mm-1475 === Subject: A little knowledge is a dangerous thing - THE HALTING PROOF For 50 years mathematicians have cowered at the concept of the halting proof. Because there exists some n, p where UTM(n, p) does not give an output and halt, for every computer theoretical idea programmers are greeted by hordes of mathematicians hollering \... IT WONT WORK....\. Say I want to make a complete list of reals using a Universal Turing Machine. All mathematicians have been capable of is this... \OH MY GOD... HE'S USING A COMPUTER\ what does that mean? \ACCORDING THE HALTING PROOF....... ............. IT WONT WORK !!!!\. It doesn't matter what algorithm you present, if its theoretical IT WONT WORK. This has been the downfall of modern mathematics, who cling to a join_ the_dots paradigm of the number line and refuting logical ordering as their price. There is no such thing as \too many to put in order\, it is at the very fundamental realm of fantasy. Put integers in order, there's still too many to count. If you can't order something you don't have it, it doesn't exist. Snap out of it you fools, cardinality is completely disproven. www.freewebs.com/namesort/numsort.html Herc === Subject: Re: A little knowledge is a dangerous thing - THE HALTING PROOF posting-account=WZMvOwwAAAC_B1TaayrVN99BJgDWQkUc > For 50 years mathematicians have > cowered at the concept of the halting > proof. > Because there exists some n, p > where UTM(n, p) does not give an output > and halt, NO, NOT because of THAT. THAT is a trivial consequence of the fact that there are some TMs that don't halt on some inputs. THAT is a trivial consequence of the fact that there is a TM that loops ON EVERY input, a TM that NEVER halts, no matter WHAT p is. There in fact exists a tm with code n that does not halt for ANY input p, that loops for ALL p. > for every computer theoretical idea programmers > are greeted by hordes of mathematicians hollering > \... IT WONT WORK....\. That's completely ridiculous; it's COMPUTER SCIENTISTS claiming that \it won't work\; the mathematicians DID NOT KNOW this UNTIL Kurt Godel INVENTED LISP AND EXPLAINED this to them. More to the point, it's NOT for EVERY \computer theoretical idea\! There are PLENTY of ideas that, since they are simple enough to be recursive, ARE describable by TMs that always halt -- or even by simpler machines. > Say I want to make a complete list of > reals using a Universal Turing > Machine. OK. \Herc is a fucking [excuse me: virgin; there's no way ANYbody THAT lame could EVER get laid] idiot.\ There. Said it. === Subject: Re: A little knowledge is a dangerous thing - THE HALTING PROOF posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L You did it last post! You said I can't formulate a list of reals using a UTM to start with. The fact the UTM scares you, its dataset has OPTIONAL DATA TYPE means you NEVER BOTHERED TO INVESTIGATE the set of outputs of all programs. If you did, you'll realise a sound completness to the set of all real numbers. McCarthy invented LISP you moron, with the very idea of hand coding a UTM like Eval function, then every other function they had been working on could just be parsed in from a library, at first he didn't believe it would work, it was an idea from a student of his. Herc === Subject: Re: A little knowledge is a dangerous thing - THE HALTING PROOF posting-account=Y88NBQ0AAAAqUB8nuxULGKuuNrRvkG6g H>> ... cardinality is completely disproven. <H>> ... cardinality is completely disproven. <Herc >Seems to be a somewhat unprofitable way of interpreting the term. >All that mathematicians are formally claiming in Cantor's diagonal >argument is that there is no algorithmic way of putting even the real >numbers that are defined as non-terminating sequences of 0's and 1's >into a 1-1 correspondence with the natural numbers. Not so. Notions of computability (\algroithmic\) do not enter into \ Cantor's diagonal argumentat all. The alleged list of all infinite binary sequences \ need not be a computable one; the diagonal argument works just the same for \ ANY alleged list, computable or not. -- --------------------------- | BBB b \\ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: A little knowledge is a dangerous thing - THE HALTING PROOF posting-account=Y88NBQ0AAAAqUB8nuxULGKuuNrRvkG6g comp.theory: BK>> The alleged list of all infinite binary sequences need not be a computable one; the diagonal argument works just the same for ANY alleged list, computable or not. < \ said: > comp.theory: > BK>> The alleged list of all infinite binary sequences need not be a > computable one; the diagonal argument works just the same for ANY > alleged list, computable or not. < Barb > Sorry, I should have clarified that I was referring only to arguments > based on functions f(n) whose domain is the natural numbers, and where > the premise is that, given any natural number n, there is an effective > method of determining a well-defined value for f(n). I would consider > such a definition algorithmic by definition. Cantor's diagonal argument > seems to appeal to such a definition. You are just totally wrong about that. Frankly, it beggars imagination how it comes to \seem\ otherwise to you based upon *any* presentation of the argument. > However, Cantor's diagonal argument pertains to the particular case > where the set S is the set of natural numbers. The argument does, then, > reduce to an algorithmic definition No, it doesn't. > if, by the dictum of Occam's razor, we do not appeal to the axioms of > a set theory. Beg pardon? Chris Menzel === Subject: Re: A little knowledge is a dangerous thing - THE HALTING PROOF posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L he / she didn't say that you moron. he defined the list exactly as you redefined it, non terminating sequences. This is EXACLTY what I\m talking about, you see the term compute and balk. What he was referring to with algorithmic was 'precise and specific' regarding the nature of the ordering, not the numbers. Herc === Subject: Re: A little knowledge is a dangerous thing - THE HALTING PROOF posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L look mormans! this is all very nice so long as you ignore my disproof of uncountability. www.freewebs.com/namesort/numsort.html don't come back until you've pressed the buttons 10 times each. Herc === Subject: Re: A little knowledge is a dangerous thing - THE HALTING PROOF posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > For 50 years mathematicians have cowered at the concept > of the halting proof. Well, not all the time. It _does_ help (human) computer programmers keep their jobs, since their jobs can't be automated. > Because there exists some n, p where UTM(n, p) does not > give an output and halt, for every computer theoretical > idea programmers are greeted by hordes of mathematicians > hollering \... IT WONT WORK....\. > Say I want to make a complete list of reals using a > Universal Turing Machine. Then you will need an infinite amount of time, even if you are only after ONE real number like pi. > All mathematicians have been capable of is this... > \OH MY GOD... HE'S USING A COMPUTER\ > what does that mean? I don't know. Actually, a complete list is possible (if you assume the Axiom of Choice), but it's the question of whether a _countable_ list is [ossible that's the issue. And the answer to this question is no. > \ACCORDING THE HALTING PROOF....... > ............. IT WONT WORK !!!!\. > It doesn't matter what algorithm you present, if its > theoretical IT WONT WORK. What do you mean by a \theoretical\ algorithm? > This has been the downfall of modern mathematics, who > cling to a join_the_dots paradigm of the number line > and refuting logical ordering as their price. > There is no such thing as \too many to put in order\, > it is at the very fundamental realm of fantasy. Anyone who is vaguely familiar with ordinals knows this. > Put integers in order, there's still too many to count. What you really mean here is: It takes an infinite amount of work. > If you can't order something you don't have it, it > doesn't exist. Cardinality and ordering are two different things. There IS a way to order the real numbers, and any mathematician will tell you so. It's denoted by the symbol <. How big the real numbers is is a separate issue. There is no way to match up the integers in a 1-1 correspondence with the integers. But this doesn't mean that they can't be ordered. > Snap out of it you fools, cardinality > is completely disproven. > www.freewebs.com/namesort/numsort.html Snap out of it you fool, you don't know what you're talking about. --- Christopher Heckman === Subject: Re: Taylor Polynomials Using a1*X^0.5+a2*X1.5+...an*Xn^n.5 >>We all know what the formula for Taylor's P is. However, has there >>been any attempt to approximate curves using the forms: >>A1*x^-1 + A2*x^-2 + A3*x^-3 + A4*x^-4 + ... An*x^-n or >> >\Laurent series\ >>a1*X^0.5+a2*X1.5+...an*Xn^n.5 ? >> >\Puiseaux series\ For those looking for info (as I did), it looks like that should be \Puiseux.\ -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Taylor Polynomials Using a1*X^0.5+a2*X1.5+...an*Xn^n.5 >>\Puiseaux series\ >For those looking for info (as I did), it looks like that should be >\Puiseux.\ or something. Puiseux was giving series solutions for functions defined by algebraic varieties. For example to solve y^2 = x + x^3 for y, start by writing x = u^2 so that y^2 = u^2 ( 1 + u^4 ). There are two solutions, opposite in sign, one of which is y = u sqrt(1 + u^4) = u ( 1 + u^4/2 - u^8/8 + ... ) So y is now a power series in u, i.e. in x^(1/2) . dave === Subject: Re: Energy without energy. posting-account=Gcwsaw0AAAAAKhQ5fitf1OoggBoNgn9r So, Jack, what does this all mean? Are we doomed, or what? === Subject: Re: Matrix Differential Calculus e. === Subject: Re: Example poset but not lattice: explanation? X-RFC2646: Format=Flowed; Original Excuse me for messing up the arbitrary union notation, the criticism of it by previous replying writers was right. But I found out where the snag was causing me not getting a lattice but a mere poset. I like thinking in terms of how information suitably coded can be optimally \ used to code a finite set of distinct definitions. It works here too. If we define a \ universe U and two subsets A[1] and A[2] of it, with Intersection(A[1],A[2]) nonempty, then \ we can divide U in 4 disjoint sets (U\\Union(A[1],A[2]), A[1]\\(Intersection(A[1],A[2])), A[2]\\(Intersection(A[1],A[2])), and Intersection(A[1],A[2])). All the sets \ that can now be defined in U, including U itself and the empty set, can be defined as combinations of these four disjoint sets, of which there are 2^4=16. We can code each combination by a \ bitstring of length 4, in which 0 and 1 in a particular position represent respectively absence and presence of one of the sets in a combination. Defining the order < on the collection of these 16 strings as x < y iff y has 1's in all positions where \ x has 1's plus in one or more additional positions, defines a lattice. This lattice is known as the 4-hypercube that you get when you, in graph-theoretical terms, connect any two of its 16 vertices with an edge if their bitstrings can be converted into each other by changing only a single bit. I suppose the same procedure can be used to get a lattice for P[n] generally. The upshot is that my original problem shows that the two operators Union and Intersection cannot specify every combination of sets if they are not all pairwise disjoint. In the latter case e.g. the difference operator '\\' must be included. If they are, of course, their only the Union operator is needed. Peter van Emburg, Leiden. === Subject: Re: Example poset but not lattice: explanation? posting-account=ixf50QwAAABFW6BQ4fQR2F0f49XaC1bE |I suppose the same procedure can be used to get a lattice for P[n] |generally. One structure you should consider is the set of n-place boolean functions f(p_1,...,p_n) where p_1,...,p_n are boolean variables (either true or false) and the value of f is also boolean. There are 2^n possible sets of values for p_1,...,p_n. For each assignment of values to p_1,...,p_n there are two possible values of f, true or false. So there are 2^(2^n) possible functions f. These 2^(2^n) functions naturally form a lattice if we take the natural ordering on them, where f<=g if f(p_1,...,p_n) <=g(p_1,...,p_n) for each combination p_1,...,p_n. Here I take false < true. If we have n sets A_1,...,A_n and a function f as above, we can consider a set B_f associated with f, where x is in B_f if f(x is in A_1, x is in A_2, ..., x is in A_n) is true. Here for \x is in A_1\ we substitute the value \true\ if x is in A_1 and \false\ if x is not in A_1. This gives us a lattice mapping from the original big lattice of order 2^(2^n) to some lattice with 2^m elements, for m<=2^n. Here m is the number of combinations of membership or nonmembership in A_1,...,A_n that actually occur. If all 2^n possible combinations of membership or nonmembership in A_1,...,A_n occur, then the sets B_f are distinct. A standard Venn diagram with three sets is arranged so all eight possibilities occur in the diagram, for example. But we could have three sets that divide up the plane into some smaller number of regions... in a sense by just leaving some of the eight possibilities empty. At one point, you mentioned \bounded\ regions appearing in the diagram. This just means leaving out the unbounded region outside of all the original sets, which leaves us with 2^(2^n-1) possibilities. |The upshot is that my original problem shows that the |two operators Union and Intersection cannot specify every combination |of sets if they are not all pairwise disjoint. |In the latter case e.g. the difference operator '\\' must be included. |If they are, of course, their only the Union operator is needed. I would give a somewhat different explanation of why you didn't get a lattice to begin with. You still get a lattice using just unions and intersections if you allow the formation of unions of intersections (or intersections of unions). It's possible to stick to just one of those two, either just intersections of unions or just unions of intersections, by using the distributive laws to put one of the two operations on the \outside\. These correspond to the monotone boolean functions, the ones where if p_1<=q_1, p_2<=q_2, ..., p_n<=q_n then f(p_1,...,p_n) <=f(q_1,...,q_n). It's not hard to prove that they form a lattice, with \f or g\ and \f and g\ as the lattice operations. If n=3, for instance, we have: p1 FTFFTTFT p2 FFTFTFTT p3 FFFTFTTT FFFFFFFF F (corresponding to the empty set) FFFFFFFT p1&p2&p3 FFFFFFTT p2&p3 FFFFFTFT p1&p3 FFFFTFFT p1&p2 FFFFFTTT (p1&p3) or (p2&p3) = (p1 or p2) and p3 FFFFTFTT (p1&p2) or (p2&p3) = (p1 or p3) and p2 FFFFTTFT (p1&p2) or (p1&p3) = (p2 or p3) and p1 FFFFTTTT (p1&p2) or (p1&p3) or (p2&p3) = (p1 or p2)&(p1 or p3)&(p2 or p3) FFFTFTTT p3 FFTFTFTT p2 FTFFTTFT p1 FFFTTTTT p3 or (p1&p2) = (p1 or p3)&(p2 or p3) FFTFTTTT p2 or (p1&p3) = (p1 or p2)&(p2 or p3) FTFFTTTT p1 or (p2&p3) = (p1 or p2)&(p1 or p3) FFTTTTTT p2 or p3 FTFTTTTT p1 or p3 FTTFTTTT p1 or p2 FTTTTTTT p1 or p2 or p3 TTTTTTTT T (corresponding to the universal set). You only get 20 out of the 256=2^(2^3) elements in the original lattice. You do have to include combinations like p1 or (p2&p3) to get a lattice, though. Keith Ramsay === Subject: Re: Example poset but not lattice: explanation? X-RFC2646: Format=Flowed; Original > These correspond to the monotone boolean functions, the ones > where if p_1<=q_1, p_2<=q_2, ..., p_n<=q_n then f(p_1,...,p_n) > <=f(q_1,...,q_n). It's not hard to prove that they form a > lattice, with \f or g\ and \f and g\ as the lattice operations. Can I infer from this that the lattice of monotone boolean functions is a sublattice in the lattice of all n-variable boolean functions, be it that they are not to be ordered pointwise but by the lattice operations \f or \ g\ and \f and g\? > You only get 20 out of the 256=2^(2^3) elements in the > original lattice. You do have to include combinations > like p1 or (p2&p3) to get a lattice, though. My original poset was obtained by considering the 12 combinations of three sets A, B and C, with A&B&C nonempty ( I'll use '&' for intersection and 'or' for union from here on), that can be defined using & and or only as 1<=i<=n-ary relational operators, including the empty set, and excluding the \ universe. Then you get: {0, A, B, C, A&B, A&C, B&C, A&B&C, AorB, AorC, BorC, \ AorBorC). I was actually quite content to have spotted that this set ordered by set inclusion was not a lattice but a poset, currently studying Davey and Priestley's \Lattices and Order\! Having got to the chapter on Formal Concept Analysis, I looked up a references to the subject. Their content confirmed my suspicion that I had finally succeeded in identifying subjects \ in mathematics that are essential for an exact study of a central problem of \ scientific theorists that concerns their thinking as well as their exposition of its results, namely, the problem of how to proceed from the abstract and general to the concrete and special in 'subitizable' steps. In \ this process, drawings are often instrumental. I had concluded some time ago \ that the simplicity of the most used types of drawing for this purpose is deceptive. My presentation of the 12-member set above can be seen as a product of my having received only a fragmentary treatment of topics in discrete mathematics as part of my formal education, in which the familiar three-set Venn diagram figured prominently, drawings of it having at most the above 12 sets labeled to illustrate the set-operations of union and intersection. The reader may conclude that I at least have been confused by \ this treatment. My original posting is likely the result of the clash of my \ misconception with my taking up math seriously again after many years significantly motivated by the desire to become fully aware of what I am doing when, as a theorist, I want to make a helpful drawing. (Another example of a problem in schematic drawing in which its mathematical \ background is relevant is that of wanting to label all connections between n \ different entities with textual definitions which, for obvious reasons, one \ wants to avoid have to fit to intersecting paths. Kuratowski's Planar Graph \ Theorem says that for n>=5 this is impossible [a similar impossibility exists for graphs containing subgraphs homeomorphic with K3,3].) Peter van Emburg, Leiden. === Subject: JSH: Letting it drop posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY It's been fun, this hobby of mine of fiddling with simple math equations, and arguing about my work, but I'm increasingly concerned that maybe it's gotten a bit out of hand. Like, I actually had a family member who didn't realize I'd deleted the mathforprofit blog and someone took it over, who actually thought that maybe it was me, ranting and raving on the blog, like that I'd completely lost it. And yes, I am an intense person, and I've often taken this Quixotic quest of pushing against \math society\ and finding my own major discoveries very seriously, but I've also had the luxury of looking at it as an odd hobby, possible because of our high tech tools. But increasingly I'm getting a feeling that others have gone way overboard, with the webpages, and especially with the blog thing, and it's just not nearly as much fun as it used to be, and not nearly as easy for me to just dismiss Usenet antics as just, Usenet antics. The other fear, of course, is that others won't just let go, but I can't control what other people do, but I can control what I do, and to me, it looks like it's time to hang up the \JSH\, move on to other things, like focusing more on open source, and on other personal projects, to occupy my free time. Math was fun for a while, occupied my attention for years, but now, it's \been there, done that\ time. It has been a wild ride. It has been often truly crazy, as in, actually crazy, with a bizarre cast of characters, and some massive adventures as far as I'm concerned, from meetings with mathematicians (yes I've met with more than one) to contacts with journals around the world, and wacky editor replies (and you didn't get them all) to hearing from people from other countries who have been, yes, at times, very oddly hostile--especially Brits and Aussies, as well as people from the Netherlands for some odd reason--to some very nice people who have been very supportive. Actually there have been quite a few very supportive people over the years, from quite a few countries, which just goes to show you how connected the world actually is, not just as an abstraction, but as easy as going out on Usenet, and posting. It is an addictive experience to be sure, and one that doesn't often seem to be substantive, like I often wonder what I was really doing. But it is something to do. You know? And thinking about stepping away doesn't bother me, at all, while I wonder if I'll ever look back, and see it as something more. Was there really much to it at all? Did it all really matter? Who knows? Now it's on to more adventures. Maybe I'll find some other Quixotic quest to satisfy my thirst for intellectual adventure, and maybe I'll find some other people to argue with, as I do love to argue. Adventure is where you find it. The future is out there, and my path moves on, beyond... James Harris === Subject: Re: JSH: Letting it drop posting-account=EH2x8QsAAABu84CuyjstkC4nRyQ1ZHKW um ... good synopsis! thus quoth: Like, I actually had a family member who didn't realize I'd deleted the mathforprofit blog and someone took it over, who actually thought that maybe it was me, ranting and raving on the blog, like that I'd completely lost it. --Martha? http://tarpley.net/bush12.htm http://larouchepub.com http://members.tripod.com/american_almanac === Subject: Re: JSH: Letting it drop >It's been fun, this hobby of mine of fiddling with simple math >equations, and arguing about my work, but I'm increasingly concerned >that maybe it's gotten a bit out of hand. >Like, I actually had a family member who didn't realize I'd deleted the >mathforprofit blog and someone took it over, who actually thought that >maybe it was me, ranting and raving on the blog, like that I'd >completely lost it. Guffaw. Presumably said family member has not been following your posts on sci.math. >[...] >Was there really much to it at all? No. >Did it all really matter? No. >Who knows? But it's curious to see _you_ expressing doubts about whether it all really mattered. Does this mean that all your statements about how the world was going to end if we didn't acknowledge your genius were wrong? Just curious. >Now it's on to more adventures. Maybe I'll find some other Quixotic >quest to satisfy my thirst for intellectual adventure, and maybe I'll >find some other people to argue with, as I do love to argue. >Adventure is where you find it. >The future is out there, and my path moves on, beyond... >James Harris ************************ David C. Ullrich === Subject: Re: JSH: Letting it drop Keywords: not-serious days. My association with the Department is that of an alumnus. >But it's curious to see _you_ expressing doubts about whether >it all really mattered. Does this mean that all your statements >about how the world was going to end if we didn't acknowledge >your genius were wrong? Just curious. To borrow from a certain poster's recent musing: How curious that a certain poster just happens to announce he will stop posting to usenet, exactly when certain comments of his are Original is gone from Google, but you can find it in mathforum: http://mathforum.org/kb/message.jspa?messageID=412528&tstart=0 (mathforum moved recently; in case it moves again, you can see it quoted in: ) Or perhaps: Can it truly be coincidence that he decides to announce he will stop posting ->now<-? Really? -- \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: Letting it drop You know, with utter certainty, that you will be back. === ] Subject: JSH: Ok, I'm a loser ] ] It's finally settled in that I'm just some pathetic loser. If I ] weren't so pathetic I'd just go away gracefully, but I'll send one ] more post, or who am I kidding, my patheticness is so great that I'll ] probably post yet again. ] ] I'm disgusting. I'm just a pile of shit. I should just die like so ] many of you have said. I hate myself. I despise this life. I'm ] nothing but a sick joke to be made fun of by those of you who have ] real educations. People who actually know something, when I know ] nothing. I'm just nothing. ] ] If I hadn't been such a disgusting human being I'd have come to this ] realization years ago instead of wasting your time. ] ] My life is nothing. I know nothing. I'm worth nothing. I'm just ] shit. ] ] Please forgive me. All your attacks were justified. ] ] James Harris -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Letting it drop <3e0u71FkloeU1@individual.net> posting-account=P77RgA0AAACc6AiMlTdskcmY8pXoJG6q You know, with utter certainty, that you will be back. It's finally settled in that I'm just some pathetic loser. If I ] weren't so pathetic I'd just go away gracefully, but I'll send one ] more post, or who am I kidding, my patheticness is so great that I'll ] probably post yet again. snip a lot of James going on about how worthless he is... Oh, for once Harris _was_ right. No, seriously James, it's better to go out when you are on top. Perhaps now you can turn your awesome intellect to solving world hunger or global peace. Good luck === Subject: Re: JSH: Letting it drop > It's been fun, this hobby of mine of fiddling with simple math > equations, and arguing about my work, but I'm increasingly concerned > that maybe it's gotten a bit out of hand. Yes it has. For about ten years now. > Like, I actually had a family member who didn't realize I'd deleted the > mathforprofit blog and someone took it over, who actually thought that > maybe it was me, ranting and raving on the blog, like that I'd > completely lost it. Then you've explained that although that blog is not yours anymore, the texts it contains were actually written by you and the he decided that, yes, you *really* had completely lost it. > The other fear, of course, is that others won't just let go, but I > can't control what other people do, but I can control what I do, and to > me, it looks like it's time to hang up the \JSH\, move on to other > things, like focusing more on open source, and on other personal > projects, to occupy my free time. I can already imagine your posts in which you say that your typesetting system is indeed the best one available, althought you're unable even to create a single page that says \Hello World!\. :-) > It has been a wild ride. It has been often truly crazy, as in, > actually crazy, with a bizarre cast of characters, and some massive > adventures as far as I'm concerned, from meetings with mathematicians > (yes I've met with more than one) to contacts with journals around the > world, and wacky editor replies (and you didn't get them all) to > hearing from people from other countries who have been, yes, at times, > very oddly hostile--especially Brits and Aussies, as well as people > from the Netherlands for some odd reason--to some very nice people who > have been very supportive. Don't forget a certain very annoying guy from Portugal. > It is an addictive experience to be sure, and one that doesn't often > seem to be substantive, like I often wonder what I was really doing. You were being a first-rate crank. > Was there really much to it at all? Did it all really matter? No. > Who knows? Anyone but you. > Now it's on to more adventures. Maybe I'll find some other Quixotic > quest to satisfy my thirst for intellectual adventure, and maybe I'll > find some other people to argue with, as I do love to argue. I guess that you meant \rant\ instead of \argue\. Jose Carlos Santos === Subject: Re: JSH: Letting it drop > It's been fun, this hobby of mine of fiddling with simple math > equations, and arguing about my work, but I'm increasingly concerned > that maybe it's gotten a bit out of hand. > Like, I actually had a family member who didn't realize I'd deleted the > mathforprofit blog and someone took it over, who actually thought that > maybe it was me, ranting and raving on the blog, like that I'd > completely lost it. But after reading your stuff on Usenet, they're happy you're sane? In any event, could you walk us through the factorization 15 = 3x5 using your SFT method, in order to demonstrate that your method can factor a relatively small number. === Subject: Re: JSH: Letting it drop posting-account=Z3s1ZQwAAADpT8ng5YkWX5bGb68lNpli > In any event, could you walk us through the factorization 15 = 3x5 using > your SFT method, in order to demonstrate that your method can factor a > relatively small number. Actually could you maybe do a slightly larger number like 11*19 = 209 so we could at least see a few iterations? === Subject: Re: JSH: Letting it drop : Now it's on to more adventures. Maybe I'll find some other Quixotic : quest to satisfy my thirst for intellectual adventure, and maybe I'll : find some other people to argue with, as I do love to argue. Always more windmills on the horizon, little damage you do to them! Justin ps. We'll let it drop; Your name hasn't been mentioned once since you posted last. === Subject: Re: JSH: Letting it drop posting-account=0oMoGgwAAAA45r-W1MV-8N0QRNWHaDRI Good decision, only about 10 yrs. late. Better late than never though. I would suggest you try out the life of a total recluse, cut yourself off from the world. That would be best for everyone. === Subject: Re: Letting it drop X-RFC2646: Format=Flowed; Original > The other fear, of course, is that others won't just let go, but I > can't control what other people do, but I can control what I do, and to > me, it looks like it's time to hang up the \JSH\, move on to other > things, like focusing more on open source, and on other personal > projects, to occupy my free time. That is very good news. All the best to you in your new endeavors. > Math was fun for a while, occupied my attention for years, but now, > it's \been there, done that\ time. So I guess that means you got your rejection letter from the Annals. > And thinking about stepping away doesn't bother me, at all, while I > wonder if I'll ever look back, and see it as something more. I should think not. > Was there really much to it at all? Did it all really matter? Nope. === Subject: Re: Letting it drop X-RFC2646: Format=Flowed; Original >...hearing from people from other countries who have been, > yes, at times, very oddly hostile--especially Brits and Aussies,... There are cultural differences. During the cold war, Russian spies, caught in America, were instructed to plead not guilty. Those, caught in England, were instructed to plead guilty. === Subject: Re: JSH: Letting it drop posting-account=UtgH7gwAAACpBhTelVPOXNP7RAfbtQrK > It's been fun, this hobby of mine of fiddling with simple math > equations, and arguing about my work, but I'm increasingly concerned > that maybe it's gotten a bit out of hand. > Like, I actually had a family member who didn't realize I'd deleted the > mathforprofit blog and someone took it over, who actually thought that > maybe it was me, ranting and raving on the blog, like that I'd > completely lost it. > And yes, I am an intense person, and I've often taken this Quixotic > quest of pushing against \math society\ and finding my own major > discoveries very seriously, but I've also had the luxury of looking at > it as an odd hobby, possible because of our high tech tools. > But increasingly I'm getting a feeling that others have gone way > overboard, with the webpages, and especially with the blog thing, and > it's just not nearly as much fun as it used to be, and not nearly as > easy for me to just dismiss Usenet antics as just, Usenet antics. > The other fear, of course, is that others won't just let go, but I > can't control what other people do, but I can control what I do, and to > me, it looks like it's time to hang up the \JSH\, move on to other > things, like focusing more on open source, and on other personal > projects, to occupy my free time. > Math was fun for a while, occupied my attention for years, but now, > it's \been there, done that\ time. > It has been a wild ride. It has been often truly crazy, as in, > actually crazy, with a bizarre cast of characters, and some massive > adventures as far as I'm concerned, from meetings with mathematicians > (yes I've met with more than one) to contacts with journals around the > world, and wacky editor replies (and you didn't get them all) to > hearing from people from other countries who have been, yes, at times, > very oddly hostile--especially Brits and Aussies, as well as people > from the Netherlands for some odd reason--to some very nice people who > have been very supportive. > Actually there have been quite a few very supportive people over the > years, from quite a few countries, which just goes to show you how > connected the world actually is, not just as an abstraction, but as > easy as going out on Usenet, and posting. > It is an addictive experience to be sure, and one that doesn't often > seem to be substantive, like I often wonder what I was really doing. > But it is something to do. You know? > And thinking about stepping away doesn't bother me, at all, while I > wonder if I'll ever look back, and see it as something more. > Was there really much to it at all? Did it all really matter? > Who knows? > Now it's on to more adventures. Maybe I'll find some other Quixotic > quest to satisfy my thirst for intellectual adventure, and maybe I'll > find some other people to argue with, as I do love to argue. > Adventure is where you find it. > The future is out there, and my path moves on, beyond... > James Harris Don't let the door hit you in the ass on the way out. === Subject: Re: JSH: Letting it drop posting-account=Kb0T_QwAAACc9B9LpxfLjH0hHHYjPxft bye and good riddance... intellectual adventure? surely you jest. what i see from you is lack of intellect. === Subject: Re: JSH: Letting it drop > bye and good riddance... intellectual adventure? surely you jest. what > i see from you is lack of intellect. he'll be back within 4 weeks. Dirk Vdm === Subject: Re: JSH: Letting it drop >>bye and good riddance... intellectual adventure? surely you jest. what >>i see from you is lack of intellect. > he'll be back within 4 weeks. 2 weeks === newsspool2.news.pas.earthlink.net!stamper.news.pas.earthlink.net!elnk-nf2-pa\ s!newsfeed.earthlink.net!newshub.sdsu.edu!headwall.stanford.edu!pookiehead.da\ tabasix.com!not-for-mail Cujo DeSockpuppet alt.astrology, sci.astro, sci.math, sci.physics, sci.skeptic === Subject: Re: Inflationary Theory ; I'm confused DataBasix - Give us a chance to ridicule you and your stupid beliefs. 24 null@databasix.com <8N5Hd.11696$wZ2.4737@newssvr13.news.prodigy.com> <420175C5.37FE@earthlink.net> <42084878.764D@earthlink.net> <421154C5.647A@earthlink.net> <421F63EE.704E@earthlink.net> <07fUd.18623$yr.13017@okepread05> <427A48D6.2A10@earthlink.net> herekittykitty.databasix.com herekittykitty.databasix.com 1115313963 1996 abuse@databasix.com Xnews/5.04.25 X-Kook-Cancel: http://www.shorn/cancel.html X-Raytard-Failure: http://www3.primushost.com/~a/raymurphy/reication.html X-Wollkook-Quiz: http://www.shore.net/~a/bin/wollmann/qe.cgi X-Powerless-Whiner: http://www.sech.com/ed X-Kooky-Kwotes: http://www.petitmorte.net/phoenix/wollma.html X-Wollkook-Plagiarism: http://www.spinicnet/plagiarism.html X-Wollmann-Is-A-Spammer: http://www.rahul.n/falk/quickrefs.html#W X-SDSU-Spankard: http://www.shoreet/~a/wollmann/spank.html http://www.databasix.com/officialcharterst/astrology/metapsych/charter.html X-Message-to-Edmo: http://internettracom/users/spamster/ X-Edlish-Dictionary: http://www.petitmoe.net/cujo/edlish.html X-Wollmann-Kook-Awards: http://www.lart.com/a/whiners.html X-Wollman-Sexist-Archive: http://home.earlink.net/~sigyn/suepiggy/ X-Wollkook-The-Stupid: http://www.angelfe.com/ego2/edmowollmann/ X-Wollkook-Whinefest-Tactics: http://www.soninet/scott/sheesh/ed.txt X-Tholenizer: http://www.mdpub.cotholen/tholenizer.html X-Turi-Fraud: http://www.free.cts.com/crash/r/ripr/turi.htm $LJ2_oby,npJ}VpKzpM{hmU8*\XbhAlL7Y00a~cS2b{L\ *xw#TTgAOP/*CN[8@D\\(*O@^x%*CK|2uYF@+LtyWq3`H|$%h>%@m X-Fucknozzle-FAQ: http://www.geocities.com/drjosemarihi/jay_faq.html#bb X-Chucknozzle-FAQ: http://www.pitmorte.net/cujo/chucknozzle/ X-Mop-Jockey-Kooky-Kwotes: http://www.petitmorte.net/cujo/Robert_Woe.html X-Spamming-Mary-FAQ: http://www.petitmorte.net/cujo/mary/spammimary.html X-bRay-Kazooski-FAQ: http://www.petitmorte.net/cujo/kao/kazoo.html X-Wollkook-Wikipedia: http://en.wikipedia.org/wiki/ X-Ilya-Shambot-Wanker: http://tiurcom/6mtcw X-Owner-Of-Herc: Cujo news.earthlink.net alt.astrology:673823 sci.astro:454952 sci.math:755102 sci.physics:1049713 sci.skeptic:771061 (newsspool2.news.pas.earthlink.net) @earthlink.net: >> > TomGee 02/26/05 >> Space comes from the internal energy of matter. > I disagree, space is an ASPECT of matter. That's the same reasoning that had you declare Orion a star too, isn't it? It didn't help you lied about it afterwards. -- Cujo - The Official Overseer of Kooks and Trolls in dfw.*, alt.paranormal, alt.astrology and alt.astrology.metapsych. Colonel of the Fanatic Legion. FL# 555-PLNTY Motto: ABUNDANCE!. Charter Member - Digital Brownshirts and Library Gestapo. \It is truly KOOKY to be so obsessed with someone's success and your inability to defeat them, that you parade around as them.\ Edmo, explaining his use of all the spinics who act as sockpuppets. -- Edmond H. Wollmann P.M.A.F.A. © 2005 Altair Publications, SAN 299-5603 Astrological Consulting http://www.astroconsulting.com/ Artworks http://www.astroconsulting.com/personal/ http://home.earthlink.net/~arcturianone/ === Subject: Wollmannizer nocem 03909 @@NCM X-FAQ: http://www.smbtech.com/ed/ X-Listing: http://www.rahul.net/falk/quickrefs.html#W Mail-copies-to: Never X-FAQ: http://www.smbtech.com/ed/ http://www.smbtech.com/ed/ http://www.nocem.org/ http://www.rahul.net/falk/quickrefs.html#W -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 http://www.smbtech.com/ed/ @@BEGIN NCM HEADERS Version: 0.9 Issuer: wollmannizer@spam.free Type: spew Action: hide Count: 1 Notice-ID: Wollmannizer03909 @BEGIN NCM BODY <427AAD5C.4300@earthlink.net> alt.astrology sci.astro sci.math sci.physics sci.skeptic \ news.admin.net-abuse.usenet @END NCM BODY -----BEGIN PGP SIGNATURE----- Version: PGP 6.5.8 iQA/AwUBQnqvDCqwMoRBNDxCEQJY8ACgy9rMKgSkuHtdMABJ8zXk+vxyVSkAoMYT sWqoRry9/u2DABm0ZRWD9fTD =4e0H -----END PGP SIGNATURE----- === Subject: Re: What Lester is on about ... > On Wed, 04 May 2005 16:25:57 -0400, Will Twentyman >On Tue, 03 May 2005 19:27:45 -0400, Will Twentyman >> >> >On 3 May 2005 12:51:55 -0700, imaginatorium@despammed.com in > > > >> >> >> >On 2 May 2005 12:49:40 -0700, imaginatorium@despammed.com in >> >> If two numbers (curves?) are \pointed >>out\ on different curves, do you have any sort of formally \ specified >>procedure for identifying when these numbers (curves?) are the \ same? > >No but I imagine the straight line radius of curvature in a plane >would suffice for plane curves. >> >>What is \the straight line radius of curvature\ of a general curve? \ A >>sine wave, for example (y = sin x in the x-y plane). (I am >>understanding correctly that Zcurves are not necessarily arcs of >>circles, yes?) > > >I'm not familiar with Zcurves but, no, curves in general are not >necessarily arcs of circles. I have no idea what the \straight line >radius of curvature\ means exactly but I have seen the phrase and it >strikes me that most curves would have one or they wouldn't be curved >and we couldn't define the curvature of the curves that are curved. >> >>Zcurves are what you mean by \curves\. Short for Zick's curves, I \ suppose. >I was being facetious. Zick's curves are pretty much what curves are >in general. They just don't happen to be pointable outable on straight >line segments. It's why they're curves to begin with. >>If so, you talk about Zcurves in such an odd manner that it is easy to >>suspect you have something either different or more limited in mind. > No, not intentionally. I might be wrong. Maybe there are curves which > can be pointed out exactly on straight lines segments. But I don't > think so. The fact that you use the phrase \curves which can be pointed out exactly on straight lines segments\ is part of why I suspect you have something else in mind. I'm not sure what it means, so it is unlikely that it is a consideration in my definition of a curve. I don't know what you mean by a curve. I know you've offered explanations in the past, but all it convinced me of was that you were discussing something that was likely to be different from my understanding of the term. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: What Lester is on about ... On Thu, 05 May 2005 19:43:33 -0400, Will Twentyman >> On Wed, 04 May 2005 16:25:57 -0400, Will Twentyman >>On Tue, 03 May 2005 19:27:45 -0400, Will Twentyman >> >> >> > > >>On 3 May 2005 12:51:55 -0700, imaginatorium@despammed.com in >> >> >> > > > >>On 2 May 2005 12:49:40 -0700, imaginatorium@despammed.com in > > If two numbers (curves?) are \pointed >out\ on different curves, do you have any sort of formally \ specified >procedure for identifying when these numbers (curves?) are the \ same? >> >>No but I imagine the straight line radius of curvature in a plane >>would suffice for plane curves. > >What is \the straight line radius of curvature\ of a general curve? \ A >sine wave, for example (y = sin x in the x-y plane). (I am >understanding correctly that Zcurves are not necessarily arcs of >circles, yes?) >> >> >>I'm not familiar with Zcurves but, no, curves in general are not >>necessarily arcs of circles. I have no idea what the \straight line >>radius of curvature\ means exactly but I have seen the phrase and it >>strikes me that most curves would have one or they wouldn't be curved >>and we couldn't define the curvature of the curves that are curved. > >Zcurves are what you mean by \curves\. Short for Zick's curves, I \ suppose. >> >> >>I was being facetious. Zick's curves are pretty much what curves are >>in general. They just don't happen to be pointable outable on straight >>line segments. It's why they're curves to begin with. >If so, you talk about Zcurves in such an odd manner that it is easy to >suspect you have something either different or more limited in mind. >> No, not intentionally. I might be wrong. Maybe there are curves which >> can be pointed out exactly on straight lines segments. But I don't >> think so. >The fact that you use the phrase \curves which can be pointed out >exactly on straight lines segments\ is part of why I suspect you have >something else in mind. Hey, mathematikers are the ones claiming curves can be pointed out on straight lines, not me. Get your references straight. I'm claiming the opposite, that curves cannot be pointed out on straight line segments. > I'm not sure what it means, so it is \ unlikely >that it is a consideration in my definition of a curve. Well having explained the idea on different occasions I can readily understand why it isn't a consideration in your definition of a curve. >I don't know what you mean by a curve. I know you've offered >explanations in the past, but all it convinced me of was that you were >discussing something that was likely to be different from my >understanding of the term. Could well be if you expect to be able to point out (0,pi) on a straight line segment. === Subject: Re: What Lester is on about ... > I don't know what you mean by a curve. I know you've offered > explanations in the past, but all it convinced me of was that you were > discussing something that was likely to be different from my > understanding of the term. You are making the charitable assumption that there is coherent thought behind Lester's words. You are too charitable and you are incorrect in that assumption. Lester is either a troll or an incoherent deluded psuedo intellectual who has incoherent babble running around inside his head instead of coherent thought. Bob Kolker === Subject: Re: What Lester is on about ... On Thu, 05 May 2005 21:10:41 -0400, Robert Kolker >> I don't know what you mean by a curve. I know you've offered >> explanations in the past, but all it convinced me of was that you were >> discussing something that was likely to be different from my >> understanding of the term. >You are making the charitable assumption that there is coherent thought >behind Lester's words. You are too charitable and you are incorrect in >that assumption. Lester is either a troll or an incoherent deluded >psuedo intellectual who has incoherent babble running around inside his >head instead of coherent thought. Well of course there is a third alternative, that I'm right. === Subject: Re: What Lester is on about ... <1114545353.dc6565966f539572e370810b160ba62e@teranews> <42766ea5.42450668@netnews.att.net> <4276d003.56664926@netnews.att.net> <4277e550.73038770@netnews.att.net> <42780949$1_4@newsfeed.slurp.net> <427923c6.89187994@netnews.att.net> <4279302e$1_4@newsfeed.slurp.net> <42793759.91949834@netnews.att.net> <427aaffc$1_5@newsfeed.slurp.net> <7rOdnWvm1duGWeffRVn-2Q@comcast.com> <427bc455.8289356@netnews.att.net> posting-account=alQKkAwAAAA82xoCXcVIo1q-o-rv2IW- > On Thu, 05 May 2005 21:10:41 -0400, Robert Kolker > >> > >> I don't know what you mean by a curve. I know you've offered > >> explanations in the past, but all it convinced me of was that you were > >> discussing something that was likely to be different from my > >> understanding of the term. > >You are making the charitable assumption that there is coherent thought > >behind Lester's words. You are too charitable and you are incorrect in > >that assumption. Lester is either a troll or an incoherent deluded > >psuedo intellectual who has incoherent babble running around inside his > >head instead of coherent thought. > Well of course there is a third alternative, that I'm right. No there isn't. Technically, this is known as a totally regressed bifurcation, and by the Throaks-Lemming theorem, the number of alternatives cannot exceed the 2^p, where p is the cardinality of the set of points of significance, which in this instance is zero. (This is true, by the way, so it's no good wasting time trying to refute it. Only make yourself look a bit silly.) Brian Chandler http://imaginatorium.org === Subject: Re: What Lester is on about ... On 6 May 2005 13:01:04 -0700, imaginatorium@despammed.com in >> On Thu, 05 May 2005 21:10:41 -0400, Robert Kolker >> > >> >> >> >> I don't know what you mean by a curve. I know you've offered >> >> explanations in the past, but all it convinced me of was that you >were >> >> discussing something that was likely to be different from my >> >> understanding of the term. >> > >> >You are making the charitable assumption that there is coherent >thought >> >behind Lester's words. You are too charitable and you are incorrect >> >that assumption. Lester is either a troll or an incoherent deluded >> >psuedo intellectual who has incoherent babble running around inside >his >> >head instead of coherent thought. >> Well of course there is a third alternative, that I'm right. >No there isn't. Technically, this is known as a totally regressed >bifurcation, and by the Throaks-Lemming theorem, the number of >alternatives cannot exceed the 2^p, where p is the cardinality of the >set of points of significance, which in this instance is zero. If you say so, Brian. Of course you also said straight lines are curves so I don't quite know how to value your claims. But your faith in Bob's claims is quite touching. >(This is true, by the way, so it's no good wasting time trying to >refute it. Only make yourself look a bit silly.) Why would I try to refute it? I have your word on it. And your word is all mathematics needs. As a matter of fact all modern math seems to have is the word of modern mathematikers on a variety of topics. === Subject: The importance of copyediting a scientific paper Springer's copyediting of a couple of my papers over the past month has inspired me to write an essay thanking them, and the publishing world generally, for their essential contributions to the progress of science: http://cr.yp.to/bib/20050504-copyediting.txt ---D. J. Bernstein, Associate Professor, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago === Subject: Re: The importance of copyediting a scientific paper >Springer's copyediting of a couple of my papers over the past month has >inspired me to write an essay thanking them, and the publishing world >generally, for their essential contributions to the progress of science: > http://cr.yp.to/bib/20050504-copyediting.txt Editors have done much worse than anything you complain about there. For example I've had commas added that made, the grammar simply wrong. Without being, informed in advance. >---D. J. Bernstein, Associate Professor, Department of Mathematics, >Statistics, and Computer Science, University of Illinois at Chicago ************************ David C. Ullrich === Subject: Re: The importance of copyediting a scientific paper > Editors have done much worse than anything you complain about there. > For example I've had commas added that made, the grammar simply wrong. > Without being, informed in advance. My favorite galleys are still the ones where the publisher managed to eliminate every p at the beginning of a word. The ublisher, I mean. But there are larger issues here. Scientists send publishers thousands Do we really want to be paying vast sums of money for new capitalization and new line breaks? Don't we have better things to do with the money? ---D. J. Bernstein, Associate Professor, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago === Subject: Re: The importance of copyediting a scientific paper X-RFC2646: Format=Flowed; Original > But there are larger issues here. Scientists send > What exactly are we getting for this money? > Do we really want to be paying vast sums of > money for new capitalization and new line breaks? > Don't we have better things to do with the money? This is a tempest in a teapot. You're tenured. You know punctuation, anyway? People may read the preprints you send them. If you want correctly edited copy, send it out that way yourself. If you are upset, sue Springer for hurting your reputation and for pain and suffering. At least get very tough with them, and stop whining over page charges. People may think you are cheap. === Subject: Re: The importance of copyediting a scientific paper On Fri, 06 May 2005 08:36:01 -0500, David C. Ullrich >For example I've had commas added that made, the grammar simply wrong. >Without being, informed in advance. Doesn't that, like, frost you when your right and they make mistake's like that with you're stuff? --Lynn === Subject: Re: The importance of copyediting a scientific paper X-RFC2646: Format=Flowed; Original > Doesn't that, like, frost you when your right and they make mistake's > like that with you're stuff? > --Lynn Cool, but what difference does it make whether you're on \your right\ or \ on your left ? === Subject: Re: The importance of copyediting a scientific paper posting-account=0QrkrwwAAABDyQGPKX7NtkkaKfngvovA I wonder if some of Springer's \copyediting\ might be the result of computerized typesetting. David Ames === Subject: Extension Field Problem posting-account=_J0onQ0AAABRfM0aDgEV-8ddGrUSdEnN I have two problems: 1) Suppose F_p^l is some extension field of F_p defined by a polynomial f(x). Given a constant term c in F_p, does there exist an element g(x) in F_p^l such that for all h(x) in F_p^l, the following holds: c*h(x) and g(x)*h(x) have the same constat term for all h(x). 2) Given any c and g(x), how many g(x) in F_p^l will lead to the same constant for c*h(x) and g(x)*h(x)? === Subject: Re: Extension Field Problem > 1) Suppose F_p^l is some extension field of F_p defined by a polynomial > f(x). Given a constant term c in F_p, does there exist an element g(x) > in F_p^l such that for all h(x) in F_p^l, the following holds: > c*h(x) and g(x)*h(x) have the same constat term for all h(x). > 2) Given any c and g(x), how many g(x) in F_p^l will lead to the same > constant for c*h(x) and g(x)*h(x)? Your notation's a bit puzzling. I'll take it that f(x) is in F_p [x] and irreducible over F_p, that F_p^l is F_p [x] / (f), and that the element g(x) + (f(x)) is what you mean by the element g(x) of F_p^l. In 2), you either mean Given any c and g(x), how many h(x) .... or Given any c and h(x), how many g(x) .... but I'm not sure which. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Extension Field Problem posting-account=_J0onQ0AAABRfM0aDgEV-8ddGrUSdEnN > > 1) Suppose F_p^l is some extension field of F_p defined by a polynomial > > f(x). Given a constant term c in F_p, does there exist an element g(x) > > in F_p^l such that for all h(x) in F_p^l, the following holds: > > c*h(x) and g(x)*h(x) have the same constat term for all h(x). > > 2) Given any c and g(x), how many g(x) in F_p^l will lead to the same > > constant for c*h(x) and g(x)*h(x)? > Your notation's a bit puzzling. I am so sorry for my layman notations as I get used to computer programming notations. I tend to think no, but don't know how to disprove it. > I'll take it that f(x) is in F_p [x] and irreducible over F_p, > that F_p^l is F_p [x] / (f), and that the element g(x) + (f(x)) > is what you mean by the element g(x) of F_p^l. You're right. > In 2), you either mean > Given any c and g(x), how many h(x) .... > or > Given any c and h(x), how many g(x) .... > but I'm not sure which. Sorry again, I meant given any c and g(x) in F_p[x]/(f), how many h(x) in F_p[x]/(f) the following would hold: c * h(x) and g(x) * h(x) have the same constant term. I guess it is 1/p * cardinality of F_p[x]/(f), but don't know how to give a rigorous argument. > -- > Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Can you calculate the complete set of anti-diagonals from this \ sublist ? posting-account=WZMvOwwAAAC_B1TaayrVN99BJgDWQkUc > There is no antidiag. There's ALWAYS an anti-diag. > For every candidate antidiag ALL those candidates DO get elected. They are ALL anti-diags. > just add it to the list. Won't help. The new-added list STILL has an anti-diag. EVERY list ALWAYS has an anti-diag. > Spot the Flaw. The flaw is that you think that the process you are outlining above will get you something without an anti-diag. It won't. EVERY list generated by your process above HAS an anti-diag. > (Hint: how many antidiags are there?) You can't give that as a hint; you personally have no idea how many. It's a lot. A lot more than you concede exist. Specifically, it's exactly the same number as the number of reals, since EVERY real is the anti-diag of a GREAT MANY lists (of more lists than there ARE reals, in fact). === Subject: Re: Can you calculate the complete set of anti-diagonals from this \ sublist ? In sci.logic, george on 5 May 2005 17:19:52 -0700 >> There is no antidiag. > There's ALWAYS an anti-diag. >> For every candidate antidiag > ALL those candidates DO get elected. > They are ALL anti-diags. >> just add it to the list. > Won't help. The new-added list STILL > has an anti-diag. EVERY list ALWAYS has > an anti-diag. >> Spot the Flaw. > The flaw is that you think that the process you are outlining > above will get you something without an anti-diag. > It won't. EVERY list generated by your process above > HAS an anti-diag. >> (Hint: how many antidiags are there?) > You can't give that as a hint; you personally have > no idea how many. It's a lot. I actually do. HERC -- well, make up your own mind. :-) Take the original list, call it L0. L0 maps N to R. The mapping is alleged 1-1 and onto. (1-1 isn't too hard. Onto...well, see below.) We construct a D0_0 using the more or less usual procedure D0_0(k) = 3 if L0(k,k) = 4 = 4 otherwise In fact, I can go further. Construct an infinite number of values: D0_n(k) = 3 if floor(n/2^k) is even and L0(k,k) = 4 4 if floor(n/2^k) is even and L0(k,k) != 4 5 if floor(n/2^k) is odd and L0(k,k) = 6 6 if floor(n/2^k) is odd and L0(k,k) != 6 I now have *another* infinite list D0. Every element in D0 is not in L0. Now I compute X: X(k) = 2 if L0(k,k) = 7 7 if L0(k) != 7 It's pretty clear that X is not in L0. *It's also nowhere in D0.* Therefore the collection of candidate diagonals, each one proving that the reals are uncountable, is itself uncountable. > A lot more than you > concede exist. Specifically, it's exactly the same > number as the number of reals, since EVERY real is the > anti-diag of a GREAT MANY lists (of more lists than there > ARE reals, in fact). Well, at this point I can also drag out Lebesgue measure and simply point out that the image of L0 is a measure 0 set (since the domain is denumerable). Since the interval [0,1) has measure 1, there's a lot of candidates. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Can you calculate the complete set of anti-diagonals from this \ sublist ? posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L you are quoting Ghosts argument against me and disputing it moron. Herc === Subject: HERE'S A TIP ! posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L Instead of skimming each post and categorising it into 1/ Cardinality 2/ Halt 3/ Godel .. then seeing who can give the best immitation of a parrot in their reply TRY UNDERSTANDING WHAT YOU ARE REPLYING TO FIRST. Herc === Subject: Re: Subnormal quasisimple subgroups On 5-May-2005, mareg@mimosa.csv.warwick.ac.uk () > >On 5-May-2005, mareg@mimosa.csv.warwick.ac.uk () > >> > > >> >On 4-May-2005, mareg@mimosa.csv.warwick.ac.uk () > >> > > >> >> > >> >> >On 2-May-2005, mareg@mimosa.csv.warwick.ac.uk () > >> >> > > >> >> >> > >> >> >> >Inspired by a recent post of Derek Holt's about Schur > >> >> >> >multipliers, I > >> >> >> >decided to see how far I could get in chapter 11 \The > >> >> >> >generalized > >> >> >> >Fitting subgroup\ of Aschbacher's _Finite group theory_. > >> >> >> > > >> >> >> >It wasn't very far. I'm already stuck on (31.6) \Let L in > >> >> >> >Comp(G) (=the > >> >> >> >set of subnormal quasisimple subgroups of G) and H an > >> >> >> >L-invariant > >> >> >> >subgroup of G. Then either L in Comp(H) or [L,H] = 1.\ > >> >> >> > > >> >> >> >Supposedly this follows from (31.4) \Let L in Comp(G) and > >> >> >> >H <| <| G > >> >> >> >(=H is subnormal in G). Then either L in Comp(H) or [L,H] = \ 1.\ > >> >> >> > > >> >> >> >But I don't see it. I'm probably missing something obvious... [...] > Maybe we don't really need induction. > Let L in Comp(G) and let L normalize H. > Then L in Comp(), and since H is normal in , we can apply > (31.4) to to deduce that either L in Comp(H) or [L,H] = 1. > But that is exactly what we are trying to prove to get (31.6). Yes, yes, yes! I finally figured that out on my own, but you beat me to occurred to me to show that L is subnormal in and then apply (31.4) to the latter. -- Jim Heckman === Subject: Re: Epistemology 202: Advanced Topics > On Wed, 04 May 2005 14:46:54 -0400, Will Twentyman >> >> > > >>I would say the basic concept of set theory is the concept of a set. >> > >Can you explain that concept to me, keeping in mind that I am a >programmer and not a mathematician? The only set concept that I am >aware of is that of a container, meaning to me a closed boundary >isolating the things inside from the things outside, and containing >no duplicates. >> >>That closely corresponds to my notion of a set. I use the example of >>a dice bag when introducing the concept, where dice are the elements >>of the set, and the bag with its dice is the set. >> >How does one place a boundary around that which is boundless? >>The example of the dice bag is just that, an example to illustrate an >>abstract concept. Sets do not actually involve boundaries. > Oh sure they do in the form of definition for what's in the bag. > That's what the bag is. I interpretted Wagner's comment as \physical boundaries\, thus his comment about boundless. Definitions are, of course, boundaries. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Epistemology 202: Advanced Topics On Thu, 05 May 2005 21:46:12 -0400, Will Twentyman >> On Wed, 04 May 2005 14:46:54 -0400, Will Twentyman >> > > >> >> >I would say the basic concept of set theory is the concept of a set. > >> >>Can you explain that concept to me, keeping in mind that I am a >>programmer and not a mathematician? The only set concept that I am >>aware of is that of a container, meaning to me a closed boundary >>isolating the things inside from the things outside, and containing >>no duplicates. > >That closely corresponds to my notion of a set. I use the example of >a dice bag when introducing the concept, where dice are the elements >of the set, and the bag with its dice is the set. > >> >>How does one place a boundary around that which is boundless? >The example of the dice bag is just that, an example to illustrate an >abstract concept. Sets do not actually involve boundaries. >> Oh sure they do in the form of definition for what's in the bag. >> That's what the bag is. >I interpretted Wagner's comment as \physical boundaries\, thus his >comment about boundless. Definitions are, of course, boundaries. Yeah, I figured there was some kind of misunderstanding. No problem. === Subject: Re: Epistemology 202: Advanced Topics >> On Wed, 04 May 2005 14:46:54 -0400, Will Twentyman >> > > >> >> > I would say the basic concept of set theory is the concept of a \ set. > >> >> Can you explain that concept to me, keeping in mind that I am a >> programmer and not a mathematician? The only set concept that I >> am aware of is that of a container, meaning to me a closed >> boundary isolating the things inside from the things outside, and >> containing no duplicates. > > > That closely corresponds to my notion of a set. I use the example > of a dice bag when introducing the concept, where dice are the > elements of the set, and the bag with its dice is the set. > >> >> How does one place a boundary around that which is boundless? > The example of the dice bag is just that, an example to illustrate an > abstract concept. Sets do not actually involve boundaries. >> Oh sure they do in the form of definition for what's in the bag. >> That's what the bag is. > I interpretted Wagner's comment as \physical boundaries\, You were wrong. > thus his comment about boundless. A plane is boundless. A closed figure on that plane is a boundary. Both are abstract. Definitions are, of course, boundaries. Definitions are definitions. Nothing more. You can define a boundary, but to define a boundless boundary is a self-contradiction and an absurdity. -- \...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them.\ Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: ed Topics >> >Your claiming that humans can't conceive infinity would be as good as >a claim from me that humans can't tell apart red and green. >>Well, a common evolution insures that, except in rare cases, >>humans can tell red and green apart as a natural byproduct of >>having eyes and a brain in the world. >>Infinity however has never been experienced by humans in any form >>at any time, nor has our survival ever depended on detecting and >>acting on such a quality, So I am baffled as to on what such a >>conceptualization would be based. But I am not a cognitive >>scientist, nor a mathematician. I only hold beliefs. I cannot >>conceive of 'infinity', 'eternity', or 'forever' or any such >>'concepts'. The brains of mathematicians, however, may be >>special in this regard. > I would dare say that an intuitive of 'infinity' and related concepts > can actually be grasped by the human brain, in forms where the basic > concept is 'more'. For example, that you can count and count and count > on without reaching a biggest number. As far as I can tell, my grandson was able to make sense of this in kindergarden last year. He proudly announced that he could count to 100. OK, I said, count from ninety to one hundred. Which he did. What's the next number? I asked. Ummmmm .... a hundred and one? he said. Yes, I said. And the one after that is? 102! Can you think of a bigger number than that? I asked. Twenty hundred and one! And I said, I can count a bigger number than that: twenty hundred and two. I know a bigger number, he said, twenty hundred thousand hundred and two! And I can add one to that, I said. He caught on, and we spent a few minutes counting ever bigger numbers. Do you think you can keep counting for the rest of the day? I asked. Yes, he said. For the rest of the week? Yes, he said. Till your next birthday? I have to stop to eat, he said. If you don't have to stop to eat, can you count forever? Yes, he said. Case proven. === Subject: Re: ed Topics > >>Your claiming that humans can't conceive infinity would be as good as >>a claim from me that humans can't tell apart red and green. >> >Well, a common evolution insures that, except in rare cases, >humans can tell red and green apart as a natural byproduct of >having eyes and a brain in the world. >Infinity however has never been experienced by humans in any form >at any time, nor has our survival ever depended on detecting and >acting on such a quality, So I am baffled as to on what such a >conceptualization would be based. But I am not a cognitive >scientist, nor a mathematician. I only hold beliefs. I cannot >conceive of 'infinity', 'eternity', or 'forever' or any such >'concepts'. The brains of mathematicians, however, may be >special in this regard. >> I would dare say that an intuitive of 'infinity' and related concepts >> can actually be grasped by the human brain, in forms where the basic >> concept is 'more'. For example, that you can count and count and count >> on without reaching a biggest number. >As far as I can tell, my grandson was able to make sense of this in >kindergarden last year. He proudly announced that he could count to 100. >OK, I said, count from ninety to one hundred. Which he did. What's the >next number? I asked. Ummmmm .... a hundred and one? he said. Yes, I >said. And the one after that is? 102! Can you think of a bigger number >than that? I asked. Twenty hundred and one! And I said, I can count a >bigger number than that: twenty hundred and two. I know a bigger number, >he said, twenty hundred thousand hundred and two! And I can add one to >that, I said. He caught on, and we spent a few minutes counting ever >bigger numbers. Do you think you can keep counting for the rest of the >day? I asked. Yes, he said. For the rest of the week? Yes, he said. Till >your next birthday? I have to stop to eat, he said. If you don't have to >stop to eat, can you count forever? Yes, he said. >Case proven. It makes no difference whether you stop or not, it can't be done. By definition. Louis Savain The Silver Bullet: Why Software Is Bad and What We Can Do to Fix it http://users.adelphia.net/~lilavois/Cosas/Reliability.htm === Subject: Re: ed Topics [...] >I would dare say that an intuitive of 'infinity' and related concepts >can actually be grasped by the human brain, in forms where the basic >concept is 'more'. For example, that you can count and count and count >on without reaching a biggest number. >>As far as I can tell, my grandson was able to make sense of this in >>kindergarden last year. He proudly announced that he could count to 100. >>OK, I said, count from ninety to one hundred. Which he did. What's the >>next number? I asked. Ummmmm .... a hundred and one? he said. Yes, I >>said. And the one after that is? 102! Can you think of a bigger number >>than that? I asked. Twenty hundred and one! And I said, I can count a >>bigger number than that: twenty hundred and two. I know a bigger number, >>he said, twenty hundred thousand hundred and two! And I can add one to >>that, I said. He caught on, and we spent a few minutes counting ever >>bigger numbers. Do you think you can keep counting for the rest of the >>day? I asked. Yes, he said. For the rest of the week? Yes, he said. Till >>your next birthday? I have to stop to eat, he said. If you don't have to >>stop to eat, can you count forever? Yes, he said. >>Case proven. > It makes no difference whether you stop or not, it can't be done. By > definition. > Louis Savain Sure, but apprehending the impossibility is equivalent to grasping the notion of infinity, I think. My grandson had an intuitive grasp of \forever\, IMO. Later on, he will no doubt examine the notion more carefully, and eventually appreciate the subtle problems of conceptualsing it. Or not. Depends on what he chooses to do with his life. === Subject: Re: ed Topics > that, I said. He caught on, and we spent a few minutes counting ever > bigger numbers. Do you think you can keep counting for the rest of the > day? I asked. Yes, he said. For the rest of the week? Yes, he said. Till > your next birthday? I have to stop to eat, he said. If you don't have to > stop to eat, can you count forever? Yes, he said. > Case proven. Indeed. Humans are clever and little kids, the cleverest of all. The only thing they lack is experience. I am convinced that all little kids are geniuses before they reach the age of ten. The only reason why they are not geniuses at the age of twenty is because their teachers and/or parents have browbeaten it out of them. I spent time with my granddaughter for her second birthday party and in the weekend I was at her house she picked up at least two dozen new words. It is absolutely mind boggling watching a kid grow his/her first language. It is like zuchini squash. Four inches long at evening, one foot long by morning. Bob Kolker === Subject: Re: ed Topics >> that, I said. He caught on, and we spent a few minutes counting ever >> bigger numbers. Do you think you can keep counting for the rest of the >> day? I asked. Yes, he said. For the rest of the week? Yes, he said. >> Till your next birthday? I have to stop to eat, he said. If you don't >> have to stop to eat, can you count forever? Yes, he said. >> Case proven. > Indeed. Humans are clever and little kids, the cleverest of all. The > only thing they lack is experience. I am convinced that all little kids > are geniuses before they reach the age of ten. The only reason why they > are not geniuses at the age of twenty is because their teachers and/or > parents have browbeaten it out of them. Actually, it's puberty that does it. During puberty, about 30% of the synapses are disconnected, and new ones built. The young adult ends up with about 10% fewer synapses than the prepubescent child. Or so I have read. The new wiring also redirects the young human's interests in the outer world, from things and neat things to with them them to persons and pleasurable ways to interact with them. > I spent time with my granddaughter for her second birthday party and in > the weekend I was at her house she picked up at least two dozen new > words. It is absolutely mind boggling watching a kid grow his/her first > language. It is like zuchini squash. Four inches long at evening, one > foot long by morning. > Bob Kolker Heck, I think zuchini is _slow_ compared to toddlers. You can hear their synpases clicking together - just put your ears up against their heads. Hear that lowpitched hum? That's the sound of very busy brain. :-) === Subject: Re: ed Topics > Heck, I think zuchini is _slow_ compared to toddlers. You can hear their > synpases clicking together - just put your ears up against their heads. > Hear that lowpitched hum? That's the sound of very busy brain. :-) Indeed. The little wheeles are turning round and round. Bob Kolker === Subject: Re: The logical structure of calculus, request for help. > .... > * minimal knowledge of vector spaces. > - Definition of a vector space. > - Definition of a subspace. > - Theorems involving subspace. > - Definition of a norm. > * minimal knowledge of differential calculus. > - Definition of a limit. > - Theorems involving combinations of limits. > - Definition of continuity at a point. > - Definition of continuity on an interval. > - Theorem: The intermediate value. > - Theorems involving combinations of continuous functions. > - Definition of derivative at a point. > - Definition of derivative on an interval. > - Theorem that differentiable at b implies continuous at b. > - Theorems for computing derivatives; power law, etc. > - Theorem of the chain rule. > The above should be the bare minimum and a logical structure for > teaching calculus? .... Absolutely not! A good pedagogical order of topics is hardly ever the same as the logical order. If it were, then we'd have to teach two-year-olds the Peano axioms before we could let them start counting \One, two, three, ...\ Advanced mathematics is often difficult, but so are foundations. The best place to begin learning is often in between. Someone who has built up a bit of experience at using naive calculus may then start to ask searching questions about its foundations, and that's the time to introduce topics from your list. If I wanted to offer extra insight to a beginner in calculus, my own preference would be historical, as in the little book by Otto Toeplitz, \The Calculus: a Genetic Approach.\ But of course other people could well do it in other ways. Ken Pledger. === Subject: Re: The logical structure of calculus, request for help. >>.... >>* minimal knowledge of vector spaces. >>- Definition of a vector space. >>- Definition of a subspace. >>- Theorems involving subspace. >>- Definition of a norm. >>* minimal knowledge of differential calculus. >>- Definition of a limit. >>- Theorems involving combinations of limits. >>- Definition of continuity at a point. >>- Definition of continuity on an interval. >>- Theorem: The intermediate value. >>- Theorems involving combinations of continuous functions. >>- Definition of derivative at a point. >>- Definition of derivative on an interval. >>- Theorem that differentiable at b implies continuous at b. >>- Theorems for computing derivatives; power law, etc. >>- Theorem of the chain rule. >>The above should be the bare minimum and a logical structure for >>teaching calculus? .... > Absolutely not! A good pedagogical order of topics is hardly ever > the same as the logical order. If it were, then we'd have to teach > two-year-olds the Peano axioms before we could let them start counting > \One, two, three, ...\ > Advanced mathematics is often difficult, but so are foundations. > The best place to begin learning is often in between. Someone who has > built up a bit of experience at using naive calculus may then start to > ask searching questions about its foundations, and that's the time to > introduce topics from your list. > If I wanted to offer extra insight to a beginner in calculus, my > own preference would be historical, as in the little book by Otto > Toeplitz, \The Calculus: a Genetic Approach.\ But of course other > people could well do it in other ways. > Ken Pledger. Are you saying that my outline was not the logical way to learn the \ subject? My goal is to go step-by-step through the theory and with no hand waving. Adam. === Subject: Re: The logical structure of calculus, request for help. \A. Boom\ > Consider calculus of a real variable. > Is the following the logical structure for at least differential > calculus, assuming knowledge of algebra, functions, set theory? IMO the word \calculus\ is not very well defined, but you might like to look over the table of contents of Dieudonne's _Treatise on Analysis_. That set of books also has quite a worthwhile set of appendices on mainstream (but not trivial) algebra. LH === Subject: Re: Complex Analysis > This is almost surely not the historical development (is it?), but if > asked to define e, the most a natural way for me is as follows: For > positive a and real x, one defines a^x as a limit of a^q, > where q approaches x through the rationals. One notes that if > f(x) = a^x, then f'(x) is proportional to f(x). We define e to be > that base a for which the constant of proportionality is unity. > Again, the definition invokes ther relationship f' - f = 0 in a very > essential way. This is how I approach it with beginning students. The students are already familiar with the exponential functions a^x. It's natural to ask what their derivatives are. One easily shows da^x/dx = C(a)*a^x, where C(a) is a constant depending only on a. It's then and there that you ask yourself: How can I make C(a) disappear? Ie, for which a is C(a) = 1? Which leads directly to e^x. But let's be clear: The point of view is not: Let's find a function such that f' = f. Rather it's: What is the derivative of a^x? === Subject: RUN LIKE HECKMAN! posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L stop teaching this newbie jerk your attack dog antics on the 'cranks'. the idiot posts up tonnes of rubbish then always nicks off, 4 times now every thread of mine he jumps in boasting this that and the other, GIVE HIM A QUESTION AND HE BOLTS!! Herc === Subject: Re: RUN LIKE HECKMAN! posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 P.S. The real reason you shouldn't reply to HERC777's posts is because HERC777 is actually a program written by me in the area of what may be termed \Artificial Stupidity.\ One of the first examples of programs in this area was known as B1FF, a linguistic filter designed to simulate a teenager typing at his brother's Commodore 64.* The program HERC777 was designed to mimic particular attributes of James Harris, including being able to ignore any form of intelligence or logical rebuttal, and to plow ahead with its own \logical conclusions\, incorporating certain keywords (like \hyperbola\ and \non-trivial\) which are entered into its database at random times. Recently, the program HERC777 has developed a virus-like program to spread itself around the Net. Every time you respond to one of its posts, you are strengthening the connection between your computer and another host computer which is already infected. If everyone ignores HERC777, it will not spread any further. Yes, in that case, I violated my own rule, but I needed to pass on the warning to the readers of sci.math before HERC777 gets a hold of a large number of computers. The consequences could be catastrophic. Naturally you can expect HERC777 to deny that it's a person and not a mindless machine, but you know better than to believe that, right? --- Christopher Heckman * The motto of the Artificial Stupidity field is: \If AI can't simulate an idiot, how do you expect it to simulate a genius?\ === Subject: Re: RUN LIKE HECKMAN! posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L prose of the coward. >If everyone ignores HERC777, it will not spread any further. >Yes, in that case, I violated my own rule, but I needed to pass on the >warning to the readers of sci.math before HERC777 gets a hold of a give up on your rhyming 1st post being quantum entangles, he shuts up give up on your 1 is not prime diatribe, he shuts up after I disprove him give up on your F U C K I N G P U N Y I N F I N I T E S I M A L observation that LOOOOK everybody, I disproved the anticantor proof, the list2 has D(L2) = r. DONT WORRY ABOUT IT PEOPLE. HE FOUND THE DIAGONAL AFTER ALL!!!!!! what a fucking tosspot. Herc === Subject: Re: RUN LIKE HECKMAN! posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L >or logical rebuttal??????? D(L2) = r ??????? THATS YOUR REBUTALL OF THE ANTI-ANTIDIAGONALISABLE LIST? YOUR A FUCKING MORON we know there's a sequence of digits down the diaonal your fuckwit. WHAT NUMBER DOES IT MAKE? clue : www.freewebs.com/namesort/numsort.html Herc LOL what a conceited moron === Subject: Re: RUN LIKE HECKMAN! posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > stop teaching this newbie Newbie!?!?!?! I've been posting for 10 years now. > jerk your attack dog antics on the 'cranks'. > the idiot posts up tonnes of rubbish Has anyone OTHER THAN YOU ever said that's what it is? I think not. > then always nicks off, 4 times now > every thread of mine he jumps in boasting this that and the other, > GIVE HIM A QUESTION AND HE BOLTS!! > Herc ATTENTION PEOPLE OF USENET: Stop paying attention to this newbie jerk named HERC777. The idiot posts up tons of garbage and then changes topics when you respond to him. 12 times now when people have responded in a rational manner he jumps in with a non sequitor or two. GIVE HIM A QUESTION AND HE IGNORES IT! --- Christopher Heckman === Subject: Re: RUN LIKE HECKMAN! \Proginoskes\ >> [crap snipped] > ATTENTION PEOPLE OF USENET: HERC777 aka Herc aka |-|erc is a well-known psycho. And anyway when you hear the words Javascript and Cantor in the same posting, the crank alarm should sound. === Subject: Re: RUN LIKE HECKMAN! posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L only a blind idiot could run the program on this website by clicking the button a few times www.freewebs.com/namesort/numsort.html and proclaim LOOK D(L2) = r WHAT IS r? The claim is that List1 and List2 don't have any set diagonal. I know you're favorite blackboard trick is to run your finger down the board on a skew and flutter your eyes at the students that you this awesome comprehension of superinfinty, but try to look at the facts here, the list DOESNT HAVE ANY DIAGONAL. GIVE THE MAN A LIST! Herc === Subject: Re: RUN LIKE HECKMAN! > stop teaching this newbie jerk your attack dog antics on the 'cranks'. > the idiot posts up tonnes of rubbish then always nicks off, 4 times now > every thread of mine he jumps in boasting this that and the other, I dunno. He's an instructor at my university, and I've never heard anyone ever say anything bad about him, especially about his knowledge of math (he teaches a problem-solving course wherein the material is tangent to virtually every field of mathematics). You, on the other hand .... Jason === Subject: Re: RUN LIKE HECKMAN! posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L you wouldn't know what a field of mathematics is. He is not a well known mathematician, he never will be and he never aspires to be. Do you know how I know? HE HASN'T DONE ANY MATHS. What's his theorem? He's never uncovered a single insight. You know how I know? Because he spends his day sitting in his office behind the computer calling the entrepreneurs of the maths world trolls and cranks for their input. How would you like to discover a new theory with world ramifications and you take it to your beloved prof and he MOCKS YOU BECAUSE ITS NEW. He doesn't understand it and he gets paid to keep the new stuff down. teach your dog some new tricks, like learning the pecking order. Results 1 - 10 of 1,540 for group:sci.logic author:\|-|erc\. Results 1 - 10 of 1,570 for author:\|-|erc\ group:sci.math Results 1 - 10 of 298 for group:sci.math author:proginoskes Herc === Subject: Re: RUN LIKE HECKMAN! posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > you wouldn't know what a field of mathematics is. > He is not a well known mathematician, he never will be and he never > aspires to be. I will admit that it's only been 5 years since I got my doctorate, and most of the time since then has been devoted to teaching. That's a situation I want to fix. So I'm not a well-known mathematician, but may be some day. > Do you know how I know? HE HASN'T DONE ANY MATHS. Now THAT is a joke. Look up the following paper: (1) Heckman, C., and Thomas, R. \A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs\, Discrete Mathematics, Vol 233/1-3, pp. 233-237 (or check out http://math.asu.edu/~checkman/514.ps.gz ) I also have several papers related to this result which are being reviewed. One solves a conjecture from 1976 which some \big names\ had trouble with. (2) Heckman, C., and Thomas, R. \Independent Sets In Triangle-Free Cubic Planar Graphs\, submitted to Journal of Combinatorial Theory, Series B. This is the paper that solves the conjecture. The paper was sent back for the second round of refereeing; the first round led to a dramatic reduction in the size of the paper (from 24 pages to 16). It can be downloaded from http://math.asu.edu/~checkman/new3-8.pdf (3) Heckman, C. \On the Tightness of the 5/14 Independence Ratio\, awaiting refereeing at Discrete Mathematics. Analyzes cases of equality in paper (1), which turns up a lot of interesting graphs, some of which show up in (2). Available at http://math.asu.edu/~checkman/Eq514.pdf (4) Heckman, C. \Independence Ratios and Matching Ratios\, being written up. This is an actual improvement from my thesis, replacing a constant (3/11) by a better one (7/25). My thesis itself is available at the Georgia Tech library. (I've authored a book!) https://gil.gatech.edu/cgi-bin/Pwebrecon.cgi?v1=7&ti=1,7&FT=%2BHeckman&CNT=2\ 5+records+per+page&phrase_type=3&SUBMIT=Go&PID=11545&SEQ=20050506022119&SID=1\ I'm also a footnote in the new Four Color Theorem Proof: Robertson, Neil; Sanders, Daniel P.; Seymour, Paul; Thomas, Robin, \A new proof of the four-colour theorem.\ Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 1, 17--25. You can download the paper itself from http://www.math.gatech.edu/~thomas/PAP/npfc.pdf I guess I didn't do so much mathematics as computer programming. And my doctrate isn't in mathematics; it's in mathematics, computer science, and industrial engineering. I'm a graduate of the ACO program at Georgia Tech. You can read more about it here if you want: http://www.math.gatech.edu/aco/ So far, it's been all research. I've also provided a glimpse of linear programming, with some interesting insights, called \Linear Programming: Beyond 4.2 (The Simplex Method)\. THAT can be found at http://math.asu.edu/~checkman/Beyond4_2.pdf There are some other papers which show how to do certain things on graphing calculators. Instead of boring you with all of this, I'll just direct you to http://math.asu.edu/~checkman/research.html#papers . > What's his theorem? He's never uncovered a single insight. Read the papers. > You know how I know? This has gotta be good for a laugh. > Because he spends his day sitting in his office That's a laugh, for two reasons; first of all, I don't have an office; I only have a cubicle. Second of all, I spend about 12 hours a week teaching. You can see what my schedule for Spring 2005 was by going to http://math.asu.edu/~checkman/teach.html#sched . > behind the computer calling the entrepreneurs of the > maths world trolls and cranks for their input. The only person I've called a \crank\ is James Harris. Are you suggesting that he is some sort of bright light that I'm trying to snuff out? Again, you haven't done your research. Look up the posts where I reply to James Harris. I am urging him along and trying to guide him into useful aspects of Surrogate Factoring. I even started a thread called \Talking Rationally About Surrogate Factoring\, which took some of his ideas and established a formal mathematical basis for them. I even told him where the productive area(s) of SF are, advice he has eventually taken. And when he simply wouldn't budge intellectually or back away from things like his \50% claim\, yes, I called him a crank. > How would you like to discover a new theory with world > ramifications and you take it to your beloved prof and > he MOCKS YOU BECAUSE ITS NEW. Again, I am assuming you're talking about James Harris here. I actually looked at his work (the prime-counting function and SF) with an open mind. The prime-counting function was not new; it's a rewriting of some formulas which have been used elsewhere. SF I've already talked about. I have not mocked anyone for any work they've done. In fact, for evidence to the contrary (you don't seem to have a lot of evidence on your side; have you noticed that?), a student wanted to show me a proof she had found that the logarithm of 5 base 2 is irrational. It was nice and short and didn't need a whole lot of number theory (just the Unique Factorization Theorem), and I told her so. I also suggested that she try to generalize it. I've responded to people who have posted alleged proofs of the 4CT at LANL's website. I debunked two of them last September or October, and and accepted it gracefully. > He doesn't understand it and he gets paid to keep the > new stuff down. If I get paid to \keep the new stuff down\, I'm not being paid well enough. As to not understanding things, well, I don't understand spiral colorings (a possible source for a new computer-free (!) proof of the 4CT) yet, but I feel there's enough there to look at and will take time to read it. But if you ARE talking about Harris, I got a larger understanding of SF than what he presented. I saw a practical use of his complicated \Surrogate Factoring Transformation\, a viable direction, and told him to seek it out. So once again, you seem to think you have some insight into my psychological character, whereas you haven't even checked out what I've done. > teach your dog some new tricks, like learning the pecking order. What pecking order? My advisor kept on insisting that I call him \Robin.\ Or learning how to communicate mathematically. 90% of the people of the world think mathematics is only algebra. 9% of the people of the world think mathematics is only calculus. A very small number of people have seen mathematics as a logical method which establishes correct conclusions from assumptions. If I make an analogy of mathematics to literature, 90% of the people in the world would think that writing a book amounts to printing letters at random. They would not know how to write words, sentences, or a whole book. The \new tricks\ are teaching people how to spell, how to organize thought and get it on paper, and how to build a major result. Of course, you won't be reading any of this, which is a shame. I have to stop here and finish writing up a Final Exam. --- Christopher Heckman === Subject: Re: RUN LIKE HECKMAN! On 5 May 2005 23:45:43 -0700, \Proginoskes\ >Of course, you won't be reading any of this, which is a shame. But Christopher, of *course* you know he won't be reading (understanding) that until you upgrade your artificial stupidity program with comprehension. :-) > I have to stop here and finish writing up a Final Exam. Exams! Yes indeed. How I miss preparing and grading exams. --Lynn === Subject: Re: RUN LIKE HECKMAN! posting-account=W2DCTA0AAAAlbhDMl3GrysSnPy1IK_7f Christopher, you shouldn't let HERC (or anyone, as a matter of fact) get to you like that. There was no need for you to list your achievements --- it is obvious to anyone but HERC that you are more of a mathematician than he could ever be in his wildest dreams. And to him it wouldn't make a blind bit of difference if you'd won a fields medal. HERC seems to have less understanding of mathematics than even JSH does. Lasse === Subject: Re: RUN LIKE HECKMAN! posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L YOU'RE HIRED! but can you describe in 2 or 3 sentences what this program demonstrates? www.freewebs.com/namesort/numsort.html because if you think r = D(L2) is a missing real you're about 85 - 90 iq. Herc === Subject: Re: Who first proved that card(P(N)) = card(R)? > Or is this still an unresolved question? > Sketch of proof: > Let f : P(N) -> [0,1) be the mapping that takes an element S of P(N) > (or, equivalently, a subset S of N) and computes the value > f(S) = sum{s in S}(2^-s). This is onto, but not quite 1-1 > (.0111... = .1000...), though it's close. Note that the number of such is countable. So making it 1-1 is the same as mapping two countable sets on a single countable set. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, \ +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Who first proved that card(P(N)) = card(R)? In sci.math, Dik T. Winter : > > Or is this still an unresolved question? > > > > Sketch of proof: > > > > Let f : P(N) -> [0,1) be the mapping that takes an element S of P(N) > > (or, equivalently, a subset S of N) and computes the value > > f(S) = sum{s in S}(2^-s). This is onto, but not quite 1-1 > > (.0111... = .1000...), though it's close. > Note that the number of such is countable. So making it 1-1 is the same > as mapping two countable sets on a single countable set. Agreed; the range of all failures is in fact T_2; the domain is the reverse mapping of T2 and the reverse mapping of T2 deconstructed in such a way as to prefer the ...111... form as opposed to the ...000... form. Hence my term \close\. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Who first proved that card(P(N)) = card(R)? In sci.math, Dave Seaman : >> Or is this still an unresolved question? > Definitely not. Just making sure. :-) >> Sketch of proof: >> Let f : P(N) -> [0,1) be the mapping that takes an element S of P(N) >> (or, equivalently, a subset S of N) and computes the value >> f(S) = sum{s in S}(2^-s). This is onto, but not quite 1-1 >> (.0111... = .1000...), though it's close. >> Google isn't horribly helpful. > A while ago there was a thread that discussed who first showed that the > Cantor set is uncountable by considering base-3 expansions that contain > no 1's. The verdict was that Cantor himself gave this argument. Notice > that there is an obvious bijection between the set of all base-3 strings > consisting entirely of 0's and 2's, and P(N). > The rest of the argument follows from the Cantor-Schroeder-Bernstein > theorem. Hmmm.... http://mathworld.wolfram.com/Schroeder-BernsteinTheorem.html -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Who first proved that card(P(N)) = card(R)? > In sci.math, Dave Seaman > >: >> A while ago there was a thread that discussed who first showed that the >> Cantor set is uncountable by considering base-3 expansions that contain >> no 1's. The verdict was that Cantor himself gave this argument. Notice >> that there is an obvious bijection between the set of all base-3 strings >> consisting entirely of 0's and 2's, and P(N). >> The rest of the argument follows from the Cantor-Schroeder-Bernstein >> theorem. > Hmmm.... > http://mathworld.wolfram.com/Schroeder-BernsteinTheorem.html Yes, that's the one. The attribution sometimes varies. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Tuesday 10 May 2005 Lisp NYC: Tim Daly on the Large Computer \ Algebra System Axiom Keywords: available, hackable, freely redistributable source; GPL, BSDL, \ ArtisticL, XSL Tim Daly, head of the Axiom Project, will address Lisp NYC, starting at 1900 hours, Tuesday 10 May 2005, in Trinity Lutheran Church, at Ninth \ Street and Avenue B on the Island of the Manahattoes. See below my signature for meeting details and directions. Axiom is a large, serious, old growth computer algebra system. Axiom is free software and almost all of Axiom's source code has been converted to \literate code\. Much of Axiom's base programming system is Lisp. History Axiom has been in development since 1971. At that time, it was called Scratchpad. Scratchpad was a large, general purpose computer algebra \ system that was originally developed by IBM under the direction of Richard Jenks. The project started in 1971 and evolved slowly. Barry Trager was \ key to the technical direction of the project. Scratchpad developed over a \ 20 year stretch and was basically considered as a research platform for developing new ideas in computational mathematics. In the 1990s, as \ IBM's fortunes slid, the Scratchpad project was renamed to Axiom, sold to the Numerical Algorithms Group (NAG) in England and became a commercial system. As part of the Scratchpad project at IBM in Yorktown Tim Daly worked on all aspects of the system and eventually helped transfer the product to NAG. For a variety of reasons it never became a financial success and NAG withdrew it from the market in October, 2001. Open \ Source NAG agreed to release Axiom as free software, under this license. The \ basic motivation was that Axiom represents something different from other programs in a lot of ways. Primarily because of its foundation in mathematics the Axiom system will potentially be useful 30 years from now. In its current state it represents about 30 years and 300 man-years \ of research work. To strive to keep such a large collection of knowledge \ alive seems a worthwhile goal. Tim Daly on one of the central problems of computerdom: My goal isn't to solve physics/math problems. My goal is to build a \ system that will be used by computational mathematicians 30 years from now. \ Once this is the stated goal several things become clear. This is from a proper rant, to be found at For the official Lisp NYC announcement, including directions to the gathering place, see below. Jay Sulzberger Corresponding Secretary LXNY LXNY is New York's Free Computing Organization. http://www.lxny.org
Please join us for our next meeting on Tuesday, May 10th from 7:00 to 9:00 at Trinity Lutheran Church. Timothy Daly, published author, academic researcher, open source programmer and lead developer of Axiom will be presenting about his role as the driving force behind Axiom. With over 70 developers and 200 researchers worldwide it can best be described as: Axiom is a general purpose Computer Algebra system. It is useful for research and development of mathematical algorithms providing a very high level way to express abstract mathematical concepts. The Axiom Library defines over 1,000 strongly-typed mathematical domains and categories. Axiom consists of an interpreter and compiler, a browser, a graphical interface, and a new online wiki that allows users to create web pages that inline computations. Axiom is built upon Common Lisp. For more Axiom information: http://savannah.nongnu.org/projects/axiom http://page.axiom-developer.org/ http://page.axiom-developer.org/zope/mathaction/ReduceWiki Directions to Trinity: Trinity Lutheran 602 E. 9th St. & Ave B., on Thomkins Square Park http://trinitylowereastside.org/ Walk East 4 blocks on St. Marks, cross Thomkins Square Park. Walk E one or two blocks, turn north for 8 short blocks Walk E one block, turn sounth for 5 short blocks The M9 bus line drops you off at the doorstep and the M15 is near get off on St. Marks & 1st) To get there by car, take the FDR (East River Drive) to Houston then go NW till you're at 9th & B. Week-night parking isn't bad at all, but if you're paranoid about your Caddy or in a hurry, there is a parking garage on 9th between 1st and 3rd Ave. _______________________________________________ Lisp mailing list Lisp@lispnyc.org http://www.lispnyc.org/mailman/listinfo.cgi/lisp
=== Subject: Are two Young's Inequality the same? posting-account=1mXT6A0AAADM2fUgi354VGkfoNS1n7nb I have a question about \Young's Inequality\. There is a version of this on mathworld and planetmath which I will not bother to restate, but I read about another inequality that also calls itself \Young's Inequality\ which goes as follows: Let f be a convex functional on a convex set C in a normed space and let [f*,C*] = [f,C]* (here the notation [f,C] means the \epigraph\ of f, i.e. the region \above\ the graph of f, and C* and f*(x*) denote the conjugate of C and the conjugate of f* at x* respectively). If x is in C and x* is in C*, then Young's inequality states that x*(x) <= f(x) + f*(x*) Are these the same inequality??? They do not seem related to me. Shin === Subject: Re: Are two Young's Inequality the same? posting-account=itv7GQ0AAABsfqw8ZBgJBF5mLAWSacCN I know of two inequalities attributed to Young which are not the same. The first and more common of the two states that the L^r norm of the convolution of two functions f and g is bounded by the product of the L^p norm of f and the L^q norm of g provided 1/r+1=1/p+1/q. For a reference see \Fourier Analysis\ by J. Duoandikoetxea, page 17. The second Young's Inequality (that I know) is realted to convex functions. What you describe certainly sounds like a generalization. For the more concrete case see A. Zygmund's \Trigonometric Series\ V1. pg 16. > I have a question about \Young's Inequality\. There is a version of > this on mathworld and planetmath which I will not bother to restate, > but I read about another inequality that also calls itself \Young's > Inequality\ which goes as follows: > Let f be a convex functional on a convex set C in a normed space and > let [f*,C*] = [f,C]* (here the notation [f,C] means the \epigraph\ of > f, i.e. the region \above\ the graph of f, and C* and f*(x*) denote the > conjugate of C and the conjugate of f* at x* respectively). If x is in > C and x* is in C*, then Young's inequality states that > x*(x) <= f(x) + f*(x*) > Are these the same inequality??? They do not seem related to me. > Shin === Subject: Re: curious properties of functions >> ... >> Hm, maybe I should have suggested he further consider what happens >> when applying the definition of the derivative to functions defined on >> the quaternions? >Yes, you should have. Is there a \quaternion analysis\? How should we begin? Do we still want f'(a) = lim (f(z)-f(a))/(z-a) ? Or should it be f'(a) = lim (z-a)^{-1} (f(z)-f(a)) ? -- don't forget that quaternion multiplication is noncommutative. Suppose f(z) = z^2 and a = i. The difference quotient is then (z^2 + 1)/(z - i). Calculus students know the next step is to factor and cancel. Unfortunately in this case we don't have (z^2+1) = (z+i)(z-i) for all z so there is no cancellation. Surely if z is a real multiple of i then this algebra is valid so the difference quotient tends to 2i. But what if z = i + t j for real t ? Then z^2+1 = t(ij + ji) - t^2 = \ -t^2 while (z-i)^{-1} = 1/(tj) = (-1/t) j, so the difference quotient is t j which tends to zero. So we have to conclude that the derivative in the quaternionic sense simply does not exist. This sort of puts the kibosh on the idea of quaternionic analysis. >And if there is, does it have triply and quadruply period functions? One can of course view H as C^2, and yes, there are quadruply- periodic functions on this complex manifold, but that's sort of cheating. >Presumably, before we get on to quaternion analysis we need to sort out >quaternion algebra. For example, how is a quaternion determinant >defined? I don't know why one needs to answer this question first, but it's not going to work as nicely as one might hope. Certainly one may form the rings M_n( H ) and the groups GL_n( H ) of their invertible elements, but there is no homomorphism GL_n(H) -->H which sends special diagonal matrices diag(1, ..., a, 1, ...) to a. (That's easy: there are two natural factorizations of diag(i,j) into such factors, but ij does not equal ji .) You can embed H into M_2(C) and therefore M_n(H) into M_{2n}(C), and then compose this embedding with the determinant, but this composite is trivial. dave === Subject: Re: curious properties of functions > >> > >> ... > >> > >> Hm, maybe I should have suggested he further consider what happens > >> when applying the definition of the derivative to functions defined on > >> the quaternions? > >Yes, you should have. Is there a \quaternion analysis\? > How should we begin? Do we still want f'(a) = lim (f(z)-f(a))/(z-a) ? > Or should it be f'(a) = lim (z-a)^{-1} (f(z)-f(a)) ? -- don't forget > that quaternion multiplication is noncommutative. > Suppose f(z) = z^2 and a = i. The difference quotient is then > (z^2 + 1)/(z - i). Calculus students know the next step is to factor > and cancel. Unfortunately in this case we don't have (z^2+1) = \ (z+i)(z-i) > for all z so there is no cancellation. Surely if z is a real multiple > of i then this algebra is valid so the difference quotient tends to \ 2i. > But what if z = i + t j for real t ? Then z^2+1 = t(ij + ji) - t^2 = \ -t^2 > while (z-i)^{-1} = 1/(tj) = (-1/t) j, so the difference quotient is > t j which tends to zero. So we have to conclude that the derivative > in the quaternionic sense simply does not exist. > This sort of puts the kibosh on the idea of quaternionic analysis. I rather thought so! > >And if there is, does it have triply and quadruply period functions? > One can of course view H as C^2, and yes, there are quadruply- > periodic functions on this complex manifold, but that's sort of cheating. > >Presumably, before we get on to quaternion analysis we need to sort out > >quaternion algebra. For example, how is a quaternion determinant > >defined? > I don't know why one needs to answer this question first, but it's not > going to work as nicely as one might hope. Certainly one may form the > rings M_n( H ) and the groups GL_n( H ) of their invertible elements, > but there is no homomorphism GL_n(H) -->H which sends special diagonal > matrices diag(1, ..., a, 1, ...) to a. (That's easy: there are two > natural factorizations of diag(i,j) into such factors, but ij does \ not > equal ji .) You can embed H into M_2(C) and therefore M_n(H) into > M_{2n}(C), and then compose this embedding with the determinant, but > this composite is trivial. === Subject: Re: A curious parallel >> Why do you lie? >Where do I lie? \You've made your bed; now lie in it.\ === Subject: RUUUN CHRISTOPHERRR.................... RUN WITH ALL YOUR MIGHT!! posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L Take the losers with you. Herc === Subject: Re: RUUUN CHRISTOPHERRR.................... RUN WITH ALL YOUR \ MIGHT!! posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > Take the losers with you. > Herc Well, I guess there's no way to reason with an automaton that spits these things out, so I'll have to take the other approach and lean back and laugh at all the stuff you come up with. I, of course will be reminded of that old saying: \Usenet is like a herd of performing elephants with diarrhea -- massive, difficult to redirect, awe-inspiring, entertaining, and a source of mind-boggling amounts of excrement when you least expect it.\ --- Gene (\spaf\) Spafford And realize that one elephant by itself can create a lot of excrement. --- Christopher Heckman P.S. At least I have the guts to sign my real name on my posts. (That ought to be good for a couple bucketsful 8-) 8-) 8-). ) === Subject: Re: RUUUN CHRISTOPHERRR.................... RUN WITH ALL YOUR \ MIGHT!! On 6 May 2005 00:24:23 -0700, \Proginoskes\ >Well, I guess there's no way to reason with an automaton that spits >these things out, so I'll have to take the other approach and lean back >and laugh at all the stuff you come up with. I, of course will be >reminded of that old saying: > \Usenet is like a herd of performing elephants with diarrhea > -- massive, difficult to redirect, awe-inspiring, > entertaining, and a source of mind-boggling amounts of > excrement when you least expect it.\ --- Gene (\spaf\) > Spafford One other saying that may apply here: \Never wrestle with a pig. You both get muddy and the pig enjoys it.\ --Lynn === Subject: Re: www.freewebs.com/namesort/numsort.html DISPROOF OF HIGHER \ INFINITY In sci.logic, HERC777 on 5 May 2005 19:36:45 -0700 > www.freewebs.com/namesort/numsort.html DISPROOF OF HIGHER INFINITY > Just click the 2 buttons a few times, its OBVIOUS that the antidiagonal > doesn't exist. > Herc It's equally obvious your javascript still doesn't have a clue. Fix it, will you? Input list: 1 0.12011102011012221202 2 0.11000110121111200221 3 0.01000100210012120020 4 0.01120200110222211000 5 0.12201122121120102122 6 0.21012020222101011122 7 0.00211122122001112121 8 0.11022011220202221120 9 0.10201111000120021121 10 0.00002020111111010212 11 0.12120201001020212220 12 0.21002011110000111221 13 0.22220220200220221210 14 0.12202000100222122020 15 0.11101202200100201012 16 0.01010010111010101002 17 0.20200211110000122011 18 0.02110022121122201200 19 0.22111021011010001010 20 0.10002120211002221210 Output, erm, ... 3 0. 0 1000100210012120020 1 0.1 2 011102011012221202 5 0.12 2 01122121120102122 6 0.210 1 2020222101011122 2 0.1100 0 110121111200221 8 0.11022 0 11220202221120 4 0.011202 0 0110222211000 9 0.1020111 1 000120021121 7 0.00211122 1 2200111212110 0.000020201 1 111101021211 0.1212020100 1 02021222015 0.11101202200 1 0020101212 0.210020111100 0 011122113 0.2222022020022 0 22121014 0.12202000100222 1 22020 -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: www.freewebs.com/namesort/numsort.html DISPROOF OF HIGHER \ INFINITY posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L 3 0. 0 1000100210012120020 1 0.1 2 011102011012221202 5 0.12 2 01122121120102122 6 0.210 1 2020222101011122 2 0.1100 0 110121111200221 8 0.11022 0 11220202221120 4 0.011202 0 0110222211000 9 0.1020111 1 000120021121 7 0.00211122 1 22001112121 10 0.000020201 1 1111010212 11 0.1212020100 1 020212220 15 0.11101202200 1 00201012 12 0.210020111100 0 0111221 13 0.2222022020022 0 221210 14 0.12202000100222 1 22020 just your linefeed character, I'll try adding another \\\n\ but it will double space it for everyone else so try www.freewebs.com/namesort/linux.html heard of dual boot? Herc === Subject: Re: www.freewebs.com/namesort/numsort.html DISPROOF OF HIGHER \ INFINITY posting-account=Kb0T_QwAAACc9B9LpxfLjH0hHHYjPxft hey knucklehead, so you're still adamant that the area of a 1 by 1 square is 0? === Subject: Re: www.freewebs.com/namesort/numsort.html DISPROOF OF HIGHER \ INFINITY posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L I said \adamantly dictate\ 3 days ago now there's an echo. like saying \fantastic\ at a party... www.freewebs.com/namesort/numsort.html (linux too) what's your take on what the algorithm shows dickwad? Herc === Subject: Re: www.freewebs.com/namesort/numsort.html DISPROOF OF HIGHER \ INFINITY posting-account=Kb0T_QwAAACc9B9LpxfLjH0hHHYjPxft i don't even need to look at them as if what you argue is true it implies that the area of a 1 by 1 square is 0. obviously you must believe that. === Subject: Lattice completeness: lack of least upper bound? I need help with lattices. Apparently, a lattice can be called complete if every subset of it has both a least upper bound [supremum] and a greatest lower bound [infinum]. absolutely no mention of the exceptions. What kind of lattice has a subset which does NOT have a supremum and/or infinum? Specifically, I was asked to prove the following textbook problem wrong: \Show that every nonempty subset of a lattice has a least upper bound and a greatest lower bound\ (Rosen, Discrete Mathematics and its Applications) Don't worry, the assignment has already been turned in, I just want to make sure I know what's going on before the final exam... === Subject: Re: Lattice completeness: lack of least upper bound? > I need help with lattices. Apparently, a lattice can be called complete > if every subset of it has both a least upper bound [supremum] and a > greatest lower bound [infinum]. > absolutely no mention of the exceptions. What kind of lattice has a > subset which does NOT have a supremum and/or infinum? Linear ordered lattices Z, Q and R. === Subject: weak form and strong form of Reductio Ad Absurdum Re: Euclid & \ Aristotle <42752913.8793B970@iw.net> <4277BAA6.FEA62A19@iw.net> <1115144201.422187@athnrd02> <427870FF.458AC519@iw.net> <4279C53C.C58C2189@iw.net> <427A50E4.F87597F8@iw.net> > >> It appears that mathematically/logically, a reductio is a method of > > proof by which, assuming p, you prove ~p. Since p & ~p can't both be > > true, p must be false. It is NOT except loosely what I thought it > was, > > that is, the derivation of an absurdity - something that violates > your > > logical system's axioms and propositions - from an assumption. But > > rather one specific absurdity - that p is both true and not true. > And of course, what the site actually does is derive p from ~p, a > contradiction, thus proving p. The opposite (but equivalent) to what > I said. > Ken > PS: the issue of whether or not the ancient greeks used the rigorous > form of reductio obviously has little to do with my own stumblings and > rumblings and bumblings. > Ken I am quoting Ioannis Galidakis in his exact words so as not to make the fallacy of putting my own words into his mouth : >Although they were >taking the consistency of their system for granted, they could easily have >used contradiction as a \device\ which allowed them to differentiate truth >from falsity. >In particular, if they arrived at it, using deductive reasoning, this meant >that their *reasoning* was faulty, and not necessarily that their >axiomatic >system was. The \atopon\ sentence p/\\~p, could still have been seen as >simply a false statement which could have resulted from >ludicrous/mistaken >reasoning.\ Ken, perhaps you stumbled and bumbled onto a good new thing. Which can happen sometime. After reading Ioannis's message I had the feeling of saying their are two or several species of Reductio Ad Absurdum. Call one of them the Weak form of Reductio ad Absurdum and call the full fledged reductio ad absurdum as Strong Form Reductio Ad Absurdum. A week ago I would have said reductio ad absurdum comes and exists in only one form. But today I am convinced there are other forms. In the weak form there is no premissa step of \Suppose false\. Perhaps the Ancient Greeks never had the Strong Form of Reductio Ad Absurdum and had only the Weak Form. I have not looked at Aristotle's Baroco and Bocardo to Barbara and analyzed it for whether the argument is clearly reductio ad absurdum or whether it is hidden and vaguely reductio ad absurdum. My guess is that it is hidden and vague and open to interpretation and translation. The reason I say this is because to have the Strong Form of Reductio Ad Absurdum that clearly started with Saccheri because he had a consistent logical full set-- Euclidean geometry. Whereas Aristotle had only forms and syllogisms and not a consistent all encompassing set. So that Aristotle, like what Ioaniss is saying that they deduced truths from axioms and when they hit onto a contradiction they merely said a above statement was \per the impossible\. The important facts are that Euclid's Infinitude of Primes proof was not Reductio ad Absurdum and that Aristotle's Barbara becomes the strongest case of reductio ad absurdum but if Aristotle's Barbara is vague and hidden and open to contention, then the Ancient Greeks did not have Reductio Ad Absurdum and was discovered by Saccheri in the 18th century. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: weak form and strong form of Reductio Ad Absurdum Re: Euclid & \ Aristotle à \Archimedes Plutonium\ \ Ûçòáãå \ óôï \ íÜîùíá > I am quoting Ioannis Galidakis in his exact words so as not to make the > fallacy of putting my own words into his mouth : > >Although they were > >taking the consistency of their system for granted, they could easily > have > >used contradiction as a \device\ which allowed them to differentiate > truth > >from falsity. > >In particular, if they arrived at it, using deductive reasoning, this > meant > >that their *reasoning* was faulty, and not necessarily that their > >axiomatic > >system was. The \atopon\ sentence p/\\~p, could still have been seen \ as > >simply a false statement which could have resulted from > >ludicrous/mistaken > >reasoning.\ > Ken, perhaps you stumbled and bumbled onto a good new thing. Which can > happen sometime. > After reading Ioannis's message I had the feeling of saying their are two > or several species of Reductio Ad Absurdum. Call one of them the Weak > form of Reductio ad Absurdum and call the full fledged reductio ad > absurdum as Strong Form Reductio Ad Absurdum. A week ago I would have > said reductio ad absurdum comes and exists in only one form. But today I > am convinced there are other forms. > In the weak form there is no premissa step of \Suppose false\. Perhaps > the Ancient Greeks never had the Strong Form of Reductio Ad Absurdum and > had only the Weak Form. > I have not looked at Aristotle's Baroco and Bocardo to Barbara and > analyzed it for whether the argument is clearly reductio ad absurdum or > whether it is hidden and vaguely reductio ad absurdum. My guess is that > it is hidden and vague and open to interpretation and translation. The > reason I say this is because to have the Strong Form of Reductio Ad > Absurdum that clearly started with Saccheri because he had a consistent > logical full set-- Euclidean geometry. Whereas Aristotle had only forms > and syllogisms and not a consistent all encompassing set. So that > Aristotle, like what Ioaniss is saying that they deduced truths from > axioms and when they hit onto a contradiction they merely said a above > statement was \per the impossible\. > The important facts are that Euclid's Infinitude of Primes proof was not > Reductio ad Absurdum and that Aristotle's Barbara becomes the strongest > case of reductio ad absurdum but if Aristotle's Barbara is vague and > hidden and open to contention, then the Ancient Greeks did not have > Reductio Ad Absurdum and was discovered by Saccheri in the 18th century. Don't quote me, I asked today professor Michael Lambrou of the University \ of Crete, who specializes in Ancient Greek Mathematics and its history, among many other things: http://www.math.uoc.gr/dept/persons/lambrou.html First, he said that Aristotle in his \Prior Analytics\ discusses \ *various* forms of contradictions and as an example, he presents the proof that sqrt(2) is not in Q. Indeed his proof in Euclid's Elements X, 117, is by contradiction. We not only checked the above in the ancient text, but also performed a search for the words \oper atopon\ (thus contradiction) in Euclid's \ original texts (he had it in his computer) and found _at least_ 27 occurences of the words, in Elements *only*. The very *first* proposition in Euclid's Book 1, demonstration 6, (which is the well known theorem if two opposite angles are equal in a triangle then the corresponding sides are equal) uses a direct contradiction. He also said that not only Greeks used it, but it was clearly a Greek \invention\, one step above direct reasoning. I believe this settles the matter. > Archimedes Plutonium -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable === Subject: Re: An interesting problem of algebra >I feel very sorry I made a mistake! >The orginal problem is >A set of matrix M defined :M is a set of matrix such that any A,B belong \ to >M satisfy (AB)^3=BA then the two arbitrary element of set M is \ commutative! This is false with M = {A,B}, where A and B are two matrices satisfying A^2 = B^2 = (AB)^4 = I. (i.e. A,B generate the dihedral group of order 8). Derek Holt. === Subject: Re: An interesting problem of algebra On Thu, 05 May 2005 12:25:17 +0100, Robin Chapman >Robin J. Chapman, MA PhD NBG ASBO And yet lacking the sense to know the *correct* way to answer a homework question on the interent. Here is one example: http://flyingmoose.org/tolksarc/homework.htm and a comment by me The reply-to email address is olczyk2002@yahoo.com. This is an address I ignore. To reply via email, remove 2002 and change yahoo to interaccess, ** Thaddeus L. Olczyk, PhD There is a difference between *thinking* you know something, and *knowing* you know something. === Subject: Re: An interesting problem of algebra posting-account=DaXtvAwAAACPATqzmVZ4JJbgwUGjL51g >Robin J. Chapman, MA PhD NBG ASBO >Robin J. Chapman, MA PhD NBG ASBO > What is incorrect in that response?? Yours was incorrect: since when is > it true that > (AB)^3 = (A^3)(B^3) for matrices?? > Writing nonsenses and then even accusing others of > \incorrectness\...ts,ts,ts! Hmmm. Another of my betes noires is misleading attributions :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html \Elegance is an algorithm\ Iain M. Banks, _The Algebraist_ === Subject: Re: An interesting problem of algebra > On Thu, 05 May 2005 12:25:17 +0100, Robin Chapman >>Robin J. Chapman, MA PhD NBG ASBO > And yet lacking the sense to know the *correct* way to > answer a homework question on the interent. At least I know that it is bad practice to give answers (even obviously bogus ones) to homework questions. Robin J. Chapman, MA PhD NBG ASBO BOGOF -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html \Elegance is an algorithm\ Iain M. Banks, _The Algebraist_ === Subject: Re: FLT: What is this? >> > Hi All, >> > >> > I saw this link on Goo-gle's directed advertising. >> > >> > http://www.manilatimes.net/national/2005/may/05/yehey/top_stories/20050505to\ p4.html >> > >> > >> > Is this some sick joke perpetrated by our resident > loon/crank/troll? >> > >> > It sure sounds very familiar! >> Hmmmmm. The \Wiles\ letter doesn't appear to be in his usual >> literary style :-( > My guess is that it's an email EEE received from a moderately > clever impostor -- clever enough to bamboozle EEE regarding > his identity, and to inject one juicy bit of irony apparently > undetected, but not clever enough to get the tone quite right. I wondered that if this \Wiles\ note were bogus, would Wiles have a case under law against the Manila Times? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html \Elegance is an algorithm\ Iain M. Banks, _The Algebraist_ === Subject: Re: FLT: What is this? posting-account=Kb0T_QwAAACc9B9LpxfLjH0hHHYjPxft someone who knows Andrew Wiles should let him know of the libel. === Subject: Re: FLT: What is this? ... > > Using the new real number system Escultura constructed many > > counterexamples to FLT showing that it is false. > So where are they? Proving the FLT is false is easy: Find one > counterexample and do the calculations. So give us the counterexample, > already! EEE uses the 10-adics. There are indeed counterexamples in the 10-adics. Not such a surprise, as there are two idempotent numbers a and b in the 10-adics that are not equal to 1 but with 1 as sum. So a^n + b^n = 1^n for every positive n. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, \ +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: FLT: What is this? X-RFC2646: Format=Flowed; Original [Proginoskes, on ] > The interesting (and suspicious) part is: >> Using the new real number system Escultura constructed many >> counterexamples to FLT showing that it is false. > So where are they? Proving the FLT is false is easy: Find one > counterexample and do the calculations. So give us the counterexample, > already! This is old stuff. \Integer\ doesn't mean what you (or anyone else, \ except perhaps Archimedes Plutonium) think it does. If you allow for infinite integers, all kinds of pesky problems go away . EEE gave specific \counterexamples\, for example, here: What made the difference between previous attempts and mine? I do not simply plunge into a field and solve a problem. I find out first if both the problem and the field make sense. In the case of FLT the underlying fields of foundations and the real number system were defective as a consequence of which FLT itself did not make sense. As a remedy it was necessary to make a thorough critique of both fields, reconstruct them under new sets of axioms and suitably reformulate the problem. AP constructed clearer \counterexamples\ a decade ago, effectively using infinite p-adics. === Subject: Triangulation of a space (help please) posting-account=X_8L9A0AAAARdmkhIWn8tPwPAzzWlP9F Hello I'm trying to understand the concept of triangulation of a topological space. I'd really appreciate if you can help to understand this idea. So far I've seen how to triangulate a torus with 10 vertices, I think I understand this procedure, you just identity the opposite edges of the square in the same direction. What I _dont_ understand at all is how to find a triangulation of the torus with _seven_ vertices. I know how the picture looks like since the author has already solved this problem, but why I don't understand is how to construct it. I have some questions: 1) How do you know from where to where connect the lines from the distinct vertices?? 2) What determines to join them in such a way? 3) Once you outlined the construction of the simplex how do you prove that the underlying space is actually homeomorphic to the torus? For instance in the case of 10 vertices, you construct a natural triangulation of the unit square I^2 and then extend linearly each vertex to the vertices of the torus, which is clear. 4) But for this example (that is the triangulation of the torus with seven vertices) how do you know that such triangulation does in fact works? 5) What do you have to check to know that it is in fact a triangulation?? The book says \each simplex is uniquely determined by its vertices\ I don't really understand what does this means. What are the steps to check that your triangulation really works? Finally what is the usual approach to construct a triangulation for a given space? === Subject: Re: Triangulation of a space (help please) > What I _dont_ understand at all is how to find > a triangulation of the torus with _seven_ vertices. The torus can be represented as the plane factored out by the action of a free abelian group of rank two, that is one chooses two linearly independent vectors v and w in the plane and identifies each x with x + mv + nw for each integers m and w. You can triangulate the plane with equilateral triangles. Now you can choose v and w in such a way that if x is a vertex then so are x + v and x + w and such that there are seven triangles T_1, ... , T_7 such that every triangle is T_j + mv + nw for some integers m and n. By identifying the boundaries of these seven triangles you get the torus! -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html \Elegance is an algorithm\ Iain M. Banks, _The Algebraist_ === Subject: Re: A cyclic inequality schrieb sangjeong.kim@gmail.com : > Let a, b and c be positive real numbers. > Then show that the following inequality holds. > 1/7 < a/(7a+b) + b/(7b+c) + c/(7c+a) <= 3/8 One of the inequalities is equivalent to 685*a*b*c+49*b*a^2+49*a*c^2+49*b^2*c > 0, the other to 35*b*a^2+35*a*c^2+13*c*a^2+35*b^2*c+13*b^2*a+13*b*c^2 >= 144*a*b*c. To show the latter inequality it might be good to calculate what sum the coefficients on the left-hand side (3x35, 3x13) add up to... === Subject: Re: A cyclic inequality posting-account=9lv2eA0AAABuI_cYMyJe4ryuJyhklxho The leftmost inequality is actually trivial. We have a/(7a+b)>a/(7a+7b+7c). Now sum. === Subject: Re: A cyclic inequality posting-account=9lv2eA0AAABuI_cYMyJe4ryuJyhklxho This is an odd inequality. Most classical inequalities of this type have the sign facing the other way, making it easier to apply, say, Cauchy or Jensen's. Thus it's possible that the best we may hope for inequality of Schur as well. However, expanding the sums makes the problems trivial. We need only show that 13 Sum_{cyc} a^2c+35 Sum_{cyc} a^2b \\geq 144 abc abc. This yields the desired result. I will think about better methods later. === Subject: Re: Polynomial problem > Assume f is a polynomial R^2 --> R defined on the unit square > I = [0,1] x [0,1] s.t. f is symmetric about the other diagonal (i.e., > f(x, y) = f(1-y, 1-x)) Characterize all such f. Here is another idea: Use that the transformation T: (x,y) |-> (1-y,1-x) is an INVOLUTION: T(T(x,y)) = (x,y). This makes that the linear map P: f |--> P(f) P(f)(x,y) := 1/2 f(x,y) + 1/2 f(T(x,y)) = 1/2 f(x,y) + 1/2 f(1-y,1-x) on the vector space of polynomials is a projection: P(P(f)) = P(f). You are asking for those polynomials f that lie in the kernel of the complementary projection (Id-P), (Id-P): f |--> [(x,y) |--> 1/2 f(x,y) - 1/2 f(1-y,1-x)]. Basic linear algebra now tells you that ker(Id-P) = Im(P) = { P(f) : f polynomial } = LinearSpan{ P(x^k*y^l) : k,l >=0 } = LinearSpan{ x^k*y^l + (1-y)^k*(1-x)^l : k >= l >= 0 } So you are looking for polynomials that are linear combinations of expressions of the form [x^k*y^l + (1-y)^k*(1-x)^l]. === Subject: Re: Is there a solution? [ ... nonsense ... ] > I already know what they are. The way I'm trying to solve my problem is > that I have about a million computer-generated sets of (a,b,c,d). There > are an infinite number of (x,y) pairs that will solve these equations. > I'm trying to find the (x,y) pair where both are integers. You should not really bother with the answers which are obviously wrong. Try to understand Dave Rusin's answer instead! === Subject: Re: Probability distribution puzzle posting-account=gldHlQ0AAACaHbY1ozOxtCBLXWaEOE_k Bravo! I take my hat off to you. But what are the Gamma and Psi functions that you mention? === Subject: Re: Probability distribution puzzle >But what are the Gamma and Psi functions that you mention? For Re(z) > 0, Gamma(z) = int_0^infty exp(-t) t^(z-1) dt. Extend by analytic continuation to the rest of the complex plane, except for poles at the nonpositive integers. For positive integers n, Gamma(n) = (n-1)!. Psi(z) = d/dz ln(Gamma(z)) I must retract my statement | For large n and m a normal approximation with that mean and variance | should be fairly good. As Herman Rubin pointed out, the Central Limit Theorem doesn't apply here, as the variance of the time to get the last sticker is a significant \ fraction of the total variance. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: can U solve this simple functional equation? More generally when f(x,y) is invariant for (x,y) ->( x+a*m(),y-c*m() ) , m() a several variables or a constant: then f(x,y)=g(x+a*y/c) ,g a real function , is a solution. Example: f(x,y)=f(x-y+x^2,3y-2x^2) , m()=m(x,y)=x^2-y , a=1, c=-2 . f(x,y)=g(x+y/2) ... I want to propose an other case when we we keep 'a little' away from invariance. ALAIN. === Subject: what's the relationship among various logics? I'm interested in learning how the following subjects / logics / proof systems relate to each other: Propositional Calculus Predicate Calculus (aka First-Order Logic) Higher-Order Logic (aka Church's \Set Theory of Types\) Sequent Calculus Intuitionistic Logic Natural Deduction Constructive Mathematics ZFC Set Theory Proof Theory I can easily find literature on any of these topics, but it is often \ written in a way as if others don't exist, and figuring out whether a certain system is a special case of; is more general than; is isomorphic to; can encode; can be encoded in etc. other systems is left as an exercise to the reader. === Subject: Re: what's the relationship among various logics? You ask a lot. There are also Syllogisms logic: http://www.duniho.com/fergus/sillysyllogisms.html Formal logic is maybe propositonal logic, a course book Predicate logic. A way of talking, Quantor logic with quantots all the upside doen A en the there exist the \ turned E. the v as or and the & as and. and all kind of rules. Basics of \ SQL. Intuitionistic Logic has to win. The tertium non datur. But a door can stand \ ajar Not open and not closed. Theree is Fuzy logic, partly true. Closer to \ the reality. ZFC Set theory is set theory a rather big subject. But also to learn. Proof theory is only great with G\Odsel all theories are incompletee \ mathematicall proved. A great subject. You asked another question Alex. But thos one I can give you. Don't you have someone in you neigborhood who can tell you which books are \ fine? It could depend on the country you live in. I am from old Europe. === Subject: Re: what's the relationship among various logics? > ... > I am from old Europe. And your post reminded me of Professor Stanley Unwin--a very funny man, === Subject: Re: what's the relationship among various logics? > I'm interested in learning how the following subjects / logics / proof > systems relate to each other: > Propositional Calculus > Predicate Calculus (aka First-Order Logic) > Higher-Order Logic (aka Church's \Set Theory of Types\) > Sequent Calculus > Intuitionistic Logic > Natural Deduction > Constructive Mathematics > ZFC Set Theory > Proof Theory > I can easily find literature on any of these topics, but it is often \ written > in a way as if others don't exist, and figuring out whether a certain > system is a special case of; is more general than; is isomorphic to; can > encode; can be encoded in etc. other systems is left as an exercise to \ the > reader. Propositional Calculus Predicate Calculus (aka First-Order Logic) Higher-Order Logic (aka Church's \Set Theory of Types\) can all be \done\ with Sequent Calculus Natural Deduction. There are Intuitionistic Logic alternatives to Propositional Calculus Predicate Calculus (aka First-Order Logic) Higher-Order Logic (aka Church's \Set Theory of Types\) which can be done with Sequent Calculus Natural Deduction. ZFC Set Theory Higher-Order Logic (aka Church's \Set Theory of Types\) are, supposedly, \foundations\ for mathematics. Constructive Mathematics is a way of doing mathematics that may be related to Intuitionistic Logic. There is an Intuitionistic Logic alternative to ZFC Set Theory. Propositional Calculus is less inclusive than Predicate Calculus (aka First-Order Logic) which is less inclusive than Higher-Order Logic (aka Church's \Set Theory of Types\) which is comparable with ZFC Set Theory. Proof Theory is the study of proof in all or any of the above. It's a great pity, isn't it, that Church's second volume was never published. There are books that deal with more than one of the above such as Church, Mendelson, Shoenfield, Bell and Machover. Btw, your list may, depending on what you think it is a list _of_, be what Gilbert Ryle would have called a category mistake. === Subject: Re: what's the relationship among various logics? Originator: tchow@archimedes.mit.edu (Timothy Chow) >I'm interested in learning how the following subjects / logics / proof >systems relate to each other: >Propositional Calculus >Predicate Calculus (aka First-Order Logic) >Higher-Order Logic (aka Church's \Set Theory of Types\) >Sequent Calculus >Intuitionistic Logic >Natural Deduction >Constructive Mathematics >ZFC Set Theory >Proof Theory Four of these are very precise and specific terms: propositional calculus, predicate calculus, intuitionistic logic, ZFC set theory. The first three are logics. Propositional calculus is essentially the part of predicate calculus without quantifiers. Both are systems for classical logic. Intuitionistic logic is non-classical logic for mathematics. ZFC set theory is not a logic, but is a set of statements ---axioms and their predicate logic consequences. \Higher-order logic\ is a generic term that refers to various systems of logic that allow more than first-order quantification. \Sequent calculus\ is a generic and slightly vague term for any system of rules for generating one sequent (\A -> B\) from another. The rules for the most common logics can be expressed in the form of a sequent calculus. \Natural deduction\ is a generic and slightly vague term for any logical system that mimics \real-life\ logical reasoning. \Constructive mathematics\ is a vague term for any kind of mathematics \ that insists that existence proofs exhibit how to construct the object in question. Intuitionism is one form of constructive mathematics. \Proof theory\ is a vague term for the branch of mathematics that studies proofs. Perhaps now you can see why Torkel Franzen finds your project \ idiosyncratic. The items on your list are not the same type of objects. It's sort of like asking for the relationships between the following items: England the United States history the American Revolution the Battle of Bunker Hill the Declaration of Independence George Washington 1776 freedom -- 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: what's the relationship among various logics? > The items on your list are not the same type of objects. What's wrong with asking about relations between objects of different \ types? For example, it's often stated that the number theory can not be encoded in the FOL. I was asking if Proof Theory could be expressed in, say, Intuitionistic Logic [*] > ZFC set theory is not a logic, but is a set of statements > ---axioms and their predicate logic consequences As I understand, the relation between ZFC set theory and Predicate Calculus is that the former is a special case of the latter. For example, in http://au.metamath.org/mpegif/mmset.html , if you remove the 7 ZFC axioms you get Predicate Logic with equality. If you further remove axioms 8 through 16, you get Pure Predicate Logic. The difference between a \ \Logic\ and a \Not a logic\ is thus artificial, depending on which axioms you \ give special status. What I find confusing though, is how can ZFC be able to express \almost \ all of mathematics\, even though we know that Predicate Calculus is not expressive enough to talk about numbers. Expressivity can not be gained by adding axioms. === Subject: Re: what's the relationship among various logics? On Fri, 06 May 2005 09:30:33 -0700, alex goldman said: >> The items on your list are not the same type of objects. > What's wrong with asking about relations between objects of different > types? For example, it's often stated that the number theory can not > be encoded in the FOL. I was asking if Proof Theory could be expressed > in, say, Intuitionistic Logic [*] There are various notions of encoding and expressing, and they can't be adequately clarified in a sentence or two. For instance, there is a very clear sense in which number theory *can* be \encoded\ in FOL, viz., the axioms of Peano arithmetic. But there is also a very clear sense in which it cannot, viz., there is no set of first-order axioms whose models are exactly the class of natural numbers and its isomorphs. >> ZFC set theory is not a logic, but is a set of statements >> ---axioms and their predicate logic consequences > As I understand, the relation between ZFC set theory and Predicate \ Calculus > is that the former is a special case of the latter. For example, in > http://au.metamath.org/mpegif/mmset.html , if you remove the 7 ZFC axioms > you get Predicate Logic with equality. If you further remove axioms 8 > through 16, you get Pure Predicate Logic. The difference between a \ \Logic\ > and a \Not a logic\ is thus artificial, depending on which axioms you \ give > special status. That's quite false. The distinction between logic and non-logic (better, logics and non-logics) is a semantic one. But to appreciate it, you'll actually have to study some model theory. > What I find confusing though, is how can ZFC be able to express > \almost all of mathematics\, even though we know that Predicate > Calculus is not expressive enough to talk about numbers. Expressivity > can not be gained by adding axioms. It means (very roughly) that the objects and structures studied by other branches of mathematics can be defined in ZFC as sets of one sort or another, and that, under those definitions the basic (first-order) axiomatizations (hence their theorems) of most other areas of mathematics can be derived as theorems of ZFC. The notion of expressivity you are talking about above is again a model theory one. Chris Menzel === Subject: Re: what's the relationship among various logics? Originator: tchow@archimedes.mit.edu (Timothy Chow) >What's wrong with asking about relations between objects of >different types? There's nothing *wrong* about it, but it's unclear what kind of \ relationship you're asking for. If I ask for the relationship between George W. Bush \ and George H. Bush, it's pretty clear what my question is. But if I ask for \ the relationship between France and George Washington, it's not so clear what the question is. >For example, it's often stated that the number theory can not be encoded >in the FOL. This sounds like a garbled version of the statement that the first-order theory of true arithmetic has nonstandard models. >I was asking if Proof Theory could be expressed in, say, >Intuitionistic Logic [*] It is very difficult to make sense of this statement. My best guess would be, can all the theorems in a typical proof theory textbook be proved intuitionistically? The answer is that some of them can be, but probably not all. In any case, this question bears little resemblance to your other garbled statement about number theory. Getting a general \map\ of a subject is a reasonable goal, but you can't do that without understanding the subject well enough to formulate coherent statements and questions. So I think your first goal should be to learn and understand precise definitions of the terms in question. >As I understand, the relation between ZFC set theory and Predicate \ Calculus >is that the former is a special case of the latter. Not really. It's better to think of the predicate calculus as providing \ the rules of inference, and ZFC as providing the axioms. >What I find confusing though, is how can ZFC be able to express \almost \ all >of mathematics\, even though we know that Predicate Calculus is not >expressive enough to talk about numbers. Expressivity can not be gained by >adding axioms. When people talk about number theory in this context, they're usually talking about the first-order language of arithmetic. Not everything that you might want to say about number theory can be expressed in the first-order language of arithmetic. However, you *can* pretty much express everything you want to say about number theory by first converting the statement into a statement about sets, and then formalizing this statement in the first-order language of *set theory*. The other thing is that you're probably confusing expressivity with categoricity. The existence of nonstandard models of arithmetic is a separate question from the ability to express mathematical statements in this or that language. -- 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: what's the relationship among various logics? >>As I understand, the relation between ZFC set theory and Predicate >>Calculus is that the former is a special case of the latter. > Not really. It's better to think of the predicate calculus as providing > the rules of inference, and ZFC as providing the axioms. Why is it \better\ to think that way? I Metamath, subsitution is the only inference rule, everything else is axioms. I find it aesthetically appealing. > When people talk about number theory in this context, they're usually > talking about the first-order language of arithmetic. Not everything \ that > you might want to say about number theory can be expressed in the > first-order language of arithmetic. However, you *can* pretty much > express everything you want to say about number theory by first \ converting > the statement into a statement about sets, and then formalizing this > statement in the first-order language of *set theory*. What does the disturbing \pretty much\ refer to? And by first-order set theory, do you mean the predicate calculus with ZFC axioms added in? === Subject: Re: what's the relationship among various logics? > For example, it's often stated that the number theory can not be encoded \ in > the FOL. What is this supposed to mean? > I was asking if Proof Theory could be expressed in, say, > Intuitionistic Logic [*] What is this supposed to mean? > As I understand, the relation between ZFC set theory and Predicate \ Calculus > is that the former is a special case of the latter. What is this supposed to mean? > even though we know that Predicate Calculus is not > expressive enough to talk about numbers. What is this supposed to mean? === Subject: Re: what's the relationship among various logics? > > For example, it's often stated that the number theory can not be encoded \ in > > the FOL. > What is this supposed to mean? If it is indeed stated, then it's for those who state it to explain what it means, not the op. > > I was asking if Proof Theory could be expressed in, say, > > Intuitionistic Logic [*] > What is this supposed to mean? It means that the op doesn't know what proof theory is. (He may not know what intuitionistic logic is either, but that is far less certain.) > > As I understand, the relation between ZFC set theory and Predicate \ Calculus > > is that the former is a special case of the latter. > What is this supposed to mean? It means that the op doesn't know what the relation between ZFC set theory and predicate calculus is. More generally, he probably doesn't know what a first order theory is. > > even though we know that Predicate Calculus is not > > expressive enough to talk about numbers. > What is this supposed to mean? It may mean something like this: in predicate calculus with \for all\ and \not\ primitive we can define \exists\ as \not for all not\ but \ we cannot define \nought\, \one\, etc. Nor can we prove theorems about them. I admit I'm guessing on this one. If the op _doesn't_ know what number theory, encoded in, FOL, Proof Theory, Intuitionistic Logic, ZFC set theory, Predicate Calculus, special case of, expressive enough to talk about, numbers mean, then you can tell him. === Subject: Re: what's the relationship among various logics? > ... I was asking if Proof Theory could be expressed in, say, > Intuitionistic Logic This makes no sense. Proof theory is about (an aspect of) intuitionistic logic. (The aspect being, of course, proof in intuitionistic logic. > What I find confusing though, is how can ZFC be able to express \almost \ all > of mathematics\, even though we know that Predicate Calculus is not > expressive enough to talk about numbers. Expressivity can not be gained \ by > adding axioms. Expressiveness is gained by adding new predicate constants (such as \is an element of\) and then adding axioms governing those predicate constants. === Subject: Re: what's the relationship among various logics? >> What I find confusing though, is how can ZFC be able to express \almost >> all of mathematics\, even though we know that Predicate Calculus is not >> expressive enough to talk about numbers. Expressivity can not be gained >> by adding axioms. > Expressiveness is gained by adding new predicate constants (such as \is > an element of\) and then adding axioms governing those predicate > constants. A logic is a tuple of a. formation rules (grammar) b. transformation rules (inference rules) c. axioms You can then add more axioms and definitions, and then produce theorems through the transformation rules. This is how one could (ideally) express some mathematical discipline in a given logic. >> ... I was asking if Proof Theory could be expressed in, say, >> Intuitionistic Logic > This makes no sense. Proof theory is ... Proof Theory itself is a branch of mathematics, so it's sensical to ask whether it can be expressed in a given logic. What is the problem? === Subject: Re: what's the relationship among various logics? > >> What I find confusing though, is how can ZFC be able to express \ \almost > >> all of mathematics\, even though we know that Predicate Calculus is \ not > >> expressive enough to talk about numbers. Expressivity can not be \ gained > >> by adding axioms. > > Expressiveness is gained by adding new predicate constants (such as \ \is > > an element of\) and then adding axioms governing those predicate > > constants. > A logic is a tuple of > a. formation rules (grammar) > b. transformation rules (inference rules) > c. axioms > You can then add more axioms Why? If completeness is possible why not have a complete set of axioms to start with? Then you if you add more axioms you will either have redundancy or inconsistency. What you need to add is constants and axioms governing those constants. Constants are added to the language and since your a., b., and c. don't mention language, I can't think that they're relevant to my point. > and definitions, and then produce theorems > through the transformation rules. This is how one could (ideally) express > some mathematical discipline in a given logic. The point I was trying to make was this. Let P(.,.) be a predicate variable in predicate logic, then we can prove things like (x)(y)(P(x,y) or not-P(x,y) which tells us _nothing_ about P. We cannot add an axiom (x)(y)(P(x,y) --> not P(y,x)) . . . . . . (*) because P is a variable and we know nothing about it. If, informally, we think of the individual variables as denoting sets and we add a constant I with I(x,y) informally meaning \x is an element of y\, then we can add (x)(y)(I(x,y) --> not I(y,x)) as an axiom. === Subject: Re: what's the relationship among various logics? >> >> What I find confusing though, is how can ZFC be able to express >> >> \almost all of mathematics\, even though we know that Predicate >> >> Calculus is not expressive enough to talk about numbers. Expressivity >> >> can not be gained by adding axioms. >> > >> > Expressiveness is gained by adding new predicate constants (such as \ \is >> > an element of\) and then adding axioms governing those predicate >> > constants. >> A logic is a tuple of >> a. formation rules (grammar) >> b. transformation rules (inference rules) >> c. axioms >> You can then add more axioms > Why? If completeness is possible why not have a complete set of axioms > to start with? It's a question of scope and modularity. I don't want your (instuitionistic logic, say) axioms to make my system (linear algebra, say) inconsistent, so we have separate sandboxes. > The point I was trying to make was this. Let P(.,.) be a predicate > variable in predicate logic, then we can prove things like > (x)(y)(P(x,y) or not-P(x,y) > which tells us _nothing_ about P. We cannot add an axiom > (x)(y)(P(x,y) --> not P(y,x)) . . . . . . (*) > because P is a variable and we know nothing about it. If, informally, > we think of the individual variables as denoting sets and we add a > constant I with I(x,y) informally meaning \x is an element of y\, then > we can add > (x)(y)(I(x,y) --> not I(y,x)) > as an axiom. So? Are we supposed to be in some kind of disagreement here? We were talking about \a logic\ vs \a mathematical theory\. I said that \ the difference is quantitative, not qualitative (see that metamath link for example). I think you set out to prove me wrong. === Subject: Re: what's the relationship among various logics? > >> ... I was asking if Proof Theory could be expressed in, say, > >> Intuitionistic Logic > > This makes no sense. Proof theory is ... > Proof Theory itself is a branch of mathematics, so it's sensical to ask > whether it can be expressed in a given logic. > What is the problem? Proof theory is a branch of mathematics and should certainly be done logically. It could, I suppose, be formalized in set theory. The set theory chosen could be intuitionistic. To ask if proof theory could be expressed in, say, intuitionistic logic, is like asking if group theory could be expressed in, say, intuitionistic logic. One _expresses_ in a language (in that respect mathematics is no different from any other subject of study), one _proves_ in a logical calculus. So, two questions for you: (1) is the language of intuitionistic logic expressive enough to do proof theory in? And (2) is the deductive power of intuitionistic logic strong enough to prove proof theories theorems in. To answer you'll first have to decide _what_ intuitionistic logic you're taking about; propositional, functional, higher order. (Easy.) And you'll have to say _what_ proof theory is. (Harder.) === Subject: Re: what's the relationship among various logics? > of intuitionistic logic strong enough to prove proof theories theorems What does that mean? I probably meant \strong enough to prove proof theory's theorems\ === Subject: Re: what's the relationship among various logics? >> I was asking if Proof Theory could be expressed in, say, >> Intuitionistic Logic > This makes no sense. > ... > So, two questions for you: (1) is the language of intuitionistic logic > expressive enough to do proof theory in? We meant different things by \express[ive]\, but do you see the irony, though? === Subject: Re: what's the relationship among various logics? > I'm interested in learning how the following subjects / logics / proof > systems relate to each other: > Propositional Calculus > Predicate Calculus (aka First-Order Logic) > Higher-Order Logic (aka Church's \Set Theory of Types\) > Sequent Calculus > Intuitionistic Logic > Natural Deduction > Constructive Mathematics > ZFC Set Theory > Proof Theory The only way of doing that is by studying the subjects. You might look at the Handbook of Mathematical Logic, Jon Barwise, ed. === Subject: Re: what's the relationship among various logics? >> I'm interested in learning how the following subjects / logics / proof >> systems relate to each other: >> Propositional Calculus >> Predicate Calculus (aka First-Order Logic) >> Higher-Order Logic (aka Church's \Set Theory of Types\) >> Sequent Calculus >> Intuitionistic Logic >> Natural Deduction >> Constructive Mathematics >> ZFC Set Theory >> Proof Theory > The only way of doing that is by studying the subjects. Here's a counterexample: Propositional Calculus is a special case of FOL. FOL is a special case of Second-order logic, (yet) Second-order logic can be encoded in FOL. Now someone can learn about these relations without studying the subjects. === Subject: Re: what's the relationship among various logics? > Propositional Calculus is a special case of FOL. > FOL is a special case of Second-order logic, (yet) > Second-order logic can be encoded in FOL. > Now someone can learn about these relations without studying the > subjects. Certainly you can memorize all sorts of such one-liners. If that's what you have in mind in speaking of \learning how the following subjects relate to each other\, you will probably need to make up your own compendium. === Subject: Re: what's the relationship among various logics? >> Propositional Calculus is a special case of FOL. >> FOL is a special case of Second-order logic, (yet) >> Second-order logic can be encoded in FOL. >> Now someone can learn about these relations without studying the >> subjects. > Certainly you can memorize all sorts of such one-liners. If > that's what you have in mind in speaking of \learning how the > following subjects relate to each other\, Yes, I'm interested in these binary relations like can_encode and is_a_special_case_of. They take about one line to write down each. > you will probably need to make up your own compendium. I'm sorry if I angered you somehow. === Subject: Re: what's the relationship among various logics? > I'm sorry if I angered you somehow. Why should you be angry? It's not be expected that such a compendium exists, in view of its apparent pointlessness. Yours is a very special interest. === Subject: Re: what's the relationship among various logics? > Why should you be angry? Never mind. I'll chalk this up as an intercultural difference. > It's not be expected that such a compendium > exists, in view of its apparent pointlessness. It's not apparent to me, apparently. > Yours is a very special interest. I'm an AI researcher, not a logician. I'm familiar with FOL, but I can't afford to spend years studying dozens of various \logics\ just for the \ hell of it. I guess I was asking for a sort of a map. === Subject: Re: what's the relationship among various logics? > I'm an AI researcher, not a logician. I'm familiar with FOL, but I can't > afford to spend years studying dozens of various \logics\ just for the \ hell > of it. I guess I was asking for a sort of a map. What kind of map? To use how? === Subject: Re: what's the relationship among various logics? > > I'm an AI researcher, not a logician. I'm familiar with FOL, but I \ can't > > afford to spend years studying dozens of various \logics\ just for the \ hell > > of it. I guess I was asking for a sort of a map. > What kind of map? To use how? Someone furnished with a map of Britain might learn that Birmingham is a settlement to the north of London which is also a settlement. The Thames is a river which runs through London. Selly Oak is a settlement that is in Birmingham. Birmingham is in England which is not a settlement. These are all facts that someone might list in a compendium. No one would be interesting in the compendium except its compiler. Nevertheless the map would be of much interest to many people. Nobody, even if they were not interested themselves, would need to ask \What kind of map?\ or \To use how?\ Even someone who will never set foot in Britain and has no interest in it, can understand that such a map might be useful to others. So, what kind of map? A collection of texts (elsewhere I suggested some), a big pile of blank paper and some pencils. To use how? To study logic. Btw, someone furnished with a map of Britain will learn that to ask what is the relation between Birmingham, London, The Thames, Selly Oak and England _may_ be to make a category mistake, but still some kind of answer can be given. === Subject: Re: what's the relationship among various logics? > Is there a logic where A or (~A) =/= True, yet > one can prove (B or C) while either B or C is not provable individually? For this to be non-trivial, we must specify some logic that must be extended. (If there are no axioms, the disjunction property - that B or C is provable whenever B v C is provable - is vacuously satisfied.) Lucasiewicz conjectured that there is no extension of intuitionistic logic with the disjunction property, but this was disproved by Kreisel and Putnam (1957). The system KP, which extends intuitionistic logic by the schema (not-P -> Q v R) -> ((not-P -> Q) v (not-P -> R)) also has the disjunction property. \The rules of intermediate logic\ by Rosalie Iemhoff, available at http://www.illc.uva.nl/D65/iemhoff.pdf will tell you a lot more. === Subject: another kind of inverse posting-account=LPDiaQ0AAACsJLMUu95_O2ZUX0DZOvYQ Consider another kind of matrix operations. Now the identity becomes [0 0 ... 0 1] [0 0 ... 1 0] [ ......... ] [0 1 ... 0 0] [1 0 ... 0 0] Did anyone heard of it? === Subject: Re: another kind of inverse > Consider another kind of matrix operations. > Now the identity becomes > [0 0 ... 0 1] > [0 0 ... 1 0] > [ ......... ] > [0 1 ... 0 0] > [1 0 ... 0 0] Exactly what was your question? What kind of matrix product would have this as the identity? Ok, it's easy to think of a product that has this is as identity by making an obvious change to the indexing of the rows, but that amounts to simply using a different notational convention. the bottom up :) ?? notation\, where you simply read from Would you prefer to call this \oriental As this difference is notational only, the theory won't change one bit. Jyrki === Subject: Re: another kind of inverse > Consider another kind of matrix operations. > Now the identity becomes > [0 0 ... 0 1] > [0 0 ... 1 0] > [ ......... ] > [0 1 ... 0 0] > [1 0 ... 0 0] > Did anyone heard of it? I haven't. inverses, not just one. Given a non-singular matrix A, there will be the left inverse B_l and the right inverse B_r, which will be such that B_l A = J and A B_r = J. Why are you interested in this? Jose Carlos Santos === Subject: Re: another kind of inverse <3e0u0eFkeppU1@individual.net> posting-account=LPDiaQ0AAACsJLMUu95_O2ZUX0DZOvYQ Why? === Subject: Re: another kind of inverse > Why? Why? Why what? You should have quoted my post, but I guess that your question is: why there will be two inverses then? If so, there's an easy answer for that: because, in general, B_l and B_r will be distinct; in other words, in general A^{-1} J and J A^{-1} will be distinct. For instance, if A is the diagonal 2x2 matrix with the first diagonal entry equal to 1 and the second one equal to 2, then 0 1 A^{-1} J = 1/2 0 and 0 1/2 J A^{-1} = 1 0. Jose Carlos Santos === Subject: Concave-down continuous function posting-account=t1bj7g0AAABLHyo4orDn1WouenV4EMmc Let f be a concave-down continuous function on [0,1] such that f(0)=f(1)=0. Then show that the following inequality holds. 2 * int_0^1 x^2 f(x) dx < int_0^1 f(x) dx === Subject: Re: Concave-down continuous function > Let f be a concave-down continuous function on [0,1] such that > f(0)=f(1)=0. Then show that the following inequality holds. > 2 * int_0^1 x^2 f(x) dx < int_0^1 f(x) dx This is quite reminiscent of a recent Monthly problem (11133 by Paul Bracken, Feb 2005). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html \Elegance is an algorithm\ Iain M. Banks, _The Algebraist_ === <20050503141632.632$lh@newsreader.com> <0qlf71556icqlcvvasbhll6el52vb8o1a2@4ax.com> posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L Herc === Subject: Re: elementary geometry question posting-account=VOcjCQwAAAAuS03wBmk1NOZrZAVSQTY1 > A tetrahedron is given. > The lengths of its opposite edges are > m and n, u and v, p and q respectively. > The volume of this tetrahedron > is V and the circumradius is R. > Then the following formula holds true: > (24VR)^2=2(upqv)^2-(mn)^4 > +2(mnuv)^2-(pq)^4 > +2(mpnq)^2-(uv)^4. As I recall there's a similar result, involving a sum of fourth powers, when the tetrahedron is coplanar, in which case V is zero and R infinite and the above identity fails (unless VR can be proved to be finite, most likely zero). === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=iYorygwAAABh_sJqNDMvFdsA7EcFurwY > I just proved for any given random diagonal, any infinite list with the > digit saturation property can be rearranged to fit that diagonal. Yes, but exactly how does that diminish, disprove, or contradict Cantor's diagonal proof in any way whatever? *Ah*. If you rearrange the list, the same numbers should still be on the list, or not on the list. Given that any number differing from the diagonal in every digit _cannot_ be on the list, being able to put any desired number on the diagonal means that there are _no_ numbers in the list. Hence, no infinite list of reals in [0,1) with cardinality aleph-null can *posess* the digit saturation property? But even a *finite* list can posess the digit saturation property: while the list pi-3, pi-3 + 1/9, pi-3 + 2/9... pi-3 + 7/9, pi-3 - 1/9 won't quite do it, if we abolished carries. Ah. Maybe if we satisfy one digit, we will *use up* the only available copies of another digit. How about taking the fractional parts of sqrt(2), sqrt(3), sqrt(5), sqrt(7)... and then creating modified versions of their fractional parts? Thus, our list becomes: .414213562... .525324673... .636435784... .747546895... .858657906... .969768017... ... .303102451... .732050807... .843161918... ... We can clearly rearrange that list to prove that its first element is not on the list: .525324673... .621949796... ... Or, rather, we can _choose elements from_ that list to prove its first element is not on the list. We are choosing only one element from the sqrt(2) batch, one element from the sqrt(3) batch, and so on. It probably is possible to backtrack, and use unused elements from previous sqrt batches, so that we can eventually use just about every element from the list. Exactly when, however, would we choose .414213562... to place on a list that differs from that number *in every place along the diagonal*? \Rearrange\ and \use some of the numbers from\ are not synonymous. Thus, it is not true that a list, posessing the kind of digit saturation property you refer to, can be 'proven' by Cantor's diagonal proof to contain no numbers. John Savard === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L > > I just proved for any given random diagonal, any infinite list with > the > > digit saturation property can be rearranged to fit that diagonal. > Yes, but exactly how does that diminish, disprove, or contradict > Cantor's diagonal proof in any way whatever? > *Ah*. If you rearrange the list, the same numbers should still be on > the list, or not on the list. > Given that any number differing from the diagonal in every digit > _cannot_ be on the list, being able to put any desired number on the > diagonal means that there are _no_ numbers in the list. > Hence, no infinite list of reals in [0,1) with cardinality aleph-null > can *posess* the digit saturation property? > But even a *finite* list can posess the digit saturation property: > while the list pi-3, pi-3 + 1/9, pi-3 + 2/9... pi-3 + 7/9, pi-3 - 1/9 > won't quite do it, if we abolished carries. > Ah. Maybe if we satisfy one digit, we will *use up* the only available > copies of another digit. > How about taking the fractional parts of sqrt(2), sqrt(3), sqrt(5), > sqrt(7)... and then creating modified versions of their fractional > parts? > Thus, our list becomes: > .414213562... > .525324673... > .636435784... > .747546895... > .858657906... > .969768017... > ... > .303102451... > .732050807... > .843161918... > ... > We can clearly rearrange that list to prove that its first element is > not on the list: > .525324673... > .621949796... > ... > Or, rather, we can _choose elements from_ that list to prove its first > element is not on the list. We are choosing only one element from the > sqrt(2) batch, one element from the sqrt(3) batch, and so on. > It probably is possible to backtrack, and use unused elements from > previous sqrt batches, so that we can eventually use just about every > element from the list. > Exactly when, however, would we choose .414213562... to place on a list > that differs from that number *in every place along the diagonal*? > \Rearrange\ and \use some of the numbers from\ are not synonymous. > Thus, it is not true that a list, posessing the kind of digit > saturation property you refer to, can be 'proven' by Cantor's diagonal > proof to contain no numbers. > John Savard John if you set the diagonal to a number different to every digit of some element, when you shuffle the list indefinately that element is forever pushed down the queue since it never gets a digit match, that does not make an equivalent set. But, if you force a random diagonal on the list it tends to be a minor shuffling procedure. In base 3, every real has 1/3 chance of going across repeatedly until it does. You can see I construct multiple instances of (parts of) saturated sets and get back 50 to 60 elements each time out of 60. www.freewebs.com/namesort/linux.html > Given that any number differing from the diagonal in every digit > _cannot_ be on the list, being able to put any desired number on the > diagonal means that there are _no_ numbers in the list. weaken no_numbers to no_random_numbers. the function (shuffle algorithm) only works with a new random diagonal. > Hence, no infinite list of reals in [0,1) with cardinality aleph-null > can *posess* the digit saturation property? A list of random numbers contains random numbers and is saturated, therefore your premise must be flawed, any number differing from the diagonal in every digit cannot_ be on the list That is false! Here is why its false. Say your list is {0.3, 0.3333333, 0.33, 0.33333,0.333,0.3333333}. What information is there about the list without specifying a full element? The diagonal is 0.330303.... NOTICE THE DIAGONAL GIVES US CLUES ABOUT THE ORIGINAL LIST! Compare that to a saturated list, we can reorder the list to have any random number for the diagonal we like! The diagonal is 0.4289430934.. OR The diagonal is 0.3838939... According to Shannons noise theory there is no information transmitted by reporting the diagonal. That is, saturated lists contain no information in their diagonal. Herc === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=W2DCTA0AAAAlbhDMl3GrysSnPy1IK_7f > > Given that any number differing from the diagonal in every digit > > _cannot_ be on the list, being able to put any desired number on the > > diagonal means that there are _no_ numbers in the list. > weaken no_numbers to no_random_numbers. the function (shuffle > algorithm) only works with a new random diagonal. That is true --- you can easily make a list so that a number picked at random will be an admissible diagonal with probability 1. As you say, there are some numbers which are NOT admissible as a diagonal. > any number differing from the diagonal in every digit > cannot_ be on the list > That is false! No, it is true. Take ANY admissible diagonal and pick a number which differs from this diagonal in all places --- this number cannot be on the list. The fact that you can make so many possible diagonals only reinforces the fact that there are many many real numbers which are not on the list. Lasse === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > BEHOLD! A list of numbers that DOESN'T CARE WHAT ITS DIAGONAL IS!! Now, after running your Javascript program 10-12 times, I see what you're doing here. (Are you ready to talk rationally? After all, I haven't \run away.\) You have a list of 20 real numbers in base-3 notation. Then, the user enters ANY diagonal (let's call it D1), you're rearranging the items in the list so that the diagonal of that list is the diagonal D1 you give it. Actually, this isn't quite true; you get the first part of D back, and it doesn't seem to be a rare event. Your claim is that if there's an infinite number of items on the list, you get all of D back. (I'm not convinced of this last statement one way or another.) ((This suggests an interesting combinatorial problem (which has probably already been solved): How many diagonals of length 20 can there arise from a list of 20 base-3 \decimals\, out to 20 digits? Certainly, the answer is at most 20!, since you can take the diagonal of any arrangement of the 20 numbers. But it's possible that there's a lot of duplication. Hmmm... One could also ask what the expected number of digits of D that you get back is, if you choose D randomly and fairly.)) > http://www.freewebs.com/namesort/numsort.html > You saw it here 1st folks, a list of numbers that YOU CANT DIAGONALISE! > STEP 1 > start at digit 1 > STEP 2 > use a radioactive decay device to generate a random number from 0 to 9. Actually, that probably should be 0 to 2. If your first digit is 3 to 9, then there aren't any items on the new list. > STEP 3 > run through the original list of reals from real 1 onwards until you > find a real with the random digit matching that digit. > STEP 4 > remove that real from the list and add it to the second list. > STEP 5 > advance one digit position > STEP 6 > Go to STEP 2 > Now you can try it too! Make the diagonal 0000000000000 it STILL > WORKS! > http://www.freewebs.com/namesort/numsort.html Okay. Now I'm ready to explain what Cantor's argument is about. First of all, I'll establish an analogy for what's going on in a general proof. Imagine you're at a carnival, and there's a display called \Your List is Incomplete\. The barker (who is promoting the thing) is calling out \Step right on up, everybody! Provide a countable list of real numbers, and Namkceh the Magnificient will find a number not on your list! Only $1 to challenge him, and you stump him, you'll win $1000!\ Now, when someone comes up to this display, they pay their $1 and hand over an extremely long piece of paper. The barker verifies that you're playing by the rules: That for every positive integer n, you've listed exactly one real number (let's call it r(n)). The barker nods and passes the paper over to Namkceh the Magnificent. Namkceh looks over the paper and reads off the decimal expansion of a real number. The customer watches the barker write it down, so that there are no mistakes or funny business. Then Namkceh the Magnificient hands back the paper, so that the customer can check whether the announced number is in fact on the list. Now, if the set of real numbers really is not countable, then Namkceh can't lose, because he can always name some number not on the list. Granted, he may need to be an extraterrestrial or a hive-mind to pull it off, but it certainly should be possible. On the other hand, if the set of real numbers is countable, then the customer can catch the barker and Namkceh, as well as receive a sizable amount of money. This is where you enter the situation. If you think that the set of real numbers is countable, then you should be able to write a list of real numbers on the list, hand it to the barker, who okays it, who then hands it to Namkceh, who is stumped. IF the list really is complete, then Namkceh has two options: admit defeat, or bullshit by rattling of a series of digits. If he chooses the latter option, you'll find out when you go through the list. This is all you need to do to show that the set of real numbers is countable. So what's your list? --- Christopher Heckman === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L http://www.freewebs.com/namesort/linux.html any better? define saturated as a list that contains every finite prefix of digits there is a function that transforms a saturated list of numbers to another list such that 1/ the set of elements is equivalent 2/ the diagonal is any real with random digits. Conclusion: any data from the original diagonal obtained through list order is lost. the set of diagonals of any saturated set is every random-number let the antidiagonal = D where D(i) = List(i,i) + 1 mod 10 D(i) = random(10) + 1 mod 10 D(i) = random(10) Therefore, according to Cantors proof every random sequence of digits is missing from any saturated list. Herc === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=W2DCTA0AAAAlbhDMl3GrysSnPy1IK_7f > http://www.freewebs.com/namesort/linux.html > any better? > define saturated as a list that contains every finite prefix of digits > there is a function that transforms a saturated list of numbers to > another list such that > 1/ the set of elements is equivalent > 2/ the diagonal is any real with random digits. Please clarify 1/: what do you mean by 'the set of elements is equivalent'? Do you mean that the lists contain the same element? Please clarify 2/: what do you mean by 'any real with random digits'? If you just mean 'any real number', then this is wrong (with the above assumption on what you mean by 1/). If you mean 'a set of real numbers which has full Lebesgue measure', then I suppose it's correct. > Conclusion: any data from the original diagonal obtained through list > order is lost. What do you mean by this? > the set of diagonals of any saturated set is every random-number There is no such thing as 'every random number', in the way you seem to be using it here. Do you mean 'with probability 1'? Then it is true --- if you pick a random number, then the probability that you can make it by reordering my above list (which consists of infinitely many copies of just three different numbers) is 1. > let the antidiagonal = D where D(i) = List(i,i) + 1 mod 10 > D(i) = random(10) + 1 mod 10 > D(i) = random(10) > Therefore, according to Cantors proof every random sequence of digits > is missing from any saturated list. Indeed - if you are given a list of numbers, the probability that a randomly chosen real number belongs to this list is zero. Is that what you are saying? This is NOT the same thing as saying that there are NO numbers on the list: pick your favorite real number, and then choose a real number at random. The probability that it is the number you picked in the beginning is zero! (Or in other words: suppose you keep rolling a die, stopping the first time you don't roll a 6. What is the probability that you never stop? It is zero.) Lasse === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=W2DCTA0AAAAlbhDMl3GrysSnPy1IK_7f > > BEHOLD! A list of numbers that DOESN'T CARE WHAT ITS DIAGONAL IS!! > Now, after running your Javascript program 10-12 times, I see what > you're doing here. (Are you ready to talk rationally? After all, I > haven't \run away.\) > You have a list of 20 real numbers in base-3 notation. Then, the user > enters ANY diagonal (let's call it D1), you're rearranging the items in > the list so that the diagonal of that list is the diagonal D1 you give > it. > Actually, this isn't quite true; you get the first part of D back, and > it doesn't seem to be a rare event. Your claim is that if there's an > infinite number of items on the list, you get all of D back. (I'm not > convinced of this last statement one way or another.) There is always a sequence which cannot be obtained as a diagonal by reordering the list: just take a given line, and pick a number which differs in every digit with it. Wherever the given line appears in the reordering, the diagonal will differ from this number. If, in the 'reordering', you also allow leaving some lines out, then this problem disappears, and you can take the list 0.012012012012... 0.120120120120... 0.201201201201... 0.012012012012... ... which contains only three different numbers. I wonder whether, according to HERC, this means that there are only three real numbers all told ... Lasse === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* >>It doesn't take long. My favourite Cooperism is that he is adamantly >>opposed to all diagonal arguments, EXCEPT for the one that shows the >>unsolvability of the Halting Problem. Go figure. >as has been told to you TEN TIMES. Repeating nonsense doesn't make it any less nonsense. >Halt is a very specific function. It was disproved. Antidiag is not >any specific number. so why don't you GO FIGURE? Try to notice the close similarity, D******: Any alleged Halting-Problem solver is shown in fact not to be one by a diagonal argument. There is no \very specific function\, because THERE IS \ NO SUCH FUNCTION AT ALL, which is what the diagonal proof shows. Any alleged enumeration of all the reals is shown in fact not to be one by a \ diagonal argument. There is no specific enumeration, because THERE IS NO SUCH ENUMERATION AT ALL, which is what the diagonal proof shows. [snip grandiose delusion] >And I don't have any criminal record Did I say you had a criminal record? I said you were *incarcerated* for threatening to put rat poison in packaged food; if you were on remand that counts too. >but some of the repetitive abusers here should have. On what charge? Unsympathetic treatment of an abusive crazy person? -- --------------------------- | BBB b \\ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* In sci.logic, Barb Knox on Fri, 06 May 2005 13:59:45 +1200 : >It doesn't take long. My favourite Cooperism is that he is adamantly >opposed to all diagonal arguments, EXCEPT for the one that shows the >unsolvability of the Halting Problem. Go figure. >>as has been told to you TEN TIMES. > Repeating nonsense doesn't make it any less nonsense. >>Halt is a very specific function. It was disproved. Antidiag is not >>any specific number. so why don't you GO FIGURE? > Try to notice the close similarity, D******: > Any alleged Halting-Problem solver is shown in fact not to be one by a > diagonal argument. There is no \very specific function\, because THERE \ IS > NO SUCH FUNCTION AT ALL, which is what the diagonal proof shows. As a Pedant Point: For any listmapping L : N -> R, one can define a function D : (: N -> R) -> R such that D(L) cannot be equal to any L(n). In fact, one can probably define a function D2 : (R - Q) x (: N -> R) -> R (if the \seed number\'s next digit doesn't meet one's requirements, advance until one finds a digit that does, with some fuzziness if the seed doesn't contain all digits 0-9). Adding D(L) to L, of course, is rather pointless; it just creates another list L', and then one creates a D(L'). Adding that to L' creates an L\, ... What HERC *has* proved, of course, is that, given the right kind of list L, he can come arbitrarily close to any real number r. But then, so can T_10 or T_3. Big fat freaking hairy deal. :-) There are elements in T_10 close to 1/3. 1/3 is *not* in T_10. The Halting Problem is of course similar; for any machine M implementing the predicate canHalt_M I can construct another machine Weird(M) which canHalt_M() will not correctly analyze. In fact, Weird(M) will work with any input machine, although the resultant machine won't do anything horribly interesting if M doesn't implement a halting-tester predicate. But Weird : machines -> machines is a 1-1 (though not onto) mapping. The list shuffling is a new one, though all that really does is introduce composition. Briefly put, the original mapping L : N -> R is now composed with the mapping X : N -> N. X is 1-1 and onto, scrambling the list entries in a predictable way. So now what do we have? Why, M = L o X, of course; the new mapping M : N -> R can be destroyed in a manner identical to the original proof (although with a different antidiagonal D(M) instead of D(L)). Bolting a bicycle wheel to a shopping cart doesn't make it a helicopter. :-) > Any alleged enumeration of all the reals is shown in fact > not to be one by a diagonal argument. There is no specific > enumeration, because THERE IS NO SUCH ENUMERATION AT ALL, > which is what the diagonal proof shows. Of course it helps to understand the difference between (AL)(En)~(Am)(L(m) = n) and (En)(AL)~(Am)(L(m) = n) The first is true. The second, because one can pick the identity mapping L(i) = i, and then m = n, is false. [snip rest for brevity] -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L for the women >H a l t i s a v e r y s p e c i f i c f u n c t i o n. I t w a s d i s p r o v e d. A n t i d i a g i s n o t a n y s p e c i f i c n u m b e r. s o w h y d o n 't y o u G O F I G U R E ? IOW. What number did you prove can't be listed? Herc === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* In sci.logic, HERC777 on 5 May 2005 19:08:57 -0700 > for the women >>H a l t i s a v e r y s p e c i f i c f u n c t i o n. > I t w a s d i s p r o v e d. A n t i d i a g i s n o > t a n y s p e c i f i c n u m b e r. s o w h y d o n 't > y o u G O F I G U R E ? > IOW. What number did you prove can't be listed? The one that has all finite prefixes in your list, just not the number itself. Is 1/3 in the set {.3, .33, .333, ...} ? Is 1/3 in the set {b/10^k: b,k in N, b < 10^k}? > Herc -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* In sci.logic, HERC777 on 5 May 2005 14:21:01 -0700 > the algorithm has been programmed, you can't dispute it. > www.freewebs.com/namesort/nums\[CapitalEth]ort.html IT WORKS! > IT DOESNT MATTER WHAT THE DIAGONAL IS. > IT MAKES AN EQUIVALENT SET WITH ANY RANDOM DIAGONAL. > (anyone understood that yet? - its just shuffles the list until it gets > a whole new diagonal) > Atleast Ghost is actually correct that T_10 = > {0.1,0.2, 0.3, 0.4,0.5, 0.6,0.7,0.8,0.9, > 0.01,0.02,0.03....0.99, > 0.001,0.002...0.999, > fits my 'saturated' definition. Ghost I don't know netscape javascript > you'll have to view the source of the code, I gave a dump of the input > and output in this thread. > There is probably a better definition of saturated_list, considering > T_10 or its equivalent in base 3 looks nothing like T_3 would be its base-3 equivalent: T_3 = {b/3^n: b,n in N, b < 3^n} Does T_3 contain 1/2? 1/2 = .111... (base 3) >> 1 0.20022022000220220012 >> 2 0.01021011111121221010 >> 3 0.11002221100201212000 >> 4 0.11112012111210221120 >> 5 0.10011201211002202121 >> 6 0.11212102012102012011 >> 7 0.20120120012111012112 >> 8 0.22112121000101112112 >> 9 0.20121222201110100020 >> 10 0.21221200021001022212 >> 11 0.21100100120111221001 >> 12 0.01110202100202012111 >> 13 0.00121121120021110111 >> 14 0.20210010210002101100 >> 15 0.12100102000011201002 >> 16 0.21120000001211111110 >> 17 0.20221000020002122111 >> 18 0.11022102210110001200 >> 19 0.20202211010110222012 >> 20 0.10111111222110002200 > This is saturated because there are oo 1s, oo 2s and oo 3s in every > digit position, its just random and the set shuffle algorithm works > perfectly on it, even though in theory it could be handling infinite > expansions.. > George you're a dumbass. suport your claim the sets are not equal and > follow through the proof as far as possible UNTIL non halting TMs in > the real enumeration actually makes a difference. > Herc -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > Here's the 1st list, in base 3. > 1 0.20022022000220220012 > 2 0.01021011111121221010 > 3 0.11002221100201212000 > 4 0.11112012111210221120 > 5 0.10011201211002202121 > 6 0.11212102012102012011 > 7 0.20120120012111012112 > 8 0.22112121000101112112 > 9 0.20121222201110100020 > 10 0.21221200021001022212 > 11 0.21100100120111221001 > 12 0.01110202100202012111 > 13 0.00121121120021110111 > 14 0.20210010210002101100 > 15 0.12100102000011201002 > 16 0.21120000001211111110 > 17 0.20221000020002122111 > 18 0.11022102210110001200 > 19 0.20202211010110222012 > 20 0.10111111222110002200 > Here is the desired diagonal 00000011111112222222 There is no \desired diagonal\ in the proof of uncountability. The issue in Cantor's proof is that the list is not complete; that no matter how you list a set of real numbers, there is a real number not in the list. Obviously, which number it is, depends on the list, but the point is that no list is complete. The diagonalization argument is the following: Take the nth digit of the nth number and add 1 to it (modulo 3). Then combine them. Use the list above as an example. The string of digits you get is 21011121220222212210, and when you add 1 (modulo 3) to each, you get 02122202001000020021. Now, is 0.02122202001000020021 in your list of real numbers? No. So I've found a number not in your list, which means your list isn't complete. And I didn't need to mention the Halting Problem to do it. --- Christopher Heckman > [rest of post snipped] === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L You can't even admit your 1st reply to me was extraordinary. >I believe the part about the hecklers! >Chis Heckman What hope do you have of admitting the truth? Next time post up your reply when its discussed or I'll tattoo it on your head. >there is a real number not >in the list. Obviously, which number it is, depends on the list Obviously your proof relies on that assumption but that is exactly what I disproved, the diagonal doesn't depend on the list, any saturated list can have any (random) diagonal. try the program! Herc === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > [loony part of the post clipped] > >there is a real number not in the list. Obviously, > >which number it is, depends on the list > Obviously your proof relies on that assumption What proof? That there's a real number not in the list? I _prove_ that in the post; I don't make that assumption. Maybe my proof that which number is not on the list depends on the list is an assumption? No, it's an empirical observation. > but that is exactly what > I disproved, the diagonal doesn't depend on the list, any saturated > list can have any (random) diagonal. That's true; HOWEVER, the real number you get by modifying that diagonal is not on the list, and that's the important part. The good news is you're not a crank. The bad news is you're a loony. A few rungs below James Harris, I'm sad to say. --- Christopher Heckman === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* >> [loony part of the post clipped] >The good news is you're not a crank. The bad news is you're a loony. Must those be mutually exclusive? I'd say he's both. >A few rungs below James Harris, I'm sad to say. That depends what ladder one uses, n'est-ce pas. Harris certainly has more mathematical knowledge and skill, but Cooper has far-more-grandiose delusions. -- --------------------------- | BBB b \\ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L Barb, every post where I ask a particular question you post over it. None of my questions get answered! Support your idiocy, or go away. thems the rules. [Herc] the diagonal doesn't depend on the list, any saturated list can have any (random) diagonal. [Prognoskis] That's true; HOWEVER, the real number you get by modifying that diagonal is not on the list, and that's the important part. What real is THAT specifically? Herc === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* >Barb, every post where I ask a particular question you post over it. >None of my questions get answered! >Support your idiocy, or go away. thems the rules. >[Herc] >the diagonal doesn't depend on the list, any saturated >list can have any (random) diagonal. >[Prognoskis] >That's true; HOWEVER, the real number you get by modifying that >diagonal is not on the list, and that's the important part. >What real is THAT specifically? That depends on the particular alleged listing function (from N -> reals), D******; just as the \halts iff it doesn't\ antidagonal depends on the particular alleged halting-solver function. -- --------------------------- | BBB b \\ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L [Herc] the diagonal doesn't depend on the list, any saturated list can have any (random) diagonal. [Prognoskis] That's true [Herc] What real is THAT diagonal specifically? [Barb] That depends on the particular alleged listing function (from N -> reals), Try again! go back 3 squares and read the postt. Herc === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L looks like Heckman has run for cover again!! you're not fit to lick James ass mate. Herc === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > looks like Heckman has run for cover again!! What evidence do you have of this? The fact that I didn't respond to a post that was only up for 18 minutes before you posted what's above? Unlike you, I have a life. (Did I really say that?) I'm not online 24/7; I have an actual job, as a Lecturer at Arizona State University. I've been teaching here for five years, which forces me away froms sci.math at times. It only LOOKS like I'm running for cover, because you're running in the opposite direction. And did you even bother to THINK about what I posted? Here's one of your last posts: > Idiot WROTE No matter how much James Harris was upset by what I said, he never called me an idiot. That puts you even lower on the ladder. (BTW, I haven't sunk to your level in doing the same thing.) > which number it is, depends on the list > NO THATS WRONG Your response is wrong! Consider the diagonalization function D, which for every list L provides a real number R; D(L)=R, got that? What my post said is equivalent to \D(L) is not a constant function\; that is, there are lists L1 and L2 such that D(L1) is not D(L2). > list can have any (random) diagonal. What did I mean by that? I meant that \D is an onto function\; that is, if r is any real number, then there is a list L such that D(L) = r. > That's true > Try not to contradict yourself. There is no contradiction! All I've said about the function D is that D(L) is not always the same, and that the range of D is the set of all real numbers. No contradiction. > the real number you get by modifying that diagonal > What is THAT referring to? Further evidence you haven't read my post, because I told you exactly how to modify D(L) to get a real number which is not on the list L. THINK BEFORE YOU POST. > Specifically, it doesn't depend on the list Yes, it does. > so tell me what real number THAT actually is here? > If you haven't run the diagonal reconstruction program at > www.freewebs.com/namesort/numsort.html > then you have NO IDEA what you are talking about. > Herc I have better things to do with my time. --- Christopher Heckman P.S. If I respond to this before you do, then you thereby admit you are permanently a weiner lunatic who should be locked up for his own good. === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > [snipped] > P.S. If I respond to this before you do, then you thereby admit you are > permanently a weiner lunatic who should be locked up for his own good. HA HA HA! YOU'RE A WEINER LUNATIC WHO SHOULD BE LOCKED UP FOR YOUR OWN GOOD! AND YOU ADMITTED IT YOURSELF!!!! HA HA HA! === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L I know list 2 has a diagonal you fuckwit. WHAT IS IT? ANYTHING you fucking moron Herc === Subject: Re: BEHOLD....... PROOF THAT *CARDINALITY IS BONKERS* posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L Idiot WROTE which number it is, depends on the list NO THATS WRONG list can have any (random) diagonal. That's true Try not to contradict yourself. the real number you get by modifying that diagonal What is THAT referring to? Specifically, it doesn't depend on the list so tell me what real number THAT actually is here? If you haven't run the diagonal reconstruction program at www.freewebs.com/namesort/numsort.html then you have NO IDEA what you are talking about. Herc === Subject: Weak and strong convergence in D'(X) posting-account=FrZeUw0AAAB4_H1glZ6NiTuSekPD85_E Let X be an open set of R^n. I've learned the weak/strong topology in term of seminorms: - Weak: We give D'(X) or E'(X) - viewed as the dual of C_0^\\infty(X) or C^\\infty(X) - the topology induced by the seminorms p_A = max_{x \\in A} , where A \\subset C^\\infty is finite - Weak: We give D'(X) or E'(X) - viewed as the dual of C_0^\\infty(X) or C^\\infty(X) - the topology induced by the seminorms p_B = sup_{x \\in B} , where A \\subset C^\\infty is bounded To test my skills with these definitions, I'm looking for a sequence of distributions, which converges in the weak topology on D'(X) or E'(X), but not in the strong. === Subject: Re: Weak and strong convergence in D'(X) There's a lot here that needs clarification: >Let X be an open set of R^n. >I've learned the weak/strong topology in term of seminorms: >- Weak: We give D'(X) or E'(X) - viewed as the dual of C_0^\\infty(X) The least of my worries is that the notation \C_0\ varies: do the functions in your C_0^\\infty(X) have compact support? >C^\\infty(X) - the topology induced by the seminorms >p_A = max_{x \\in A} , where A \\subset C^\\infty is finite Here's where I get very confused: The p_A you define here is not a seminorm - for example it's not real-valued. >- Weak: We give D'(X) or E'(X) - viewed as the dual of C_0^\\infty(X) or >C^\\infty(X) - the topology induced by the seminorms >p_B = sup_{x \\in B} , where A \\subset C^\\infty is bounded Similar problem here, also a typo. Another problem with this definition is that what it means for a subset of C_0^\\infty(X) to be \bounded\ depends on what topology we're giving the space... >To test my skills with these definitions, I'm looking for a sequence of >distributions, which converges in the weak topology on D'(X) or E'(X), >but not in the strong. A person could just assume that you meant the usual definitions, except that the definitions above have nothing to do with any usual definitions that I've ever seen! In particular, assuming as seems likely that D'(X) is supposed to be the space of distributions on X, then (i) I've never seen a definition of a \strong topology\ on D'(X), and (ii) the usual weak topology on D'(X) is not the topology induced by a family of seminorms. ************************ David C. Ullrich === Subject: Re: Weak and strong convergence in D'(X) posting-account=FrZeUw0AAAB4_H1glZ6NiTuSekPD85_E > There's a lot here that needs clarification: guessed, I'm new to the subject of distributions and locally convex vector spaces, so its possible I got things messed up. I used the notes from http://www.math.ku.dk/~grubb/distribution.htm, and the definition of strong/weak topology given by my prof. By the way, this is my first time on usenet; please tell me if I'm doing something not right. > >Let X be an open set of R^n. > >I've learned the weak/strong topology in term of seminorms: > >- Weak: We give D'(X) or E'(X) - viewed as the dual of C_0^\\infty(X) > The least of my worries is that the notation \C_0\ varies: > do the functions in your C_0^\\infty(X) have compact support? I was not aware of this variation: yes, C_0^\\infty means infinite differentiable with compact support. > >or > >C^\\infty(X) - the topology induced by the seminorms > >p_A = max_{x \\in A} , where A \\subset C^\\infty is finite > Here's where I get very confused: The p_A you define here > is not a seminorm - for example it's not real-valued. Okay, I should have been more careful. What I meant was the topology induced by the seminorms p_A : D'(X) --> R p_A(u) = max_{\\phi \\in A} |u(\\phi)| with A a finite set of elements of C_0^\\infty, that is, a finite number of functions. > >- Weak: We give D'(X) or E'(X) - viewed as the dual of C_0^\\infty(X) or > >C^\\infty(X) - the topology induced by the seminorms > >p_B = sup_{x \\in B} , where A \\subset C^\\infty is bounded > Similar problem here, also a typo. I'm sorry, what I meant: p_B : D'(X) --> R p_B(u) = sup_{\\phi \\in B} |u(\\phi)| with B a bounded set of elements of C_0^\\infty. > Another problem with this definition is that what it means > for a subset of C_0^\\infty(X) to be \bounded\ depends on > what topology we're giving the space... Again, you are completely right. Boundedness of B in C^\\infty(X) and C_0^\\infty(X) means that p(B) is bounded (in R) for each seminorm p. The seminorms on C^\\infty(X) are p_{k,K}(\\phi) = \\max_{|\\alpha| \\leq k} \\sup_{x \\in K} \ |\\partial^\\alpha phi(x)| C_0^\\infty(X) is viewed as \\bigcup_{j=1}^\\infty C_{K_j}^\\infty(X), with K_j a sequence of compact sets absorbing X, and given the inductive limit topology. > >To test my skills with these definitions, I'm looking for a sequence of > >distributions, which converges in the weak topology on D'(X) or E'(X), > >but not in the strong. > A person could just assume that you meant the usual definitions, > except that the definitions above have nothing to do with any > usual definitions that I've ever seen! > In particular, assuming as seems likely that D'(X) is supposed > to be the space of distributions on X, then (i) I've never > seen a definition of a \strong topology\ on D'(X), and > (ii) the usual weak topology on D'(X) is not the topology > induced by a family of seminorms. If I'm not mistaking, strong/weak convergence and topology is a general way of giving the dual space a topology. It would be very nice if you could explain to me what the usual weak topology on D'(X) is, if it isn't what I thought. I hope you have the time to respond, as I really would like to learn this subject. Ciro Bonano. === Subject: Re: Weak and strong convergence in D'(X) >> There's a lot here that needs clarification: I was hoping to post this retraction before you saw my original reply. More or less everything I said was wrong, sorry. (I was thinking about the topology on D(X), not D'(X).) The question makes perfect sense, not sure offhand what the answer is.) >As you might have >guessed, I'm new to the subject of distributions and locally convex >vector spaces, so its possible I got things messed up. I used the notes >from http://www.math.ku.dk/~grubb/distribution.htm, and the definition >of strong/weak topology given by my prof. >By the way, this is my first time on usenet; please tell me if I'm >doing something not right. No, you did everything right, including being very polite to a very stupid reply. >> >Let X be an open set of R^n. >> > >> >I've learned the weak/strong topology in term of seminorms: >> >- Weak: We give D'(X) or E'(X) - viewed as the dual of C_0^\\infty(X) >> The least of my worries is that the notation \C_0\ varies: >> do the functions in your C_0^\\infty(X) have compact support? >I was not aware of this variation: yes, C_0^\\infty means infinite >differentiable with compact support. >> >or >> >C^\\infty(X) - the topology induced by the seminorms >> >p_A = max_{x \\in A} , where A \\subset C^\\infty is finite >> Here's where I get very confused: The p_A you define here >> is not a seminorm - for example it's not real-valued. >Okay, I should have been more careful. What I meant was the topology >induced by the seminorms >p_A : D'(X) --> R >p_A(u) = max_{\\phi \\in A} |u(\\phi)| >with A a finite set of elements of C_0^\\infty, that is, a finite number >of functions. >> >- Weak: We give D'(X) or E'(X) - viewed as the dual of C_0^\\infty(X) >> >C^\\infty(X) - the topology induced by the seminorms >> >p_B = sup_{x \\in B} , where A \\subset C^\\infty is bounded >> Similar problem here, also a typo. >I'm sorry, what I meant: >p_B : D'(X) --> R >p_B(u) = sup_{\\phi \\in B} |u(\\phi)| >with B a bounded set of elements of C_0^\\infty. >> Another problem with this definition is that what it means >> for a subset of C_0^\\infty(X) to be \bounded\ depends on >> what topology we're giving the space... >Again, you are completely right. Boundedness of B in C^\\infty(X) and >C_0^\\infty(X) means that p(B) is bounded (in R) for each seminorm p. >The seminorms on C^\\infty(X) are >p_{k,K}(\\phi) = \\max_{|\\alpha| \\leq k} \\sup_{x \\in K} \ |\\partial^\\alpha >phi(x)| >C_0^\\infty(X) is viewed as \\bigcup_{j=1}^\\infty C_{K_j}^\\infty(X), \ with >K_j a sequence of compact sets absorbing X, and given the inductive >limit topology. >> >To test my skills with these definitions, I'm looking for a sequence >> >distributions, which converges in the weak topology on D'(X) or >E'(X), >> >but not in the strong. >> A person could just assume that you meant the usual definitions, >> except that the definitions above have nothing to do with any >> usual definitions that I've ever seen! >> In particular, assuming as seems likely that D'(X) is supposed >> to be the space of distributions on X, then (i) I've never >> seen a definition of a \strong topology\ on D'(X), and >> (ii) the usual weak topology on D'(X) is not the topology >> induced by a family of seminorms. >If I'm not mistaking, strong/weak convergence and topology is a general >way of giving the dual space a topology. It would be very nice if you >could explain to me what the usual weak topology on D'(X) is, if it >isn't what I thought. >I hope you have the time to respond, as I really would like to learn >this subject. >Ciro Bonano. ************************ David C. Ullrich === Subject: Re: Bivariate/Multivariate Joint PDF MKnight123@gmail.com: | I posted earlier in reference to a similar problem. | | I'm trying to model a similar problem without any sample data. | If given, | f(x|a)~Normal | f(y|b)~Normal | corr(x,y|a,b) | | and I state the assumption that | f(x|a,b)=f(x|a) | f(y|a,b)=f(y|b) | | How incorrect would it be to assume that f(x,y|a,b)~bivariate | normal ? I do not understand your notation, as you use the values a and b in an unusual (redundant) way. For example, what does f(y|a,b) or corr(x,y|a,b) mean? -- Sebastian Stern Freedom is the freedom to say (= (+ 2 2) 4). If that is granted, all else follows. === Subject: Re: SFT: Experimentation starts >ullrich@math.okstate.edu says... >> >oooh, those are some strong words David, don't think you'd want your >> >students to read such things.... >> Right. They've never read words like that. >Which doesn't make it any more appropriate, Huh? I didn't say that that made it appropriate. The fact that Harris is a fucking asshole is what makes it appropriate to call him a fucking asshole. You're free to disapprove, and to express your disapproval. Even though doing so makes you sound awfully childish (in a \Mommy, Dave said a bad word!\ sort of way.) >even in this case. >I'm sure your employer would be somewhat less than laudatory >if they were aware of it. Taking that literally, well duh, of course I wouldn't expect praise from my employer for writing what I But you seem to be suggesting that my employer would express disapproval - that's certainly possible, or that my employer would try to initiate some sort of punitive action. If you're suggesting the second, all I can say is that I wish they would, I could use the money. (You might note that the comment of mine that you're complaining about was not made in the classroom, not in the office, not directed at a student, etc. What I say at home on my own time is my business.) If anyone has a complaint here it's Harris. And any complaints he makes are going to be laughed out of whatever court he makes them in when I quote some of the things he's said to me. (To (sigh) save a little space in the thread: Yes, Mom, I understand that two wrongs don't make a right. Why don't you grow up a little, and take a look at the way (many) real people actually speak these days.) ************************ David C. Ullrich === Subject: Re: SFT: Experimentation starts X-RFC2646: Format=Flowed; Original > Which doesn't make it any more appropriate, even in this case. > I'm sure your employer would be somewhat less than laudatory > if they were aware of it. Bullshit! His freedom of speach is guaranteed by his government. His school \ cannot do much about it. === Subject: Re: SFT: Experimentation starts >> Which doesn't make it any more appropriate, even in this case. >> I'm sure your employer would be somewhat less than laudatory >> if they were aware of it. > Bullshit! His freedom of speach is guaranteed by his government. His > school cannot do much about it. Sure the school can if it wants to. J -- __________________________________________ When will Bush be tried for war crimes? \Our enemies are innovative and resourceful, and so are we. They never stop thinking about new ways to harm our country and our people, and neither do we.\ --G. W. B. Joe Peschel D.O.E. SysWorks http://members.aol.com/jpeschel/index.htm __________________________________________ === Subject: Re: SFT: Experimentation starts X-RFC2646: Format=Flowed; Original >> Bullshit! His freedom of speach is guaranteed by his government. His >> school cannot do much about it. > Sure the school can if it wants to. What do you figure the school could/would do? === Subject: Re: SFT: Experimentation starts ig@whidbeytel.com: > Bullshit! His freedom of speach is guaranteed by his government. His > school cannot do much about it. >> Sure the school can if it wants to. > What do you figure the school could/would do? It can do pretty much whatever it wants. J -- __________________________________________ When will Bush be tried for war crimes? \Our enemies are innovative and resourceful, and so are we. They never stop thinking about new ways to harm our country and our people, and neither do we.\ --G. W. B. Joe Peschel D.O.E. SysWorks http://members.aol.com/jpeschel/index.htm __________________________________________ === Subject: Re: SFT: Experimentation starts X-RFC2646: Format=Flowed; Original > It can do pretty much whatever it wants. I am talking almost from first hand experience. My father was bit outspoken \ (to put it mildly), and the university where he taught tried to do several things against him, but failed. There was litigation from time to time. But \ my father never lost anything as a result. He had tenure, and he was a department head, so he may have had more protection than some would. But I still figure that a prof can say pretty much what ever they want to. And in \ the current case, a rather mild expletive was used outside of the confines of the school. So what do you think? === Subject: Re: SFT: Experimentation starts posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > Well I finally couldn't resist the impulse to check out > the SFT, and hey, the equations actually do work! This is actually a response to a post in the Surrogate Factoring group. I can't post there because I dare question your posts. This is (was?) your May 4 post. > Well I finally actually checked the equations in the SFT > and they are correct. Which is not news for the rest of sci.math. No one was doubting the derivation. > So that's a good feeling for me, as it means the > derivations are correct and the signs and all the > details are correct. > I also noticed that using integers in your factorization > of the surrogate does not work well for factoring. This has been pointed out repeatedly by people, but you just called them \liars\. > However, the SFT is over rationals, and that just > means you need to use factors of the surrogate that > are fractions versus using integers. > Yes, integers are easier, but easy doesn't work here. > But how do you pick the fractions? > Good question. > That's where the research is. I could SWEAR I said the same thing earlier ... Aha! The original post of the \Talking Rationally About Surrogate Factoring\ thread, about 2 weeks ago. Which, BTW, is still in the Google Groups Archive. --- Christopher Heckman === Subject: Re: maple bug posting-account=DooOggwAAADdcqtzKMrYoQ_DNGq8lCTf === B> Subject: maple bug B> int(exp(-abs(x-xp))*tanh(xp), xp=-infinity..infinity); Yes, this is yet another Maple bug not fixed over years, catch with us; it has been added to an internal version of Maple Bugs Encyclopaedia. The integral cannot be 'undefined' as abs(tanh(z)) -> 1 as z-> infinity while exp(-abs(a-z)), for any complex- valued a, decays wildly to 0 as z-> -infinity or z-> infinity. Thus, the integral converges. Also, if abs(a) <> 0, the general case you presented, the integrand is not an odd function, so the integral cannot be = 0. int(exp(-abs(a-z))*tanh(z), z=-infinity..infinity); undefined undefined undefined -------------------- (2002) Maple 8 -------------------------- undefined -------------------- (2001) Maple 7 -------------------------- 0 -------------------- (2000) Maple 6 -------------------------- int(exp(-abs(a-z))*tanh(z),z = -infinity .. infinity) -------------------- (1997) Maple V Rel 5 -------------------- int(exp(-abs(a-z))*tanh(z),z = -infinity .. infinity) -------------------- (1995) Maple V Rel 4 -------------------- int(exp(-abs(a-z))*tanh(z),z = -infinity .. infinity) -------------------- (1994) Maple V Rel 3 -------------------- int(exp(-abs(a-z))*tanh(z),z = -infinity .. infinity) --------------------------------------------------------------- Best wishes, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director Cyber Tester, LLC 13 Dekabristov Str, Simferopol Crimea 95000, Ukraine tel: +38-(0652)-447325 tel: +38-(0652)-230243 tel: +38-(0652)-523144 fax: +38-(0652)-510700 http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ === Subject: Re: maple bug -> int(exp(-abs(a-z)), z=-infinity..infinity); posting-account=DooOggwAAADdcqtzKMrYoQ_DNGq8lCTf === B> Subject: maple bug B> int(exp(-abs(x-xp))*tanh(xp), xp=-infinity..infinity); Here comes yet another bug the long-liver of a kind similar to yours. In this case, the sheer absurdity of the answer Maple yields over a decade, is obvious. restart; int(exp(-abs(a-z)), z=-infinity..infinity); 2 2 2 -------------------- (2002) Maple 8 -------------------------- 2 -------------------- (2001) Maple 7 -------------------------- 2 -------------------- (2000) Maple 6 -------------------------- 2 -------------------- (1997) Maple V Rel 5 -------------------- 2 -------------------- (1995) Maple V Rel 4 -------------------- 2 -------------------- (1994) Maple V Rel 3 -------------------- int(exp(-abs(a-z)),z = -infinity .. infinity) --------------------------------------------------------------- Let a = I . evalf(int(exp(-abs(I-z)), z=-infinity..infinity)); 1.203814460 Best wishes, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director Cyber Tester, LLC 13 Dekabristov Str, Simferopol Crimea 95000, Ukraine tel: +38-(0652)-447325 tel: +38-(0652)-230243 tel: +38-(0652)-523144 fax: +38-(0652)-510700 http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ === Subject: Simple Abstact Algebra Question posting-account=UsMTzhMAAAAITFFPlj9rUB2S3BtXg0OQGdaE3UVKRbgY8da85xMcNA I know this question has been asked many a time but I don't understand the answers that I find. Let R be a finite commutative ring with an identity. I'm trying to prove that the intersection of all the prime ideals is an ideal which contains exactly all the nilpotent items. I know how to prove that if x is nilpotent it belongs in the intersection. Now let's take an x that is not nilpotent. I would like to find a prime ideal that doesn't contain x but am having a very hard time. === Subject: Re: Simple Abstact Algebra Question days. My association with the Department is that of an alumnus. >I know this question has been asked many a time but I don't understand >the answers that I find. >Let R be a finite commutative ring with an identity. I'm trying to >prove that the intersection of all the prime ideals is an ideal which >contains exactly all the nilpotent items. >I know how to prove that if x is nilpotent it belongs in the >intersection. >Now let's take an x that is not nilpotent. I would like to find a prime >ideal that doesn't contain x but am having a very hard time. Consider the multiplicative subset S = {1,x, x^2, x^3,...} Then consider the set of all ideals that intersect S trivially. Since x is not nilpotent, {0} is one such ideal. They form a partially ordered set under inclusion, so there is a maximal ideal intersecting S trivially. Call it P. I claim that P is a prime ideal. Suppose not. Then there are a,b not in P such that ab is in P. Then there is some power of x, x^j = ra + p for r in R, p in P, j>= 0. x^i = sb + q for s in R, q in P, i>=0. Then x^{i+j} = (ra+p)(sb+q) = rsab + raq + sbp + pq and the right hand side is contained in P, contradicting that P is disjoint from S. Therefore P is a prime ideal, and it intersects S trivially. -- \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: Simple Abstact Algebra Question posting-account=DaXtvAwAAACPATqzmVZ4JJbgwUGjL51g > I know this question has been asked many a time but I don't understand > the answers that I find. > Let R be a finite commutative ring with an identity. I'm trying to > prove that the intersection of all the prime ideals is an ideal which > contains exactly all the nilpotent items. > I know how to prove that if x is nilpotent it belongs in the > intersection. > Now let's take an x that is not nilpotent. I would like to find a prime > ideal that doesn't contain x but am having a very hard time. ***************************** Hi: You'll need Zorn's lemma for this one. Let x be non-nilpotent, and define S:= {x^n; n a natural number}. Then S does not contain zero ==> if G is the set of all ideals that don't contain any element of S ordered by inclusion of set, then G is non-empty (why?) and is inductive, meaning: every chain (subset of G fully ordered) has an upper bound ==> by Zorn's lemma G has a maximal element M. If M is a non-prime ideal then there exist a,b in R-M s.t. ab is in M ==> since M < M + aR, M + bR, by maximality of M there exist natural n,m s.t. x^n in M + aR, x^m in M + bR. Now check that x^(n+m) is in M and this is contradictio. Q.E.D. Tonio === Subject: Re: Simple Abstact Algebra Question > You'll need Zorn's lemma for this one. No he won't. > Let x be non-nilpotent, and define > S:= {x^n; n a natural number}. Then S does not contain zero ==> if G is > the set of all ideals that don't contain any element of S ordered by > inclusion of set, then G is non-empty (why?) and is inductive, meaning: > every chain (subset of G fully ordered) has an upper bound ==> by > Zorn's lemma G has a maximal element M. R finite => G finite => G has a maximal element. Jose Carlos Santos === Subject: Re: Simple Abstact Algebra Question <3e1clhFmhugU1@individual.net> posting-account=DaXtvAwAAACPATqzmVZ4JJbgwUGjL51g Hehe. Completely missed the \finite\ part...anyway, he's now the general stuff. Tonio === Subject: Re: Interesting curve - is it ellipse? > Hello mathematicians! > I have the following parametric definition of a curve: > x = A cos(t), y = B cos(t+C), > where A, B and C are constants. > I am almost certain that this curve is an ellipse, rotated > by a certain angle from its normal position. I have succeeded > to write this curve in polar coordinates and to find the angles > of maximum and minimum radii from the center of the coordinate > system and its appropriate radii values. So if I draw a > rotated ellipse with these parameters and they graphically seem > to fit perfectly. > I would like to know if there is a way to PROVE that this is > really an ellipse? [...snip...] > Marko Pinteric (physicist) Others have answered your question pretty completely, I think. But let me add just a bit and ask if someone can 'explain' my observations. After eliminating t as suggested by Jyrki Lahtonen and others I get the Cartesian equation: B^2*x^2 - 2*A*B*cos(C)*x*y + A^2*y^2 = A^2*B^2*sin(C)^2 Now this is an ellipse iff the discriminant -4*A^2*B^2*sin(C)^2 is negative, i.e. providing A, B, sin(C) are all non-zero. The ellipse degenerates to a line segment if A,B are non-zero and sin(C)=0. The ellipse is a circle when A=+-B and cos(C)=0. Now here is my observation, which I would like insight for: Consider the triangle with A*y and B*x two sides and C the angle between them. Then the third side squared is just the left side of the equation above, by the law of cosines. Also, the right side of the equation looks like a cross product. In fact, it is the square of the norm the cross product of two vectors of lengths A and B separated by angle C. So what is going on here? Is this just coincidence or is there maybe a vector approach to this problem that makes things more transparent? All comments appreciated. -- Jim Buddenhagen === Subject: Towards New Math Correctness Standards for Computer Algebra Systems \ - 6 posting-account=DooOggwAAADdcqtzKMrYoQ_DNGq8lCTf http://www.cas-testing.org/img/times4.gif ................................................................. Towards New Math Correctness Standards for Computer Algebra Systems - 1 to 5 ................................................................. === Subject: Re: Technology for Maple testing ------------6FEC13FFBEA52D LB> First, I must apologize for taking as much time to get back to you. Maybe, it is me who must apologize for writing to you at so hot time. Actually, you are on the eve of the official release. And, still, you have found the time to reply me. I contacted you because I hope that I could help Waterloo Maple Inc. make your Maple even stronger and more attractive for the customers. LB> we strive to increase Maple's quality on an ongoing basis and LB> your applying your methodology at this point might allow us LB> more time to fix arising problems for a future release. Hopefully, it is the case. LB> May I ask for a little more detail on your method? Sure. In a single sentence, I emulate working of the users having various experience, from beginner to expert level, and analyze the corresponding output using the system and my own (sometimes, rather sophisticated) procedures. The approach is split into 3 parts: * automated black box testing (hundreds mathematical functions, including int, limit, series, sum, product, and much more) * additional manual black box testing (plotting and documentation) Right this day, I still have no satisfactory approach to automated plotting testing, but I keep working on this problem * white box testing (I read carefully the source code, and try to find the suspicious lines and analyze them) A. Automated testing starts from a precomputed database # 1 of several thousand entries involving several dozen most common mathematical functions. In military terms, record adjustment. It is here where I make a choice of the further testing strategy. B. Secondly, 800+ heuristics and 100+ metaheuristics direct the further processing. In particular, the monitor tries, if possible, to identify the minimal (in a sense) form which still causes the concrete problem. Step by step, I add manually the new heuristics to the database # 2. In the future, though it is not a very trivial task, I hope to generate these new heuristics automatically. In Mathematica, I also intercept the system messages, and by analyzing them can efficiently identify new problems. C. At last, I am in the process of scanning and converting of several selected problem books and handbooks into the proper form, so later I can use this stuff (DB #3) to reinforce the approach. All wrong and suspicious outputs are added to the database # 4. I had implemented a part of such an approach in the Mathematica language. In the attached file Sample02.nb, please ignore the messages in green, they are a kind of my internal debugging information. The Mathematica's bugs in red look somewhat messy, because * I still did not implement the automatic computation of the 'minimal' form, for the given bug * I did not sort the bugs automatically in a sequence which was the easiest to read by human (e.g., thematically) * The general outlay should be improved, too. These points are quite manageable, I am working on that. About the performance. In the shown file, 425 unique bug manifestations were identified in 22 hours, on an Athlon Some of these problems, as I suspect, might be not \linearly independent\. LB> In particular, can you give a few examples of the types of LB> problems your method has uncovered in the past, either in LB> Maple or in a competitor's product? I have applied this approach to Derive, Mathematica and MuPAD. The attached notebooks file Sample02.nb demonstrates how approximately it looks THIS moment. MathReader is a free application for viewing Mathematica notebooks. You can download it at www.wolfram.com/mathreader. The file Sample01.txt gives more examples as well as demonstrates the types of the problems which I can identify with the help of the technology. The second competitor is MuPAD by SciFace Software GmbH. Using the following authorization username: \vvb\ password: \wl2hfy\ you may wish to find out the information on MuPAD 2.5 beta testing http://www.mupad.de/site/bt25/contest/bug_query_beta.shtml You can ask Bugzilla to show the problems identified by myself. Here is the official estimation of my work http://www.mupad.de/site/bt25/contest/winner.shtml With MuPAD, I, in fact, used the same technology, but I was proposed to participate in the beta testing all of a sudden, and, being busy with the other projects, simply had not enough time to implement the full-fledged monitor in the MuPAD language. Instead, I only used a couple of simple procedures based on the same ideas. I would hope, this, so-to-say, large-scale computational approach, that is running Maple around the clock would almost certainly reveal soon more than one new problem (as it was with Mathematica). I have learnt about the existence of Maple in 1992, at Glushkov Institute of Cybernetics (Kiev) during a seminar lead by Prof Letichevsky. Since then, my (pleasant) experience with Maple has convinced myself that your symbolic computer algebra system is rather reliable - and could be improved, too. The Maple users might like these improvements. Once, during ISSAC'94 in Oxford I showed to Michael Monagan a couple of examples. In fact, I used that time a very very weak version of the technology I am pleased to propose you to focus the efforts of your team on the actual research and coding. At last, I am in the process of exploring a new layer of ideas, which, I hope, will at least double the performance of the technology on the same resource. I expect that I will finish this preliminary investigation within 6 months, and report you the details of my experimenting. As far as I know, possibly, nobody has tried this somewhat unusual path. Naturally, I realize that you have an enviable computational experience. That is why it would be extremely interesting to hear any comments from you. Let me thank you cordially for your interest in my ideas. If you have any kind of questions, please feel absolutely comfortably to let me know about it. Vladimir Bondarenko Applied mathematician Email: vvb@mail.strace.net Voice: (380)-652-447325 76 Zalesskaya St., Apt 29. Simferopol Crimea 95044 Ukraine .................................................................. As I'd learnt from McNamara, the discipline of writing something down is the first step toward making it happen. In conversation, you can get away with all kinds of vagueness and nonsense, often without even realizing it. But there's something about putting your thoughts on paper that forces you to get down to specifics. That way, it's harder to deceive yourself - or anybody else. Lee Iacocca An Autobiography .................................................................. === Subject: Re: a field extension problem posting-account=dtjpKg0AAADxSYvJeHZBzr0U4Xoj_aRZ > >Let F in E (both fields) be a field extension and let a,b be in E. > >1. Show that (F(a))(b)=(F(b))(a) > Later you write: > > I thought part 1 was almost by definition > >but I must have misread it. > Depends on your definition. If \F(r)\ means \the smallest subfield of > E which contains F and contains r\, then it is by definition. > Otherwise, you may have to do a bit of work: If F(a) consists of all > polnomials in a with coefficients in b, then it only requires you to > note that you can rewrite any polynomial in a and b with coefficients > in F as either a polynomial in b with coefficients in F(a), or a > polynomial in a with coefficients in F(b). You're right. I'm going to have to do work on it. I haven't gotten it yet though. It looked easier than it was. > >2. Assuming a and be are transcendental over F show that b is > >algebraic over F(a) if and only if a is algebraic over F(b). > Are we missing \b\ after the \and\? If not, there are obvious counterexamples. > If so, use part (1): (F(b))(a) = (F(a))(b). And both F(a) and F(b) are > isomorphic to F(x). So (F(b))(a) is isomorphic to a finite extension > of F(x). I'm sorry. It was supposed to read: 2. Assuming a and b are transcendental over F show that b is algebraic over F(a) if and only if a is algebraic over F(b). James > \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: a field extension problem days. My association with the Department is that of an alumnus. >> >Let F in E (both fields) be a field extension and let a,b be in E. >> > >> >1. Show that (F(a))(b)=(F(b))(a) >> Later you write: >> > I thought part 1 was almost by definition >> >but I must have misread it. >> Depends on your definition. If \F(r)\ means \the smallest subfield of >> E which contains F and contains r\, then it is by definition. >> Otherwise, you may have to do a bit of work: If F(a) consists of all >> polnomials in a with coefficients in b, then it only requires you to >> note that you can rewrite any polynomial in a and b with coefficients >> in F as either a polynomial in b with coefficients in F(a), or a >> polynomial in a with coefficients in F(b). >You're right. I'm going to have to do work on it. I haven't gotten it >yet though. It looked easier than it was. It is NOT hard, but you need to know what your definition of F(r) is. >> >2. Assuming a and be are transcendental over F show that b is >> >algebraic over F(a) if and only if a is algebraic over F(b). >> Are we missing \b\ after the \and\? If not, there are obvious > >counterexamples. >> If so, use part (1): (F(b))(a) = (F(a))(b). And both F(a) and F(b) > >are isomorphic to F(x). So (F(b))(a) is isomorphic to a finite extension >> of F(x). >I'm sorry. It was supposed to read: >2. Assuming a and b are transcendental over F show that b is algebraic >over F(a) if and only if a is algebraic over F(b). Right; so my answer tells you what to do. You already know that F(a,b) is equal to both (F(a))(b) and to (F(b))(a); if a and b are both transcendental, then F(a) and F(b) are both isomorphic to F(x). Since (F(x))(b) is isomorphic to (F(a))(b) which is isomorphic to (F(b))(a) which is isomorphic to (F(x))(a), then the degree of b over F(a) is the same as the degree of a over F(b). -- \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: the New ERA is at hand, we are preparing it. That's Brian, Great! Your explanation about sets. We only have to be honest to ourselves. I like your photo's Brian. I also paint and you can expect I like to use \ techniques that are providing me with staircase and bannister to come \ further. ACCEPTANCE And I see other people photograph and paint in another way.. That is not \ really my way. But I don't say it's not my way. I have another approach... \ \That painting fits in that other environment\. A great horse for the \ stockbreeding of horses. For the environment of the clubhouse. And together \ we can be enthusiast about it. Do you understand me. You can ask me \ everything. But please don't put that painting in my environment. So he is my friend. \ And we differ a little bit. But in friendship we grow to each other. I can accept very much Tony. Useful in the environment of computers. And \ what a development is there. I would he could accept me in another \ environment. It's just a matter of environment what we appreciate. Axioms are \ rules we can obey or we prefer other rules. And now the wider view, accepting your own intelligence and all intelligent \ people. You have the brains for it. Discrete mathematics is introducing the notion form computer people of steps \ in the time. Set theory is discrete. The Turing machine is discrete. The \ natural numbers are discrete. Quantum behavior like the photo-electric effect \ is discrete. We walk with steps. We breathe with steps. Circles are also \ steps. Just one orbit. We have already a lot. And the great Alan Church has proved the Turing machine with states is \ equivalent to the lambda-calculus and so equivalent to set theory. And we go further on this way introducing more notions from the scientists. \ Wolfram and Chaitin are big names. And I am building on papers from famous scientists and we have forgotten \ those papers. While they were so right. In a next posting I will tell about those forgotten papers. Or only a few \ people know of it. 1. Niels Bohr 2. Henri Jules Poincar\.8e (1899) (the earth has layers) 3. G\.9adel (known by to few people) (theories are incomplete). 4. The early Santa Fe group (1967) (everywhere layers with physical emergence, that means with local gains and local loss. And not knowing exactly what is happening. Thus we introduce margins and we are getting then a lot for free.). 5. Popper (known by too few people) 6. others And I have the idea. People are on running trains never looking anymore at \ the beginning. Where are they heading to? Mathematicians can be critical. And \ then we are extremely needed. Do you know what comes in our reach. Power outages. Earth quakes and other \ big catastrophes. Flight simulators, also games. Top mathematicans have tried to predict the financial courses. And they have \ failed. You can't predict a bad acccident or a happy surprise. The simulation tools are really needed. But then for sunspots and all the \ sciences. For we are concentrating on simulation tools. And we need to combine it with \ analyzing tools. So we are only building some further. But we need everybody. \ Brain researches need us now as well. Simulation tools can teach the brain. \ Everywhere we are needed. I will tell you about the notion PANDAC. We do it all the time \preparing \ and acting\. But Pandac we see everywhere as soon as we are awake for it. \ Molecules have some not so interesting path and then the chemical reaction \ takes place. Only the chemical reaction is important the way to it is nearly \ circumstantial The same happens with shooting positrons on electrons for annihilation. \ Physicists know that. Such notion could be a kind of eye openers. That is of \ coarse typical for learning. Understanding breaks through. We are getting \ grip on it. That is also a PANDAC. And we see here what we also are doing. \ The preparation is pretty rough. and we are already thinking inductively. \ That is the same gathering scientist do as well. Let's talk about that some \ more another time. We use PANDAC to look at the traveling salesman again. Now with a wider \ view. I will tell you also about the wider view. We extend the mathematical domains with measurements And we make theories \ now in versions that can grow. That is what scientist want as well. We also widen our image space or solution space. We accept also numbers \ given by the computer. We can make graphs from it that are not exactly \ analytical .but so what we can locally approach them with easy formulas. An \ everybody is happy for some time So we join with the scientists; listening to \ each other. Accepting each other peculiarities. And always I learn form that \ as well. We are certainly going to a new ERA. And if you want it you are a part of \ it. Look this marvelous idea. I am sending now charm emails to political \ parties, papers and organizations. Also about language and other things. \ Mathematicians have hobbies, for sure. For the hearts of the people. Nice \ postings. Establishing people. And another marvelous idea. We are organizing a huge stage play. Already a \ rough script on the open net. We are working together with the cultural \ people. And like a feast that stage play will go through the streets. I can also tell more about that. Some people we never see. But they exist. Google once for -child abuse. And we are frightened. I have on the site \ solutions for those people as well. It's the L+F relationship, that suits \ every one. Please use your brains. We can make it. In Canada an ERA movement is growing. Most cultural people. But they are \ good as well. And when we are listening to others then we are needed everywhere. Listening is then accepting the other with his or her peculiarities in his \ or her environment. And then we can join. We are building in a PANDAC way. We are preparing the New ERA. We are the \ founders. And I promote everyone who like to become active ed van der meulen ameulen@users.berlios.de === Subject: Re: the New ERA is at hand, we are preparing it. > And I am building on papers from famous scientists and we have > forgotten those papers. While they were so right. Don't forget those great physicists who can do an infinite number of things between eating breakfast and going to work! === Subject: Re: the New ERA is at hand, we are preparing it. I am not against the notion infinity Torkel I ally tell ther is also \ disctrete mathematics. Taht is very productive. Tell me please Torkel what is \ your background. Infinty tools are often for ananlyzing things. So they are not bad. And \ discrete tools are for simulating and teaching. Also not bad and the have to \ work together. You stress infinity. That is special. have a good day ed === Subject: differential equation posting-account=AadtSA0AAACGiBbjZIh9MUmRCuFemK4V I have conducted a shear stress vs. strain test on a sample. The strain was at a constant rate i.e. dStrain/dt = const. I belive my material is deforming in a way that can be modeled by a mass, damper, strain-softening spring, where the strain softening spring is of the form Ko*Sin(Je^-D*strain) ie. the equation is of the form: Md2q/dStrain2 + Cdq/dStrain + Ko*Sin(Je^-D*strain)q = F(strain) where F(strain) is such that dStrain/dt = Constant second order, linear, non-homogenous, non-autonomous equation with one varying coefficient (the spring). My question is this: How do I solve this equation? Is an analytical solution possible? or if not, given that I have data available to curve fit, how would I do a numerical solution? Paul === Subject: Re: differential equation >I have conducted a shear stress vs. strain test on a sample. The >strain was at a constant rate i.e. dStrain/dt = const. >I belive my material is deforming in a way that can be modeled by a >mass, damper, strain-softening spring, where the strain softening >spring is of the form Ko*Sin(Je^-D*strain) I'm not sure how to parse that, but I suppose you mean Ko sin(J exp(-D \ strain)) >ie. the equation is of the form: >Md2q/dStrain2 + Cdq/dStrain + Ko*Sin(Je^-D*strain)q = F(strain) >where F(strain) is such that dStrain/dt = Constant In slightly more pleasant notation, M q'' + C q' + K sin(J exp(-D x)) q = F(x) where the independent variable is x (= strain). If dx/dt = constant, it doesn't matter whether you consider x or t as the independent variable (of course it changes the constants). >second order, linear, non-homogenous, non-autonomous equation with one >varying coefficient (the spring). >My question is this: How do I solve this equation? Is an analytical >solution possible? or if not, given that I have data available to >curve fit, how would I do a numerical solution? To curve fit what? Which of the constants and the function F(x) are known, and which are unknown? If F is unknown, you should note that for any twice-differentiable q(x) and any values of the constants, you can get an F that works (just make it equal to the left side of the equation after plugging in q(x)). If you have numerical values for the constants and a particular function F(x), you \do\ a numerical solution by entering the differential equation \ and initial conditions in to a numerical differential equation solver in your favourite mathematical software package (e.g. dsolve(..., numeric) in Maple). The homogeneous equation M q'' + C q' + K sin(J exp(-D x)) q = 0 probably doesn't have closed-form solutions. Given two linearly independent \ solutions of the homogeneous equation, you can use variation of parameters to express \ the general solution of the non-homogeneous equation in terms of integrals. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: RE: Which book on calculus? posting-account=KBDKKw0AAABybm0yocoQfeeytV5xjywG This is a response to an old thread, entitled \Which book on calculus?\ what about G.H. Hardy's book, A Course of Pure Mathematics? It should be available freely from most public libraries of any worth. And because it is so old, you can usually find at least an early edition not checked-out. If you are lucky, maybe you cant find an OCRd version on the web. >Leonardus Hutabarat Sep 20 2002, 4:18 am show options this >author >Local: Fri,Sep 20 2002 4:18 am === >Subject: Which book on calculus? >Reply to Author | Forward | Print | Individual Message | Show original | Report >Abuse >Anyone got any idea as to good book on calculus? Can I learn calculus >in the internet? Is there a web site to learn calculus? === Subject: Harmonic Analysis questions (1) I'm going through Katznelson's book and a few assertions are left to the reader without proof - which makes the book rather tiring to read at first reading, but it's really a good book -. For some of them references are given - so I suppose their proofs are not trivial -, and for others no reference is given. I will write the assertions below, and would appreciate some help (if one of the assertions is easy to prove, knowing I still have weak complex analysis background, you can just mention it so I can try to think about it significantly). n denotes an integer. 1¡) The existence of y such that lim (h->0) 1/h*int(abs[(f(t0+u)+f(t0-u))/2 - y], u = 0 .. h), where f is in L1[T] and \ T = R/(2PiZ), is satisfied for almost every t0 in T. Reference: Zygmund, Trigonometric series. Note that the weaker assertion with the integral inside the abs(.) is easy to prove (and it's done later in the book). 2¡) There are examples of trigonometric series converging \ to 0 almost everywhere without being identically zero. 3¡) A trigonometric converging to 0 everywhere is \ identically 0. Reference: Ensembles parfaits et s\.8eries trigonom\.8etriques, \ Kahane/Salem. 4¡) In order to prove Lebesgue's theorem with the Poisson's \ summability kernel, all we need is the existence of y such that: int([f(t0+u)+f(t0-u)]/2 - y, u = 0 .. h) = o(h), which is weaker than \ 1¡) (1¡) being necessary in the proof of Lebesgue's theorem \ with the Fejer kernel). I could not complete the proof correctly with such a weaker hypothesis. 5¡) A sufficient condition for f to be continuous on T is \ that sum(fourrier_coeff_f(n)) < +oo, and a necessary condition is that sum(fourrier_coeff(n)^2) < +oo (I suspect those assertions are easy to prove, if someone can confirm that, I will try them as an exercise). 6¡) If f has bounded variations on T, then: abs(fourrier_coeff_f(n)) <= var(f)/(2*Pi*abs(n)). The book uses Stieltjes integral doing an integration by parts (it takes half a line) but I don't understand the process (I'm even not so familiar with those integrals, just know how to construct them). -- Julien Santini === Subject: Re: Harmonic Analysis questions (1) On Fri, 6 May 2005 17:14:53 +0200, \Julien Santini\ >I'm going through Katznelson's book and a few assertions are left to the >reader without proof - which makes the book rather tiring to read at first >reading, but it's really a good book -. For some of them references are >given - so I suppose their proofs are not trivial -, and for others no >reference is given. I will write the assertions below, and would \ appreciate >some help (if one of the assertions is easy to prove, knowing I still have >weak complex analysis background, you can just mention it so I can try to >think about it significantly). >n denotes an integer. >1¡) The existence of y such that lim (h->0) >1/h*int(abs[(f(t0+u)+f(t0-u))/2 - y], u = 0 .. h), where f is in L1[T] and \ T >= R/(2PiZ), is satisfied for almost every t0 in T. >Reference: Zygmund, Trigonometric series. >Note that the weaker assertion with the integral inside the abs(.) is easy >to prove (and it's done later in the book). You mean the weaker assertion with the abs _outside_ the integral? If you give the \right\ proof using maximal functions (see for example Folland \Real Analysis\ in the chapter on differentiation) the two proofs are more or less the same. _Or_ you can actually derive the stronger assertion from the weaker one by a trick - I think that trick is also in Folland. >2¡) There are examples of trigonometric series converging \ to 0 almost >everywhere without being identically zero. >3¡) A trigonometric converging to 0 everywhere is \ identically 0. >Reference: Ensembles parfaits et s\.8eries trigonom\.8etriques, \ Kahane/Salem. These two are certainly not trivial exercises, but they're both somewhere in Zygmund. >4¡) In order to prove Lebesgue's theorem with the \ Poisson's summability >kernel, all we need is the existence of y such that: >int([f(t0+u)+f(t0-u)]/2 - y, u = 0 .. h) = o(h), which is weaker than \ 1¡) >(1¡) being necessary in the proof of Lebesgue's theorem \ with the Fejer >kernel). I could not complete the proof correctly with such a weaker >hypothesis. Which \Lebesgue's theorem\ are we trying to prove here? >5¡) A sufficient condition for f to be continuous on T is \ that >sum(fourrier_coeff_f(n)) < +oo, and a necessary condition is that >sum(fourrier_coeff(n)^2) < +oo (I suspect those assertions are easy to >prove, if someone can confirm that, I will try them as an exercise). Assuming that you really meant sum abs(fourrier_coeff_f(n)) < +oo these are very easy exercises. (If you meant sum(fourrier_coeff_f(n)) < +oo the result is false.) >6¡) If f has bounded variations on T, then: >abs(fourrier_coeff_f(n)) <= var(f)/(2*Pi*abs(n)). The book uses Stieltjes >integral doing an integration by parts (it takes half a line) but I don't >understand the process (I'm even not so familiar with those integrals, \ just >know how to construct them). The same proof in different language (and assuming that 2 pi = 1): Since f has bounded variation there is a (complex) measure mu such that f(x) - f(0) = mu((0,x)) for almost all x (cf Folland on functions of bounded variation). Then var(f) = ||mu|| = |mu|(T), and f periodic implies that mu(T) = 0. Say f(0) = 0 to simplify the notation. Write f(x) = int_0^x d mu(t). Insert that into the formula for f^(n), apply Fubini's theorem, and then use the fact that mu(T) = 0, and you see that mu^(n) = in f^(n) (possibly with a minus sign). Since |mu^(n)| <= ||mu|| = var(f) the inequality follows. (Or later, when you know about distributions: mu is the distribution-wise derivative of f, hence mu^(n) = in f^(n).) ************************ David C. Ullrich === Subject: de Rham space of R^n minus 2 points question.... X-RFC2646: Format=Flowed; Original I am trying to compute the de Rham cohomology space of R^n minus 2 points. Please let me know if I am correct : First, let M denote R^n minus 2 \ points. Consider R^2 and suppose the points were at (-2,0) and (2,0) WLOG. Draw infinite dotted vertical lines at x = -1 and a dotted vertical line at x = 1. Now, let U denote everything to the right of x = -1 (except the point (2,0) ) and let V denote everything to the left of x = 1 (except the point (-2,0) ). Then U and V are open subsets of M, and U union V equals M (in the case n = 2). The Mayer Vietoris sequence gives an exact sequence H^(p-1) (U) (+) H^(p-1) (V) ----> H^(p-1) (U /\\ V) ---> H^p (M) ----> H^p (U) (+) H^p (V) I will calculate from left to right. Since R^n minus one point is homotopic \ to S^(n-1), the first term is R x R if p = 0 or p = n. Otherwise it is 0 if \ 0 < p < n. The next term is H^(p-1) of a star-shaped open in R^n, so it is 0. The next \ term is what we need to calculate, and the final term is now a little different than the first term. It is R x R if p = 0 or p = n-1 (since U is \ homotopic to S^(n-1) and we are dealing with p and not p-1 here). Thus, I have that the first and last term are never the same since p and p-1 \ are different integers (and since in general, H^p(S^n) = R if p = 0 or p = n, and 0 otherwise) So my sequence is : If p = n, R x R ----> 0 ----> H^p(M) ----> 0. So H^p(M) = 0. But if I recopy the sequence above and lower p by 1, I get H^(p-2) (U) (+) H^(p-2) (V) ----> H^(p-2) (U /\\ V) ---> H^(p-1) (M) ----> H^(p-1) (U) (+) H^(p-1) (V) When p = n , I now get 0 x 0 -----> 0 ----> H^(p-1) (M) ---> R x R So H^(p-1) (M) = R x R. Is this right? So I think I am getting that H^(p) (R^n minus 2 points) = R \ x R when p = n - 1. I might however be confused. Any comments are highly appreciated. James === Subject: \ A simple functional equation g(2x, y/2+1/(2x) )=g(x,y) /(1+g(x,y)) \ The last functional equations we have dealt with suggest me this one. g is R^2->R x,y real . Bon courage, Alain. === Subject: Re: \ A simple functional equation g(2x, y/2+1/(2x) )=g(x,y) /(1+g(x,y)) \ >The last functional equations we have dealt with >suggest me this one. >g is R^2->R x,y real . Changing variable to z = xy - log_2(x), we have h(x, z) = h(2x, z). So the general solution to the original equation for x > 0 is given by g(x, y) = f(log_2(x), xy - log_2(x)) where f(x, y) is periodic in x with period 1. Mike Guy === Subject: Stuck on this simple algebraic expansion derive Heron's formula. The part I am stuck on is expanding this: b^2 - [(b^2 + c^2 - a^2)^2]/4c^2 into [(a+b+c)(b+c-a)(a+b-c)(a-b+c)]/4c^2 any hints? Mike === Subject: Re: Stuck on this simple algebraic expansion > derive Heron's formula. The part I am stuck on is expanding this: > b^2 - [(b^2 + c^2 - a^2)^2]/4c^2 > into > [(a+b+c)(b+c-a)(a+b-c)(a-b+c)]/4c^2 > any hints? > Mike How about the direct approach of fully expanding both? To motivate this, you could notice what happens to the first formula if a = b + c . -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Re: Stuck on this simple algebraic expansion posting-account=9lv2eA0AAABuI_cYMyJe4ryuJyhklxho Hi Mike, Heron's formula can be proved in the following way: K=1/2*ab*sinC cosC=(a^2+b^2-c^2)/(2ab) sin^2C=1-cos^2C=1-(a^2+b^2-c^2)^2/(2ab)^2=(2ab+a^2+b^2-c^2)/(2ab)*(2ab-a^2-b\ ^2+c^2) Now, 2ab+a^2+b^2-c^2=(a+b)^2-c^2=(a+b+c)(a+b-c) 2ab-a^2-b^2+c^2=c^2-(a-b)^2=(c+a-b)(b+c-a) You should be able to finish it from here. === Subject: Re: Stuck on this simple algebraic expansion posting-account=9lv2eA0AAABuI_cYMyJe4ryuJyhklxho (small mistake, the line containing sin^2C-1cos^2C=....... should end in /(2ab) ) Sorry. === Subject: Prove that V+J=k[x]...? J=sum from i=1->n of V={polynomials f with deg in X_i < q) k is a field with q elements. k[X] is the polynomial ring in \n\ \ variables\. i am supposed to prove that V+J=k[x]. i was told that this could be done with induction for the number \k\, by \ using the following notion.given a polynomial f E k[x]: k=the sum of L_j. j=1...\number of monomials\. L_j is the sum of of the \degrees\ of the \indeterminates/variables\ \ which are >q in monomial \j\. so L_j=sum of all s_i, where s_i > q. s_i are the exponents of the monomial \ which are >q. so somehow i have to show that this is true for My company develop Scientific Letter software ( >http://www.sciletter.com/ ). Scientific Letter is an equation mailer >that allows users to create mail messages including mathematical >equations. I wish to discuss available mathematical and chemical >symbols. What additional symbols are necessary for normal dialogue by >mail? >Slava Shevtsov >http://www.sciletter.com/ - mailer with equations You have heard of TeX, .pdf files, and MicroSoft Word haven't you? The world already knows how to email mathematics files. --Lynn === Subject: Re: Math symbols in e-mail messages > My company develop Scientific Letter software ( > http://www.sciletter.com/ ). Scientific Letter is an equation mailer > that allows users to create mail messages including mathematical > equations. I wish to discuss available mathematical and chemical > symbols. What additional symbols are necessary for normal dialogue by > mail? > Slava Shevtsov > http://www.sciletter.com/ - mailer with equations This should get you started... http://www.fi.uib.no/Fysisk/Teori/KURS/WRK/TeX/symALL.html ...of course mathematics uses even more than this... -- G. A. Edgar \ http://www.math.ohio-state.edu/~edgar/ X-PGP-KEY: http://www.courier-mta.org/KEYS.bin === Subject: Re: Math symbols in e-mail messages boundary=\=_mimegpg-commodore.email-scan.com-7548-1115396949-0002\; micalg=pgp-sha1; protocol=\application/pgp-signature\ --------------------------------------------------------------------- > My company develop Scientific Letter software ( > http://www.sciletter.com/ ). Scientific Letter is an equation mailer > that allows users to create mail messages including mathematical > equations. I wish to discuss available mathematical and chemical > symbols. What additional symbols are necessary for normal dialogue by > mail? If you can express the mathematical formulas with characters from the UTF-8 \ character set (probably borrowing bits and pieces of Latin and Greek alphabets), you should be able to get something done by generating UTF-8 mail. If not, you'll probably have to generate multipart/related content with HTML \ and inline images. --------------------------------------------------------------------- -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.2.4 (GNU/Linux) iD8DBQBCe5tVx9p3GYHlUOIRAnajAJ9Et1g91bcN1PhAkktQAe0CwHGvCgCfS1gY V3yUn9BhxiS+jPDMTikCfQc= =c23Y -----END PGP SIGNATURE----- === Subject: Re: Math symbols in e-mail messages X-Newsreader-location: 1.5.6.3 (c) 'LIGHTSPEED' off line news reader for the \ Linux platform NewsFleX homepage: http://panteltje.com/panteltje/newsflex/ and ftp \ download ftp://sunsite.unc.edu/pub/linux/system/news/readers/ X-Abuse-Notes: Abuse reports must be submited via the usenetabuse.com portal \ listed above. X-Abuse-Notes2: Reports sent via any other method will not be processed. On a sunny day (6 May 2005 08:58:20 -0700) it happened slava@sciletter.com >My company develop Scientific Letter software ( >http://www.sciletter.com/ ). Scientific Letter is an equation mailer >that allows users to create mail messages including mathematical >equations. I wish to discuss available mathematical and chemical >symbols. What additional symbols are necessary for normal dialogue by >mail? What symbol do you have for a raised middle finger? === Subject: Re: Math symbols in e-mail messages <1115395700.ec53079279307c9772a01093d1dea491@teranews> posting-account=mmiamAwAAAATmtxVYi9NGy7tls83IKO3 > [...] > What symbol do you have for a raised middle finger? I don't think you can TeX that, but you should be able to Metafont it. --- Christopher Heckman === Subject: Re: Correction: Can any lattice with top and bottom be made \ complete? > Typo: > > where a meet over any (also infinite) set exists? > Should be: > where a join over any (also infinite) set exists? It is a standard result, that the existence of all meets is equivalent to the existence of all joins. > Addition: > The lattices I consider are semirings with idempotent join = + and > partially ordered by \<=\. Is the meet operation relatet to the semiring structure as well? Did you check the books on Semirings by Golan or by Hebisch/Weinert ? Marc === Subject: Re: Can any lattice with top and bottom be made complete? > I'm looking for any result in the direction, whether any lattice > (L,<=, join, meet, top, bottom) > can be extended to a lattice where a meet over any (also infinite) set > exists? No problem. For example you could consider the set Idl(L) of all ideals of L. (i.e. down-sets, closed under finite joins). This is a complete lattice and the map L ---> Idl(L) given by x -> { y in L | y <= x } is a lattice homomorphism. In fact, every system of subsets of P(L) which is stable under arbitrary intersections is a complete lattice; if it contains the principle ideals of L, you have an order embedding from L to that system, which at least preserves meets. > Could anyone direct me to any results? Any of the standard texts on lattice theory will probably do. I guess you can read German, so you might like chapter 8 of the course 'Ordnungs- und Verbandstheorie' from the FU Hagen, which is based on the Book of M.Erne. Marc === Subject: Re: Can any lattice with top and bottom be made complete? === Subject: Can any lattice with top and bottom be made complete? Will you keep the same subject thruout a thread? Will you also post a complete copy instead of patching? As it was, your posts showed up in two distince locations and it was only by chance that I matched the two together. > I'm looking for any result in the direction, whether any lattice > (L,<=, join, meet, top, bottom) > can be extended to a lattice > where a meet over any (also infinite) set exists? > Typo > where a join over any (also infinite) set exists? Yes, order embed any partial order into the collections of down sets, ie lower sets. This is meet, ie intersection, complete. Then use the standard construction for the join of a meet complete order. cf, description of this process below applied to algebra. > The lattices I consider are semirings with idempotent join = + Huh? Aren't joins and meets idempotent? > and partially ordered by \<=\. Do you mean by subset inclusion? > Could anyone direct me to any results? It is standard construction, for example within an algebraic variety, for the subset ordered lattice of sub-algebra, which are infinite intersection, ie meet closed, for the infinite join to be the meet of all the sub-algebra that include all the elements of sub-algebras of join. That is the join of a collection of sub-algebras is the sub-algebra generated by the collection. ---- === Subject: Re: Can any lattice with top and bottom be made complete? >[...] >> I'm looking for any result in the direction, whether any lattice >> (L,<=, join, meet, top, bottom) >> can be extended to a lattice >> where a meet over any (also infinite) set exists? >> Typo >> where a join over any (also infinite) set exists? > Yes, order embed any partial order into the collections of down sets, ie > lower sets. This is meet, ie intersection, complete. Then use the > standard construction for the join of a meet complete order. cf, > description of this process below applied to algebra. disadvantage that it does not preserve the finite joins that already exist in L. For example, the for element lattice L 1 / \\ a b \\ / 0 yields the down-sets L | {a,b} / \\ {a} {b} \\ / 0 It is better to work with lattice ideals instead. Marc === Subject: On the non countability of the measurable real functions image \ set. posting-account=Sj5OqQ0AAADnmKQCmz7UhvJgjMGLNk0Z let's define the following limit function f: E -->/R+ (= R+ U {+oO}): f(x)= lim_n fn(x) (E and /R+ have the required topology such that the lim symbol is well defined) with fn(x)= sum_{i:1 to n} a_in 1_Ain(x): * Ain and Ajn disjoint subsets of /R+ for any i,j such that i=/=j. * 1_Ain(x): function E --> /R+ where 1_Ain(x)=1 if x in Ain otherwise 0. * a_in are positive real numbers We have for any n, fn(E) is a countable subset of /R+ (at most n different values). Question: how to demonstrate that f(E) may be a non countable subset of /R+? Seratend. P.S. We can consider the case where we replace /R+ by /R (this point is not really important I think). === Subject: Re: On the non countability of the measurable real functions image \ set. > let's define the following limit function f: E -->/R+ (= R+ U {+oO}): > f(x)= lim_n fn(x) > (E and /R+ have the required topology such that the lim symbol is well > defined) > with fn(x)= sum_{i:1 to n} a_in 1_Ain(x): > * Ain and Ajn disjoint subsets of /R+ for any i,j such that i=/=j. > * 1_Ain(x): function E --> /R+ where 1_Ain(x)=1 if x in Ain otherwise > 0. > * a_in are positive real numbers > We have for any n, fn(E) is a countable subset of /R+ (at most n > different values). > Question: how to demonstrate that f(E) may be a non countable subset of > /R+? > Seratend. > P.S. We can consider the case where we replace /R+ by /R (this point is > not really important I think). Hint: Back in calculus you learned how to approximate a continuous function by step functions. === Subject: Re: On the non countability of the measurable real functions image \ set. posting-account=Sj5OqQ0AAADnmKQCmz7UhvJgjMGLNk0Z I mean P.S. We can consider the case where we replace /R+ by R (this point is not really important I think). === Subject: declare Yesterday I had post a message which the title was \An interesting problem \ of algebra \ I feel very sorry I made a mistake! The orginal problem is A set of matrix M defined :M is a set of matrix such that any A,B belong to \ M satisfy (AB)^3=BA then the two arbitrary element of set M is commutative! === Subject: Re: declare posting-account=SAlCkQwAAADOQfb8GuNFkWcQtC01OYCg It doesn't hold. Just take: / 1 0 \\ A =( ) \\ 0 0 / and / 0 1 \\ B =( ) \\ 0 1 / It is easy to compute that: / 0 1 \\ A.B =( )!=0 \\ 0 0 / and B.A=0. (I.e. these two matrices do not commute.) But (AB).(AB)=0 and hence also (AB)^3=0 => (AB)^3=BA. But if you assume moreover that I is in M (I stands for identity matrix) and M is closed under multiplication then by putting B:=I we get: (1) A^3=A (for all A in M) And by considering (1) for the matrix A.B we get that A.B=(A.B)^3=B.A. Maybe in the original post there were more assumptions but I couldn't find it. Martin === Subject: Re: how to generate random process of any abitary power density \ function? posting-account=vfiyOQ0AAADmkm9Qpytqg7PDZYh7k9f9 I apologize, I meant what you have said. T. === Subject: Re: New postage stamp honors von Neumann > von Neumann is >the most problematic of the group for me, mostly because he seems to have >been nastier than his brilliance would excuse. I don't think brilliance ever excuses any amount of nastiness. On the other hand, nastiness doesn't cancel out brilliance. They are on the stamps for their scientific achievements, not for being nice people. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: New postage stamp honors von Neumann >> von Neumann is >>the most problematic of the group for me, mostly because he seems to have \ >>been nastier than his brilliance would excuse. >I don't think brilliance ever excuses any amount of nastiness. On the >other hand, nastiness doesn't cancel out brilliance. They are on the >stamps for their scientific achievements, not for being nice people. Was von Neumann nasty? I did not get that impression at all in my brief association with him. Now Norbert Wiener was quite nasty. -- 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: New postage stamp honors von Neumann posting-account=EH2x8QsAAABu84CuyjstkC4nRyQ1ZHKW may be a follow-up could be a stamp for the Philadelphia Experiment, ha-ha. --Martha? http://tarpley.net/bush12.htm http://larouchepub.com http://members.tripod.com/american_almanac === Subject: Re: New postage stamp honors von Neumann X-RFC2646: Format=Flowed; Original >> von Neumann is >>the most problematic of the group for me, mostly because he seems to have >>been nastier than his brilliance would excuse. > I don't think brilliance ever excuses any amount of nastiness. On the > other hand, nastiness doesn't cancel out brilliance. They are on the > stamps for their scientific achievements, not for being nice people. Well, the other part of that is that the things he did that I know about stopped after he left them (many of them continued after a gap, when others \ picked up new angles or came back to them from a different direction). I attribute this to reluctance of people to try to get coaching from him as a \ pioneer, or to tread on anything he might still consider his turf. I'm thinking in particular about game theory, the theory of self-reproducing automata and his approaches to the theory of computation and to computer engineering. A new beginning, followed by 5-40 years of avoidance, seems close to a wash. === Subject: Re: New postage stamp honors von Neumann \MrPepper11\ > ASSOCIATED PRESS / May 4, 2005 > New Postage Stamps Honor Four Scientists > WASHINGTON (AP) -- The post office turned its attention to science > Wednesday, issuing four new stamps honoring pioneering American > scientists. > \These are some of the greatest scientists of our time; their > pioneering discoveries still influence our lives today,\ John F. Walsh, > a member of the U.S. Postal Service's board of governors, said in a > statement. > Featured on the 37-cent stamps: > -- Josiah Willard Gibbs, who lived from 1839 to 1903, was a pioneer in > the study of vector analysis, electromagnetic theory, statistical > analysis and thermodynamics. He earned the first doctorate in > engineering to be conferred in the United States. He taught at Yale > University and was the author of several books and scientific papers. > -- Barbara McClintock won the 1983 Nobel Prize in medicine for her > discoveries in genetics. She was among the first scientists to study > the way genetic material controls the development of an organism. > -- John von Neumann was one of the top mathematicians of the 20th > century. He helped develop a machine that became a model for modern > computers, worked with Albert Einstein at the Institute for Advanced > Study and was a consultant in the project to build the first atomic > bomb. > -- Richard P. Feynman won the Nobel Prize in physics in 1965 for work > in quantum electrodynamics. His work included diagrams that help > First day of issue ceremonies were being held at Yale University in New > Haven, Conn., with the 37-cent stamps going on sale nationwide on > Thursday. > ^------ > On the Net: > U.S. Postal Service: http://www.usps.com This 1983 Soviet stamp carries a portrait of Muhammad al-Khwarizmi, from whose name the words \algorism\ and (later) \algorithm\ descend: http://www3.telus.net/ldh/khwarizmi.jpg Nice to see that J.W.Gibbs is still remembered. === Subject: Re: New postage stamp honors von Neumann posting-account=v4vmWA0AAABbhj-NDihq94ZXLn1e-fBh Although a physicist, Gibbs is certainly remembered by the U.S. mathematical community. The Josiah Willard Gibbs lecture is a highlight of the annual Joint Mathematical meetings, and many of the lecturers have been non-mathematicians with important mathematical insights, for example Roger Penrose a few years ago. Actually, the worst Gibbs lecturer I ever heard was David Mumford, who ignored the instruction that the talk be accessible and lost 90% of the audience within five minutes. It's interesting that of the four scientists, three made at least tangential contributions to mathematics. === Subject: Re: New postage stamp honors von Neumann <4279a040$0$16226$bb4e3ad8@newscene.com> \ > > Upon correspondence with US Treasury, I was told that the penny could > > not be removed from US coinage because that would cause inflation. > LOL. A nickel buys less today than a penny did in the 1950s and > inflation was lower then. What? How dare you laugh out loud about rationing of government brains! You are not true believing patriot. === Subject: How to compute the inverse of Toeplitz Hermitian Matrix I would like to compute the inverse of Toeplitz Hermitian matrix such as row 1: a_0 a_1 a_2 ... a_(M-1) row 2: a*_1 a_0 a_1 ... a_(M-2) row 3: a*_2 a*_1 a_0 ... a_(M-3) row M-1: a*_(M-2) a*_(M-3) ... a_0 a_1 row M: a* _(M-1) a*_(M-2) ... a*_1 a_0 Does a closed form inverse of this matrix exist? If not, is there any algorithm to compute it? === Subject: Re: How to compute the inverse of Toeplitz Hermitian Matrix X-RFC2646: Format=Flowed; Original >I would like to compute the inverse of Toeplitz Hermitian matrix such as > row 1: a_0 a_1 a_2 ... a_(M-1) > row 2: a*_1 a_0 a_1 ... a_(M-2) > row 3: a*_2 a*_1 a_0 ... a_(M-3) > row M-1: a*_(M-2) a*_(M-3) ... a_0 a_1 > row M: a* _(M-1) a*_(M-2) ... a*_1 a_0 > Does a closed form inverse of this matrix exist? > If not, is there any algorithm to compute it? The algorithm of W F Trench used to be state of the art but this paper is more recent: === Subject: Re: How to compute the inverse of Toeplitz Hermitian Matrix >I would like to compute the inverse of Toeplitz Hermitian matrix such as >row 1: a_0 a_1 a_2 ... a_(M-1) >row 2: a*_1 a_0 a_1 ... a_(M-2) >row 3: a*_2 a*_1 a_0 ... a_(M-3) >row M-1: a*_(M-2) a*_(M-3) ... a_0 a_1 >row M: a* _(M-1) a*_(M-2) ... a*_1 a_0 >Does a closed form inverse of this matrix exist? No. >If not, is there any algorithm to compute it? The usual Cholesky decomposition will work. However, there are ways of doing it in O(M^2) operaations. The book by Golub and Van Loan is good on problems like this. -- 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: \ =?iso-8859-1?q?Good_morning_or_good_evening_depending_upon_your_location._I_w\ ant_to_ask_you_the_most_important_question_of_your_life._Your_joy_or_sorrow_f\ or_all_eternity_depends_upon_your_answer._The_question_is:_Are_you_saved=3F_I\ t_is_not_a_question_of_how_good_you_are,_nor_if_you_are_a_church_member,_but_\ are_you_saved=3F_Are_you_sure_you_will_go_to_Heaven_when_you_die=3F_GOOGLE=B7\ NEWSGROUP=B7POST=B7156?= posting-account=AugKOA0AAAC7mkWFGvSTFu6997IBLHCp This is the most important question of your life. The question is: Are you saved? It is not a question of how good you are, nor if you are a church member, but are you saved? Are you sure you will go to Heaven when you die? The reason some people don't know for sure if they are going to Heaven when they die is because they just don't know. The good news is that you can know for sure that you are going to Heaven. The Holy Bible describes Heaven as a beautiful place with no death, sorrow, sickness or pain. God tells us in the Holy Bible how simple it is to be saved so that we can live forever with Him in Heaven. \For if you confess with your mouth Jesus is Lord and believe in your heart that God raised Him from the dead, you WILL BE SAVED.\ (Romans 10:9) Over 2000 years ago God came from Heaven to earth in the person of Jesus Christ to shed His blood and die on a cross to pay our sin debt in full. Jesus Christ was born in Israel supernaturally to a virgin Jewish woman named Mary and lived a sinless life for thirty-three years. At the age of thirty-three Jesus was scourged and had a crown of thorns pressed onto His head then Jesus was crucified. Three days after Jesus died on a cross and was placed in a grave Jesus rose from the dead as Jesus said would happen before Jesus died. If someone tells you that they are going to die and then three days later come back to life and it actually happens then this person must be the real deal. Jesus Christ is the only person that ever lived a perfect sinless life. This is why Jesus is able to cover our sins(misdeeds) with His own blood because Jesus is sinless. The Holy Bible says, \In Him(Jesus) we have redemption through His blood, the forgiveness of sins...\ (Ephesians 1:7) If you would like God to forgive you of your past, present and future sins just ask Jesus Christ to be your Lord and Saviour. It doesn't matter how old you are or how many bad things that you have done in your life including lying and stealing all the way up to murder. Just pray the prayer below with your mouth and mean it from your heart and God will hear you and save you. have a home in Heaven with You when I die. I agree with You that I am a sinner. I believe that You love me and want to save me. I believe that You bled and died on the cross to pay the penalty for my sins and that You rose from the dead. Please forgive my sins and come into my heart and be my Lord and Saviour. me through Your merciful grace. Amen. Welcome to the family of God if you just allowed God to save you. Now you are a real Christian and you can know for sure that you will live in Heaven forever when this life comes to an end. As a child of God we are to avoid sin(wrongdoing), but if you do sin the Holy Bible says, \My dear children, I write this to you so that you will not sin. But if anybody does sin, we have one who speaks to the Father in our defense Jesus Christ, the Righteous One.\ Those of you that have not yet decided to place your trust in the Lord Jesus Christ may never get another chance to do so because you do not know when you will die. Jesus said, \I am the way, the truth and the life: no one can come to the Father(God)(in Heaven), but by me.\ (John 14:6) This means that if you die without trusting in Jesus Christ as your Lord and Saviour you will die in your sins and be forever separated from the love of God in a place called Hell. The Holy Bible descibes Hell as a place of eternal torment, suffering, pain and agony for all those who have rejected Jesus Christ. The good news is that you can avoid Hell by allowing Jesus Christ to save you today. Only then will you have true peace in your life knowing that no matter what happens you are on your way to Heaven. Praise the Lord! Servant of the Lord Jesus Christ Ronald L. Grossi * Show this to your family and friends so they can also be saved. * This message may get deleted so you may want to print a copy. * Just press the [Ctrl][P] keys on your keyboard to print this page. `¡\.bcá...á\.bc¡`\[DownExc\ lamation]\.bcá...á\.bc¡`¡\2\ 74á.º..á\.bc¡`¡\.bc\[AAc\ ute]...á\.bc¡`¡\.bcá..\[AAc\ ute]\.bc¡` Got Questions? http://www.gotquestions.org/archive.html Other Languages http://www.godssimpleplan.org/gsps.html Free Movie: To Hell and Back http://www.tbn.org/index.php/8/1.html Animation http://www.gieson.com/Library/projects/animations/walk/index.html The Passion Of The Christ http://www.thepassionofthechrist.com Beware Of Cults http://www.carm.org/cults/cultlist.htm About Hell http://www.equip.org/free/DH198.htm Is Jesus God? http://www.powertochange.com/questions/qna2.html Free Online Bible http://www.biblegateway.com `¡\.bcá...á\.bc¡`\[DownExc\ lamation]\.bcá...á\.bc¡`¡\2\ 74á.º..á\.bc¡`¡\.bc\[AAc\ ute]...á\.bc¡`¡\.bcá..\[AAc\ ute]\.bc¡` === Subject: Re: When I was studying about Mobius transformation.... <5660989.1115108841499.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=tXLCfg0AAABZr7Br-ZfAljtQnu4aB2xD Sorry Alain. I wasn't at home for several days, and so I couldn't use internet. My question was in the complex plane but yours were in the real coefficients. And notations are a little bit complicate to read at the first reading However if I read it carefully again, I think I can get the right ideas to solve my question. By the way, I solved my question by myself using induction. It was the simple exercise about the induction. Anyway thank you so much your reply and help with all my heart... agentmath. === Subject: Re: Courage? On Thu, 05 May 2005 17:34:20 -0400, Robert Kolker >> Is that anything like having to say numbers of what in order to >> analyze numbers? >No. But when talking of sets it is often necessary to say what the >cardinality of the set under discussion is. Cardinality pertains to >sets, not to elementary individuals which are not sets. Horseshit. Cardinality has a long history of application to things such as the cardinal integers other than sets. === Subject: Re: Courage? On Thu, 05 May 2005 17:32:45 -0400, Robert Kolker >> If tautologies are always true then anything outside tautologies are >> always false. Ergo problematic empirical observations which can be >> true can only be true in tautological terms. QED. >Wrong. A non-tautology may be sometimes true or sometimes false >depending on the truth value of the elemental propositions that make up >the formula. For example: p .and. q is true when p and q are true and >false otherwise. Empirical propositions are contingently true. For >example: The United States has fifty states. This is currently true. It >was not always true. Back in 1947 the United States had forty eight \ states. Well perhaps you would be good enough to point out how something outside a tautology can be true other than merely asserting its truth. === Subject: Re: Courage? <427124bd.77027148@netnews.att.net> <42717466_1@newsfeed.slurp.net> <4272780f.1910249@netnews.att.net> <4276827d$1_4@newsfeed.slurp.net> <42769a32.51174051@netnews.att.net> <4276b62a$1_5@newsfeed.slurp.net> <4277bc04.62464519@netnews.att.net> <4277d25d$1_1@newsfeed.slurp.net> <4277f3bd.76732260@netnews.att.net> <427817b8$1_1@newsfeed.slurp.net> <4278edf8.84801655@netnews.att.net> <1lq7f218oz4jf$.1em9dnykdvl4y$.dlg@40tude.net> <42794459.95278474@netnews.att.net> <1odrwbzmv2rs8$.oudwceyi21o9$.dlg@40tude.net> <427a3d79.104002676@netnews.att.net> <2JWdnQ2D-ZzY3OffRVn-og@comcast.com> <427a87d3.109099213@netnews.att.net> <427ba770.891235@netnews.att.net> posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs > Well perhaps you would be good enough to point out how something > outside a tautology can be true other than merely asserting its truth. There is a cup of coffee on my desk. This is true. You don't need me to just assert it, other observers can walk by and make the same observation. This is not a tautology. Though some observers might believe otherwise, there actually are sizable intervals when there is NOT a cup of coffee on my desk. For instance, when I am in a meeting and have taken my coffee mug with me. - Randy === Subject: Re: Courage? On 6 May 2005 10:53:06 -0700, \Randy Poe\ in >> Well perhaps you would be good enough to point out how something >> outside a tautology can be true other than merely asserting its >truth. >There is a cup of coffee on my desk. This is true. So you're saying tautologies don't encompass all possibilities for truth? >You don't need me to just assert it, other observers >can walk by and make the same observation. They don't even need to walk by to make the same observation. >This is not a tautology. Though some observers might >believe otherwise, there actually are sizable intervals >when there is NOT a cup of coffee on my desk. For >instance, when I am in a meeting and have taken my >coffee mug with me. Well as fascinating as your caffeine habits might appear to some they don't really address the question of how something outside a tautology can be true if tautologies exhaust all possibilities for truth. Unless of course tautologies don't exhaust all possibilities for truth in which case it's curious that tautologies are considered always true and it remains to be seen exactly which truths tautologies are not equipped to handle. Bob seems to think facts are true but not part of tautologies. But then Bob thinks he can integrate points into lines. === Subject: Re: Courage? <4276827d$1_4@newsfeed.slurp.net> <42769a32.51174051@netnews.att.net> <4276b62a$1_5@newsfeed.slurp.net> <4277bc04.62464519@netnews.att.net> <4277d25d$1_1@newsfeed.slurp.net> <4277f3bd.76732260@netnews.att.net> <427817b8$1_1@newsfeed.slurp.net> <4278edf8.84801655@netnews.att.net> <1lq7f218oz4jf$.1em9dnykdvl4y$.dlg@40tude.net> <42794459.95278474@netnews.att.net> <1odrwbzmv2rs8$.oudwceyi21o9$.dlg@40tude.net> <427a3d79.104002676@netnews.att.net> <2JWdnQ2D-ZzY3OffRVn-og@comcast.com> <427a87d3.109099213@netnews.att.net> <427ba770.891235@netnews.att.net> <427bda4d.10216155@netnews.att.net> posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs > On 6 May 2005 10:53:06 -0700, \Randy Poe\ in > >There is a cup of coffee on my desk. This is true. > So you're saying tautologies don't encompass all possibilities for > truth? Yes, since \there is a cup of coffee on my desk\ is true, but not tautological. - Randy === Subject: Re: Courage? > Well perhaps you would be good enough to point out how something > outside a tautology can be true other than merely asserting its truth. A statement may be true because it asserts a fact, not because it is true in every possible world. Thus my example about the number states. True now, false at another time etc etc. Tautologies are -always- true, which is why they assert nothing about the world. Bob Kolker === Subject: Re: Courage? On Fri, 06 May 2005 13:48:54 -0400, Robert Kolker >> Well perhaps you would be good enough to point out how something >> outside a tautology can be true other than merely asserting its truth. >A statement may be true because it asserts a fact, not because it is >true in every possible world. So facts are not true. You are indeed a true fount of wisdom. Not in all possible worlds perhaps, but in your own little island unto itself. > Thus my example about the number \ states. >True now, false at another time etc etc. Tautologies are -always- true, >which is why they assert nothing about the world. Yes, but the burning question of the moment is whether those things which were true then were actually true then. There are no time dependencies in tautologies that I'm aware of. So you're saying that for any tautology t:[subject][not subject] we need to recast it along the lines of t:[subject][not subject]f(t) to accommodate facts? === Subject: Re: Courage? On Thu, 05 May 2005 17:24:17 -0400, Robert Kolker >> Exactly part of my original comment. Interesting to speculate on the >> nature of that missing operand. Might be a mind, mightn't it. >Do you know what an operator or an operand is in the mathematical sense? >Apparently you don't. One more indication of your abyssimal ignorance of >mathematics. >Let me give you an illustration: let * mean numerical multiplication. >What does the expression *b*c mean where b,c are numbers. Answer: >nothing. It is an ill formed expression. An expression with two >multiplications in it would have to look like a*(b*c) or (a*b)*c or >a*b*c assuming the associative law. You will notice two operator >instances, three operands. An operarand is what an operator operates >upon. * is a binary operator and it acts on two operands as in a*b. So what are you saying, that logical operations have no conjunction to anthing but the specified operand. === Subject: Re: Courage? <10y3736s524lr$.qbugiyw2prs3$.dlg@40tude.net> <427a3f8d.104534641@netnews.att.net> <2JWdnQ6D-ZyO3-ffRVn-og@comcast.com> <427a8a5c.109748242@netnews.att.net> posting-account=jEbKGQ0AAADZF1UpkDsHa5gkWBqABnUE > On Thu, 05 May 2005 11:55:56 -0400, Robert Kolker > >> > >> Jesus, Bob, infinity just means undefined. A lot less than 22 words. > >Wrong. Infinite of cardinals is well defined. When you talk about > >infinity you have to say infinity of what. > Is that anything like having to say numbers of what in order to > analyze numbers? It is more like having to say volumes of what in order to analyze volumes. === Subject: Re: Courage? On 6 May 2005 08:18:45 -0700, \guenther vonKnakspot\ >> On Thu, 05 May 2005 11:55:56 -0400, Robert Kolker >> > >> >> >> >> Jesus, Bob, infinity just means undefined. A lot less than 22 >words. >> > >> >Wrong. Infinite of cardinals is well defined. When you talk about >> >infinity you have to say infinity of what. >> Is that anything like having to say numbers of what in order to >> analyze numbers? >It is more like having to say volumes of what in order to analyze >volumes. Of course. You've finally said something relevant to the subject at hand. === Subject: Re: Courage? <10y3736s524lr$.qbugiyw2prs3$.dlg@40tude.net> <1s59e4m7nc0wc.duikpv3fminl$.dlg@40tude.net> <427a4177.105024132@netnews.att.net> <427a8d21.110458027@netnews.att.net> posting-account=jEbKGQ0AAADZF1UpkDsHa5gkWBqABnUE > On 5 May 2005 10:25:19 -0700, \guenther vonKnakspot\ > >> > >> > > >> > Yea, Zick, and more importantly only if you can grasp finitude. You > >are > >> > being boringly byzantine. Next you'll be talking about angels and > >> > pinheads. > >> > >> How many angels can dance on Lester's head? > >> > >> Bob Kolker > >Beats me. Must be the low quality of the pencil shavings I buy. > Or the fact that you smoke them. Of course, they are to dry to drink and to coarse to snort === Subject: Re: Courage? <10y3736s524lr$.qbugiyw2prs3$.dlg@40tude.net> <1s59e4m7nc0wc.duikpv3fminl$.dlg@40tude.net> <427a4177.105024132@netnews.att.net> <427a8bc8.110112377@netnews.att.net> posting-account=jEbKGQ0AAADZF1UpkDsHa5gkWBqABnUE > On 5 May 2005 09:08:42 -0700, \guenther vonKnakspot\ > >> On Thu, 05 May 2005 10:40:40 -0400, Robert Kolker > >> > >> > > >> >> (just for the record, I personally meant to make no statement to > >wether > >> >> the human mind can actually experience infinity) > >> > > >> >If the human mind can grasp finitude, it can grasp infinitude by > >simple > >> >negation. > >> > >> Only if one can grasp negation first. > >> > >Yea, Zick, and more importantly only if you can grasp finitude. You are > >being boringly byzantine. Next you'll be talking about angels and > >pinheads. > And spoil your conversation, pinhead? Not hardly. You know Zick that I truly appreciate your civility. === Subject: Re: Courage? >>On 5 May 2005 09:08:42 -0700, \guenther vonKnakspot\ >>On Thu, 05 May 2005 10:40:40 -0400, Robert Kolker >> >> > > >>(just for the record, I personally meant to make no statement > to >wether >>the human mind can actually experience infinity) > >If the human mind can grasp finitude, it can grasp infinitude by >simple >negation. >> >>Only if one can grasp negation first. >> >Yea, Zick, and more importantly only if you can grasp finitude. You > are >being boringly byzantine. Next you'll be talking about angels and >pinheads. >>And spoil your conversation, pinhead? Not hardly. > You know Zick that I truly appreciate your civility. It seems to be, on average, at least on a par with your's. Unless, you perceive your remarks above as not intentionally insulting simply because they are not peppered with curse words. Tell me, is a pickpocket who works wearing gloves innocent of theft? -- \...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them.\ Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Courage? <10y3736s524lr$.qbugiyw2prs3$.dlg@40tude.net> <1s59e4m7nc0wc.duikpv3fminl$.dlg@40tude.net> <427a4177.105024132@netnews.att.net> <427a8bc8.110112377@netnews.att.net> posting-account=jEbKGQ0AAADZF1UpkDsHa5gkWBqABnUE > >>On 5 May 2005 09:08:42 -0700, \guenther vonKnakspot\ > >> > >> > > > >>On Thu, 05 May 2005 10:40:40 -0400, Robert Kolker > >> > >> > > > > > >>(just for the record, I personally meant to make no statement > > to > >wether > > > >>the human mind can actually experience infinity) > > > >If the human mind can grasp finitude, it can grasp infinitude by > > > >simple > > > >negation. > >> > >>Only if one can grasp negation first. > >> > > > >Yea, Zick, and more importantly only if you can grasp finitude. You > > are > >being boringly byzantine. Next you'll be talking about angels and > >pinheads. > >> > >>And spoil your conversation, pinhead? Not hardly. > > You know Zick that I truly appreciate your civility. > It seems to be, on average, at least on a par with your's. Hey, I was honestly responding to Zick's concern about the quality of my conversation and avoiding to be upset by the term \pinhead\ which is quite a funny pun in the context. > Unless, you perceive your remarks above as not intentionally > insulting simply because they are not peppered with curse words. > Tell me, is a pickpocket who works wearing gloves innocent of > theft? I should be quite offended if you are implying that I am unaware of how insulting my posts can be. === Subject: Re: Courage? <1115003931.528589e77a7bcd768e5fcdf677333b7d@teranews> <1uj2dr4skbzk7$.1l5yt0jp1kxfu.dlg@40tude.net> <15t915tfg6cr1.4c3paiuaoj23$.dlg@40tude.net> <1w7s7s1ga6shx$.1rhqy5k6ux9xf.dlg@40tude.net> <1a4unxdlm1673$.qfp4h4tghugj.dlg@40tude.net> posting-account=jEbKGQ0AAADZF1UpkDsHa5gkWBqABnUE > >> > > > > > > > >> > >> > > > > > > > >>You're simply a contentious asshole. > > > >Ran out of arguments so soon? > > > >> > >>No. You simply have not seriously replied to my previous > >>arguments. Why throw pearls before swine? > > > > > >Since there was no argument, I could scarcely reply in any way. > > Until > >you provide an actual logical derivation which shows the > >contradictions, you simply have *no* arguments, just claims. > > > >> > >>Yes, on reconsideration, you are correct. I now believe that > >>there is no way to make such an argument, because there are no > >>allowed words to even describe what I had hoped to refute, which > >>was that measurable points that take up space are inconsistent > >>with Hilbert's axioms. It was, however, made clear to me that > >>the rules of inference did not even allow talking of such > >>objects, and without the ability to talk of such objects and > >>their properties (length, diameter, inches, etc.), it is > > Unless you previously define them. > I suppose I am confused as to who the burden of defining belongs. > Does it belong to the one who postulates 5\ diameter points, > without defining '5', 'inches' or 'diameter'? Or does it fall on > the one who disagrees to define what the first person claimed but > failed to define? > > That is why I was entreating you to > > try and make a proof using the axioms with the word 'points' > > substituted for twinkies, the absurdity of it would have made it clear > > that the relationships established by the axioms hold regardless of the > > flavour or creaminess of fillings. > Sorry, Guenther. I'm just to scattered in my thinking to make > such a silly substitution. > > If you wish to talk about points > > with diameter in a meaningfull way, you have to define the meaning of > > diameter within the context of the discussion. > Oh, I see. So the burden of definition falls on the one who > wishes to disprove the claim of another, even when that other was > the one who introduced 5\ diameter points. Well, I freely admit > that I am totally baffled as to how this can ever lead to a > consensus. For I have no way of knowing whether or not my > definitions of '5', 'inches', or 'diameter' are what the proposer > intended, in which case there is simply no possible way to prove > consistency or inconsistency. > > If you had added the > > pertinent postulates then you could have talked of the properties you > > had defined. And then you may have come to the conclusion that the > > system proced by Hilbert's and your Axioms is inconsistent. > But that is not the same as proving Hilbert's system alone > inconsistent. I was not interested in proving a new system > inconsistent with itself. The claim was made to me that 5\ > diameter points were not inconsistent with Hilbert's system. This whole matter of the 5'' points is very unfortunate. Without postulates that introduce formally new features into the system, the system remains unchanged wether you are talking of hickory smoked , honey roasted or square points. As soon as you introduce a new postulate, you have to check whether the new system is consistent. Your contention that Hilbert's system is only consistent with dimensionless points is not valid, because it contains no postulate that permits us to speak about any properties whatsoever of points. I am sorry to insist, but if you had gone on to give the whole thing a try with twinkies, maybe you would have seen the problem. I also must say something in your favor, when the discussion about the 5'' points went on to include such terms as overlapping, the other guys should have taken into consideration that instead of clarifying things for a laymann it would confuse things even more and I was also under the impression that some things were asserted without proof. > >>impossible to refute or verify anything whatsoever about them. > >>So I have decided to do as you do and simply accept on the > >>authority of sci.mathers that anything they say regarding > >>inference from axioms is correct. > > I guess this is simply sarcasm, because no one here seems to feel > > bounded by the authority of fellow sci.mathers. Actually, dealing with > > formal systems is part of what you learn at a mathematics faculty. The > > study of mathematics is hard work. > Yes. It would appear that it is. > > You have to learn to avoid the > > pitfals of confusing natural and formal language.You have to learn to > > abstract, to handle complex thoughts, to train your intuition so you > > may make the right choices instead of following endless blind alleys. > No doubt. I found these same qualities and skills were required > in my programming. But, of course, /everything/ was defined. > What you call systems of axioms seem to me to be merely empty > shells that consist of natural language statements with all > meaningful context-giving words labeled 'undefined'. Yes, but having some experience in this area myself (though not such a long one as your's) I will permit myself to ask you wether you remember how when confronted with new languages or even \paradigms\, as they say, one would be baffled at features that seemed all but consistent with what one considered reasonable. And then, maybe the fact that programming languages are more removed from natural languages, makes it easier to deal with them on an abstract level. Maybe you should read some good book that shows how the body of mathematics is derived from a basic set of axioms, maybe someone here can recommend a good one. If the writing isn't abstruse, such reading can be quite fun and interesting. > -- > \...how an individual invents a new way of giving order > to data now all assembled must here remain inscrutable > and may be permanently so... Almost always the men > who achieve these fundamental inventions of new > paradigm have either been very young or very new > to the field whose paradigm they change... > (they) are particularly likely to see that those rules > no longer define a playable game and to conceive > another set that can replace them.\ > Thomas Kuhn > The Structure of Scientific Revolutions === Subject: Re: Courage? >>What you call systems of axioms seem to me to be merely empty >>shells that consist of natural language statements with all >>meaningful context-giving words labeled 'undefined'. > Yes, but having some experience in this area myself (though not such a > long one as your's) I will permit myself to ask you wether you remember > how when confronted with new languages or even \paradigms\, as they > say, one would be baffled at features that seemed all but consistent > with what one considered reasonable. And then, maybe the fact that > programming languages are more removed from natural languages, makes it > easier to deal with them on an abstract level. Maybe you should read > some good book that shows how the body of mathematics is derived from a > basic set of axioms, maybe someone here can recommend a good one. If > the writing isn't abstruse, such reading can be quite fun and > interesting. just above, my experiences in programming and new programming paradigms was not quite as you imagine. My first 5 years in programming were spent doing mainframe assembler, initially with no real operating system so that much of what is today done in hardware and firmware had to be specifically programmed in either assembler or something lower yet. The result was a very low level understanding of the phenomena actually invoked in the hardware by my statements. Adding layers of abstraction on top of that understanding was essentially effortless and freeing. New paradigms when they arose caused me only delight and a sense of instant recognition in that I had already been involved in attempts to transcend the older paradigms on my own. I don't take well to jumping in at the top of a new subject and then having to accept the lower level foundational material on faith and authority. A careful examination of my posts I think will reveal this pattern. (Ignoring, of course, all of the anger and frustration when my push for lower level understanding were thwarted by insistence that I not question the foundations too seriously or closely.) I arrived here with very highly developed reasoning skills and a healthy dose of skepticism, which was misinterpreted as contentiousness and stupidity. I fully understood much of what was assumed that I didn't understand, but discovered that the foundational material that I sought was simply not forthcoming. I have no idea why, but I have begun to suspect that many on sci.math simply don't understand the foundations and assumptions of their own profession. Perhaps, I am at this point more interested in the philosophy and history of math than in math itself. -- \...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them.\ Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Courage? > I have no idea why, but I have begun to suspect that many on sci.math > simply don't understand the foundations and assumptions of their own > profession. Perhaps, I am at this point more interested in the > philosophy and history of math than in math itself. And many understand far more than you do. Your understanding of mathematics is rather inadequate. Bob Kolker === Subject: Re: Courage? On Fri, 06 May 2005 12:32:53 -0400, Robert Kolker >> I have no idea why, but I have begun to suspect that many on sci.math >> simply don't understand the foundations and assumptions of their own >> profession. Perhaps, I am at this point more interested in the >> philosophy and history of math than in math itself. >And many understand far more than you do. Your understanding of >mathematics is rather inadequate. As is your understanding of universal truth. === Subject: Re: Courage? >> What you call systems of axioms seem to me to be merely empty shells >> that consist of natural language statements with all meaningful >> context-giving words labeled 'undefined'. > One can associate with an axiomatic system some suggestions that act as > intuition pumps. I have no idea what such suggestions would look like or what rules would control their content. > But these are not a formal or proper part of the system > itself. They function as heuristics to aid thinking. Thus we tend of > think of points as dots without extent. Just pure location. Why 'thus'? I fail to follow how the first sentence justifies the second. However, I agree that points are without extend, just pure location. > But in one model of geometry, points are n-tuples of elements from a an > algebraic ring (a technical term, not something that a phone does or > what one wears on a finger). Do these elements from a ring function as coordinates when placed into a point n-tuple? -- \...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them.\ Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Courage? >> I am not passively accepting anything, Bob. I remain convinced that >> only dimensionless points are consistent with Hilbert's axioms. > Define \dimension\ as used in this context. 'Dimension' wasn't used in this context; the word used was 'dimensionless', the opposite of 'dimensioned'. Dimensionless: having zero dimensions, e.g having no length, width or height. Dimensioned: having two or more dimensions, e.g. having length and/or width and/or height. -- \...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them.\ Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Courage? >> publishing a fixed version. >> In fact, I may be wrong, but wasn't something like that >> the reason Hilbert decided to improve on Euclid? >> Or perhaps it was just that Euclid's foundation >> wasn't rigorous enough. > Euclid smuggled in assumptions not explicitly stated with his > postulates. Hilbert fixed that defect. He appears to me to have done *much* more. Simply stating the unstated would have been a simpler fix. >> We aren't saying axioms MUST be constructed consistently, >> we're just saying that the sets we're working with >> in this discussion, laid down by some smart people >> and reasoned over for many decades, have been >> thoroughly vetted. > Well, an inconsistent axiom system is not terribly useful or interesting > since any well formed statement can be proven. As to determining whether > an axiom system is consistent or not, this is generally not provable > within the system in question. The matter of consistency then becomes a > metamathematical question. If you can show an axiom system has a finite > instance of model, this is the strongest proof of its consistency. This > is not always possible. Some axiom systems do not have finite instances > or models. > Bob Kolker -- \...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them.\ Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Courage? > On Wed, 04 May 2005 16:48:23 -0400, Robert Kolker >together. I don't see any way you could have a larger sentence >containing \red car\ in it that would be readible when \and red and \ car\ >is substituted for it. >>Try this: if an object is both red and car it is a red car. You think \if an object is both red and car it is a and red and car\ makes sense? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Courage? On Thu, 05 May 2005 21:35:12 -0400, Will Twentyman >> On Wed, 04 May 2005 16:48:23 -0400, Robert Kolker >>together. I don't see any way you could have a larger sentence >>containing \red car\ in it that would be readible when \and red and \ car\ >>is substituted for it. >Try this: if an object is both red and car it is a red car. >You think \if an object is both red and car it is a and red and car\ >makes sense? I don't think describing a \red car\ as \red and car\ makes any more or less sense than saying \and red and car\. === Subject: Re: Courage? <726ce.367$Uz4.364@okepread04> <4276414a.37528703@netnews.att.net> <42767b7f.44683089@netnews.att.net> <4277f0cd.75979997@netnews.att.net> <42781bbf_1@newsfeed.slurp.net> <4278ea7f.83912370@netnews.att.net> <427917e3_1@newsfeed.slurp.net> <42793dbc.93585123@netnews.att.net> <427aca27$1_5@newsfeed.slurp.net> <427bb82b.5174566@netnews.att.net> posting-account=alQKkAwAAAA82xoCXcVIo1q-o-rv2IW- > I don't think describing a \red car\ as \red and car\ makes any more > or less sense than saying \and red and car\. And stuff and nonsense. Brian Chandler http://imaginatorium.org === Subject: Re: Courage? On 6 May 2005 12:20:51 -0700, imaginatorium@despammed.com in > >> I don't think describing a \red car\ as \red and car\ makes any more >> or less sense than saying \and red and car\. >And stuff and nonsense. You really need to get back to pointing out curves on straight lines if you've quite gotten over your little mal de tete, Brian, you know for your self esteem if nothing else. === Subject: Re: Courage? > On Thu, 05 May 2005 11:50:35 -0400, Robert Kolker >So what's wrong with my saying \red car\ means \red and car\ or >\and red and car\? >>\and red and car\ is ungrammatical. > Much of math is ungrammatical in terms of generic language. However, math has a grammar which must be followed, just as when writing proper English. Failure to follow the basic grammar renders any attempt to communicate difficult at best, nonsense at worst. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Courage? On Thu, 05 May 2005 21:33:57 -0400, Will Twentyman >> On Thu, 05 May 2005 11:50:35 -0400, Robert Kolker >>So what's wrong with my saying \red car\ means \red and car\ or >>\and red and car\? >\and red and car\ is ungrammatical. >> Much of math is ungrammatical in terms of generic language. >However, math has a grammar which must be followed, just as when writing >proper English. Failure to follow the basic grammar renders any attempt >to communicate difficult at best, nonsense at worst. I don't see \red car\ is any the less exact for not reflecting conventional math grammar \red and car\. In fact I think exactly the same lack of sanity you imputed to \and red and car\ would attach to \red and car\ used in generic speech to mean what is reflected in \red car\. So we're faced with generic speech which is more concise than math logic equivalent but no less exact. === Subject: Re: Courage? > car\. So we're faced with generic speech which is more concise than > math logic equivalent but no less exact. Balls! \and\ is a binary connective and must occur -between- two terms. That is why \and x and y\ is ill formed. The first \and\ does not occur \ between two terms. Bob Kolker === Subject: Re: Courage? On Fri, 06 May 2005 13:47:12 -0400, Robert Kolker >> car\. So we're faced with generic speech which is more concise than >> math logic equivalent but no less exact. >Balls! \and\ is a binary connective and must occur -between- two terms. >That is why \and x and y\ is ill formed. The first \and\ does not occur \ >between two terms. And how do you know that? === Subject: Re: Courage? > >>Well for several years on the usenet I worked with the phrase >>\differences and different from differences\ but people just >>complained that they couldn't understand what differences meant or >>what differences I was referring to. So I thought I'd use [not] and >>[not not] instead because at least [not] seemed pretty obviously >>tautologically regressible to self contradictory alternatives in ways >>people should be familiar with. > Apart from acolytes who decline to anything other than repeat your > words, nobody has yet professed to understand what \tautologically > and couldn't find a combination of 'tautolog(icall)y' and 'regressible' > that wasn't one of your posts. And you seriously claim to believe > people \should be familiar with\ this expression? I think I'm close to understanding what he means, but I would never use that phrase to express the concept. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Courage? On Thu, 05 May 2005 21:13:11 -0400, Will Twentyman >> >Well for several years on the usenet I worked with the phrase >\differences and different from differences\ but people just >complained that they couldn't understand what differences meant or >what differences I was referring to. So I thought I'd use [not] and >[not not] instead because at least [not] seemed pretty obviously >tautologically regressible to self contradictory alternatives in ways >people should be familiar with. >> Apart from acolytes who decline to anything other than repeat your >> words, nobody has yet professed to understand what \tautologically >> and couldn't find a combination of 'tautolog(icall)y' and 'regressible' >> that wasn't one of your posts. And you seriously claim to believe >> people \should be familiar with\ this expression? >I think I'm close to understanding what he means, but I would never use >that phrase to express the concept. Why? The phrase means exactly what I mean in using the phrase. === Subject: Re: Courage? > On Wed, 04 May 2005 14:26:23 -0400, Will Twentyman >On Tue, 03 May 2005 20:34:00 -0400, Will Twentyman >> >> >On Tue, 03 May 2005 15:39:47 -0400, Will Twentyman > > >> >> >On Mon, 02 May 2005 19:21:41 -0400, Will Twentyman > > >> >> >On Mon, 02 May 2005 15:42:00 -0400, Will Twentyman > > >>A linear space can consist of polynomials, vectors, matrices, or \ other >>objects. A linear space does not imply lines. > > >I suspect a linear space does imply lines or objects composed of \ lines >or it wouldn't be called linear. Otherwise I can't make any sense \ of >Bob's contention that lines are subsets of linear space. >> >>The simplest example of a linear space is lines. There are others \ that >>are made of planes in R^3, etc. > > >And planes are defined by the intersection of straight lines. >> >>And some linear spaces are made up of 2x3 matrices over the reals. > > >Which I would imagine are also defined by the intersection of straight >lines except that the transcendental part of the real would be defined >on curves instead of straight lines so I don't expect transcendental >parts of the reals could be defined in linear space. >> >>You imagine incorrectly. Do you know what a matrix is? >I imagine what incorrectly? >>Imagining that the linear space of 2x3 matrices is defined by lines is >>incorrect. No curves or lines are involved. > Good. Then I'm sure you won't need rectangles to define matrices. Quite right. >I have no idea what a matrix is; >>A matrix is a rectangular arrangement of numbers. > Oops. So you do need rectangles. And those rectangles are defined in > terms of straight line segments and right angles and not in terms of > curves. I wonder if there are non Euclidean matrices? I just don't > think non Euclidean matrices can exist in linear space. Actually, I can just say that a 2x3 matrix is 6 numbers indexed as follows: matrix A has elements a_11, a_12, a_13, a_21, a_22, a_23. It is just convenient to represent them as: A = [ a_11 a_12 a_13 ] [ a_21 a_22 a_23 ] which is clearly arranged in a rectangular \ fashion. >> \ In this example, it >>contains 6 numbers in 2 rows and 3 columns. The operations of scalar >>multiplication, as well as matrix addition and additive inverse are >>defined on them in such a way as to satisfy the axioms of a linear >>(vector) space. >I fell >off a truck last night. Do you have any idea what universal truth is? >>As you use it, I think it has to do with finding all the options except >>those which would result in a logical contradiction. > Actually the reverse. I find all those things which result in self > contradiction and infer what's left over is universally true. Works > for me. I'm not sure how that's different from what I said. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Courage? On Thu, 05 May 2005 20:16:52 -0400, Will Twentyman >> On Wed, 04 May 2005 14:26:23 -0400, Will Twentyman >>On Tue, 03 May 2005 20:34:00 -0400, Will Twentyman >> > > >>On Tue, 03 May 2005 15:39:47 -0400, Will Twentyman >> >> > > >>On Mon, 02 May 2005 19:21:41 -0400, Will Twentyman >> >> > > >>On Mon, 02 May 2005 15:42:00 -0400, Will Twentyman >> >> >>>A linear space can consist of polynomials, vectors, matrices, or \ other >>>objects. A linear space does not imply lines. >> >> >>I suspect a linear space does imply lines or objects composed of \ lines >>or it wouldn't be called linear. Otherwise I can't make any sense \ of >>Bob's contention that lines are subsets of linear space. > >The simplest example of a linear space is lines. There are others \ that >are made of planes in R^3, etc. >> >> >>And planes are defined by the intersection of straight lines. > >And some linear spaces are made up of 2x3 matrices over the reals. >> >> >>Which I would imagine are also defined by the intersection of \ straight >>lines except that the transcendental part of the real would be \ defined >>on curves instead of straight lines so I don't expect transcendental >>parts of the reals could be defined in linear space. > >You imagine incorrectly. Do you know what a matrix is? >> >> >>I imagine what incorrectly? >Imagining that the linear space of 2x3 matrices is defined by lines is >incorrect. No curves or lines are involved. >> Good. Then I'm sure you won't need rectangles to define matrices. >Quite right. >>I have no idea what a matrix is; >A matrix is a rectangular arrangement of numbers. >> Oops. So you do need rectangles. And those rectangles are defined in >> terms of straight line segments and right angles and not in terms of >> curves. I wonder if there are non Euclidean matrices? I just don't >> think non Euclidean matrices can exist in linear space. >Actually, I can just say that a 2x3 matrix is 6 numbers indexed as >follows: matrix A has elements a_11, a_12, a_13, a_21, a_22, a_23. It >is just convenient to represent them as: >A = [ a_11 a_12 a_13 ] > [ a_21 a_22 a_23 ] which is clearly arranged in a rectangular \ fashion. Yes, well it's certainly good you don't have to use the word rectangle. > \ In this example, it >contains 6 numbers in 2 rows and 3 columns. The operations of scalar >multiplication, as well as matrix addition and additive inverse are >defined on them in such a way as to satisfy the axioms of a linear >(vector) space. >>I fell >>off a truck last night. Do you have any idea what universal truth is? >As you use it, I think it has to do with finding all the options except >those which would result in a logical contradiction. >> Actually the reverse. I find all those things which result in self >> contradiction and infer what's left over is universally true. Works >> for me. >I'm not sure how that's different from what I said. And I'm still not sure what you meant by what you said. === Subject: Re: Courage? > On Wed, 04 May 2005 16:43:52 -0400, Robert Kolker >How about the empirical observation [not]? That's a non trivial >observation proven universally true through tautological regression to >self contradictory alternatives. >>[not] is not an observation, emprical or otherwise. It is an operator >>which when applied to a proposition negates the proposition. It is a >>truth value flipper. true to false, false to true. \not\ standing by >>itself is meaningless. > And you know this how exactly? Did it come to you in dream or did you > consult an oracle? He consulted the definition. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Courage? On Thu, 05 May 2005 20:10:27 -0400, Will Twentyman >> On Wed, 04 May 2005 16:43:52 -0400, Robert Kolker >> >>How about the empirical observation [not]? That's a non trivial >>observation proven universally true through tautological regression to >>self contradictory alternatives. >[not] is not an observation, emprical or otherwise. It is an operator >which when applied to a proposition negates the proposition. It is a >truth value flipper. true to false, false to true. \not\ standing by >itself is meaningless. >> And you know this how exactly? Did it come to you in dream or did you >> consult an oracle? >He consulted the definition. Which is arbitrary? Educated guess, nothing more. === Subject: Re: Courage? > On Wed, 04 May 2005 16:44:04 -0400, Will Twentyman >On Wed, 04 May 2005 07:57:56 -0400, Will Twentyman >> >> >On Tue, 03 May 2005 15:27:11 -0400, Will Twentyman > > >> >> >> >[...] > > > > >>But since \X not X\ is a self contradiction, we know it to be \ false. >>Does it really add anything then? > >After carefully reading Lester's explanations to you twice, I think \ he's >saying that if you can describe empirical reality with a logically >consistent model such that the negation of the model is inconsistent \ >(self contradictory), then the model must be true. Which is IMO \ obvious. >> >>I was hoping that the question will help Lester find ways that his >>theory would be non-trivial. It would be interesting if his theory \ can >>produce something new, but as it stands now, I suspect that it lacks \ the >>power to say anything. > >So a demonstrable standard for universal truth is trivial and not new? >> >>As I understand your presentation, yes. For example: it appears that >>you would assert: [there is a unique parallel line through a point not >>on a line][there is not a unique parallel line through a point not on a \ >>line] >> >>Well, yes, but so what? It's not until you stand behind one or the >>other that you can start doing something interesting. >What you describe is a simple tautology that doesn't necessarily >regress to anything self contradictory. What I described by [not] >was a tautology that does necessarily regress to self contradictory >alternatives [not not]. In the absence of further regressions what you >describe is only problematic whereas what I describe is necessarily >true universally. >>Ok, I don't see why my example regresses. Please explain further. > You regress the empirical observation \[there is a unique parallel > line through a point not on a line]\ through a tautology to form > \[there is not a unique parallel line through a point not on a line]\ > at least as close as I can read. That is a tautological regression. Perhaps I should have said: I don't see why my example regresses *further* than what I already did. >Trouble is, there's a hidden hook. Such models have the general form \ \IF >Model, THEN Observations\, or [M-->O]. However, reality has a nasty \ >habit of throwing up some Observations' [O'] that M does not imply, \ or >worse, whose negations are implied by M, so the premise [M] is >falsified. That doesn't falsify the conclusion, however: [O] will \ still >be true. But [M] will no longer be the only logically possible reason \ >for their being true. Some new Model' will now imply bothe the old \ and >the new observation, or [M'-->(O'+O)]. -- These comments amount to a \ >view of how scientific/empirical theorising proceeds. Newton's Model \ >implied many observations, but failed for others. A new model implies \ >all the observations his model implies, as well as others. Hence >Newton's model is no longer true in the sense that it does not hold \ for >many observations. Yet it is still true for a restricted domain of >observations, and because it's simpler, is used in that domain (eg, \ in >mechanical and civil engineering.) > >If Lester does in fact intend something like what I've outlined \ above, >he's unfortunately not original. OTOH, he seems to want to say that \ he >has discovered a way of generating the form \IFF Model THEN >Obsrevations\, or [M==O]. If that's his position, I think he's just \ >plain wrong. >> >>Since the hook would reframe Lester's work as empiricism with a new >>notation, I don't think it's what he has in mind. > > >Of course not. > > > > >> \ My understanding is >>that he wants to avoid empiricism completely and deduce the laws of >>science from a purely logical perspective without having to conduct >>experiments. > > >Not true. Regression to universal truth don't say what is true of >what. It just says what can't be true of what. >> >>That's not entirely different. How is limiting options not the same as \ >>asserting truth? There is a theorem in neutral geometry (it doesn't >>have a parallel axiom) which says there is at least one parallel through \ >>a point not on a given line. It excludes the possibility of no >>parallels, without determining if the parallels are unique or not. To >>say something cannot be true is itself a claim. >But not necessarily a universally true regressible claim. Limiting >options doesn't indicate whether undelimited options are true or false >or could be delimited under further tautological regession. To say >something cannot be true is a claim but a claim which can be true if >regressed to self contradictory alternatives. >>How do you know which claims can be regressed and which cannot? Can you >>give an example of regressing a simple claim in ordinary english? > All problematic claims, that is those which can be true, can be > regressed tautologically. It's tougher to tell which can be regressed > to self contradictory alternatives because, as I explained yesterday > or the day before, regressing compound predicates is an inherently > ambiguous process depending on how predicates are negated. That's > why its referred to as subjective mechanics. That's why I asked for a *simple* claim as an example. You know, something like \the car is red\ or \The pizza has pepperoni and \ sausage\. >> Unfortunately, you can generate perfectly \ reasonable >>systems for physics that are self-consistent but do not correspond to \ >>reality (heavier objects fall faster, for instance). > >I don't understand what these perfectly reasonable self consistent >systems have to do with universal truth. >> >>I don't see that your universal truth will exclude them. As a result, I \ >>don't see how your universal truth will help us do science that is \ correct. >It's universal truth not my universal truth. >>Since you are the only proponent of this description of universal truth, >>I'd just as soon link it to you somehow. > I'm the only demonstrator of this or any description of universal > truth. So I'd prefer it be called universal truth and not > particularized as if there were other possible versions of universal > truth and this just happened to be my go at it. There are other claims of universal truth out there. Just talk to a member of most any religion. >Universal truth will only >exclude things which are not regressible to the basis for universal >truth. It excludes things which are regressed to non universal truth. >>Can you give an example of non universal truth? > Well there are certainly any number of problematically non universal > truths in the sense that whether 2+2=4 is actually universally true or > only empirically so But the only actual example of non universal truth > is self contradiction because it is universally false. It is true because of the definitions of 2, +, 4, =. >Of course the linkage between self consistent systems and universal >truth is problematic and remains to be determined. But the idea that >self consistent systems are true is incorrect. >>If Lester is looking for a way to reframe empiricism without changing \ >>methodologies, I'm not sure he'll have actually added anything to how \ we >>do science. > >Of course not. I can't imagine where Wolf gets these ideas. >> >>I was just listing possibilities. I'm not clear on what your goal is. >I thought I had explained that. My direct goal is a reformation of >conepts of truth underlying science in general. For example Einstein's >idea that dimensional frames of reference shorten in the direction of >travel is demonstrably incorrect. The idea that there is any single >real number line is incorrect. The idea that points can be integrated >into lines is incorrect. The idea that there can be more than three >spatial dimensions is incorrect. The idea that transcendentals can be >pointed out on straight lines is incorrect. The idea that irrationals >include transcendentals is incorrect. Etc. etc. etc. >>If you are going to say that *definitions* are incorrect, you will not >>get very far with your ideas, at least among mathematicians. > I don't have any specific reason to say that the definitions are > incorrect only that the grouping of irrationals and transcendentals is > incorrect. Your claim of incorrect grouping contradicts the results of the definitions. Worse, your definitions are completely unrelated to the standard definitions. You say the definitions are incorrect by refusing to use them or accept the immediate conclusions of them. >>know enough about Einstein's idea to know if you stand a chance of >>getting physicists on board with your idea. Definitions are the basis >>of communication between mathematicians. You can say we used the wrong >>name for something, but the name doesn't affect the content of what is \ said. > It's the functional grouping for concepts that's at stake and not just > names. But you want to use the names for different concepts. >>It sounds like you are taking after Don Quixote. > Yeah but at least I understand what I'm tilting at and why and I'm not > sure Don Q. did. That's debateable.