... http://www.cbau.freeserve.co.uk/ :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Question inspired by JSH ... In sci.math, Nora Baron odd integers. > However, might there be algebraic integers A and B such that > (2*A + 1)*(2*B + 1) = 2 ? > Nora B. Hmm...lessee. If we assume algebraic numbers a and b are the roots of an equation where a*b = 2, then we can find [1] (x - a) * (x - b) = 0 and [2] x^2 + cx + 2 = 0 for a rational number c. Write a = 2*A + 1, b = 2*B + 1. If we assume A and B are algebraic numbers as well (if they're not the problem's not all that interesting anyway :-) ), they are roots of another equation, [3] (y - A) * (y - B) = 0. If we set x = 2*y + 1 and substitute in [2], we get [4] (2*y+1)^2 + c(2*y + 1) + 2 = 0 = 4*y^2 + 4*y + 1 + 2*c*y + c + 2 = 4*y^2 + (2*c+4)*y + c + 3 Divide this by 4, and we get [5] y^2 + (c/2+1)*y + (c+3)/4 = 0 In order for A and B to be algebraic integers, both the coefficients in [5] must be integers. The x coefficient requires c to be an even integer. The constant coefficient requires c to be an odd integer. Therefore, the answer is no -- if I've not made an excruciatingly bone-headed error somewhere in the above logic. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: The pure math lie, In sci.math, James Harris of it, for its purity. Yet I found one of the most pure and > beautfiful expressions ever, and they're trying to ignore it. > Mathematicians are cons, liars, sham artists, and I have the math to > prove it. > See the pure math they're ignoring and run the Java program. > http://mathforprofit.blogspot.com/ > I'll admit it. I don't go in for the pure math hooey, and am > honestly in it for the money. I get mathematicians to acknowledge my > discovery, get in the papers, sell my story, get paid. > I found the prime counting function, it's pure math; it's short and > beautiful, so they must acknowledge it. > James Harris > My math discoveries, found for profit > http://mathforprofit.blogspot.com/ *cough* Applet *cough* *hack* JNLP/Java Web Start *hack* *wheeze* BigInteger *wheeze* Of course, I could sponsor another contest in Java, if you really really want.... :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Is this an NP complete problem? time and attention first. The problem I am thinking is as follows: Given an undirected graph. Every edge in this graph will be in a specific color. Now I want to find a subset of edges that contains the least kinds of colors. Is it an NP complete problem? Helen [ comp.ai is moderated. To submit, just post and be patient, or if ] [ ask your news administrator to fix the problems with your system. ] === Subject: Re: Is this an NP complete problem? >Given an undirected graph. Every edge in this graph will be in a >specific color. Now I want to find a subset of edges that contains >the least kinds of colors. The graph with no edges will have 0 colors. Perhaps you underspecified the problem? What is the difference between colors and kinds of colors? What are *all* of the requirements on the output (obviously what you provided is only one of the requirements)? -- Mark Ping emarkp@soda.CSUA.Berkeley.EDU [ comp.ai is moderated. To submit, just post and be patient, or if ] [ ask your news administrator to fix the problems with your system. ] === Subject: Re: Is this an NP complete problem? # # time and attention first. # # The problem I am thinking is as follows: # # Given an undirected graph. Every edge in this graph will be in a # specific color. Now I want to find a subset of edges that contains the # least kinds of colors. # # Is it an NP complete problem? No, probably not. Just take the empty set (in polynomial time). It contains zero colors, and it will be hard to find something better. # # Helen # --Gerhard __________________________________________________________________ Gerhard J. Woeginger http://wwwhome.cs.utwente.nl/~woegingergj/ [ comp.ai is moderated. To submit, just post and be patient, or if ] [ ask your news administrator to fix the problems with your system. ] === Subject: Two diophantine eq'ns Another thread made me wonder about these two. I think x^4 - 1 = 2y^2 has no solution except with y=0, and x^4 + 1 = 2y^2 has no solution except with y=1 or -1. Any easy way? TIA, Larry === Subject: Re: Two diophantine eq'ns >Another thread made me wonder about these two. I think >x^4 - 1 = 2y^2 >has no solution except with y=0, and x^4 - 1 = (x^2-1)(x^2+1), and gcd(x^2-1,x^2+1)= gcd(2,x^2-1). Clearly x must be odd if y <> 0, so x^2+1 == 2 mod 4, and thus x^2+1 = 2 u^2 where u is odd. Then x^2-1 = (y/u)^2 must be a square. The rest is easy... >x^4 + 1 = 2y^2 >has no solution except with y=1 or -1. Any easy way? Not sure. It might be doable with Gaussian integers. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Two diophantine eq'ns Robert Israel >Another thread made me wonder about these two. I think >x^4 - 1 = 2y^2 >has no solution except with y=0, and > x^4 - 1 = (x^2-1)(x^2+1), and gcd(x^2-1,x^2+1)= gcd(2,x^2-1). > Clearly x must be odd if y <> 0, so x^2+1 == 2 mod 4, and thus > x^2+1 = 2 u^2 where u is odd. Then x^2-1 = (y/u)^2 must be a square. > The rest is easy... Aha. >x^4 + 1 = 2y^2 >has no solution except with y=1 or -1. Any easy way? > Not sure. It might be doable with Gaussian integers. I suspect so. Some cubics can be done that way. === Subject: Re: Sets vs. categories as a foundation Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow) >Perhaps. I came into this thread secondhand. Why don't you tell us >what you mean when you write, all categories are in a sense >imitations of the category of sets, the objects being imitations of >sets and the morphisms being imitations of functions. It's very >plausible that I don't know what the heck you mean. Seems pretty clear to me. The fact that set theory and category theory can be used as foundations for mathematics is irrelevant to james dolan's point, and is being dragged in uninvited. Suppose one had said instead, All homology and cohomology theories are in a sense imitations of simplicial (co)homology. Presumably nobody is tempted to use simplicial homology as a foundation for mathematics, and hence nobody is tempted into failing to understand this statement. The fact that there are other ways of thinking about categories is why he used the phrase in a sense. It's just an informal and suggestive statement. -- 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: Sets vs. categories as a foundation : > How exactly one would one even DEFINE the category of sets if one : > were NOT STARTING with ZFC or some other rich set theory as a : > foundation? My point is simply that if you have ZFC, what do you : > need categories for? Sets are already adequate as a foundation; you : > can do EVERYthing, INCLUDING categories, AS sets. The category of : > sets starts to get viciously circular. But if you don't have a set : > theory, if you are using categories as a foundation instead, then : > the category of sets is simply nowhere in evidence: how do you : > even DEFINE set? : And here, I think George goes too far (or I don't get his point). : Some folks want category theory as an alternate foundation, it's true. : Others just like it for its unifying qualities and ability to make : apparently disparate phenomena particular instances of a common : structure. I don't see why there's any particular issue for talking : about the category of sets as a particular category for *either* : group. Well, until individuals assert their claim to represent their respective camps, anything said by the rest of us risks mis-characterizing their actual positions. A core point is that there is more than one way to axiomatize either of category theory OR set theory. For some of these issues, it does actually matter which axiomatization you pick. Any treatment that alleges that a category needs to have a set of objects is going to get accused of begging the question. There is a simple (in classical FOL with equality) axiom-set for category theory in which every element of the domain is an arrow, the objects are all&only the subclass of arrows that go from themselves to themselves, and every arrow has to go from an object to an object. That treatment identifies each category with a whole model of the axiom-set, which pushes some of the interesting questions into your preferred model theory, which might not be category-theoretic. It might not be set-theoretic either, but then again it might be. If it is then there is a sense in which you haven't really picked a side, in which you are simultaneously demanding the necessity of both foundations. Ensuring that a category defined by THAT axiom-set grows into a category of sets basically requires adding enough axioms to restrict the models to topoi. The necessity of that MUCH work IS an issue for talking about the category of sets, because intuitively, sets seem simpler than that -- so MUCH simpler that you could in fact have defined categories in terms of them. There are textbooks that introduce the category of sets (and functions) in the first chapter, and don't get around to topoi until the last. On the other side, if you are using set-theoretic foundations, then one obvious issue in talking about the category of all sets is that in such axiomatizations of set theory as ZFC, it is too large to exist at all. That IS DEFINITELY an issue in talking about it. -- --- It's difficult ... you need to be united to have any strength, but internal issues have to be addressed. --- E. Ray Lewis, on liberalism in America === Subject: Re: Sets vs. categories as a foundation : at 12:50 PM, greeneg@cs.unc.edu (George Greene) said: : : >This is a very set-centric view. : >I thought the whole purpose of bothering with : >category theory in the first place was to escape this view. : : No. Category theory is a tool for other branches of Mathematics, : whether you take categories as fundamental or take sets as : fundamental. ANYthing that you take as fundamental, purely BY VIRTUE of the fact that you have taken it as fundamental, becomes MORE THAN JUST a tool for other branches of mathematics. If your attempt take it as fundamental has actually succeeded, then it is, BY DEFINITION of fundamental, a tool for ALL other branches of mathematics. This thread was started by the allegation that all categories are in some deep sense imitative of the category of sets. That is, I REiterate, a very set-centric view. It is a view of sets as the fundamentally deeper concept, as somehow unavoidable, EVEN when you are doing something explicitly designed to avoid them (i.e. attempting to take something *else* as fundamental, something like categories). : >My point is simply that if you have ZFC, what do : >you need categories for? : : Because you can prove things once in Category Theory and then apply : them to various branches of Mathematics. Your question is like asking : what you need groups for. No, it isn't. There are a lot of things you CAN'T DO with groups. Groups are unusually defined and limiting. That's why you need semigroups, quasigroups, loops, and algebras. Groups don't even PRETEND, don't even ASPIRE, to be able to do it ALL. But sets and categories DO. So my question is NOT like asking that. : >The category of sets starts to get viciously circular. : Why is that an issue? Because the vicious circle principle is a matter of basic intellectual hygiene, that's why. Also because circular definitions risk sheer irrelevance and meaninglessness. : You can do Set Theory without the Axiom of Foundation. Sure, but that's Work. It TAKES Work to make sure that your circles aren't vicious. If you are going to talk about a category of all sets as something SIMPLE (as OPPOSED to as something as COMPLEX as a topos) then that work is going to become overwhelming. My point is: a) if sets are foundational then the category of all sets is just irrelevantly circular. There are a whole lot of other NON-categorical set-structures in that world and they are every bit as important as the categorical ones. To choose to focus on categories at all is to throw THOSE structures OUT of the universe of discourse. Saying that you have somehow gotten them all back in because you mean the category of all sets is just hubris. b) if categories are foundational and sets are not, then the whole question of how to even define a set is COMPLICATED -- it is at least as complicated as the topos axioms. It therefore becomes hard to maintain that *simpler* categories -- ones that that come nowhere near topoi -- are *still* imitative of the category of all sets. How, for example, could any category with a finite number of objects be imitative of one with an infinite number? And why is it even meaningful to speak of ONE category of all sets when THE DISCRETE category of all sets is ALSO a category of all sets? I mean, there is more than 1 category whose object-domain is the class of all sets (or the class of all identity-functions from a set to the same set). Or are you just going to allege that any set encoding a surjective function on one set onto another is automatically an arrow from the one to the other, regardless of whether any of them likes it or not? If so, then, since that representation-of-a-function-as-a-set is itself a set, there are going to be unavoidable arrows from it (arrows from arrows to arrows, as OPPOSED to from&to objects). This is all simply trying to say that a category of all sets, whether with topoi or without, must necessarily be a complicated thing. Given that there are simple categories out there, they should not be alleged to be participating, not even by analogy, in all that -- IRRELEVANT, FOR THEM -- complexity. -- --- It's difficult ... you need to be united to have any strength, but internal issues have to be addressed. --- E. Ray Lewis, on liberalism in America === Subject: Re: Sets vs. categories as a foundation > : >> so part of what's going on here is this: first, all categories are in > : >> a sense imitations of the category of sets, the objects being > : >> imitations of sets and the morphisms being imitations of functions (or > : >> maps or mappings or whatever you call them). > : > This is a very set-centric view. > : > I thought the whole purpose of bothering with > : > category theory in the first place was to escape this view. > : That is a bit exaggerated. > : My impression is, that categorists have no problem with sets as such, > : but would like to regard function as the primitive concept instead > : of elementhood, because this is closer to mathematical practice. > Well, maybe Marc and James should conduct competing polls > of the category of all categorists; I mean, I doubt it > can simultaneously be the case that both all categories > are imitations of the category of sets AND that function, > as opposed to element, should be the primitive concept. These two aspects do not appear mutually exclusive to me. I can not read minds, but I guess, James emphasized in his comment, that the category of sets is one of the leading examples, that motivated abstract categories; so, if you are working with some abstract category, it _does_ help to apply your intuition from previous experience with sets, in order to shape concepts, question and answers. To me, this is no more surprising than the idea, that you should try to think about an abstract ring, as if it was a ring of functions, even when it is not. Certainly, I would add Monoids and Preorders as the other two leading examples, from where you should import concepts and ideas into the category at hand. When I mentioned mathematical practice I was actually thinking of those mathematicians outside foundational work. So far I have met exactly _one_ mathematician who really needed *axiomatic* set theory --- he happened to work in that field. Everyone else was happy with a more or less naive idea about sets, and with the knowledge, that you could in principle found everything needed on axiomatic set theory. But nobody ever thought, that the question how many elements has pi? had any mathematical importance. Nobody believed, that number theory would really depend on whether one defines the sucessor of a natural number n as {n} or n union {n} (this example of McLarty is taken from the Barr paper). >[...] > : > if you are using categories as a foundation instead, then > : > the category of sets is simply nowhere in evidence: > : > how do you even DEFINE set? > : You would define set like categories. > Just how hard is that? Don't you need to add, to the basic axioms, > some definitions describing which categories are categories- > of-categories, and then some more definitions, describing > which of THOSE categories have sets-as-their-objects? > That's a lot of layers up, for something that was allegedly > supposed to be the foundational template. A category is set like when it satisfies the so-and-so axioms. Why would you need to have the category of all categories available at that stage? > : First you can add the topos axioms, then you can add more axioms > : to narrow down the candidates. This is already sketched in the appendix > : of the 2nd edition of MacLanes CWM; > Well, sure, topoi are categories with sets in them. > But how does J.Random Topos relate to the category of > all sets? There are a big new bunch of topos axioms (above > and beyond the handful defining a category), and isn't it simply > ABSURD to allege that all categories are imitations of the > category of all sets, when a great many of those categories > need FAR FEWER additional axioms than are needed by topoi? For topoi you just need three little additional axioms. (one can cut this down even more, it is a matter of taste) Additional work has to be done in order to arrive at the _more_special_ case. One can argue, that this specializing to (traditional) Sets is not really important. > I mean, topoi are complicated. The category of all sets > is, in at least SOME people's opinion, simple -- simple enough > to be analogous to a lot of other simple small categories. What exactly is so complicated? Marc === Subject: Re: Sets vs. categories as a foundation : > Well, sure, topoi are categories with sets in them. : > But how does J.Random Topos relate to the category of : > all sets? There are a big new bunch of topos axioms (above : > and beyond the handful defining a category), and isn't it simply : > ABSURD to allege that all categories are imitations of the : > category of all sets, when a great many of those categories : > need FAR FEWER additional axioms than are needed by topoi? : For topoi you just need three little additional axioms. : (one can cut this down even more, it is a matter of taste) One little, two little, three little axioms can easily become four little, five little, six little axioms, once you try to make things clear to new undergrads. And if you cut down to three little ones, you're so small that you are *far* less powerful than sets. : Additional work has to be done in order to arrive at the _more_special_ case. Right, to get you to some sets you can actually use. : One can argue, that this specializing to (traditional) Sets : is not really important. One can also argue that it is Really Important. I am about to quote somebody from Ohio State arguing that, to the FOM list. : > I mean, topoi are complicated. The category of all sets : > is, in at least SOME people's opinion, simple -- simple enough : > to be analogous to a lot of other simple small categories. : : What exactly is so complicated? Well, I am not an expert. It's complicated to *me* simply because I haven't figured it out yet. But as for opinions that it might be complicated in general: [ this from Jan.23,1998: ] > The point of this posting is to give some entirely conventional axioms for > set theory that are a bit simpler than many that are normally given. They > are fully formal. By comparison, look at the axioms of elementary topoi > that are given in MacLane/Moerdijk, Sheaves in Geometry and Logic, A first > Introduction to Topos Theory, Springer-Verlag, 1994, on p. 163. The > difference in complexity is strikingly grotesque. Maybe MacLane hadn't seen your 3 little axioms. > Topos theory [with natural number object] is insufficient to develop > undergraduate real analysis - although many fom postings conceal this fact. > One has to add to topoi: [not only a natural number object, but] well pointedness, > and choice. The latter two are nothing more than slavish translations of set > theory into the topos framework. The idea is to take a fatally flawed > foundational idea and force it to work by transporting important ideas > from set theoretic foundations. This was 5 years ago so I don't know for sure that well-pointedness *remains* dismissable as a slavish translation of set theory into the topos framework; maybe something more elegant has been discovered since. > But the axioms of elementary topoi are already incomparably more > complicated than the axioms for set theory presented here. Vs. this I have to continue to plead ignorance since I have seen neither of your 3 little axioms nor MacLane's grotesquely complex ones from the Sheaves text. The post from which I am quoting goes on to give a simple 5-axiom-set for finite set-theory and a simple 8-axiom treatment of ZFC. With these in hand, it triumphantly declares, > This allows such a simple axiomatization in purely primitive > notation. Try to write down the axioms > of a topos with natural number object, well pointed, and choice, in simple > primitive notation!! Good luck. > As is well known, ZFC is practically complete. > The version in the book on p.163 - which does not even include natural > number object, well pointedness, or choice - requires a very substantial > amount of preliminary abbreviations in order to formalize. The whole point of all this is that a category of all sets is hard, but I am not being dogmatic about it -- if you know the easy version, feel free to share. -- --- It's difficult ... you need to be united to have any strength, but internal issues have to be addressed. --- E. Ray Lewis, on liberalism in America === Subject: Re: Sets vs. categories as a foundation > |I think I agree with George here. One can take the set-theoretic > |intuitions too far. What about posets as categories? The arrows > |aren't imitations of functions, are they? > sure they are; specifically, of inclusion functions between subsets > (is one way to think of it). A category can be thought of in many ways; for instance, that it's nothing more than a typed monoid, or that it's an automaton whose states are its objects and state transitions its arrows. Trying to force the idea in one mould completely misses the point that it's all these things, and therefore none of them at all. Besides, everyone knows that categories are actually automata and automata are categories. === Subject: Goedels completeness theorem.. Hi everyone! Is there anyone who can tell me where to find resources on Goedels completeness theorem? /Anders === Subject: Re: Gegenbauer polynomials mod 2? Have you tried Erdelyi, A., et al (1953), Higher Trancendental Functions? The volume on Orthogonal polynomials has a section on Gegenbauer Polynomials and Functions. In particular, IIRC, C_n^1(x) are the eigenfunctions of the angular momentum operator in 4 dimensions. So they've been played with fairly extensively. > I'm looking for any information about the Gegenbauer polynomials > (specifically, C_n^{(1)}(x)) in F_2[x] (F_2 = finite field with two > elements). I'll take anything I can get, but I'm particularly > interested in roots (in the algebraic closure of F_2), factors, etc. > I'd take any information about Legendre polys mod 2, since I might be > able to generalize. > TIA! > Lot-o-fun === Subject: Re: Gegenbauer polynomials mod 2? > I'm looking for any information about the Gegenbauer polynomials > (specifically, C_n^{(1)}(x)) in F_2[x] (F_2 = finite field with two > elements). I'll take anything I can get, but I'm particularly > interested in roots (in the algebraic closure of F_2), factors, etc. > I'd take any information about Legendre polys mod 2, since I might be > able to generalize. Actually, I'm interested in C(n,1,x/2) mod 2, not C(n,1,x) mod 2. Here's some Mathematica output that shows the polynomials I mean: In[8]:= PolynomialMod[Table[GegenbauerC[n,1,x/2],{n,0,10}],2]//InputForm Out[8]//InputForm= {1, x, 1 + x^2, x^3, 1 + x^2 + x^4, x + x^5, 1 + x^4 + x^6, x^7, 1 + x^4 + x^6 + x^8, x + x^5 + x^9, 1 + x^2 + x^4 + x^8 + x^10} === Subject: Re: Gegenbauer polynomials mod 2? >> I'm looking for any information about the Gegenbauer polynomials >> (specifically, C_n^{(1)}(x)) in F_2[x] (F_2 = finite field with two >> elements). I'll take anything I can get, but I'm particularly >> interested in roots (in the algebraic closure of F_2), factors, etc. >> I'd take any information about Legendre polys mod 2, since I might be >> able to generalize. >Actually, I'm interested in C(n,1,x/2) mod 2, not C(n,1,x) mod 2. >Here's some Mathematica output that shows the polynomials I mean: >In[8]:= PolynomialMod[Table[GegenbauerC[n,1,x/2],{n,0,10}],2]//InputForm >Out[8]//InputForm= >{1, x, 1 + x^2, x^3, 1 + x^2 + x^4, x + x^5, 1 + x^4 + x^6, x^7, > 1 + x^4 + x^6 + x^8, x + x^5 + x^9, 1 + x^2 + x^4 + x^8 + x^10} It's useful to make a shift of index. The recurrence relation for f_n(x) = C(n-1,1,x/2) (taking C(-1,1,x/2) = 0) is f_0(x) = 0 f_1(x) = 1 f_n(x) = x*f_(n-1)(x) - f_(n-2)(x) and this is still valid over F_2. For any p(x) in F_2[x], if p(x) divides f_n(x), then f_(n+k)(x) = f_(n+1)(x) f_k(x) mod p(x) for all k and p(x) divides f_{nj}(x) for all j. Moreover, since the recurrence is reversible, if n is the least positive integer such that p(x) divides f_n(x), then p(x) divides f_m(x) if and only if m is a multiple of n. If n is composite, f_n(x) always has factors, namely f_k(x) for all factors k of n. And every polynomial in F_2[x] is a factor of some f_n(x). It turns out f_n(x) is a square in F_2[x] if n is odd and x times a square if n is even. The proof is by induction: if f_n(x) = x g_n(x)^2 for even n and g_n(x)^2 for odd n, then if n is even f_{n+2}(x) = x g_{n+1}(x)^2 + x g_n(x)^2 = x (g_{n+1}(x)+g_n(x))^2 while if n is odd f_{n+2}(x) = x^2 g_{n+1}(x)^2 + g_n(x)^2 = (x g_{n+1}(x) + g_n(x))^2 Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Gauss and superstitious belief > ............ Newton, on the other hand, was quite involved in > mystical speculations, alchemy etc. - does that make him any less > of a mathematician? > I don't believe this is strictly correct. Newton's beliefs seem > quirky to us, but they were not mystical in the sense that astrological > and alchemical beliefs are usually understood. > He believed that there was a former age which had obtained superior > knowledge of nature, and that this knowledge was somehow encoded > in alchemical and other ancient writings. This is similar to Simon > Stevin's belief in an age of wisdom ( siecle sage, ) I think, and > I suppose it was widely held. > Newton's own approach to natural philosphy was consistently > materialistic and rationalistic, however. Cf. his General Scholium > at the end of the Principia: > [ re nervous activity in animals ] > But these are things which cannot be explained in few > words, nor are we furnished with that sufficiency of > experiments which is required to an accurate determination > of the laws by which this electric and elastic spirit operates. > As to his mathematics, a perusal of various sections of the > Principia leads me to opine that Newton was not only a Genius, > he was pretty smart, too. > Lew Mammel, Jr. It is said genius is 1 percent inspiration and 99 percent perspiration. Cerebral perspiration it would appear, is a teasing creative process driven by an underlying/ unifying belief of proper geometrical/chemical/dynamical order getting established among premises --- experience coming convincingly first and explanations gradually following to fit it in. === Subject: Re: Gauss and superstitious belief ... > superstitious? Newton, on the other hand, was quite involved in > mystical speculations, alchemy etc. - does that make him any less > of a mathematician? I think you are quite right regarding the independance of Newtons mathematical prowess from his other interests. For that matter there are several other mathematicians who held beliefs which by modern standards would be considered as false or unscientific, like John Dee, Pythagoras to name two. Many approaches to knowledge which are not in accordance to modern science did contain elements of the proper scientific method. It is only through the progress of science that they became invalid, not because they were bad science but because the assumptions they worked on, turned out to be wrong. Take Alchemy as an example. Although there were of course many who pursued it only for greed, there are also many instances of Alchemists who were totally serious about it. The belief that matter can be transmuted was based on the belief that matter consisted of four elements. Thus, practically all substances were thought to be transformable by altering the proportions of the component elements, and this applied for gold also, which was not considered to be one of the four elements. It was only until about Newtons time, that scientists started to discover the true nature of things, the existance of several elements, and the elemental nature of many substances. And nowadays we know that matter can be transmuted, only the processes involved are quite other than those available at the time of Alchemy. === Subject: Probability continuity and countable additivity Can someone please explain the importance of the equivalence of continuity and countable additivity in the development of prob. theory? It would appear that some people can accept continuity but not count. additivity, but I can't understand the controversy, given that the 2 are mathematically equivalent. Any help appreciated. === Subject: statistical mechanics and ensemble. newbie question. i don't even know if this is a proper forum for this question. what is statistical mechanics and ensemble method? ensemble method fits a data set with errors to a model? how? I guess a post will not do the justice for the topic... does anyone know of any concrete examples for something like that? perhaps a website? i have very little math background. last math oriented class was quantum chem i took in college which was 10 yrs ago. any thoughts or insights is appreciated. john === Subject: Proof by induction On a un n-gone (polygone .88 n c.99t.8es) r.8egulier dont les sommets sont num.8erot.8es 0, 1, ... , n-1, o.9d le sommet i suit le sommet i-1 dans l'ordre anti-horaire. Pour tout entier positif t ≤ n, une rotation anti-horaire par un angle de 360t/n degr.8es envoie le sommet 0 vers le sommet t. D.8emontrez que le plus petit nombre strictement positif de rotations anti-horaires par 360t/n degr.8es qui envoie le sommet 0 vers lui-m.90me est n/pgcd(t,n). En d.8eduire que n rotations anti-horaires par 360t/n degr.8es sont n.8ecessaires pour envoyer le sommet 0 vers lui-m.90me si et seulement si pgcd(t,n)=1. pgcd = plus grand commun diviseur j'aimerai avoir vos opinions..... === Subject: Re: Proof by induction Michel grava .88 la saucisse et au marteau: > j'aimerai avoir vos opinions..... The first one is that you must not post in French (enven though I did it once) in this newsgroup. So ask your question in english or post it on fr.education.entraide.maths -- Nicolas === Subject: Re: Proof by induction > j'aimerai avoir vos opinions..... > The first one is that you must not post in French (enven though I did it > once) in this newsgroup. So ask your question in english or post it on > fr.education.entraide.maths Notice in passing that it is much more common to see an English-speaking person being told to speak in French on French newgroups, than a non-English speaking person being required to talk in English on sci.math... which is kinda paradoxical, since English is commonly used as the language of science. More: most people on sci.math actually try to answer questions without relentlessly complaining about language; I find it a far better behaviour than ours on the French newsgroup!! -- J.S === Subject: [HS] Re: Proof by induction Julien Santini grava .88 la saucisse et au marteau: > Notice in passing I'd be flabbergasted if this were real English :) > More: most people on sci.math actually try to answer questions without > relentlessly complaining about language; I find it a far better behaviour > than ours on the French newsgroup!! The reason is different. We are from far the best mathematicians of the world. As a consequence, one does not need to ask for help to non-french speaking persons, it's useless. If we can't do it, who can ? -- Nicolas, who tries to be as dumb and patriotic as Bob Kolker, but I guess there's a lot more to do. === Subject: ideal of noninvertible elements of a ring If R is a ring with 1 satisfying: for each r in R, either r or 1-r is invertible, then the set M of noninvertible elements of R forms an ideal. I'm having trouble showing that M is closed under addition. Any suggestions? dan === Subject: Re: ideal of noninvertible elements of a ring >If R is a ring with 1 satisfying: >for each r in R, either r or 1-r is invertible, >then the set M of noninvertible elements of R forms an ideal. >I'm having trouble showing that M is closed under addition. Any suggestions? Hint: if a+b is invertible, say (a+b)^(-1)=c, then ac or 1-ac = bc is invertible. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: How to use Central Limit Theorem here? > since it doesn't seem to be depending of the number of elements > in the examined set. Er, yes it does. If pi is the population proportion s.e.(p) = sqrt(pi (1-pi)/n) Glen === Subject: Re: Prime numbers, my find, and discovery I think, just before you started the 10-millionth item on your mathforfun&profit, someone gave an account of your work as like Legendre's method, but streamlined. if you are unhappy with that, why?... if not, then you should by all means begin to teach it to students, if his other guage of this method was accurate. ah, but what would Halton Arp have said?... for the record, a LOT of postings can also be read, virutally ALL of the postings, and we are your only (apparent) fan club -- unless the FBI pays you throught the blog. > I should be a rather happy guy. After all, over 18 months ago I found > this partial difference equation I call dS(x,y), and the sum of dS > from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to and > including x. > Now you may have seen a LOT of postings from people trying to attack > the worth of my find, which can be a healthy process--if they stick > with the facts. > Maybe that's part of the problem as we know that architects require a > lot of schooling beyond just art, as they need to know physics, like > materials science, and engineering, among many other things. --ils duces d'Enron! === Subject: Re: Prime numbers, my find, and discovery > I should be a rather happy guy. After all, over 18 months ago I found > this partial difference equation I call dS(x,y), and the sum of dS > from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to and > including x. Al your psoted spew has been soundly roundly discredited. Take your psychosis elsewhere. Hey stooopid loud troll James Harris, put up or shut up, http://www.rsasecurity.com/rsalabs/challenges/factoring/faq.html http://www.rsasecurity.com/rsalabs/challenges/factoring/numbers.html http://www.crank.net/harris.html It's not every braying jackass that gets a whole page at crank.net -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Prime numbers, my find, and discovery ...snip... >But hey, they're only human. >James Harris >My math discoveries, found for profit >http://mathforprofit.blogspot.com/ And you are obviously not. But you are very entertaining, whatever non-human critter you are. Hey, James, you haven't taken up my offer regarding that bridge between Manhattan and New Jersey. Too bad, the offer is now withdrawn. We coulda made loadsadough. And attracted many excellent babes. Sigh. -- Wolf Kirchmeir, Blind River ON Canada Nature does not deal in rewards or punishments, but only in consequences. (Robert Ingersoll) === Subject: Re: Prime numbers, my find, and discovery > I should be a rather happy guy. Considering the amount of medications you must take, that is quite likely. === Subject: Re: Prime numbers, my find, and discovery > Gauss wondered what the discrete count of prime numbers could have to > do with continuous functions like x/ln x, and while mathematicians > made progress in finding relations that gave limits, like Chebyshev's > use of the zeta function discovered by Euler, they never found a > reason why. > I may have found that reason. What is it? You keep talking as if you are about to present it, but never do. > As I've found a partial difference equation, it leads to a partial > differential equation. That partial differential equation may answer > many questions. What partial differential equation? You have never posted it. You claim it may provide answers to many questions, but neither post it nor the answers. It's one thing to claim magnificent properties for a specific result, but how can you claim them for an undetermined result? Either post this so-called partial differential equation and show the connection with prime counting. Your failure to do so will be taken as conclusive proof that you CANNOT DO SO. [snip tiresome, paranoid, rambling, repetitive, unsupported diatribe against academia] > dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1, > sqrt(y-1))], > S(x,1) = 0, p(x, y) = floor(x) - S(x, y) - 1, > and S(x,y) is the sum of dS from dS(x,2) to dS(x,y). > And p(x,sqrt(x)) is the count of prime numbers up to and including x. > That's pure knowledge. Information discovered by me, and hey, it > wasn't like it just jumped in my lap you know. There's a value to > cheering on discovery, and not attacking it. OK. 2 + 3 = 5. I demand that you cheer this discovery unless, of course, you are a hypocrite. > The value is hope for the future. Hope that there may be answers out > there from unlikely sources. Hope that every person can be valuable. The obvious purpose of your posts is to demand that *your* work be considered valuable. > Maybe mathematicians want a reality that has them ordained as the only > route for new mathematical knowledge. Possibly they wish control over > the creative process, and total dominion over mathematical discovery. > But hey, they're only human. Wacky, isn't it? But hey, it's only basic math. Yup, yup, yup! -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: Prime numbers, my find, and discovery >I should be a rather happy guy. After all, over 18 months ago I found >this partial difference equation I call dS(x,y), and the sum of dS >from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to and >including x. >Afer talking with mathematicians all over the world by email and >Usenet, and searching math references, both bought and on the >Internet, I know that I have a first-find. >Somehow, I am the first human being in recorded human history to find >a partial difference equation that sums to give the count of prime >numbers. Not true. Won't become true through repetition. See http://mathworld.wolfram.com/LegendresFormula.html >This post is about some of the significance of that beyond >it being a first-find. Really? Curious that it's such a long post, then. >Prime numbers have fascinated people for some time, and mathematicians >especially. The great mathematician Karl Gauss is credited with >making an important hypothesis in the field of prime numbers, as he'd >noticed something. >Gauss noticed that the count of primes numbers could be approximated >by x/ln x, for instance, the count of primes up to 1000 is 168, and >1000/ln 1000 approximately is 144.76. The count of primes up to 10000 >is 1229, and 10000/ln 10000 is approximately 1085.73, which is a >closeness that continues as you go higher. >Gauss wondered what the discrete count of prime numbers could have to >do with continuous functions like x/ln x, and while mathematicians >made progress in finding relations that gave limits, like Chebyshev's >use of the zeta function discovered by Euler, they never found a >reason why. Not true. A reason why (that is, a proof of the Prime Number Theorem) was found long ago, I think in the 1890's. More or less simultaneously by two people, who I think are the people whose names I think are spelled something like Hadamard and de-Vallee Poisson. >I may have found that reason. Nothing you've ever posted gives any explanation for this. >Not surprisingly, a first-find in the area of prime numbers *should* >be a big deal, but despite the ease with which I link my discovery to >some of the biggest names and biggest issues in mathematics, there is >the value to society of the discoverer. >Since when has modern society decided that discoverers should be >attacked instead of cheered? >Now you may have seen a LOT of postings from people trying to attack >the worth of my find, which can be a healthy process--if they stick >with the facts. >Unfortunately posters have shown a dismaying tendency to lie, but >that's minor to the problem I've faced where mainstream mathematicians >have tried to ignore or downplay my result. >I have a first-find in the area of prime numbers, and my not being a >mathematician does not mean that mathematicians can just deny the >reality if it suits them. While they may feel they have many reasons >to attack the value of an important find from a non-mathematician, >those reasons cannot be in the best interests of society. >If Gauss were alive today, would he cheer me? >I like to think he would, as he was someone interested in asking >questions *and* in getting answers. First and foremost I think he >would have been driven to find out just where my discovery led, and if >it was the answer to the question that intrigued him. >As I've found a partial difference equation, it leads to a partial >differential equation. That partial differential equation may answer >many questions. May indeed. You've never shown why the solution to the partial differential equation has anything to do with pi(x), and you've ignored explanations of why that seems unlikely, based on analogy with other difference equations and the corresponding differential equations. >Or more importantly, it should raise many more. >You should not allow mathematicians to continue to pervert a process >that has helped humanity for so many thousands of years. You must not >show a loss of faith in the future of humanity, as if discoverers are >no longer needed. >Academic institutions can no more constrain who can make a major >discovery, than they could limit who will be a great painter, >composer, or architect. >Maybe that's part of the problem as we know that architects require a >lot of schooling beyond just art, as they need to know physics, like >materials science, and engineering, among many other things. >So it's easy to assume that a great building can only come from >someone heavily trained in academia who can manage a huge structure. >However, sometimes something a little smaller in terms of physical >size can be huge in terms of social value, and the person who built >it, might be someone from just around the corner, outside of academia. >Maybe I'm pushing the analogy, but I hope that you'll agree that at >the end of the day, what's important is the *information* and petty >squabbles and personal attacks are irrelevant, and often forgotten >over history anyway. >It's the knowledge that remains--pure. >dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1, >sqrt(y-1))], >S(x,1) = 0, p(x, y) = floor(x) - S(x, y) - 1, >and S(x,y) is the sum of dS from dS(x,2) to dS(x,y). >And p(x,sqrt(x)) is the count of prime numbers up to and including x. >That's pure knowledge. Information discovered by me, and hey, it >wasn't like it just jumped in my lap you know. There's a value to >cheering on discovery, and not attacking it. >The value is hope for the future. Hope that there may be answers out >there from unlikely sources. Hope that every person can be valuable. >Maybe mathematicians want a reality that has them ordained as the only >route for new mathematical knowledge. Possibly they wish control over >the creative process, and total dominion over mathematical discovery. Or possibly you're just a megalomaniac idiot. Don't want to overlook that possibility... >But hey, they're only human. >James Harris >My math discoveries, found for profit >http://mathforprofit.blogspot.com/ ************************ David C. Ullrich === Subject: Re: Prime numbers, my find, and discovery > I should be a rather happy guy. After all, over 18 months ago I found > this partial difference equation I call dS(x,y), and the sum of dS > from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to and > including x. Yeah, it must suck to find out that nobody cares because it was such a trivial exercise, huh? -jcr === Subject: Re: Prime numbers, my find, and discovery > If Gauss were alive today, would he cheer me? Gauss's motto was Few, but ripe. He would probably call your contributions, Many, and rotten. === Subject: Re: Prime issue challenge for the record, I'm a big fan of the new math school of Bourbaki, may he rest in pieces. for, I was actaully taught set-theory in the 3rd grading. like the man said, Death to the triangle! > Throughout this discussion, all sets will be comprised of > positive integers and q will always refer to a (positive) > prime integer. |A| will denote the cardinality of the set A > and for a positive integer n, nA will denote the set > {n*a | a in A}. We will use the notation op [cond : term] > to denote the iteration of the operator op, while some > condition _cond_ is true, over the terms _term_, so for > example sum [0 < i <= n : i] = 1 + 2 + ... + n. > Define p(x, y) to be the cardinality of the (disjoint) union > of P(x, y) and C(x, y), where > P(x, y) = {q | q <= min(x, y)} > C(x, y) = {n | 1 < n <= x and > n is not divisible by any q in P(x, y)}. > We see immediately that p(x, 1) = x - 1 and > p(x, floor(sqrt(x)) = the number of primes <= x. > Theorem. If x >= y, then > x - 1 - p(x, y) = sum [q <= y : p(floor(x/q), q-1) > - p(q-1, floor(sqrt(q-1))] > Proof of theorem. We may regard x - 1 - p(x, y) as the > cardinality of the set {2, 3, 4, ..., x} - P(x, y) - C(x, y), > which is the set of all numbers between 2 and x inclusive > that are not primes <= y and are divisible by some prime <= y. > This set is equal to the disjoint union > union [q <= y : D(x, q)] [1] > where D(x, q) is the set of composites 1 < c <= x that are > divisible by the prime q and not divisible by any prime < q. > Now we observe that the union in [1] may be expressed as > union [q <= y : qC(floor(x/q), q - 1)] [2] > and finally that P(x, q-1) = P(q-1, floor(sqrt(q-1)), when x >= q, > from which with [2], we conclude the equality in the theorem > statement. > Is it better than Legendre's method? Well, it does not require > knowing the primes, since they computed on the fly. Unfortunately, --ils duces d'Enron! === Subject: Re: Prime issue challenge why do you do this with set theory -- is that what it is? is this a tutorial for set theory?... I mean, Jimi's explanation was shorter! > However, I know exactly how my formula works, so it seems to me that a > good check of others is to see if they do as well. > Throughout this discussion, all sets will be comprised of > positive integers and q will always refer to a (positive) > prime integer. |A| will denote the cardinality of the set A > and for a positive integer n, nA will denote the set > {n*a | a in A}. We will use the notation op [cond : term] > to denote the iteration of the operator op, while some > condition _cond_ is true, over the terms _term_, so for > example sum [0 < i <= n : i] = 1 + 2 + ... + n. > In what follows, I've interspersed the exposition with > parenthetical examples, delimited HTML-style with > .... These may be eliminated without > affecting the demonstartion. > Define p(x, y) to be the cardinality of the (disjoint) union > of P(x, y) and C(x, y), where > P(x, y) = {q | q <= min(x, y)} > C(x, y) = {n | 1 < n <= x and > n is not divisible by any q in P(x, y)}. > C(50, 3) = {5, 7, 11, 13, 17, 19, 23, 25, > 29, 31, 35, 37, 41, 43, 47, 49} > so > p(50, 3) = 18 > p(x, floor(sqrt(x)) = the number of primes <= x. > Theorem. If x >= y, then > x - 1 - p(x, y) = sum [q <= y : p(floor(x/q), q-1) > - p(q-1, floor(sqrt(q-1))] > 49 - p(50, 3) = 49 - 18 = 31 > = (p(50/2, 1) - p(1, 1)) + (p(floor(50/3), 2) - p(2, 1)) > = (p(25, 1) - p(1, 1)) + (p(16, 2) - p(2, 1)) > = (25 - 1 - 0) + (|P(16, 2) union C(16, 2)| - 1) > = 24 + |{2} union {3, 5, 7, 9, 11, 13, 15}| -1 > = 24 + 1 + 7 - 1 > = 31 > cardinality of the set {2, 3, 4, ..., x} - P(x, y) - C(x, y), > which is the set of all numbers between 2 and x inclusive > that are not primes <= y and are divisible by some prime <= y. > This set is equal to the disjoint union > union [q <= y : D(x, q)] [1] > where D(x, q) is the set of composites 1 < c <= x that are > divisible by the prime q and not divisible by any prime < q. > {2, ..., 50} - {2, 3} - C(50,3) (listed above) > = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, > 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50} > = {2*2, 2*3, ..., 2*25} union {3*3, 3*5, ..., 3*15} > = D(50,2) union D(50, 3) > union [q <= y : qC(floor(x/q), q - 1)] [2] > D(50, 3) = 3{3, 5, 7, 9, 11, 13, 15} = 3C(floor(50/3), 2) > = q, > from which with [2], we conclude the equality in the theorem > statement. --ils duces d'Enron! === Subject: Re: Almost an Integer - e^? > Also: (Pi*Sqrt(163))^Exp(1)= 22806.9992386 Do you have any reason to believe that is more than a coincidence? === Subject: Re: Almost an Integer - e^? > > I remember there is an explanation for these numbers. > Anyone knows where to find the reference (books better)? > > E.g. J.Silverman, Advanced Topics in the Arithmetic of Elliptic > Curves or something like that (Springer GTM series) has the > explanation involving class fields and their relations to the > j-invariant. There may be other explanations. > Jyrki Lahtonen On the much lighter side --- Another interesting fact about e^(pi * sqrt(163)) is --- If you add this large composite reciprocal to 163 which is a (29) digit long integer --- e^(pi * sqrt(163 + (1/43072298941682041177938098750))) = 262537412648768743.999999999999999999999999999999999999999998219574092.. Gives (41) 9's in its decimal expansion. More interesting is -- e^(1/43072298941682041177938098750) = 1.000000000000000000000000000023216777942... Where if you -1 from the above --- = 1/43072298941682041177938098750 ;-) Probably the largest integer reciprocal that could be added to 163 giving the same floor (value) of the original (e^(pi * sqrt(163))) and producing a limit of (41) 9ës in the decimal expansion compared to (12) 9's in the original. Dan === Subject: What does .sig mean? I have been told to get rid of .sig in my responses. What does that mean? Lurch >Maybe I'm missing something, too, but I think you're right: The >condition we need is that the inverse image of N w.r.t. T is contained >in N, rather than the way the problem is stated. > > But then T' would not be well-defined. I think you need the > condition T(N) = N for this question to make much sense. > Ah, quite right. that the problem is with the original question then I don't feel so bad about not getting it. John Harrison