mm-365 === Subject: : Differential equation Let z be a complex number, z* it's complex conjugate, d the derivative w.r.t. z and d* w.r.t z*. Consider the differential equation d d* ln(zz*) = 2pi delta(z)delta(z*) I want to integrate w.r.t. z and z* to show that the delta distribution is normalized properly. Using ln(zz*) = lnz + lnz*, the divergence theorem and the residual theorem, I get Int(d^2z, ln(zz*)) = 4pi which gives a normalization of 2 for the delta distribution. I used d d* ln(zz*) = dd*(lnz + lnz*) = d*(1/z) + d(1/z*). On the other Hand, d d* ln(zz*) = d(1/z*) = d*(1/z). Integrating this gives the correct normalization Int(delta dist.) = 1. What is wrong with my [CapitalThorn]rst calculation? Ren.8e. -- Ren.8e Meyer Student of Physics & Mathematics Zhejiang University, Hangzhou, China === Subject: : Re: Differential equation >Let z be a complex number, z* it's complex conjugate, d the derivative >w.r.t. z and d* w.r.t z*. Consider the differential equation >d d* ln(zz*) = 2pi delta(z)delta(z*) ??? Ignoring the question of whether this is a differential equation: Presumably delta denotes a Dirac delta, ie a point mass at the origin. If so I have no idea what delta(z)delta(z*) is supposed to be. In fact that Laplacian of ln(zz*) is some multiple of _delta_. >I want to integrate w.r.t. z and z* to show that the delta >distribution is >normalized properly. Using ln(zz*) = lnz + lnz*, the divergence >theorem and the residual theorem, I get >Int(d^2z, ln(zz*)) = 4pi >which gives a normalization of 2 for the delta distribution. >I used d d* ln(zz*) = dd*(lnz + lnz*) = d*(1/z) + d(1/z*). >On the other Hand, >d d* ln(zz*) = d(1/z*) = d*(1/z). Integrating this gives the correct >normalization Int(delta dist.) = 1. >What is wrong with my [CapitalThorn]rst calculation? Hard to say, since you didn't show the details of the actual calculation. You say you used d d* ln(zz*) = dd*(lnz + lnz*) = d*(1/z) + d(1/z*); whether that's correct or not depends on exactly what you mean, and exactly _where_ you applied this equality. (It's not true in the sense of distributions in the plane, for example...) >Ren.8e. ************************ David C. Ullrich === Subject: : Re: Differential equation >>d d* ln(zz*) = 2pi delta(z)delta(z*) > Presumably delta denotes a Dirac delta, ie a point mass > at the origin. If so I have no idea what delta(z)delta(z*) > is supposed to be. In fact that Laplacian of ln(zz*) > is some multiple of _delta_. delta(z)delta(z*) means the two-dimensional Delta distribution, which is often written as a product of two one-dimensional ones. >>Using ln(zz*) = lnz + lnz*, the divergence >>theorem and the residual theorem, I get >>Int(d^2z, ln(zz*)) = 4pi >>which gives a normalization of 2 for the delta distribution. >>I used d d* ln(zz*) = dd*(lnz + lnz*) = d*(1/z) + d(1/z*). >>On the other Hand, >>d d* ln(zz*) = d(1/z*) = d*(1/z). Integrating this gives the correct >>normalization Int(delta dist.) = 1. > Hard to say, since you didn't show the details of the > actual calculation. Look at the quotation above. I used the divergence theorem, and the theorem on residuals (hope this is the correct english term) in complex analysis to [CapitalThorn]rst make a line integral out of the integral over the complex plane and then calculate the residuals encircled by the curve of the line integral. > You say you used > d d* ln(zz*) = dd*(lnz + lnz*) = d*(1/z) + d(1/z*); > whether that's correct or not depends on exactly > what you mean, and exactly _where_ you > applied this equality. (It's not true in the > sense of distributions in the plane, for example...) OK, I write the calculation in ASCII: dd*ln(zz*) = 2pi delta^2(z,z*) <- two-dimensional Delta-Distribution Now I am using ln(ab)=ln(a)+ln(b) dd*(lnz + lnz*) = 2pi delta(z,z*) d*(1/z) + d(1/z*) = 2pi delta(z,z*) Integrating over the complex plane Int(dz, d*(1/z) + d(1/z*)) = 2pi Int(d^2z delta(z,z*)) The left Integral can be written as the line integral i Int(dz*/z* - dz/z) = - 2*2pi i with the curve encircling the origin. Calculating the Residue gives the right hand side. Now compare this with above: 2pi Int(d^2zdelta^2(z,z*)) = 2*2pi this means that the Delta distribution has normalization two. Going through the same calculation with dd*(lnzz*) = d*(1/z) = d(1/z*) one gets the correct normalization one. Ren.8e. -- Ren.8e Meyer Student of Physics & Mathematics Zhejiang University, Hangzhou, China === Subject: : Re: Eulerian path in in[CapitalThorn]nite graph > [...] > BTW, this is a particularly nice exercise, since any [CapitalThorn]nite grid > (suf[CapitalThorn]ciently large) is clearly not Eulerian (many vertices of odd > degree), but the in[CapitalThorn]nite graph is Eulerian. Nice. Grids on torus, i.e. C_m * C_n, are [CapitalThorn]nite and Eulerian. === Subject: : Re: Eulerian path in in[CapitalThorn]nite graph 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21: > BTW, this is a particularly nice exercise, since any [CapitalThorn]nite grid > (suf[CapitalThorn]ciently large) is clearly not Eulerian (many vertices of odd > degree), but the in[CapitalThorn]nite graph is Eulerian. Nice. Grids on torus, i.e. C_m * C_n, are [CapitalThorn]nite and Eulerian. More to the point for this speci[CapitalThorn]c exercise, even-sized diamond-shaped subsets of the integer grid (rather than the more obvious rectangular shape) are also [CapitalThorn]nite and Eulerian. -- David Eppstein http://www.ics.uci.edu/~eppstein/ Univ. of California, Irvine, School of Information & Computer Science === Subject: : Re: Gaussian Curve Algorithm ? >Gaussian Curve Algorithm I am trying to plot a Gaussian Curve using the formula : y=y0 * exp -[(x-x0)**2]/ 2*sigma**2 > ^^^^^^^^^^^^^ MISPRINT As an input from a chromatography instrument I have only the y0, x0. I was wondering if anyone knows if there are algorithms that can be used in >this situation ? Thanks for any info >Nik > you get the density at the max? > well , the density is > (1/sigma)*1/(sqrt(2*pi))*epx( -(x-x0)**2/(2*sigma**2) ) > so from y0 you get sigma and can now plot the curve. > hth > peter Thanks but I dont under stain how to get the sigma value from this expression Nikos === Subject: : Re: Cosets in Q...index... > Hi all I am staring at the following problem: Prove that Q (rational numbers with +) has no proper subgroup of [CapitalThorn]nite > index. I'm trying to [CapitalThorn]nd a constructive way to show it, in the sense > of explicitly constructing an in[CapitalThorn]nite list of distinct cosets. I > would rather have such a proof than a suppose x1,...,xn are are [CapitalThorn]nite > set of coset reps, here is a contradiction... proof, which I have > already. Any suggestions? Thanks! > Justin ps. As elementary as possible would be nice! ************************************************************** Hi Justin: Let us take again the hint about divisible groups that you already got, and try to make it simple: 0) If G is a [CapitalThorn]nite group, say of orden n, then for all g in G we have g^n = 1 (or ng = 0, in additive notation); i) Prove that if G is ANY group and N is a normal sbgp. of index n, then for all g in G we have g^n (or ng, in additive notation) belongs to N (use(0)); ii) Prove that if q is ANY element in Q and n is any non-zero integer, then there exists some b in Q s.t. q = nb (hint:write q = a/b, a,b integers, etc.); iii) Suppose now that N is a sbgp. of Q of [CapitalThorn]nite index, say n; If F:Q --> Q/N is the canonical homomorphism (i.e. F(q):= q + N. Pay attention to the fact that q in Q BELONGS to N iff F(q) = N...), then for ALL q in Q we'll get that F(q) belongs to N, thus getting N = Q. Q.E.D. (Hint: Since n is not zero, there exists b in Q s.t. q = nb (this is (ii)) ==> from here we get: F(q) = F(nb) = nF(b) (now use (i)) ...etc.). Good luck! Jose A. Perez === Subject: : Re: Give me that old time ontology: (was: the anticlassicalist }{ i: |It is far too easy to be dismissive of something if you have not considered |it. It seems to me that if I really haven't considered something, I tend to be noncommital about it. |For instance, in this case there is most certainly a manifestation of the |non-standard model in the presentation of mathematical literature. That |possibility did not seem to enter your thoughts at all. Excuse me if I [CapitalThorn]nd this a somewhat peculiar comment. This notion of a manifestation of [a] nonstandard model in the literature would seem broad enough to include cases of mathematicians unwittingly doing things in a manner which is best explained by constructive facts of which they're unaware, since they come up naturally for consideration only if one is thinking constructively. I've looked for such cases, since it seems like it would be useful motivation. (I'm afraid I haven't spent much time on it or had much success. It's like looking for obvious gold nuggets in streams.) In the discussion up to this point, I don't remember anybody asking me anything that would have revealed whether such thoughts are on my mind. Would having such thoughts cause me to be amazed to [CapitalThorn]nd that there is (in a certain sense) a non-Boolean algebra modelling classical propositional logic? No, I would say rather the opposite; the more familiar a person is with potential manifestations of nonstandard models, the less surprising the more mundane of them seem. So it seems like you're just sort of guessing that this is the kind of thing that I would tend not to think of. Moreover, it seems like you're using quite a lot of very generic terms here. The literature? The nonstandard model? (Which one? Nonstandard how? Presumably not as in nonstandard analysis.) Manifesting? As I see it, one of the main obstacles that a proponent of teaching nonclassical logic has, is to convince the relevant people that it's worth teaching it *as logic* rather than merely as lattice theory, topology, and so on. Perhaps the main difference between merely having a Heyting algebra appear in your work and treating it as a model of intuitionist logic (and between merely having subspaces of a Hilbert space, and treating them as a model of quantum logic) is that in the latter, one thinks of the elements as *propositions*. If you want to be amazing, try to think of a case where it actually pays to think of the elements of the algebra as propositions, and reason with them as if they were (but satisfying a nonclassical logic). [...] |:-) | |mitch Keith Ramsay :-) === Subject: : Re: Give me that old time ontology: (was: the anticlassicalist }{ i: : As I see it, one of the main obstacles that a proponent of teaching : nonclassical logic has, is to convince the relevant people that it's worth : teaching it *as logic* rather than merely as lattice theory, topology, and : so on. Perhaps the main difference between merely having a Heyting : algebra appear in your work and treating it as a model of intuitionist : logic (and between merely having subspaces of a Hilbert space, and : treating them as a model of quantum logic) is that in the latter, one : thinks of the elements as *propositions*. If you want to be amazing, : try to think of a case where it actually pays to think of the elements : of the algebra as propositions, and reason with them as if they were : (but satisfying a nonclassical logic). Well, I really though that... -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= There are Heyting algebras in a lot of [CapitalThorn]elds. They are one of the more useful logical forms for our models in these [CapitalThorn]elds. In the study of cognitive research we have models of neural interaction, and many experiments match well to the models. There is a school of thought that says concepts are attractors in the neural models dynamics. This allows us to pose dynamical propositions in the neural models. The propositional structure would then likely be Heyting in such a model. Neural organisation: structure, function, and dynamics by Arbib, Erdi, and Szentogathai The algebraic structure of sets of regions by Stell and Worboys A canonical model of the region-connection calculus by Jochen Renz A relation-algebraic approach to the region-connection calculus by Duntsch, Wang, and McCloskey Interestingly, there are also very similar things found in the logical study of the way we reason about the world. How logic emerges from the dynamics of information by Peter Gardenfors Bimodal logics for reasoning about continuous dynamics by Davoren and Gore Categorical and Kripke semantics for constructive S4 modal logic by Alechina, Mendler, de Paiva, and Ritter The algebra of topology by Tarski and McKinsey Sentential calculus and topology by Tarski And this [CapitalThorn]ts in with a particular history of research in the logics http://plato.stanford.edu/entries/logic-modal/ http://plato.stanford.edu/entries/model-theory/ http://en.wikipedia.org/wiki/Formal_language http://en.wikipedia.org/wiki/Algorithmic_information_theory which builds the context for the way we reason about the world in terms that become computable. Its at the point where the models are able to process the information in the manner of a state machine with a discrete, directed collection of states (often represented by time) that languages are formalised. The models become dynamic. This as well is interesting since the well-known Curry-Howard isomorphism describes a natural relationship between lambda calculi and Heyting algebras. Lectures on the Curry-Howard isomorphism by Morten Heine B. Sorensen and Pawel Urzyczyn Language and the dynamics of conversations can be modeled by these automatons. Representing communicative intentions in collaborative conversational agents by Matthew Stone Speech and language processing by Jurafsky and Martin Denotational semantics for agent communication languages by Frank Guerin and Jeremy Pitt Attractors in the development of communication by E. D. de Jong Autonomous formation of concepts and communication by E. de Jong And when you look at the computational ability of our models of brain activity On the computational power of neural nets by Siegelmann and Sontag we see an intimate connection forming. This is not meant to af[CapitalThorn]rm a logical linguistic programme, only to show that there are Heyting algebras associated to the [CapitalThorn]elds around such a programme from several different directions, and the theory of computation in particular has strong foundations in structures that are associated with these algebras. There are other logics also associated with turing-completeness The Turing-completeness of multimodal categorial grammars by Bob Carpenter which shows that the logical programme of linguistics can arive to equivalence to lambda calculi from different directions. There are many interesting relationships between Heyting algebras and other logics known, such as Combining possibilities and negations by Greg Restall A semantical study of orthologics by Miyazaki Yutaka Algebras and frames for modal logics And so on... Now, Heyting algebras are all over the place in mathematics. Besides the above (which also mentions the classic result of topologies de[CapitalThorn]ning Heyting algebras), there is the general theory of topoi. Topoi: the categorial analysis of logic by Robert Goldblatt http://en.wikipedia.org/wiki/Topos http://math.ucr.edu/home/baez/topos.html which shows that the study of mappings between mathematical objects often occurs in categories that are topoi. All topoi have a natural Heyting algebra structure. Many models of our world contain reasoning inside topoi, on objects that are members topoi and the transformations between them. In particular, we have evolutionary theories on graphs and digraphs (which include trees and directed trees) A guided tour in the topos of graphs by Sebastiano Vigna where we [CapitalThorn]nd both topological logics and some proposed connections with fuzzy logics, both in modeling and the inverse problem A general study on genetic fuzzy systems by Oscar Cordon A formulation of fuzzy automata and its application as a model of learning systems by Wee and Fu and we have again Heyting algebras found for the logic of fuzzy sets. Lattice of fuzzy sets by Takashi Mitsuishi and Grzegorz Bancerek A natural interpretation of fuzzy sets and fuzzy relations by Mamoru Shimoda So there's seems to be something about abstraction, computational, evolutionary logics that tend to draw Heyting models. Something connectional and topological. And still, we have the wonderful Fotini Markopoulou-Kalamara and her programme of causal sets and their Heyting structure. The internal description of a causal set: What the universe look like from the inside by Fotini Markopoulou An insider's guide to quantum causal histories by Fotini Markopoulou Dynamics of causal sets by David Rideout Causal sets and frame-valued set theory by John L. Bell Quantum spacetime as a qunatum causal set by Ioannis Raptis Topos theory and consistent histories: the internal logic of the set of all consistent sets by C. J. Isham Which ties itself up nicely with, to me, one of the neatest Heyting algebra of all, that of the operator projection calculus on a quantum Hilbert space. Operational quantum logic: an overview by Coecke, Moor, and Wilce Quantum logic in intuitionistic perception by Bob Coecke Constructive mathematics and quantum physics by Douglas Bridges and Karl Svozil and of course we now have ties with quantum computing. Locality, weak or strong anticipation and quantum computing II: constructivism with category theory by Heather and Rossiter So those are the models I see when I think about Heyting algebras. It seems to me to have something intimate to do with process. And these are all quite interesting models, about learning and our symbol systems, about calculation, evolution, the quantum. And these particular algebras mentioned are the non-Boolean kind. Some mathematicians may also be familiar with Goedel algebras (also Heyting). http://math.chapman.edu/cgi-bin/structures.pl?Goedel_algebras So, for the mathematicians, I think the thing that ties it all together is the theory of realisability. Its a well established [CapitalThorn]eld that, in many ways, works at describing these various relationships mentioned. Realizability: an historical essay by Jaap van Oosten Thats pretty much why I wanted to discuss Heyting algebras. Their beautiful things, and I thought, from my exposure to its ubiquity in all of these [CapitalThorn]elds, maybe there might be others who had seen at least a piece of it in their own [CapitalThorn]elds. I thought maybe all the of all the people out their from the various directions, maybe one or two from each [CapitalThorn]eld would be hanging around the newsgroups. Maybe a few others not active working on it, but maybe had seen some of the work. This bibliography is nothing. I didn't want to waste all my time tossing everything I could together, but just what the numbers of people working is large. But there is also Carlos Leguizamon and his programme on the algbraic-relational theory in biological systems. Review: the algebraic-relational theory and its applications by Alba Zeretsky with their pseudo-Booleans found in antigen-antibody reactions and other biological modelings. Their are many beautiful models with more exotic structures, but Heyting algebras seem to have this special universality principle that somehow seems connected to the Church-Turing thesis. So when I mentioned education in my [CapitalThorn]rst post, I thought these algebras would be the least controversial to put prior to Boolean. It pretty much [CapitalThorn]ts right there, with fewer de[CapitalThorn]ning relations, full computational ability, potential cognitive primacy. Its useful in all sorts of evolutionary models. But in some ways, its also a step to a better education about models in general, and our ability to calculate in agreemant with our observations. Because new models come out all the time. Its important to understand their logical structure to reason correctly within them. And I thought maybe this might be a good time to start thinking about teaching some of these connections I mentioned. Nearly all my friends went through logic classes as prerequisites in their undergraduate studies, and these never explained anything outside of quanti[CapitalThorn]ed classical logic (often without much algebra). And it seems to me that with just a little restructuring, such a course could include constructive logic as it builds the axiom set. All of the various research in many different directions easily provides a large example set. Interesting new cases come up all the time. An information-based theory of conditionals by Wayne Wobcke -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ...had pointed to quite a corpus with just the answers you requested. Nearly every one of those pieces was speci[CapitalThorn]cally on propositions and a logical approach. Model theory in general requires the logical connectives for sentences on its structures. You can always avoid saying things like and and or or even implies and forall or whatever when reasoning and just write everything down in some other notation or translation of the notions, but that doesn't mean the actions of reasoning you are performing are any different. -- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: : Re: Give me that old time ontology: (was: the anticlassicalist }{ i: > |It is far too easy to be dismissive of something if you have not considered > |it. > It seems to me that if I really haven't considered something, I tend to > be noncommital about it. > |For instance, in this case there is most certainly a manifestation of the > |non-standard model in the presentation of mathematical literature. That > |possibility did not seem to enter your thoughts at all. > Excuse me if I [CapitalThorn]nd this a somewhat peculiar comment. This notion of > a manifestation of [a] nonstandard model in the literature would > seem broad enough to include cases of mathematicians unwittingly > doing things in a manner which is best explained by constructive facts of > which they're unaware, since they come up naturally for consideration > only if one is thinking constructively. I've looked for such cases, since it > seems like it would be useful motivation. (I'm afraid I haven't spent much > time on it or had much success. It's like looking for obvious gold nuggets > in streams.) > In the discussion up to this point, I don't remember anybody asking me > anything that would have revealed whether such thoughts are on my mind. > Would having such thoughts cause me to be amazed to [CapitalThorn]nd that there > is (in a certain sense) a non-Boolean algebra modelling classical > propositional logic? No, I would say rather the opposite; the more familiar > a person is with potential manifestations of nonstandard models, the > less surprising the more mundane of them seem. > So it seems like you're just sort of guessing that this is the kind of thing > that I would tend not to think of. > Moreover, it seems like you're using quite a lot of very generic terms here. > The literature? The nonstandard model? (Which one? Nonstandard > how? Presumably not as in nonstandard analysis.) Manifesting? > As I see it, one of the main obstacles that a proponent of teaching > nonclassical logic has, is to convince the relevant people that it's worth > teaching it *as logic* rather than merely as lattice theory, topology, and > so on. Perhaps the main difference between merely having a Heyting > algebra appear in your work and treating it as a model of intuitionist > logic (and between merely having subspaces of a Hilbert space, and > treating them as a model of quantum logic) is that in the latter, one > thinks of the elements as *propositions*. If you want to be amazing, > try to think of a case where it actually pays to think of the elements > of the algebra as propositions, and reason with them as if they were > (but satisfying a nonclassical logic). Absolutely everything you are saying here is correct and on target Keith. I once put up a post on apartness (having only learned about it). At one point in the thread, I showed you some axioms. Your response was that it was some sort of boring version of set theory--and it is. But, my interest in apartness did not stem from non-classical mathematics. It arose in relation to semantics realized from strictly classical considerations. Experience seems to be telling me that there is simply no way for me to motivate interest in this matter. If I try to say something about set theory, I am told that sets are archetypes. If I try to point out that the questions involving descriptive set theory are the ones at issue, I am told that this is not Zermelo-Fraenkel set theory. If I try to point to... It really does not matter. The question is not one of utility, relevance, or interest. It is a question of correctness. And, it has little or no rami[CapitalThorn]cation for mathematicians unless one is simply interested in how the structure presents itself in mathematics. I tried to direct you to a paper where exclusive disjunction and logical equivalence are the only exceptions with the hope that it might give you some starting point. Unfortunately, it seems that Citeseer has removed the page for some reason. You say that I am guessing about other people's motivations. Well to be honest, not seem to see what I see or how to motivate them. And, I am sincerely sorry for anything that might have been construed badly (as well as the little ßair of anger that did reßect my frustration). So, for my part, I have failed. As for what you said about propositions, Jacek Malinowski is doing just that. He has some online papers at http://www.uni.torun.pl/~jacekm/publications.htm and you should be able to [CapitalThorn]nd something along those lines there. With regard to your statement about teaching *as logic* you might read his paper on Strawson presupposition and logical entailment. There is a sense in which there is no logic to be taught. This reßects a difference between what I have been looking at and what Galathaea wanted to accomplish. And, by the way, thanks. :-) mitch === Subject: : Re: Give me that old time ontology: (was: the anticlassicalist }{ i: > |It is far too easy to be dismissive of something if you have not considered > |it. It seems to me that if I really haven't considered something, I tend to > be noncommital about it. |For instance, in this case there is most certainly a manifestation of the > |non-standard model in the presentation of mathematical literature. That > |possibility did not seem to enter your thoughts at all. Excuse me if I [CapitalThorn]nd this a somewhat peculiar comment. This notion of > a manifestation of [a] nonstandard model in the literature would > seem broad enough to include cases of mathematicians unwittingly > doing things in a manner which is best explained by constructive facts of > which they're unaware, since they come up naturally for consideration > only if one is thinking constructively. I've looked for such cases, since it > seems like it would be useful motivation. (I'm afraid I haven't spent much > time on it or had much success. It's like looking for obvious gold nuggets > in streams.) In the discussion up to this point, I don't remember anybody asking me > anything that would have revealed whether such thoughts are on my mind. > Would having such thoughts cause me to be amazed to [CapitalThorn]nd that there > is (in a certain sense) a non-Boolean algebra modelling classical > propositional logic? No, I would say rather the opposite; the more familiar > a person is with potential manifestations of nonstandard models, the > less surprising the more mundane of them seem. Well they are surprising. They're more surprising than anything in the mathematical literature. Since it's the whole foundation of intuitionist logic. And since it's the only reason that the theory of evolution even exists, it's also most of the reason that science even exists. === Subject: : Re: Give me that old time ontology: (was: the anticlassicalist }{ i: |I must agree with this. I think that is why I felt so hurt that Keith had |dismissed my questions and focused more on attacking my legitimacy. It |doesn't change my high esteem of his helpfulness on these groups, but it |does leave me a bit sore... I think you should consider whether I really dismissed your questions. As I intended to explain at one point, whether nonclassical logic could usefully be introduced into the curriculum (outside of certain specialized courses) strikes me as interesting, but I don't see how I could offer any kind of de[CapitalThorn]nite answer to it. Don't generalize from not considering certain things legitimate to not considering certain people personally legitimate. Only a few rare individuals manage to go from making me wish they did things differently on usenet, to making me wish they'd just go away and leave us alone, and it takes a certain dedication to being a pest to get there. Keith Ramsay === Subject: : Re: ?? categoricity or naive felicity ?? |I will also add that set theory provides a similar distinction. Logic is too |weak to support strong assumptions like identity as a formal symbol of the |language or eliminable identity. But, getting anyone to consider the |rami[CapitalThorn]cations of that is nearly impossible. Perhaps because identity can be included as a formal symbol in the language, or can be eliminated in favor of the extensional de[CapitalThorn]nition of equality? Keith Ramsay === Subject: : Re: ?? categoricity or naive felicity ?? : |I will also add that set theory provides a similar distinction. Logic is too : |weak to support strong assumptions like identity as a formal symbol of the : |language or eliminable identity. But, getting anyone to consider the : |rami[CapitalThorn]cations of that is nearly impossible. : : Perhaps because identity can be included as a formal symbol in the : language, or can be eliminated in favor of the extensional de[CapitalThorn]nition : of equality? I too will not go into much detail here, because there is a corpus available if you are truly interested in such questions. However, besides the applications to [CapitalThorn]elds such as pattern recognition that I have already mentioned, and some standard applications in type theory, you are right to see a connection between the intensional and extensional. However, the use of extensional notions of identity can get one into some dif[CapitalThorn]culty in many areas of legitimate mathematical interest. Take for instance, quantum number that allows one to speak of systems of n electrons, say, but not one relationships will not count for you (and identity is not an appropriate formal symbol -- remember that logic is about naming relationships). Even proof theory makes use of the intensional notions of identity -- or else there would be no proof theory! And very much all of these various applications focus on cognitive and observational theories of the way we can reason about the world around us (which has been historically one of the major sources of intuitionistic theory). You see, there really are uses for such models. These are not the type of questions that can be swept under a rug of extensionality or relegated to the philosophy departments. The people who gave us modern formal logics have considered these issues very important, and they are more important now because of their uses. Even Birkhoff's Lattice Theory makes no such presuppositions. In fact, if you look to page 212 of Goldblatt's Topoi: the categorial analysis of logic (a book I've been mentioning a lot lately for some strange concordance of topics), the concept of intension is speci[CapitalThorn]cally linked to the notion of faithful representation in an ontology. -- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: : Re: ?? categoricity or naive felicity ?? > |I will also add that set theory provides a similar distinction. Logic is too > |weak to support strong assumptions like identity as a formal symbol of the > |language or eliminable identity. But, getting anyone to consider the > |rami[CapitalThorn]cations of that is nearly impossible. > Perhaps because identity can be included as a formal symbol in the > language, or can be eliminated in favor of the extensional de[CapitalThorn]nition > of equality? Whatever. I am really not going to pursue the same discussion that I have had for the past year elsewhere. :-) Frege is God. Mathematical ontology derives from contradiction. We are all happy idiots. :-) mitch === Subject: : Re: lim, lim sup for a function > I was reading the de[CapitalThorn]nition of the Dini derivative D^+f(x)=liminf_{h --> 0+} (f(x+h)-f(x))/h And I don't understand what does mean limsup of a function and in what > it is different from the lim. limsup_{x -> a} f(x) is equal to lim_{x -> 0} sup f(]a - x,a + x[). So, for instance, if you take f(x) = sin(1/x), lim_{x -> 0} f(x) does not exist, but limsup_{x -> 0} f(x) = 1. === Subject: : Multiplies in different bases Sorry for the confusing post, but I can't think of a better way to explain it. Short Version: In base 2^32 (or any base for that matter) is it possible that the number found by adding the low result of a multiply to the high result of a multiply will be greater then the current base EG supposing integer base Ôn', I multiply (n-1)*(n-1) to get a high and a low result - it will take 2 symbols in base Ôn' to show the result. Will the sum of these 2 results ever be greater then or equal to Ôn'? Long winded version: In base ten, suppose I want to calculate 999 * 9. in the [CapitalThorn]rst digit, I have 9 * 9 = 81. I put the 1 in the result in the leftmost place, and write the high part of the result (the 8) above the next digit - in this case, 9. I then move along to the next digit - the middle 9. I do 9*9 again, get 81, and this time add the last high result to the low result of the multiply. eg add the 8 to the 1 to obtain 9. I put this 9 in the next place of the result For the last 9, again I preform 9*9, get 81, add the previous high result (8) to the low result to get yet another 9. This time I write the high result to the end. I am currently writing a multiword multiply routine for a 32 bit processor. I'm thinking about the problem as if each 32 bit word is in base (2^32). My question is... Is there ever a case where, when adding the low result of a multiply to a previous high result the result of that will be larger then 2^32? In my above example, the situation for base ten is to is add the 8 (High result from previous multiply) to the 1 (low result from current multiply) to obtain 9.). 9 is less then ten, so there is no need to check for the presence of a high result in base ten. What about other bases? thanks very much, and this isn't homework :-) Al === Subject: : Re: Multiplies in different bases thanks for the replies... The example of 77*7 was so simple, I must have been half alseep. cheers, Al === Subject: : Re: Multiplies in different bases In-reply-to: Al Borowski In base 2^32 (or any base for that matter) is it possible that the >number found by adding the low result of a multiply to the high result >of a multiply will be greater then the current base >EG supposing integer base Ôn', I multiply (n-1)*(n-1) to get a high and >a low result - it will take 2 symbols in base Ôn' to show the result. >Will the sum of these 2 results ever be greater then or equal to Ôn'? >Long winded version: >In base ten, suppose I want to calculate 999 * 9. >in the [CapitalThorn]rst digit, I have 9 * 9 = 81. I put the 1 in the result in the >leftmost place, and write the high part of the result (the 8) above >the next digit - in this case, 9. >I then move along to the next digit - the middle 9. I do 9*9 again, get >81, and this time add the last high result to the low result of the >multiply. eg add the 8 to the 1 to obtain 9. I put this 9 in the next >place of the result >For the last 9, again I preform 9*9, get 81, add the previous high >result (8) to the low result to get yet another 9. This time I write the >high result to the end. >I am currently writing a multiword multiply routine for a 32 bit >processor. I'm thinking about the problem as if each 32 bit word is in >base (2^32). >My question is... Is there ever a case where, when adding the low >result of a multiply to a previous high result the result of that will >be larger then 2^32? >In my above example, the situation for base ten is to is add the 8 (High >result from previous multiply) to the 1 (low result from current >multiply) to obtain 9.). 9 is less then ten, so there is no need to >check for the presence of a high result in base ten. What about other bases? In base ten, consider 19*9. The high part of the product of 9*9 is 8 and the low part of the product of 1*9 is 9. 8+9 = 17 > 10. Are you concerned about this or are you only concerned about the high and low parts of the product of two digits? If the latter, then consider 7*7, whose high and low part sum to 13. Rob Johnson take out the trash before replying === Subject: : Re: Multiplies in different bases >Sorry for the confusing post, but I can't think of a better way to >explain it. >Short Version: >In base 2^32 (or any base for that matter) is it possible that the >number found by adding the low result of a multiply to the high result >of a multiply will be greater then the current base >EG supposing integer base Ôn', I multiply (n-1)*(n-1) to get a high and >a low result - it will take 2 symbols in base Ôn' to show the result. >Will the sum of these 2 results ever be greater then or equal to Ôn'? In base 10, consider 6 * 8 or 7 * 7. John Roberts-Jones === Subject: : Lagrange equation problem about reversely deducing the unknown parameter of equation under maximal value is known Dear all: I have two problems about optimazation. Problem(1) is solved. And I spend much time trying to obtain the solution of problem (2). But I still can't........... The description is as below: Problem (1) Max. X'(ADRDA)^-1 X=C S.T. X'(DWD)^-1 X=C X: n*1 vector A(is given): n*n matrix (Offdiagonal elements are zero) D(is given): n*n standard deviation matrix (Offdiagonal elements are zero) R(unknown) : n*n correlatiion matrix (diagonal elements are 1 and offdiagonal elements are unknown) W(is given): n*n correlatiion matrix (diagonal elements are 1 and offdiagonal elements are given) C(is given): constant Now, my goal is to obtain unknow R and i take advantage of Lagrange multiplier. L=X'(ADRDA)^-1 X - Z[X'(DWD)^-1 X -C] dL/dX = 2(ADRDA)^-1 X - 2Z(DWD)^-1 X =0......(1) dL/dZ =X'(DWD)^-1 X -C =0....................(2) let X'*(1) we get X'(ADRDA)^-1 X - ZX'(DWD)^-1 X=0 Z =[X'(ADRDA)^-1 X] / [X'(DWD)^-1 X] Because X'(DWD)^-1 X =C (constraint) max X'(ADRDA)^-1 X =C(is given) so maxZ =1 ...substitude into (1) and we get (ADRDA)^-1 X - (DWD)^-1 X =0 [(ADRDA)^-1-(DWD)^-1] X =0 If X have nontrival solution so that [(ADRDA)^-1-(DWD)^-1] are nonfull rank so let det((ADRDA)^-1-(DWD)^-1)=0 and then we can get unknown R. Problem(2) Max. (X-T)'(ADRDA)^-1 (X-T)=C S.T. (X-M)'(DWD)^-1 (X-M)=C X: n*1 vector T,M(is given):n*1 vector A(is given): n*n matrix (Offdiagonal elements are zero) D(is given): n*n standard deviation matrix (Offdiagonal elements are zero) R(unknown) : n*n correlatiion matrix (diagonal elements are 1 and offdiagonal elements are unknown) W(is given): n*n correlatiion matrix (diagonal elements are 1 and offdiagonal elements are given) C(is given): constant My goal is to obtain unknown R. How to do??? It is so dif[CapitalThorn]cult for me and any body can help me or give me a hint? thx very very much Jammy My Mail : R91546020@ntu.edu.tw === Subject: : easy topolgy problem~ hello........ 1. topologist's sine curve is not locally connected at (0,0) topologist's sine curve = {(x,y) | y=sin(1/x),0 hello........ 1. topologist's sine curve is not locally connected at (0,0) topologist's sine curve > = {(x,y) | y=sin(1/x),0 True 2. Y is locally connected at (0,0) Y = {(x,y) | y=sin(1/x),0 No, it's the same. As soon as (0,0) is in your space, it cannot be locally connected. It's not the vertical line that cause problem but the graph x->sin(1/x). A neighborohood of (0,0) contains disconnected fragments of this graph. You can use Z = {(x,y) | y=sin(1/x),0 2. Y is locally connected at (0,0) Y = {(x,y) | y=sin(1/x),0R f(x+y) = f(x) + f(y), show that if lim f(x+y) = L x->0 then L = 0 ---------------------------- um.....it is right problem?? === Subject: : Re: easy problem.... In sci.math, hot-girl : > f:R->R > f(x+y) = f(x) + f(y), > show that if > lim f(x+y) = L > x->0 > then L = 0 > ---------------------------- > um.....it is right problem?? I suspect you mean f(x+y) = f(x) * f(y). That would make f(x+y) some variant of exp(), if it's continuous. That problem, at least, I've seen before. Either f() is identically 0 everywhere, or f(0 + y) = f(0) * f(y) = f(y) for all y, which means f(0) = 1. If you meant lim f(x+y) = L x,y->(0,0) then it seems to me L can be 0 or 1. If you do mean f(x+y) = f(x) + f(y), then few conclusions can be drawn regarding the limit, since y is a free variable; L = L(y) is therefore some function in y, which may or may not be f(y) (although if f() is continuous they should be equal). If you mean something even more esoteric, such as f(x+y) = f(x) / f(y) (which of course has some very weird problems, chießy f(x) / f(y) = f(y) / f(x) as required by the commutativity of addition, which means f(x) = 1 everywhere) then you may want to reconsult your problem set. (Either that, or your instructor might! :-) ) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: : convex problem..... i know convex function. example....f(x) = x^2 form.... f(x) = -x^2 is not convex. because def of convex is f((1-t)x + ty) <= (1-t)f(x) + t*f(y) , 0<= t <=1 and i know that f is convex <=> f''(x) >=0 but...i think....... / / / / / / / / this is graph... f'(x) : + -> - f''(x) : + -> + if case, f'' > 0 and convex but left section is -x^2 form and right section is x^2 form... thus this graph don't satis[CapitalThorn]ed convex def. how do you think about it?? === Subject: : Re: Permutation Group Automorphisms > This is probably a trivial and/or basic question, but so far I > haven't been able to prove nor disprove it myself, nor [CapitalThorn]nd the > answer in any book I have on hand: If G is a group with an (outer) automorphism that mixes 2 > conjugacy classes of subgroups, are the images of the actions of > G on the cosets of the 2 classes necessarily permutation > isomorphic? I think the answer will be yes. Let H be a subgroup of G and let H' be the image of H under an automorphism alpha. Suppose we list the cosets of H with speci[CapitalThorn]c representatives: H*1, H*c_2,...,H*c_[G:h]. The action of the element g of G on these cosets is determined by relations of the form H*c_i*g=H*c_j. Any automorphism mapping H to H' can be applied to all these equations and sets so the actions should be permutation isomorphic... this reasoning isn't completely rigorous as it stands, but I hope it will guide you to a rigorous answer to your question. After you have one, you may want to try playing around with the symmetric group S_6, if you haven't been doing that in trying to construct counterexamples to your question. ---- David === Subject: : Re: Permutation Group Automorphisms >This is probably a trivial and/or basic question, but so far I >haven't been able to prove nor disprove it myself, nor [CapitalThorn]nd the >answer in any book I have on hand: >If G is a group with an (outer) automorphism that mixes 2 >conjugacy classes of subgroups, are the images of the actions of >G on the cosets of the 2 classes necessarily permutation >isomorphic? Let O = { H x1, ..., H xr } be the set of cosets of one of the subgroups. Then, if f is the automorphism of G, the set of cosets of the image subgroup is f(O) = { f(H)f(x1), ..., f(H)f(xr) }. Then the action of g (on the right) on O clearly corresponds to the action of f(g) on f(O), so the images of G in Sym(O) and Sym(f(O)) are permutation isomorphic. But the two actions are G-equivalent actions if and only is H and f(H) are conjugate in G. To be G-equivalent, the actions of g on O and on f(O) must correspond. In general, if the coset actions of G on subgroups H1 and H2 are G-equivalent, and the equivalence maps H1 to coset H2 a, then the stabilizer in G of H1, which is H1, is equal to the stabilizer in G on H2 a, which is a^-1 H2 a, so H1 = a^-1 H2 a. Conversely, if this is the case then there is a G-equivalence mapping H1 to H2 a. As an example, L(3,2) has two conjugacy classes of subgroups isomorphic to S4. These give rise to two inequivalent actions of G on 7 points. But the images of these actions are conjugate subgroups of S7. Derek Holt. === Subject: : Re: Half a hole, and its true! IMO > the real question in this, is why people say there's no such thing as half a > hole? is it because they didn't realized what the original was? that's what > I'm assuming. Suppose I have a rectangle with width 2, length 6. I draw a new line which cuts it in half parallel to the width, and get two rectangles with width 2 and length 3. Now, either new rectangle has half the area of the original, but does that mean it is half a rectangle and not, in fact, a rectangle? A similar issue exists with your holes. You can dig a hole with half the volume of some other hole, but it is still simply a hole, not half a hole. -- Will Twentyman email: wtwentyman at copper dot net === Subject: : Compact but not sequentially compact by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i29CwEC06366; The examples of sets that are compact but not sequentially compact are very common! However, is there a set that is sequentially compact but not compact? Thanks Diogo === Subject: : Re: Compact but not sequentially compact > The examples of sets that are compact but not sequentially compact are very > common! However, is there a set that is sequentially compact but not compact? Thanks Diogo > First uncountable ordinal, with order topology (and without a last point). -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: : Re: where to get latex help >> Not to be confused with comp.tex.text, which unfortunately exists >> but seems to be unused. >Actually, it doesn't exist, not of[CapitalThorn]cially. (That is, according to >the most of[CapitalThorn]cial of the newsgroup lists, comp.text.tex exists but >comp.tex.text doesn't.) >The fact that it may appear on some news servers even though it >doesn't of[CapitalThorn]cially exist is a technical Usenet curiousity that >probably isn't worth getting into. Suf[CapitalThorn]ce it to say that people >whose news servers mistakenly believe comp.tex.text exists may be >able to have conversations that the rest of us will never see Possibly - when I tried it I found I was just talking to myself. >(and >that Google will never archive unless they're crossposted to real >groups.) ************************ David C. Ullrich === Subject: : Re: Schedule-writing This seems like somehow it should be a problem in graph theory. > Not sure :P I am looking for an algorithm to help write a schedule: a group > of people, divided into 3 shifts, to cover several positions which > must be manned 24/7, giving people fairly regular weekends (but not > necessarily on the weekend proper)... and the algorithm should be > ßexible enough to handle both one-time requests for days off as well > as regular requests (like every other thursday). Also it should (over > long periods of time) work everyone close to an equal amount, > regardless of above mentioned requests. Any hints? Any speci[CapitalThorn]c algorithms I should look up? This is of > course no homework. I am in[CapitalThorn]nitely in your debt for any help you can > give. See Desrosiers et al, Time constrained routing and scheduling, in Ball et al, eds, Network Routing, Handbooks in OR&MS, North-Holland, Amsterdam, 1995. It's a pretty comprehensive survey about all kinds of (crew, vehicle, staff) scheduling with all bells and whistles and should cover your problem nicely. Cheers Tuomo === Subject: : Re: how many discontinuous points > Let f be any function from R to R, and de[CapitalThorn]ne g as g(x)={f(x) if f is continuous at x > {0 otherwise Can g be discontinuous at uncountably many points, or is there a proof that the number of points, where it is discontinuous, has to be countable? Consider the Cantor function h(x). now let f(x)=h(x) if x does belong to Cantor set abd f(x)=0 otherwise consider f(x) and the corresponding g(x) then. === Subject: : Re: area of a circle Ignacio Larrosa Ca.96estro > Keith A. Lewis > given a circle with 2 chords cutting perpendicularly and then > bisecting the four angles with a 45 degree angle..thus giving 8 > regions in a circle. turns out that the alternate regions areas sum > equals the other four regions areas. is there a nice proof for this. > seems like i should have learned this in my college geometry class/ > thanks in advance. > The integrals are messy, but cancel trivially ... > Is there a elementary, better euclidean, proof? I drew four lines through a point P, producing 8 equal angles, and then a circle surrounding P. If we _rotate_ all the lines by dTheta around P, one of the sum-areas changes by (AA - BB + CC - ... - HH) dTheta/2 (where A,B,...,H are lengths of segments from P to the circle) which is zero by Euclid III.35 about two intersecting chords of a circle. Thus we can integrate zero up to some theta such that the claim is trivial by a reßection symmetry. No doubt the result extends somehow to 2n segments for n other than four. LH === Subject: : Re: area of a circle > Ignacio Larrosa Ca.96estro >Keith A. Lewis >given a circle with 2 chords cutting perpendicularly and then >bisecting the four angles with a 45 degree angle..thus giving 8 >regions in a circle. turns out that the alternate regions areas sum >equals the other four regions areas. is there a nice proof for this. >seems like i should have learned this in my college geometry class/ >thanks in advance. >>The integrals are messy, but cancel trivially ... >>Is there a elementary, better euclidean, proof? I drew four lines through a point P, producing 8 equal angles, and then > a circle surrounding P. If we _rotate_ all the lines by dTheta around > P, one of the sum-areas changes by > (AA - BB + CC - ... - HH) dTheta/2 > (where A,B,...,H are lengths of segments from P to the circle) I think you mean here AA = A*A = A^2 where A = length of PA, one of the two parts of chord AE. > which is zero by Euclid III.35 about two intersecting chords of a circle. ???? _ANY_ chords ??? Not shure... the proof should be detailed a little more... If you draw 3 chords AB, CD, EF intersecting at 6 equal angles pi/3 with segments PA=a, PB=b...PF=f (vertices A,C,E,B,D,F in that order on the circle) you get : dArea = (a^2 - c^2 + e^2 - b^2 + d^2 - f^2) dTheta/2 IMHO not z.8ero. This is OK here because you have _eight_ equal angles, so the chords are rectangular by pairs. The hint is that for two _RECTANGULAR_ chords, with length AB=PA+PB=a+b and CD=PC+PD=c+d a^2 + b^2 + c^2 + d^2 = 4*R^2 (easy with some pythagoras with the center and the middles of chords) If you take two other _RECTANGULAR_ chords offset from the [CapitalThorn]rst set by angle alpha, the sum of one over two regions satis[CapitalThorn]es : dArea=(a^2 - a'^2 + b^2 - b'^2 + c^2 - c'^2 + d^2 - d'^2)*dTheta/2 = 0 because a^2 + b^2 + c^2 + d^2 = a'^2 + b'^2 + c'^2 + d'^2 (= 4*R^2) So this sum is constant when you rotate the whole pattern. Of course you choose alpha = pi/4 to get equal areas by symetry. > Thus we can integrate zero up to some theta such that the claim is > trivial by a reßection symmetry. > No doubt the result extends somehow to 2n segments for n other > than four. > LH Result doesn't extend to 2n but to 4n regions (n at least 2) (the chords pairs _must_ be rectangular). You split the regions in n sets of each 4 regions by taking one region over n. The n sets are equal areas if 4n equal angles. regards. -- philippe (chephip at free dot fr) === Subject: : Re: area of a circle > En el mensaje:c2ih22$1tr2m6$1@ID-137122.news.uni-berlin.de, >08:52:44 -0800: >>given a circle with 2 chords cutting perpendicularly and then >>bisecting the four angles with a 45 degree angle..thus giving 8 >>regions in a circle. turns out that the alternate regions areas sum >>equals the other four regions areas. is there a nice proof for this. >>seems like i should have learned this in my college geometry class/ >>It is better put the intersection point at origin, and the centre >>elsewhere, integrating in polars. But the integrals are messy. It >>seems to me that are elliptical ones. But the numeric results seem to >>agree ... The integrals are messy, but cancel trivially ... Is there a elementary, better euclidean, proof? I've found this as Pizza theorem with google An explanation (proof ?) given at http://www.maths.unsw.edu.au/~mikeh/webpapers/paper57.pdf (or the same as paper57.ps at same site) But... The [CapitalThorn]gures are missing. I guess the [CapitalThorn]g 2 is : chord AC and BD intersect in P, O is the center of the circle M and N are the middles of chords AC and BD, to satisfy the NP=|a-c|/2 and MD=(b+d)/2 statements. I don't understand the OD^3 = OM^3 + MD^3 What is true is OD^2 = OM^2 + MD^2 (pythagora) But for cubes... The proof requires this to be true for cubes ! In fact it seems there is a typo in the paper IMHO the differential area in polar coordinates is 1/2 r^2 dtheta and not 1/2 r^3 dtheta as claimed in the paper. Correcting this (changing all cubes into squares) the proof becomes OK regards -- philippe (chephip at free dot fr) === Subject: : Re: area of a circle En el mensaje:404DAC02.90709@free.invalid, philippe 92 escribi.97: >> En el mensaje:c2ih22$1tr2m6$1@ID-137122.news.uni-berlin.de, >> 08:52:44 -0800: > given a circle with 2 chords cutting perpendicularly and then > bisecting the four angles with a 45 degree angle..thus giving 8 > regions in a circle. turns out that the alternate regions areas > sum equals the other four regions areas. is there a nice proof > for this. seems like i should have learned this in my college > geometry class/ > It is better put the intersection point at origin, and the centre > elsewhere, integrating in polars. But the integrals are messy. It > seems to me that are elliptical ones. But the numeric results seem > to agree ... >> The integrals are messy, but cancel trivially ... >> Is there a elementary, better euclidean, proof? > I've found this as Pizza theorem with google > An explanation (proof ?) given at > http://www.maths.unsw.edu.au/~mikeh/webpapers/paper57.pdf > (or the same as paper57.ps at same site) > But... > The [CapitalThorn]gures are missing. > I guess the [CapitalThorn]g 2 is : > chord AC and BD intersect in P, O is the center of the circle > M and N are the middles of chords AC and BD, to satisfy the > NP=|a-c|/2 and MD=(b+d)/2 statements. M is the middpoint of BD and N of the AC. > I don't understand the > OD^3 = OM^3 + MD^3 > What is true is > OD^2 = OM^2 + MD^2 (pythagora) > But for cubes... > The proof requires this to be true for cubes ! > In fact it seems there is a typo in the paper > IMHO the differential area in polar coordinates is > 1/2 r^2 dtheta > and not 1/2 r^3 dtheta as claimed in the paper. > Correcting this (changing all cubes into squares) the proof becomes OK You must be a problem with your reader. All the exponents I see in the paper are squares. In both, the .PDF and the .PS [CapitalThorn]les. At http://www.xente.mundo-r.com/ilarrosa/Teorema_Pizza.GIF you have a version of [CapitalThorn]g 2. -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: : Re: area of a circle [...] You must be a problem with your reader. All the exponents I see in the paper > are squares. In both, the .PDF and the .PS [CapitalThorn]les. Very strange... it is the only document pdf/ps with that bug... The ps is viewed by extracting, then displaying with GSview (Ghostgum), the pdf is viewed by acrobat reader 5.0.5. At http://www.xente.mundo-r.com/ilarrosa/Teorema_Pizza.GIF you have a > version of [CapitalThorn]g 2. Thanks. It corresponds to the one I had drawn. BTW, only one equation suf[CapitalThorn]ce : R^2 = OD^2 = OM^2 + MD^2 = NP^2 + MP^2 = ((PA-PC)/2)^2 + ((PD+PB)/2)^2 = (PA^2 + PB^2 + PC^2 + PD^2)/4 + PD*PB - PA*PC But PD*PB = PA*PC (power of P), QED. -- philippe (chephip at free dot fr) === Subject: : Re: area of a circle En el mensaje:404DF6A5.1060405@free.invalid, philippe 92 escribi.97: > [...] >> You must be a problem with your reader. All the exponents I see in >> the paper are squares. In both, the .PDF and the .PS [CapitalThorn]les. > Very strange... it is the only document pdf/ps with that bug... > The ps is viewed by extracting, then displaying with GSview > (Ghostgum), the pdf is viewed by acrobat reader 5.0.5. Curious. I also use Gsview 4.5 (Ghostgum) and Acrobat Reader 6.0. >> At http://www.xente.mundo-r.com/ilarrosa/Teorema_Pizza.GIF you have a >> version of [CapitalThorn]g 2. > Thanks. It corresponds to the one I had drawn. > BTW, only one equation suf[CapitalThorn]ce : Two minor bugs: > R^2 = OD^2 = OM^2 + MD^2 = NP^2 + MP^2 = R^2 = OD^2 = OM^2 + MD^2 = NP^2 + MD^2 = > ((PA-PC)/2)^2 + ((PD+PB)/2)^2 = > (PA^2 + PB^2 + PC^2 + PD^2)/4 + PD*PB - PA*PC (PA^2 + PB^2 + PC^2 + PD^2)/4 + (PD*PB - PA*PC)/2 > But PD*PB = PA*PC (power of P), QED. A tasty theorem, certainly ... -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: : Re: Cantor Paradox :-) >[...] >I understand now that Forte Agent (which David uses) indeed has a bug. >It is based on what sendmail does do (insert a CRLF followed by a >tab), but not on what the standard allows. Luckily this is sci.math, where the concept of off topic doesn't exist... What makes you conclude it gets tabs right? Just for fun I performed a few experiments just now in 01alt.test; the thread Ullrich test 2 used linebreak/tab combinations, and it doens't look right in Agent. ************************ David C. Ullrich === Subject: : Re: Cantor Paradox :-) >Luckily this is sci.math, where the concept of off topic doesn't >exist... ITYM where the concept of Ôon topic' doesn't exist. HTH. -- I'm not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: : Re: Cantor Paradox :-) >[...] >I understand now that Forte Agent (which David uses) indeed has a bug. So I decided a few days ago. I've told them about it - remains to be seen what I win... >It is based on what sendmail does do (insert a CRLF followed by a >tab), but not on what the standard allows. ************************ David C. Ullrich === Subject: : easy algebra problem... dihedral group is nonabelian -------------------------- um......it is true or false?? === Subject: : Re: easy algebra problem... > dihedral group is nonabelian -------------------------- um......it is true or false?? True. === Subject: : Re: easy algebra problem... >> dihedral group is nonabelian >True. What about D_2? -- I'm not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: : Re: easy algebra problem... Toni Lassila write: > dihedral group is nonabelian >>True. What about D_2? I see the dihedral group D_n as the symmetry group of an n-sided regular polygon; therefore n > 2. But I agree that if we de[CapitalThorn]ne D_n as the group generated by two generators r and f together with the relations r^n = 1, f^2 = 1 and f^{-1}.r.f = r^{-1}, then this de[CapitalThorn]nition still makes sense when n = 2. In this case, of course that D_2 is abelian (and so is D_1, of course :-) ). === Subject: : Re: Ln typo or....? Adjunct Assistant Professor at the University of Montana. Cc: >>I have a problem that requires a proof that ln(a) + ln(b) = ln(a + b) >>by somehow de[CapitalThorn]ning the ln number as the area under the curve of y = >>1/x ....which makes no sense to me because ln(a) + ln(b) is not equal >>to ln(a + b). >> That should probably be ln(a) + ln(b) = ln(a x b). >> You can de[CapitalThorn]ne ln(a), for a>=1, to be the area under y=1/x from 1 to >> a. If 0> 1. >I agree, it should probably have been ln(a x b). All areas under the >curve y = 1/x for x>0 are positive and for values between 0 and 1, the >de[CapitalThorn]nite integral takes care of this even though the ln values are >negative. Int[a,b]1/x dx = ln(x)[a,b]. For a = .5, b = .8 this becomes >ln(.8) - ln(.5) = -.223 - (-.693) = .470. I think this is what you were >trying to say but your reference to minus the area is a little >confusing. Don't see why: area is positive, but ln(.5) is a negative number. By de[CapitalThorn]nition, the de[CapitalThorn]nite integral is the ->net signed area<- under the curve on the interval; that means that area is considered positive if it is above the X-axis and is measured left to right, or if it is below the X-axis and measured right to left; and considered negative if it is below the X-axis and measured right to left, or if it is below the X-axis and measured left to right. So, ln(a) is NOT the area under 1/x on the interval between a and 1, but the signed area: if a>1, you take the area under 1/x on [1,a]; if 0and multiply by -1 to get the answer<-. -- = = 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: Ln typo or....? I have a problem that requires a proof that ln(a) + ln(b) = ln(a + b) Then you're in deep doo-doo, unless for example a = b = 2. ....which makes no sense to me because ln(a) + ln(b) is not equal >to ln(a + b). Right. The proof you sketch is actually not so hard if you > switch to proving the right thing: ln(ab) = ln(a) + ln(b). Must have been a typo. Yes, but it isn't so obvious to me that the sum of 2 areas under the > curve of y = 1/x is the same as another single area, minus any circular > reasoning. Say for ln(ab) = ln(a) + ln(b) we take ln(a) as the area from > x = 1 to 2 and ln(b) as the area from 1 to 3, this is the same as ln 2 + > ln 3 = ln 6. The area from x = 1 to 6 isn't obviously the same as the > sum of the two smaller areas. What am I missing? A proof. You need to PROVE the two areas are the same, not just stare at them and hope they are! There are a number of ways of doing this. Perhaps the easiest is to use calculus. You de[CapitalThorn]ne ln(x) to be the integral from 1 to x of (1/t) with respect to t. Now consider the function f(x) = ln(ax) = integral from 0 to ax of (1/t) with respect to t. that the derivative of f(x) with respect to x is equal to the function we are integrating evaluated at the upper limit, times the derivative of the upper limit with respect to x. IN this case, we have (1/ax)*(a) = 1/x. Likewise, the derivative of ln(x) is (1/x). Therefore, by the Constant Difference Theorem, ln(x) and f(x) differ by a constant (they have the same derivative); that is, there is a constant k such that for all x, f(x) - ln(x) = k; Therefore, there is a constant k such that for all x, ln(ax) - ln(x) = k, for all x on (0,in[CapitalThorn]nity). Now substituting x = 1 into the equation we have k = ln(a) - ln(1) = ln(a). + ln(x). Arturo Magidin, sans .sig === Subject: : Re: Ln typo or....? > A proof. You need to PROVE the two areas are the same, not just stare > at them and hope they are! Yes, it's never worked for me. > There are a number of ways of doing this. Perhaps the easiest is to > use calculus. You de[CapitalThorn]ne ln(x) to be the integral from 1 to x of (1/t) > with respect to t. Now consider the function f(x) = ln(ax) = integral > from 0 to ax of (1/t) with respect to t. > that the derivative of f(x) with respect to x is equal to the function > we are integrating evaluated at the upper limit, times the derivative > of the upper limit with respect to x. IN this case, we have (1/ax)*(a) > = 1/x. > Likewise, the derivative of ln(x) is (1/x). Therefore, by the Constant > Difference Theorem, ln(x) and f(x) differ by a constant (they have the > same derivative); that is, there is a constant k such that for all x, > f(x) - ln(x) = k; > Therefore, there is a constant k such that for all x, ln(ax) - ln(x) = > k, for all x on (0,in[CapitalThorn]nity). > Now substituting x = 1 into the equation we have k = ln(a) - ln(1) = > ln(a). > + ln(x). I was going to use the substitution rule of integration by [CapitalThorn]rst de[CapitalThorn]ning g(t) = at. Then ln(ab) = Int[1,ab]1/x dx = Int[1,a]1/x dx + Int[a,ab]1/x dx = Int[1,a]1/x dx + Int[1,b]1/t dt = ln(a) + ln(b). What do you think? Phil Holman === Subject: : Re: Ln typo or....? |Yes, but it isn't so obvious to me that the sum of 2 areas under the |curve of y = 1/x is the same as another single area, minus any circular |reasoning. I hope circular reasoning doesn't make things seem obvious to you. |Say for ln(ab) = ln(a) + ln(b) we take ln(a) as the area from |x = 1 to 2 and ln(b) as the area from 1 to 3, this is the same as ln 2 + |ln 3 = ln 6. The area from x = 1 to 6 isn't obviously the same as the |sum of the two smaller areas. What am I missing? The area from 1 to 6 is obviously the sum of the areas froom 1 to 2 and from 2 to 6. So really you need to show that the area from 1 to 3 and from 2 to 6 are the same, or more generally, that the area from 1 to b and the area from a to ab are the same. There's a way to show it. Try to [CapitalThorn]nd it. Do you know any techniques of integration? Keith Ramsay === Subject: : Re: Ln typo or....? > |Yes, but it isn't so obvious to me that the sum of 2 areas under the > |curve of y = 1/x is the same as another single area, minus any circular > |reasoning. I hope circular reasoning doesn't make things seem obvious to you. |Say for ln(ab) = ln(a) + ln(b) we take ln(a) as the area from > |x = 1 to 2 and ln(b) as the area from 1 to 3, this is the same as ln 2 + > |ln 3 = ln 6. The area from x = 1 to 6 isn't obviously the same as the > |sum of the two smaller areas. What am I missing? The area from 1 to 6 is obviously the sum of the areas froom 1 to 2 > and from 2 to 6. So really you need to show that the area from 1 to 3 > and from 2 to 6 are the same, or more generally, that the area from > 1 to b and the area from a to ab are the same. There's a way to show it. Try to [CapitalThorn]nd it. Do you know any techniques > of integration? Keith Ramsay The substitution rule. Got it thanks. Phil Holman === Subject: : One Geometry algebra question I am starting to learn Geometry Algebra by myself, there is one problem Reßect line segment ( 3 e1 - 2 e2 + 4 e3) in the mirror described by e13 - e12. As far as I know planes can be described in terms of the area spanned by two directed line segments, we can refer to our mirror M as a product of two perpendicular line segments. In other words, we could write M = AB where A and B are perpendicular and have unit lengths. Any pair of unit line segments work for A and B as long as they are in the M plane and perpendicular. Lets chose A to be in the same direction as the direction of the part of the line L that is parallel to the mirror plane. More descriptive names will be assigned to A and B for this case. A becomes mpar and B becomes m_{perp}. The reßection of the line L in the mirror M is written as follows. L' = m{par}^{-1} L m_{par} I don't know how to [CapitalThorn]nd m_{par} Thanks a lot! === Subject: : Determining the Equation of a Shadow Cast onto a Surface Given a ßat circular disk (high-gain antenna), a planar surface (a solar array), and an arbitrary vector to a point light source (the Sun), I would like to determine the equation of the resultant shadow (which would be an ellipse) that would be cast onto the plane. === Subject: : Re: Determining the Equation of a Shadow Cast onto a Surface >Given a ßat circular disk (high-gain antenna), a planar surface (a >solar array), and an arbitrary vector to a point light source (the >Sun), I would like to determine the equation of the resultant shadow >(which would be an ellipse) that would be cast onto the plane. Never mind the previous post, it doesn't matter. I had a little time to kill today and you have an interesting problem here. I have a couple of Maple worksheets worked out for you. Are you looking for a general solution or do you have a speci[CapitalThorn]c set of data? And do you have access to Maple yourself? What I post for you depends on what you need. --Lynn === Subject: : Re: Determining the Equation of a Shadow Cast onto a Surface >Given a ßat circular disk (high-gain antenna), a planar surface (a >solar array), and an arbitrary vector to a point light source (the >Sun), I would like to determine the equation of the resultant shadow >(which would be an ellipse) that would be cast onto the plane. Is the antenna oriented arbitrarily or is it parallel to the plane? --Lynn === Subject: : =?ISO-8859-1?Q?=E9cart_type?= Question d'un inculte en la mati.8fre: un .8ecart type est il toujours -inf.8erieur/sup.8erieur/.8da d.8epend- au plus grand des extr.90mes? Merci. Marc. === Subject: : Re: =?ISO-8859-1?Q?=E9cart_type?= Marc a .8ecrit: > Question d'un inculte en la mati.8fre: un .8ecart type est il toujours > -inf.8erieur/sup.8erieur/.8da d.8epend- au plus grand des extr.90mes? .82a d.8epend. Par example, pour n'importe quel nombre x, l'.8ecart type de x - 1 et x + 1 c'est 1. Donc, il peut .90tre inf.8erieur ou sup.8erieur (ou .8egal) au plus grand des extr.8fmes. Ceci dit, je suppose que le newsgroup fr.sci.maths serait plus appropri.8e pour ce sujet. Cordialement, Jos.8e Carlos Santos === Subject: : Re: Topologie >Let U & V be two connected open subsets of the plane R. >If U & V have the same fondamental group, doe it mean than U & V are >homeomophs ? Yes, I think so. Here's one way to structure this question: Using the natural complex structure on R^2, U becomes a (connected) 1-dimensional complex manifold. Its universal cover X is then a simply- connected complex 1-manifold, of which there are three types: the complex plane, the unit disk, and the Riemann sphere. Of course any surjective image of the sphere would be compact, hence closed if embedded in a Euclidean space, so since U is open too, it'd have to be empty or all of R^2 (and in neither case is the Riemann sphere really its universal cover). So your question becomes, if we have two groups G, H of automorphisms of the complex plane or the unit disk, and the groups themselves are isomorphic, are the quotient spaces X/G and X/H homeomorphic? The answer isn't obviously yes (that is, for a general space, or object in another category, we can easily have subsgroups of Aut(X) which are abstractly isomorphic but not conjugate or otherwise related) but I think I have seen this result in the two cases X=C, X=Disk since we understand their groups of complex automorphisms. dave === Subject: : Re: Topologie Bcc: jdolan@math.ucr.edu,rusin >>Let U & V be two connected open subsets of the plane R. >>If U & V have the same fondamental group, doe it mean than U & V are >>homeomophs ? >Yes, I think so. Here's one way to structure this question: I would like to retract that statement until I've thought about it a little more, for two reasons, one prompted by someone in email: >Using the natural complex structure on R^2, U becomes a (connected) >1-dimensional complex manifold. Its universal cover X is I was being a little glib here. There are some technical conditions which must be satis[CapitalThorn]ed before one can make an identi[CapitalThorn]cation like U = X/G; usually one takes U to be locally connected and locally simply connected or something like that, and the OP didn't impose any such constraints. So at best my approach using covering spaces applies only to tame cases. >So your question becomes, if we have two groups G, H of automorphisms >of the complex plane or the unit disk, and the groups themselves are >isomorphic, are the quotient spaces X/G and X/H homeomorphic? [...] >but I think I have seen this result in the two cases X=C, X=Disk I'm going to have to track this down before I am willing to believe it in generality. Even in a fairly tame case this is not so clear to me right now. If for example U and V are obtained from the plane by removing a [CapitalThorn]nite set of points, then each open set has a 1-simplex as a deformation retract, from which it is clear that it has the homotopy type of a bouquet of circles, and in particular, the fundamental group is a free group on a number of generators equal to the number of points removed; if the groups are isomorphic, this requires the numbers of generators to be equal, so that the numbers of points removed from the plane must be equal, making U and V homeomorphic. But when the set of points is in[CapitalThorn]nite, I would be tempted to compute the fundamental group from a cover or something, leading to ( lim^1 ? ) computations and things which I have avoided whenever I see them coming. For example, what if U = R^2 - {1, 2, 3, ...} and V = R^2 - {0, 1, 1/2, 1/3, ...} ? In that case U and V are obviously homeomorphic (using inversion in C ). We have a family of inclusions e.g. U -> U_n := R^2 - {1, 2, ..., n} and it looks to me like pi_1(U) is the inverse limit (union) of the pi_1(U_n). Obviously the situation is the same with the homeomorphic family of spaces V_n . But if instead we start with V = R^2 - {1, 1/2, 1/3, ...}, then pi_1(V) must be larger than the union of the pi(V_n), since there are paths like f(t) = t cos(pi/t) (suitably completed to a closed loop) which are not homotopic to a [CapitalThorn]nite product of loops around single elements of R^2 - V. I don't know for sure whether in this case pi_1(V) is isomorphic to pi_1(U) (I suspect not) and I guess I also don't know for sure whether V is homeomorphic to U. (This V is not locally simply-connected.) I suppose you can make more interesting examples by removing other sets of points from R^2, and I won't hazard a guess as to what happens if, for example, you tell me to well-order R and then remove the [CapitalThorn]rst uncountable subset of R from C ... So like I said, I'm back to not really knowing the answer to the original question. dave PS -- the OP's original question may be adapted to ask, what if we know that U and V have the same [CapitalThorn]rst integer homology group? But here too it seems that we run into tricky issues regarding exactly which homology theory is used and so on. Here is a similar trap I fell into once: http://www.math-atlas.org/95/sum.vs.prod === Subject: : Re: Topologie === Subject: : Topologie >Let U & V be two connected open subsets of the plane R. >If U & V have the same fondamental group, doe it >mean than U & V are homeomophs ? No, the open half plane and the open unit square are both connected open & simply connected but not homeomorphic as one is compact and the other isn't. ---- === Subject: : Re: Topologie >Let U & V be two connected open subsets of the plane R. >If U & V have the same fondamental group, doe it >mean than U & V are homeomophs ? > No, the open half plane and the open unit square are both > connected open & simply connected but not homeomorphic as > one is compact and the other isn't. That's not true, as an open set, neither is compact. And actually, they are homeomorphic. Every simply connected open subset of the plane is homeomorphic to the open disk (that's part of the Riemannian mapping theorem). I don't know if it is true in general, can't help you there, sorry... Cheers Philipp === Subject: : Re: Topologie >Let U & V be two connected open subsets of the plane R. >If U & V have the same fondamental group, doe it >mean than U & V are homeomophs ? > No, the open half plane and the open unit square are both > connected open & simply connected but not homeomorphic as > one is compact and the other isn't. Actually: 1) None of them is compact. 2) They *are* homeomorphic. It is easy to deduce from the Riemann mapping theorem that any two connected and simply connected open subsets of the plane are homeomorphic. === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs >More importantly, there's my other point (somewhere crossthread), where I >was asking how the current mathematical models of continuous probability >will completely break down if you say that Ôzero probability' is the same >as Ôimpossible'. >Or to say it on other words: Why can't I state that if I pick a random >number, with continuous distribution on [0..1], that it's impossible to >pick the number 0.5 that way ? (Or any other given number for that matter.) >That, to me, is a far more interesting question. If you pick a random number in [0..1], every real in that range has a probability of being picked of 0. So (by your statement) it's impossible to pick ANY number in that range. So you can't pick a random number in [0..1]. That certainly breaks down the whole idea of continuous probability right there. -- Matthew T. Russotto mrussotto@speakeasy.net Extremism in defense of liberty is no vice, and moderation in pursuit of justice is no virtue. But extreme restriction of liberty in pursuit of a modicum of security is a very expensive vice. === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs >) If you pick a random number in [0..1], every real in that range has a >) probability of being picked of 0. So (by your statement) it's >) impossible to pick ANY number in that range. So you can't pick a >) random number in [0..1]. That certainly breaks down the whole idea of >) continuous probability right there. >No, I'm just saying it's impossible to pick any *given* number in that >range. Is this difference somehow hard to grasp ? It's a distinction without a difference. Whatever number is picked is a number you would have said (prior to the pick) would be impossible to be picked. -- Matthew T. Russotto mrussotto@speakeasy.net Extremism in defense of liberty is no vice, and moderation in pursuit of justice is no virtue. But extreme restriction of liberty in pursuit of a modicum of security is a very expensive vice. === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs ) It's a distinction without a difference. Whatever number is picked is ) a number you would have said (prior to the pick) would be impossible ) to be picked. You can't go back in time and make me pick exactly that number. (The fact that I won't be able to describe that number is beside the point.) If you pick two numbers randomly from [0..1], is it possible that those two numbers are the same ? SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs Supersedes: ) If you pick a random number in [0..1], every real in that range has a ) probability of being picked of 0. So (by your statement) it's ) impossible to pick ANY number in that range. So you can't pick a ) random number in [0..1]. That certainly breaks down the whole idea of ) continuous probability right there. No, I'm just saying it's impossible to pick any *given* number in that range. Is this difference somehow hard to grasp ? For example, I can easily state that if you pick a random number in [0..1], it will not be a rational number. (If I did the math right and the probability actually is 0, that is...) SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs >) If you are describing single numbers, then each rational number has a [CapitalThorn]nite >) description. If you are describing sets of numbers, then the set of rational >) numbers has a [CapitalThorn]nite description. >Yes, and if you have [CapitalThorn]nite time and space, there are rational numbers you >can not describe. There is not a single rational number that cannot be described in [CapitalThorn]nite time and space. I know you're trying to get at the idea that you can only describe a [CapitalThorn]nite number of rationals given [CapitalThorn]nite time and space, but that's an uninteresting result. -- Matthew T. Russotto mrussotto@speakeasy.net Extremism in defense of liberty is no vice, and moderation in pursuit of justice is no virtue. But extreme restriction of liberty in pursuit of a modicum of security is a very expensive vice. === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs ) There are sets of rational numbers that you can not describe, but there ) is no such thing as a rational number that you can not describe. There is if the [CapitalThorn]nite space and time are given beforehand. Which I said in the original comment: Ô... given [CapitalThorn]nite space and time.' SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs > ) There are sets of rational numbers that you can not describe, but there > ) is no such thing as a rational number that you can not describe. > There is if the [CapitalThorn]nite space and time are given beforehand. > Which I said in the original comment: Ô... given [CapitalThorn]nite space and time.' Each rational number (and some irrational numbers) can be described in [CapitalThorn]nite space and time. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs ) Each rational number (and some irrational numbers) can be described in ) [CapitalThorn]nite space and time. Why do you guys keep repeating that ? At least I try to change my statement each time to try and get the point across. I'm not disagreeing with that statement, just saying that *given* [CapitalThorn]nite space and time, there are rational numbers you can *not* describe. This is a different statement. SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs > ) Each rational number (and some irrational numbers) can be described in > ) [CapitalThorn]nite space and time. > Why do you guys keep repeating that ? At least I try to change my > statement each time to try and get the point across. I changed my statement, and I will change it again. There are real numbers that cannot be described, but there is no such thing as a rational number that cannot be described. > I'm not disagreeing with that statement, just saying that *given* [CapitalThorn]nite > space and time, there are rational numbers you can *not* describe. > This is a different statement. It's a different statement, and it's a wrong statement. Each rational number has a [CapitalThorn]nite description. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs [CapitalThorn]nite space and time. > Why do you guys keep repeating that ? At least I try to change my > statement each time to try and get the point across. Counter-argument: Why do you keep shifting around what you say your argument is? At least Dave is being honest and consistent about what he's saying. (No offense intended.) > I'm not disagreeing with that statement, just saying that *given* [CapitalThorn]nite > space and time, there are rational numbers you can *not* describe. > This is a different statement. Dave says: For each x in Q, there exists a space T such that x can be described in space T. You say: For each space T, there exists an x in Q such that x cannot be described in space T. (Both statements implicitly assume that there is given some standard method of describing x. Otherwise, the latter statement would be trivially false; just as I can devise a compression scheme in which Shakespeare's Hamlet compresses to the single bit Ô1', I can devise a describing scheme $D_x$ under which any $xin Q$ compresses to the single token Ô42'. Both statements are correct. Neither statement has anything to do with probability, or measure theory, or anything. This debate is pointless; it only shows that you, Willem, need to learn to express your ideas rigorously if you're going to talk math with people. (FWIW, I was kind of on Willem's side with the intuitive thing a few weeks ago, but I know just enough *about* measure theory -- I don't *know* measure theory, but I know *about* it -- to believe that here Dave knows what he's talking about. Besides, it looks like Willem's argument has been boiled down to This really arcane aspect of probability theory is counter-intuitive, and thus must have some sort of hole in it; and that's kind of silly.) -Arthur === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs ) Counter-argument: Why do you keep shifting around what you say ) your argument is? At least Dave is being honest and consistent ) about what he's saying. (No offense intended.) When I say Ôchange' I obviously mean Ôchange syntactically'. The semantic meaning stays the same. ) Dave says: For each x in Q, there exists a space T such that ) x can be described in space T. ) ) You say: For each space T, there exists an x in Q such that x ) cannot be described in space T. ) ) (Both statements implicitly assume that there is given some standard ) method of describing x. Otherwise, the latter statement would be ) trivially false; just as I can devise a compression scheme in which ) Shakespeare's Hamlet compresses to the single bit Ô1', I can devise ) a describing scheme $D_x$ under which any $xin Q$ compresses to ) the single token Ô42'. It's all a question of what comes [CapitalThorn]rst. You want to [CapitalThorn]x the method of description *after* I picked my number. And the math guys want to [CapitalThorn]x the space *after* I picked my number. I want to do it the other way round, picking my number last. How about this: For each space T, and method of description M, there are only [CapitalThorn]nitely many x in Q such that x can be described using method M in space T. ) Both statements are correct. Neither statement has anything to ) do with probability, or measure theory, or anything. This debate ) is pointless; it only shows that you, Willem, need to learn to ) express your ideas rigorously if you're going to talk math with ) people. I thought I was being pretty rigorous, but maybe I was wrong in assuming that people would not vigorously try to read the wrong meaning into my words, forcing me to plug every damn hole in my statement. ) (FWIW, I was kind of on Willem's side with the intuitive thing ) a few weeks ago, but I know just enough *about* measure theory -- ) I don't *know* measure theory, but I know *about* it -- to believe ) that here Dave knows what he's talking about. Besides, it looks ) like Willem's argument has been boiled down to This really arcane ) aspect of probability theory is counter-intuitive, and thus must ) have some sort of hole in it; and that's kind of silly.) I'm not even saying there's a hole in it, I'm just asking what's so wrong with saying that Ôthe probability is zero' is the same as Ôit's impossible' in everyday use. And I also happen to know a bit of probability theory, I'm just not accepting that where mathematicians draw the line is the only right way. SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs >) There are sets of rational numbers that you can not describe, but there >) is no such thing as a rational number that you can not describe. >There is if the [CapitalThorn]nite space and time are given beforehand. >Which I said in the original comment: Ô... given [CapitalThorn]nite space and time.' De[CapitalThorn]ne Ôtime'. Mathematically. Also explain why it is a variable in the proposition being made. -- I'm not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs ) There are sets of rational numbers that you can not describe, but there >) is no such thing as a rational number that you can not describe. There is if the [CapitalThorn]nite space and time are given beforehand. >Which I said in the original comment: Ô... given [CapitalThorn]nite space and time.' De[CapitalThorn]ne Ôtime'. Mathematically. Also explain why it is a variable in > the proposition being made. Turing machines, or URMs perhaps? Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs > ) If you're using [0,1] as the universal set, then the difference > ) between P(a randomly chosen value equals 0.5) and P(it equals 14) > ) is admittedly less important, since 14 will never be used for *any* > ) purpose. It still seems intuitively obvious, though, that P(it > ) equals 0.5) should be an in[CapitalThorn]nitesimally small but non-zero value - > ) which, however, is treated as zero in certain situations. > Well, no. To me it seems intuitively obvious that P(it equals 0.5) is > zero, which means that it's impossible to pick 0.5 randomly. > Impossible to pick any rational number, for that matter. I still don't see what's wrong with this. Because you're breaking the integral-calculus paradigm (of an in[CapitalThorn]nite number of in[CapitalThorn]nitesimals summing up to a [CapitalThorn]nite total) for (apparently) no good reason. Cf. why should 0! = 1?. In[CapitalThorn]nitely improbable is not the same as impossible. Cf. _Heart of Gold_. === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs )> Well, no. To me it seems intuitively obvious that P(it equals 0.5) is )> zero, which means that it's impossible to pick 0.5 randomly. )> Impossible to pick any rational number, for that matter. )> )> I still don't see what's wrong with this. ) Because you're breaking the integral-calculus paradigm (of an in[CapitalThorn]nite ) number of in[CapitalThorn]nitesimals summing up to a [CapitalThorn]nite total) for (apparently) ) no good reason. Cf. why should 0! = 1?. As I see it, I'm only breaking that if I *also* state that you can't sum up an in[CapitalThorn]nite number of impossibilities to get a [CapitalThorn]nite total other than 0. ) In[CapitalThorn]nitely improbable is not the same as impossible. Cf. _Heart of Gold_. So you're saying that it's completely okay to say: Something with probability zero doesn't have to be impossible, because you have to be able to add an in[CapitalThorn]nite number of those to get non-zero. But it's somehow completely wrong to say: Something with probability zero is impossible, but if you add an in[CapitalThorn]nite number of those you can get something other than zero. Look, if you want to say Ôit's not impossible, just in[CapitalThorn]nitely improbable' in mathematics to be able to differentiate between those that don't add to non-zero, and those that do, that's okay. But in normal, everyday use, there's no need to differentiate, and normal, everyday people tend to equate the two, and I think that's completely okay. It's just when mathematicians want to impose their exact terminology that you get discussions like this. SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT === Subject: : Re: puzzle: GCDs of In[CapitalThorn]nite Set of Integer Pairs >So you're saying that it's completely okay to say: >Something with probability zero doesn't have to be impossible, because >you have to be able to add an in[CapitalThorn]nite number of those to get non-zero. >But it's somehow completely wrong to say: >Something with probability zero is impossible, but if you add an >in[CapitalThorn]nite number of those you can get something other than zero. Adding is a [CapitalThorn]nite operation. Integrals are not the same as [CapitalThorn]nite sums. Take two functions: { 1, if x in the Cantor set f(x) = { { 0, otherwise { 1, if x is irrational between (0, 1) g(x) = { { 0, otherwise In both cases when you integrate over [0, 1] you have uncountable many non-zero points. But: Int(0->1, f(x)) = 0 Int(0->1, g(x)) = 1 >Look, if you want to say Ôit's not impossible, just in[CapitalThorn]nitely improbable' >in mathematics to be able to differentiate between those that don't add to >non-zero, and those that do, that's okay. Measure theory needs your approval? How quaint. >But in normal, everyday use, there's no need to differentiate, and normal, >everyday people tend to equate the two, and I think that's completely okay. >It's just when mathematicians want to impose their exact terminology that >you get discussions like this. I think it's a case of everyday people trying to impose real life on mathematics. At least you haven't told us yet how mathematics should strive to conform itself to the needs of useful real-life work like [CapitalThorn]zzics or ingineering. -- I'm not interested in mathematics that might have anything to do with reality. -- Russell Easterly, in sci.math === Subject: : Re: Cantor's Diagonal Argument In sci.logic, |-|erc <404cf056$0$8358$afc38c87@news.optusnet.com.au>: >> This guarantees diagonal constructions can be applied to the list of >> computable numbers. >> OK so far? >> This construction only governs computable numbers. I suspect these >> are similar to algebraic numbers in that both are countable. >> Are you specifying an explicit mapping (e.g., one can compute >> Computable(n) for all n > 0) or merely assuming one exists? > The mapping I outlined should be explicit except for the > variable m (processing grain) and the particular UTM used. What is m? Integer? Real? Random entity? Bellybutton lint? :-) > UTMs exist in real life, I could actually start producing a binary > list from a UTM from a book. I'm assuming UTM = Ôvon Neumann machine' here. (The classical UTM is not useful here, although it can emulate all other Turing machines. Note that a von Neumann machine does not have a tape, but it has number of states. Of course, a large number of states are equivalent; > UTMs applied to a data input can emulate any computer function. > So in theory there exists a [CapitalThorn]nite x for each UTM where > UTM(x,0) = 3. UTM(x,1) = 1 UTM(x,2) = 4 > UTM(x,3) = 1 UTM(x,4) = 5 adding pi to the list of computables. > I'm trying to show the set of computable numbers is a larger class > than previously thought and includes all irrationals. If it includes all irrationals that would certainly prove that a bijective mapping between the reals and the natural numbers exists. So go for it. > We need a de[CapitalThorn]nition of countable!? Countable: can be associated with a bijective mapping to the natural numbers {1, 2, ...}. In short, the more or less usual de[CapitalThorn]nition. The number could be fed on an input tape, in the more or less usual form -- a sequence of digits, highest N-power [CapitalThorn]rst. (For various interesting reasons N is not necessarily equal to 10.) > Herc -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: : Re: Different hand calculators, different results In sci.math.num-analysis, Gerry Myerson I mean round off errors, not parsing ones. For instance, >> my TI and CASIO hand calculators give different results >> for: >> 9e50*(29/9e50) - 29 > Calculator? I think it does what you want, I hope you can [CapitalThorn]nd it. I can only illustrate by example here, but for the longest time the Windows calculator would display 3.11 - 3.10 = 0.00 . This appears to be a combination of naive roundoff programming during output conversion, and loss of signi[CapitalThorn]cance. In IEE754 (?), 1.00 = 3ff0000000000000 3.11 = 4008e147ae147ae1 3.10 = 4008cccccccccccd diff = 3f847ae147ae1400 0.01 = 3f847ae147ae147b The right side is in hex. Sign bit, 11-bits in excess-0x200, hidden 1 bit, and 52 bits of mantissa. Note the rounding in 0.01 and 3.10; a true mathematical equality would have 0.01 = 3f847ae147ae147ae147ae147ae... assuming an in[CapitalThorn]nite mantissa. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: : Re: ZZCrap In sci.math, ZZBunker : >>> to isn't a verb. >> The last time I heard that to wasn't an intransitive >> verb was the last time that math wasn't spelled >> in the moronic fashion maths. >> What's math? > Nobody knows, Since like we've been telling morons > for several thousand years, now: > Maths is philosophy not science. > See Plato for philosophy and festive pillar work. Math is an abstract model, nothing more. Occasionally it's useful; we've even harnessed the Queen of Uselessness, number theory, into modern cryptography. :-) Who says primes aren't useful? :-) However, Ô1' cannot be captured in a butterßy net. Ôpi', Ôe', and i=sqrt(-1) are even more elusive. Good luck [CapitalThorn]nding a Taylor or MacLaurin series running around in the wild. There are examples of Nature taking advantage of Fibonacchi numbers (e.g., sunßower seed spirals, plant stalk branchings) but that doesn't mean Fibonacchi numbers are out there. The main requirement for math is that it be self-consistent. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: : Re: Differential topology (Sard's Theorem) Now this function F isn't suf[CapitalThorn]cient: it's not injective! > Consider then G:x |-> F(x) + x. Now this is a bijection between [0,1] > and [0,2]. It is easy to see that the Lebesgue measure of > G([0,1]C) is 1 and so G(C) has Lebesgue measure 1. Now extend > G to R by G(x) = x for x < 0 and G(x) = x + 1 for x > 1. > This is a homeomorphism R -> R with G(C) having measure 1. After thinking about this a little more, I don't really see why G(C) > has measure 1. Isn't it possible that adding x changes the measure > (as for example: Let f:R->R with f(x)=-x then g(x)=x+f(x)=0, i.e. > g([0,1])=0 although f([0,1])=1.) Doesn't the +x kind of Ôsmash' the intervall into tiny pieces, > such that the result maybe doesn't have measure 1?? Please explain! > Marc If I is one of the middle third intervals that's removed when you construct the Cantor set, the F is _constant_ on I and so G maps I one-one onto an interval such that m(G(I)) = m(I) (since the restriction of G to I is just a translation). Also, under G, distinct middle third intervals have disjoint images. By the countable additivity of m, it follows that m(G([0,1]C)) = m([0,1]C) = 1 . Since G(C) = [0,2] G([0,1]C), you can conclude that m(G(C)) = 1. === Subject: : admissible cycle cancelling in discrete mathematics Does anybody know of an algorithm called admissible cycle cancelling? It's one algorithm that solves minimum-cost circulation problems. I would like to know the basics about the algorithm. Could someone give me a pointer as to what paper/book I may get information about it (all I need is basics since I don't have a applied math background)? Your help is much appreciated. -Ed === Subject: : Re: Polynomial solutions to Pell eqn >>The Pell equation and the obvious cubic generalisation >>X^2 - DY^2 = 1 and >>X^3 + DY^3 +(D^2)Z^3 - 3DXYZ = 1 >>have simple polynomial solutions >>(n,1) for D = n^2 1 >>(n^2, n, 1) for D = n^31 >>In the square case, you can use continued >>fractions to expand expression of the form, say, >>SQRT(an^2 + bn + c), but in the cubic case >>no such option is available. >>Simple cases can be guessed >>for example >>D = n^3 3 >>X = n^6 3n^3 1 >>Y = n^5 2n^2 >>X = n^4 n >>D = n^3 +2, n even >>X = (9n^6)/4 + (9n^3)/2 +1 >>Y = (9n^5)/4 + 3n^2 >>Z = 3n(3n^3 +2)/4 >>I was wondering if there is any >>systematic approach to obtaining >>solutions ? >> Not real sure about a systematic approach, but looking at a few data points >for >> speci[CapitalThorn]c D may be useful. Consider the following: >> D=10^3+5*10 >> X=62591931727331611501 >> D=100^3+5*100 >> X=4910017588770392230083131681058781350001 >> D=1000^3+5*1000 >> X=489788270129887199471510415004171147924480921387800135000001 >> D=10000^3+5*10000 >> 00000001 >> D=100000^3+5*100000 >> X= 48977602682444006529246451274826892825824096005404164480666792 0000448092 >> 0000138780000001350000000001 >> D=1000000^3+5*1000000 >> X= 48977602561224440064012924645120074826892800258240960000540416 4480006667 >> 920000004480920000001387800000000135000000000001 >> D=10000000^3+5*10000000 >> X= 48977602560012244400640001292464512000074826892800002582409600 0000540416 >> 44800000666792000000004480920000000013878000000000013500000000 000001 >> D=100000000^3+5*100000000 >> X= 48977602560000122444006400000129246451200000074826892800000025 8240960000 >> 00005404164480000000666792000000000044809200000000001387800000 000000013500 >> 00000000000001 >> There is a pattern here that seems to indicate that a subset of {n^3+5*n} >has a >> polynomial solution to P3. I'm not sure what the exact answer is, but it >could >> probably be worked out, given suf[CapitalThorn]cient time and interest. There are lots >of >> other curiosities that can be found this way. >Yes n^3 +3n and 3n^2 +3n +1 seem easy to approximate. >n^ + Dn^3 where (n,D) >1 also seems to work. >I did try expanding cubrt(n^6 +2n^3 +4) and cubrt(n^3 +2),say >in steps >cubrt(n^6 +2n^3 +4) = n^2 + y, etc. >cubrt(n^3 +2) = n + z, etc >and truncating the two expression so that they both have >the same denominator. This sems to work >if, say, y = 1/kn where k is an integer. >This suggests continued fractions but simulataneous >approximation of polys seems specially dif[CapitalThorn]cult. FWIW it seems that [CapitalThorn]nding polynomial solutions to P3, given D=f(k), may be equivalent to [CapitalThorn]nding an integer Imax and an array s=s[0..Imax] of {0,1} so that the following procedure produces a solution to P3 for each k: 1: x[0]=1; y[0]=D^(1/3); z[0]=D^(2/3); i=0; 2: a[i]=ßoor(y[i]/x[i]); b[i]=ßoor(z[i]/x[i]); 3: x[i+1]=y[i]-a[i]*x[i]; y[i+1]=z[i]-b[i]*x[i]; z[i+1]=x[i]; 4: i=i+1; 5: if s(i) then swap(x[i],y[i]) 6: if i Dear All, I encountered a differention equation problem as below. Assume g(x) is a a differentiable scalar function of x, > and the equation is > (g'(x))^2 = 1/(1+g^2(x)), where g'(x) is the [CapitalThorn]rst order direvative of > g(x) w.r.t x. So given the above equation, how to get the > explicit form of g(x) or implicit value of g(x) give a particular value x. y'y' = 1 / (1 + yy) a solution can be formally obtained by setting y' = 1 / sqrt(1 + yy) OR y' = -1 / sqrt(1 + yy) separateing the former, sqrt(1 + yy)dy = -dx to integrate the LHS, let y = tan(T). > Thanks for your point. Fred === Subject: : Re: How to solve this differention equation? Adjunct Assistant Professor at the University of Montana. >Dear All, >I encountered a differention equation problem as below. >Assume g(x) is a a differentiable scalar function of x, >and the equation is >(g'(x))^2 = 1/(1+g^2(x)), where g'(x) is the [CapitalThorn]rst order direvative of >g(x) w.r.t x. >So given the above equation, how to get the >explicit form of g(x) or implicit value of g(x) give a particular value x. Well, you have that g'(x) = 1/sqrt(1+g^2) or g' = -1/sqrt(1+g^2). So you have two equations: g'sqrt(1+g^2) = 1 g'sqrt(1+g^2) = -1. sqrt(1+g^2)dg = dx or sqrt(1+g^2)dg = -dx at which point we can integrate. The [CapitalThorn]rst equation yields x + C = int (sqrt(1+g^2))dg = (g/2)sqrt(1+g^2) + (1/2)ln(g +sqrt(1+g^2)) and the second yields -x + C = int (sqrt(1+g^2))dg = (g/2)sqrt(1+g^2) + (1/2)ln(g +sqrt(1+g^2)) -- = = 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: How to solve this differention equation? > Dear All, > I encountered a differention equation problem as below. > Assume g(x) is a a differentiable scalar function of x, > and the equation is > (g'(x))^2 = 1/(1+g^2(x)), where g'(x) is the [CapitalThorn]rst order direvative of > g(x) w.r.t x. > So given the above equation, how to get the > explicit form of g(x) or implicit value of g(x) give a particular value x. ******************************************* Why not use the separation of variables technique? First write g'(x) = e/sqrt(1 + g^2(x)), where e = 1 or -1. _________________________________________________________ Eric J. Wingler (wingler@math.ysu.edu) Dept. of Mathematics and Statistics Youngstown State University One University Plaza Youngstown, OH 44555-0001 330-941-1817 === Subject: : Optimal Strategy for playing StockMarket VonNeumann game-theory; two crossovers for MDT & MRK Portfolio of PAF as of 9MAR04 : BCE 7,350 ~22.05 ~$162,067.50 BMY 900 ~27.50 ~$24,750.00 SBC 13,260 ~25.10 ~$332,826.00 realestate land 3APR03 of 3 lots $19,000. science-art of pictures,porcelain etc starting JAN03 for $15,206. realestate land 30JUL03 another lot $11,500. Last time I traded was in February where I was trading BCE for SBC due to a Crossover but on my last trade I got stuck with selling 1150 BCE and hoping to buy more SBC since the spread was a mere $2. to $2.50, but then SBC did a surprize move upward even though it had bought AT&T Wireless. So then I did a bad move of investing those 1,150 shares sold of BCE into BMY. A month earlier I had commented on the goodness of a switching campaign between MDT and MRK, Medtronics and Merck in that the two are bouncing around close together where Crossovers are easily and nicely achieved. This is a fault of mine and nearly everyone that plays the stockmarket in that they seldom follow their own advice very closely. Since Feb, MDT and MRK have reached 2 Crossovers and just yesterday MDT was over $50. whereas MRK was $47. when back in Feb. MRK was $48. and MDT was $47. during the time I had bought 850 BMY when I should have bought MDT instead. The recent trouble with BMY is the statin wars of cholesterol drugs. But I never viewed BMY as a heart drug company but rather as a premier cancer drug company. I think BMY ought to sell its Pravachol drug and focus primarily on its cancer drugs. Certainly Merck does not focus on both heart and cancer drugs and so why should BMY. Besides, I have always felt that cholesterol drugs are sham and phony medicine in that the crux of society is fat and overweight and instead of losing weight they stupidly run to doctors who prescribe cholesterol drugs. Smart and intelligent people do not take cholesterol drugs but instead lose weight and that the entire Cholesterol-drug industry is a sham and phony medicine that is a band-aid on the real ailment-- obesity and overweight which exacts a toll on the heart. So, instead of BMY competing with P[CapitalThorn]zer and Merck on cholesterol drugs, the entire cholesterol industry is just phony medicines and that the people of the world will wake up and simply lose weight and solve their problem. There are very very few people in the world that are 50 years or older who are slim and trim and have heart troubles. The people with heart troubles are invariably overweight, fat and allowed to eat junk foods. And these people are put on the stupid and silly regimen of taking cholesterol drugs to allow them to further continue with their stupid eating habits. BMY should sell its cholesterol drug division and focus on its realm of excellence of cancer drugs. Recently Imclone's erbitux was approved and look at what the colon cancer drug of Genentech had done for Genentech. So, BMY, why waste time on cholesterol drugs when in the [CapitalThorn]nal analysis it is all phony medicine that gives license to people to remain fat and eat fat. Archimedes Plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: : Birkhoff / Maclane Algebra - category theory i was wondering if Algebra by Birkhoff and Maclane is a good choice for a self-study book in algebra. I have had previous courses in abstact algebra, linear algebra, group theory (basic) and ring theory (noetherian rings) at undergraduate level. In fact, i kind of like this book (at least the [CapitalThorn]rst chapter) and i know it is considered as a good one but i am quite afraid of the fact that they use category theory as a basis in the rest of the book. Category theory seems a little bit rough to me, i don't want to rush into something and spend a great amount of time on it if it is too early in my learning process. I might miss quite a great amount of things in the rest of the text if i don't have a strong grasp on basic category theory. Any advice here ? other suggestions ? Thank you in advance for your help, Charles === Subject: : Re: Birkhoff / Maclane Algebra - category theory [...] : Category theory seems a little bit rough to me, i don't want to rush : into something and spend a great amount of time on it if it is too : early in my learning process. I might miss quite a great amount of : things in the rest of the text if i don't have a strong grasp on basic : category theory. : : Any advice here ? other suggestions ? Don't be afraid of category theory. Any language you have to learn, including any language of a mathematical [CapitalThorn]eld you are interested in, requires you to learn the pantheon of objects in it and their transformations. All category theory does is express this abstraction for you visually and give you a nice diagram language for expressing universal properties and such inside it. Its a very convenient abstraction. In some ways it can be seen as an alternative to set theory (they are both as powerful expressively). However, there is a sense where abstractions are easier to see, at least in my experience and in a lot of the writings by the developers of the category concept. Sometimes, what we are talking about is not mapping relationships between sets or graphs, for example, but instead general relationships in all topoi. A mapping between manifolds may really be using only topological information. Those kinds of things, where one might think a theorem is about a speci[CapitalThorn]c type of object, can often be seen quite readily to be more abstract when you express it in the language of categories. I started learning category theory quite early on in my mathematical development. Of course, you always have to come back to ideas from different angles before it all [CapitalThorn]ts in place, but I found my appreciation of category theory grew as I saw it explain for me connections between galois theory, cohomology, sheaves, logic, algebra, etc. that other approaches had not. So I really do not think that it is something you need to worry about as far as whether you are ready. Its more the type of the thing you can integrate as you go along, coming back to explore certain topics that were not clear at [CapitalThorn]rst, and something you can bring along throughout all of your studies. Getting an early taste of it may be very helpful, and Mac Lane is certainly one of those who can explain it well. My suggestion is to get the book as well as checking out a couple of other books on category theory in the process. The traditional Categories for the working mathematician (Mac Lane) is fairly straightforward, and the book Topoi: the categorial analysis of logic by Goldblatt has a very nice (and very elementary) introduction to categories in its [CapitalThorn]rst 3 chapters. You don't need to work through the extra books, just use them as references to help you [CapitalThorn]gure out the categorial notions as you work through your primary book. Good luck with your studies! =) -- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: : Re: Birkhoff / Maclane Algebra - category theory i was wondering if Algebra by Birkhoff and Maclane is a good choice > for a self-study book in algebra. I have had previous courses in > abstact algebra, linear algebra, group theory (basic) and ring theory > (noetherian rings) at undergraduate level. In fact, i kind of like > this book (at least the [CapitalThorn]rst chapter) and i know it is considered as > a good one but i am quite afraid of the fact that they use category > theory as a basis in the rest of the book. > Category theory seems a little bit rough to me, i don't want to rush > into something and spend a great amount of time on it if it is too > early in my learning process. I might miss quite a great amount of > things in the rest of the text if i don't have a strong grasp on basic > category theory. The one with category theory is MacLane-Birkhoff. I believe it is intended to be read by a student without previous knowledge of category theory. The previous one is Birkhoff-MacLane (1941); although a [CapitalThorn]ne book, you are beyond it. === Subject: : Re: Birkhoff / Maclane Algebra - category theory Adjunct Assistant Professor at the University of Montana. >i was wondering if Algebra by Birkhoff and Maclane is a good choice >for a self-study book in algebra. I have had previous courses in >abstact algebra, linear algebra, group theory (basic) and ring theory >(noetherian rings) at undergraduate level. There are two such books; one is called simply Algebra. The other is called A Brief Survey of Modern Algebra. The latter is much better than the former. Otherwise, it is probably a reasonably good introduction to abstract algebra. Another good choice IMHO is I. Herstein's Topics in Modern Algebra. >In fact, i kind of like >this book (at least the [CapitalThorn]rst chapter) and i know it is considered as >a good one but i am quite afraid of the fact that they use category >theory as a basis in the rest of the book. >Category theory seems a little bit rough to me, i don't want to rush >into something and spend a great amount of time on it if it is too >early in my learning process. I might miss quite a great amount of >things in the rest of the text if i don't have a strong grasp on basic >category theory. They develop the category theory they need; for the most part, it is used to show the parallels between a number of constructions and proofs, and to de[CapitalThorn]ne certain objects (like the product of groups/rings) by their mapping properties. I suggest going through their category chapter interpreting everything they say about objects as meaning sets, and arrows as being functions; then try it replacing Ôobjects' with Ôgroups' and Ôarrows' with Ôgroup homomorphisms'; then Ôrings' and Ôring homomorphisms'; then maybe if you know some topology Ôtopological spaces', Ôcontinuous maps'. -- = = 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: : Strange Complex Variables Problem Here is a problem I'm trying to [CapitalThorn]gure out. I tried a bunch of different things, but not sure what to use here -- the argument principle? I wonder if anyone can make heads or tails of it. You have the space H of all analytic functions on a doubly-connected domain D (it's a hilbert space under the inner product for functions, they all have an L2 norm). Then consider the subset J of H consisting of all functions which have integral 2PI*i going once around the hole in the domain. Let F be the function mapping the domain onto an annulus. Show that F'/F (logarithmic derivative) has the smallest L2 norm of all L2 norms of functions in J. --- I don't know, here we are taking the integral of |F'/F|^2 throughout the whole domain D, so can I even apply the argument principle? Since F'/F analytic there (F is analytic there) and, say, continuous on the boundary we can just go around the boundary of D? F maps these to the circles of the annulus. I guess I just don't know the relationship of F to F'/F except that if you integrate F'/F you get the change in argument of F :) -Greg === Subject: : Re: Strange Complex Variables Problem > Here is a problem I'm trying to [CapitalThorn]gure out. I tried a bunch of > different things, but not sure what to use here -- the argument > principle? I wonder if anyone can make heads or tails of it. You have the space H of all analytic functions on a doubly-connected > domain D (it's a hilbert space under the inner product for functions, > they all have an L2 norm). Then consider the subset J of H consisting > of all functions which have integral 2PI*i going once around the hole > in the domain. Let F be the function mapping the domain onto an > annulus. Show that F'/F (logarithmic derivative) has the smallest L2 > norm of all L2 norms of functions in J. Try stating the problem precisely. What are you integrating over? What do you mean the function F mapping the domain onto an annulus? Which annulus? Etc, etc. === Subject: : Re: is LU and LDU matrix factorization unique? or under what condition are they unique? > Dear all, Here is my question: is LU and LDU matrix factorization unique? (here for LDU we follow the de[CapitalThorn]nition that L is UNIT lower triangular > matrix, D is diagonal, U is UNIT upper triangular matrix) How to prove that they are unique? If they are not unique, then under what condition do they become unique? Thanks a lot, Joenyim If A is nonsingular, LU and LDU triangular decompositions are unique within a permutation (for LU, as long as one agrees on which one is unit triangular). When solving A x = b with some form of pivoting, one actually processes P A x = P b where P is a permutation matrix. Then PA = L U, and L U generally depends on P. === Subject: : Re: is LU and LDU matrix factorization unique? or under what condition are they unique? Dear all, Here is my question: is LU and LDU matrix factorization unique? (here for LDU we follow the de[CapitalThorn]nition that L is UNIT lower triangular > matrix, D is diagonal, U is UNIT upper triangular matrix) How to prove that they are unique? Yes. If L_1*D_1*U_1 = L_2*D_2*U_2 then, with all the proper hypotheses, > L_1 = L_2, D_1 = D_2 and U_1 = U_2. The proof goes something like this: U_1*U_2^(-1) = D_1^(-1)*L_1^(-1)*L_2*D_2. The LHS is upper triangular with unit diagonal. The RHS is lower > triangular. So, LHS = I. Etc. Thank you Paul! But how about LU factorization? Is there uniqueness with LU? === Subject: : Re: is LU and LDU matrix factorization unique? or under what condition are they unique? > Here is my question: is LU and LDU matrix factorization unique? (here for LDU we follow the de[CapitalThorn]nition that L is UNIT lower triangular > matrix, D is diagonal, U is UNIT upper triangular matrix) How to prove that they are unique? Yes. If L_1*D_1*U_1 = L_2*D_2*U_2 then, with all the proper hypotheses, > L_1 = L_2, D_1 = D_2 and U_1 = U_2. One of the hypotheses is that the matrix is not singular. If the matrix is singular there can be many ways. Consider: (0 0) (1 0)(0 0)(1 y) (0 1) = (x 1)(0 1)(0 1) for any x and y. > Thank you Paul! But how about LU factorization? Is there uniqueness with LU? LU factorisation is unique only if you require that U or L is a unit, and again, if the matrix is not singular. The proof is similar. -- 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 : Birkhoff / Maclane Algebra - category theory >> i was wondering if Algebra by Birkhoff and Maclane is a good choice >> for a self-study book in algebra. I have had previous courses in >> abstact algebra, linear algebra, group theory (basic) and ring theory >> (noetherian rings) at undergraduate level. > There are two such books; one is called simply Algebra. The other is > called A Brief Survey of Modern Algebra. The latter is much better > than the former. why ? can you develop a little bit more on this ? Thank you, Charles, === Subject: : Re: Re : Birkhoff / Maclane Algebra - category theory Adjunct Assistant Professor at the University of Montana. > i was wondering if Algebra by Birkhoff and Maclane is a good choice > for a self-study book in algebra. I have had previous courses in > abstact algebra, linear algebra, group theory (basic) and ring theory > (noetherian rings) at undergraduate level. >> There are two such books; one is called simply Algebra. The other is >> called A Brief Survey of Modern Algebra. The latter is much better >> than the former. >why ? can you develop a little bit more on this ? Here's the review for the [CapitalThorn]rst edition of the Survey (Brief Survey being a somewhat abbreviated version, which lacks the last four chapters, Trans[CapitalThorn]nite arithmetic, Rings and Ideals, Algebraic number [CapitalThorn]elds, and Galois Theory): Birkhoff, Garrett; MacLane, Saunders A Survey of Modern Algebra. Macmillan Company, New York, 1941. xi+450 pp. 09.1X This is a text on modern algebra that is particularly suited for a [CapitalThorn]rst year graduate course or for an advanced undergraduate course. A very striking feature of the book is its broad point of view. There are contacts with many branches of mathematics and so it can serve as an introduction to nearly the whole of modern mathematics. Thus there is a careful development of real numbers, such as Dedekind cuts, and such set-theoretic concepts as order, countability and cardinal number are discussed. Throughout the study of matrices and quadratic forms the geometric point of view is emphasized. There is also contact with the [CapitalThorn]eld of mathematical logic in the chapter on the algebra of classes and with the ideas of topology in the proof of the fundamental theorem of algebra. The following is the table of contents by chapters: I. The integers. II. Rational numbers and [CapitalThorn]elds. III. Real numbers. IV. Polynomials. V. Complex numbers. VI. Group theory. VII. Vectors and vector spaces. VIII. The algebra of matrices. IX. Linear groups. X. Rank and determinants. XI. Algebra of classes. XII. Trans[CapitalThorn]nite arithmetic. XIII. Rings and ideals. XIV. Algebraic number [CapitalThorn]elds. XV. Galois theory. As can be judged from this outline, concrete systems with which the student is familiar are studied [CapitalThorn]rst and then their properties are formulated as axioms leading thereby to the de[CapitalThorn]nition of important abstract algebraic systems. This has been done with great skill in the text and moreover the large number of excellent exercises should enable the student to keep a [CapitalThorn]rm hold on the theory. There is a wide gap between the [CapitalThorn]rst chapter dealing with elementary properties of integers and the last chapter proving the insolvability of quintic equations by radicals. One is con[CapitalThorn]dent, however, that a student who has made the journey over the intermediate chapters will have achieved the maturity needed to follow the intricacies of the Galois theory. The most serious criticism that the reviewer has to make is a rather minor one: The theory of determinants is developed only for matrices with entries in a [CapitalThorn]eld. Because of the applications we believe it would have been worthwhile to treat the general case in which the entries belong to any commutative ring. The book is very clearly written and seems to be free from error. Reviewed by N. Jacobson You'll note that there is no particular emphasis on category theory. The Algebra I had always heard as inferior to the Survey was the [CapitalThorn]rst edition (which I had access to but no longer have). There, universal constructions were introduced quickly but not very clearly. Apparently, the second edition was largely rewritten, moving it to Chapter IV; the review for the second edition claims this is ->much<- better, so perhaps the later editions are not so disparate anymore. Interestingly, the author list for Algebra has Mac Lane (the correct spelling, by the way) [CapitalThorn]rst and Birkhoff second; Survey or Brief Survey has Birhoff [CapitalThorn]rst and Mac Lane second. Almost everyone will recommend Birkhoff-Mac Lane, not Mac Lane-Birkhoff. -- = = 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: Hints for self taught topology? I am undertaking the quest to teach myself topology. Text I'm > using right now is Munkres' Topology: a [CapitalThorn]rst course, but this is a > library book (interlibrary loan) so I may or may not be able to obtain > my own copy and might have to settle for something less highly > recommended. I'm no stranger to self teaching, I've self taught myself > galois theory and just about everything leading up to that, as well as > the standard calculus package and diffeq. But I *am* a stranger to > topology and it strikes me as exceedingly subtle (which is a good > thing:) and I look forward to mastering it. > Are there any common pitfalls, etc. that I need to be ready for, > or anything else I should know for this mission? You can also try General Topology by Kelley. HTH, Felix. === Subject: : Re: Hints for self taught topology? dreamvigile > I am undertaking the quest to teach myself topology. Text I'm > using right now is Munkres' Topology: a [CapitalThorn]rst course, ... I don't know Munkres, but if it's general topology, or point set topology, then actually Bourbaki's tomes General Topology are (unlike some of his other tomes) pretty readable, with an impressive collection of exercises to boot. Larry Hammick, Knight of the Most Obsure Order of Autodidactics === Subject: : Re: Hints for self taught topology? | | I am undertaking the quest to teach myself topology. Text I'm |using right now is Munkres' Topology: a [CapitalThorn]rst course, but this is a |library book (interlibrary loan) so I may or may not be able to obtain |my own copy and might have to settle for something less highly |recommended. I'm no stranger to self teaching, I've self taught myself |galois theory and just about everything leading up to that, as well as |the standard calculus package and diffeq. But I *am* a stranger to |topology and it strikes me as exceedingly subtle (which is a good |thing:) and I look forward to mastering it. | Are there any common pitfalls, etc. that I need to be ready for, |or anything else I should know for this mission? probably the most important thing you should know is that galois theory and the so-called algebraic topology part of topology (which is a big part of what munkres's text is about) are, in a sense, secretly the same subject. -- [e-mail address jdolan@math.ucr.edu] === Subject: : Candidate looking for Employment am looking for a job. Here's a copy of my resume Name : Suresh kumar Devanathan contact : mdsuresh ___@thing__ media __.thing__ mit __.thing__ edu Experience : MIT Media Lab Rutgers CAIP Lab Awards : AP Scholar With Distinction Other : Army Reserve Worked for : Dr. Vishwani Agarwal( IEEE/ACM fellow) on VLSI ATPG Dr. Edward Fredkin( CMU Distinguished professor) on Cellular Automata/ SARS-TA Public Projects: http://www.sourceforge.net/projects/atpg ----------------------------------------- Built this project, as a PROOF of CONCEPT for statistical techniques, i developed to test chips http://www.sourceforge.net/projects/blitz ----------------------------------------- Patch work for blitz, a highly successful C++ math library in use, by more than 15000 programmers worldwide Job Skill : C++, C#, Unix, Cadence Design System, MATLAB, .NET Education : 3 years of college in Electrical Engineering Left college to jumpstart a partly successful entrepreneurship === Subject: : Re: Candidate looking for Employment am looking for a job. Here's a copy of my resume 1) Get a better handle. Nobody will consider The Lord of Chaos. Try Big Smile if you want a position. 2) Get a better resume. Who will read it, what will they want? Personnel/Human Resources/Human Factors Engineerng will see it [CapitalThorn]rst, then discard it. Personnel only hires drinking buddies. Employers want to see blind loyality, bovine silence, and pro[CapitalThorn]t. Nobody wants intelligence. They lease consultants and wash their hands after touching. At least format the thing so it plays on Usenet. Anybody who contacts you as is should be avoided. 3) Know whom to contact. Nobody is hired through the front door. If your prof is any ing godddamned good either he will set you up or he make sure you get nothing. Why in Hell did you sign up under him if he isn't useful? 4) You aren't looking for a job. You are pursuing your career. Don't be a lab nigger. Circuit boards are fungible. 5) 20 lb 96+% white paper. Crisp jet black laser printer or ink jet. One page, mostly white space. One fontface. Print large enough so a 50-year old man can read it (presbyopia). If the Liberal Arts C-average reader is not creaming in his shorts within two inches of the heading, it is a crap resume. Mail it in a manila envelope, no folds, typeset address. Brief cover letter (pen with a nib for your signature; practice) and resume. FedEx is better - open an account so you don't look like a schmuck with the paperwork. Personnel, Mister Devanathan, isn't it expensive to FedEx hundreds of resumes? Candidate, I only contact quali[CapitalThorn]ed leads. Isn't that how *you* run an ef[CapitalThorn]cient, productive of[CapitalThorn]ce? Nobody cares where you went to school, what awards you got, or how many old ladies you charitably tongue each weekend at the community center. Are you a leader? Are you a pro[CapitalThorn]t center? Can they show you in public? Do you play golf, can you order from a wine list? Do the buttons on your suit sit on the fabric or are they perched on little pillars of silk thread? If a single technical term worms its way into your interview with Personnel, you are a corpse. Candidate, I didn't get the job. You can lead a boss to water but you can't make him drink. Uncle Al, Your job is to make him thirsty. Deep thoracic voice. Paced delivery. Look the squirmy bastards right in the eye and *never* turn away when answering a question. Do something about your nails - you're going to be shaking hands, fer christsakes. The most successful fellow chemist Uncle Al has known polished his lab benches each Friday afternoon. He was a useless idiot and smarter than all the rest of us combined. But not now. Learning curve and blood. Uncle Al says, The time to negotiate compromise is after your enemies are dead. -- Uncle Al http://www.mazepath.com/uncleal/qz.pdf http://www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) === Subject: : Re: Candidate looking for Employment ... Education : 3 years of college in Electrical Engineering > Left college to jumpstart a partly successful > entrepreneurship What's an entrepreneurship? -- G.C. === Subject: : Re: Hypocritical Fortune Tellers (What are the odds?) > are you a mathematician or a numerologist? > is what you're posting anything more than a > glori[CapitalThorn]ed horoscope? > I am not looking to the future but goink back to > the past, to the day the person was born. God > provided the individual with his or her life and > also their name, back then. Horoscopes deal with > astrology, can you see any discussions of planets > and stars in my postings? The mainstream > Christian churches embrace pagan holidays that > are linked to sun whoreship and merge them in with > their Christian worship, you are likely a member > of one of these cults and now you hypocritically > condemn me and my work as being astrology. For > example, by December 25th the sun is visibly > returning from the south, calling this pagan > sunwhoreshipping holiday Christmas is a > violation of God's Third Commandment, for > it is not Christ's Mass but is instead a pagan > mass, it is the use of God's name in vain. And > for example, the mainstream Christians embrace > the pagan Easter holiday at a time based upon > the phase of the moon. Anyone who thinks Easter is only a Christian holiday hasn't looked at the de[CapitalThorn]nition: The [CapitalThorn]rst Sunday after the [CapitalThorn]rst new moon (or is it full moon) after the vernal equinox (i.e., the [CapitalThorn]rst day of spring). It's hard to get more cabalistic than that. > And then look at all the > predictions that the mainstream mathematicians > attempt, you people are tossing coins and looking > for the probababbility of them landing as heads > or tails (while disavowing Satanic inßuences that > can and do manipulate these coins). Probability (there is no such word as probababbility) has nothing to do with prediction of a single event. It will tell you what will likely happen over several repeated trials. And I wonder: Is the statement, The sun will rise tomorrow, a Satanic statement? > You mathe- > maticians are looking for the probababbility of > events on earth (while disavowing spiritual > inßuences), and then have the audacity to call > me a fortune teller. I am a mathematician who > deals with God and little primes, while the > mainstream mathematicians are Godless > numerologists who salivate when they discover > a prime that is large enough to [CapitalThorn]ll a telephone > book. Real mathematicians will only salivate if they're the one who found the new prime. All mathematicians know there are an in[CapitalThorn]nite number of primes. -- Christopher Heckman, Godless numerologist === Subject: : Re: Hypocritical Fortune Tellers (What are the odds?) > I am not looking to the future but goink back to > the past, to the day the person was born. God > provided the individual with his or her life and > also their name, back then. Horoscopes deal with > astrology, can you see any discussions of planets > and stars in my postings? The mainstream > Christian churches embrace pagan holidays that > are linked to sun whoreship and merge them in with > their Christian worship, you are likely a member > of one of these cults and now you hypocritically > condemn me and my work as being astrology. For > example, by December 25th the sun is visibly > returning from the south, calling this pagan > sunwhoreshipping holiday Christmas is a > violation of God's Third Commandment, for > it is not Christ's Mass but is instead a pagan > mass, it is the use of God's name in vain. And > for example, the mainstream Christians embrace > the pagan Easter holiday at a time based upon > the phase of the moon. And then look at all the > predictions that the mainstream mathematicians > attempt, you people are tossing coins and looking > for the probababbility of them landing as heads > or tails (while disavowing Satanic inßuences that > can and do manipulate these coins). You mathe- > maticians are looking for the probababbility of > events on earth (while disavowing spiritual > inßuences), and then have the audacity to call > me a fortune teller. I am a mathematician who > deals with God and little primes, while the > mainstream mathematicians are Godless > numerologists who salivate when they discover > a prime that is large enough to [CapitalThorn]ll a telephone > book. -Daryl S. Kabatoff i am not a member of any cult, but how typical of you to assume so. good job. you want to talk about being pagan and being part of cultish practices? i would like to see you go over to the forums at www.christianity.com and watch as they laugh you out of there. and what does satan care about a coin toss result? if you have a problem with probablility theory, then why not tell us which parts exactly, and we can discuss it... your posts here are completely devoid of any kind of mathematics. disagree? then describe your methods and show your proofs. until then, you're simply a numerologist in denial who uses his faith as a shield === Subject: : Re: Hypocritical Fortune Tellers (What are the odds?) > are you a mathematician or a numerologist? > is what you're posting anything more than a > glori[CapitalThorn]ed horoscope? DARYL: > I am not looking to the future but goink back to > the past, to the day the person was born. God > provided the individual with his or her life and > also their name, back then. Horoscopes deal with > astrology, can you see any discussions of planets > and stars in my postings? The mainstream > Christian churches embrace pagan holidays that > are linked to sun whoreship and merge them in with > their Christian worship, you are likely a member > of one of these cults and now you hypocritically > condemn me and my work as being astrology. For > example, by December 25th the sun is visibly > returning from the south, calling this pagan > sunwhoreshipping holiday Christmas is a > violation of God's Third Commandment, for > it is not Christ's Mass but is instead a pagan > mass, it is the use of God's name in vain. And > for example, the mainstream Christians embrace > the pagan Easter holiday at a time based upon > the phase of the moon. And then look at all the > predictions that the mainstream mathematicians > attempt, you people are tossing coins and looking > for the probababbility of them landing as heads > or tails (while disavowing Satanic inßuences that > can and do manipulate these coins). You mathe- > maticians are looking for the probababbility of > events on earth (while disavowing spiritual > inßuences), and then have the audacity to call > me a fortune teller. I am a mathematician who > deals with God and little primes, while the > mainstream mathematicians are Godless > numerologists who salivate when they discover > a prime that is large enough to [CapitalThorn]ll a telephone > book. -Daryl S. Kabatoff > i am not a member of any cult, but how typical of you to assume so. good > job. you want to talk about being pagan and being part of cultish > practices? i would like to see you go over to the forums at > www.christianity.com and watch as they laugh you out of there. DARYL: When you accused my work of being a glori[CapitalThorn]ed horoscope, was there not an implicit underlying message that my material was cultish? And I don't really care if 100% of the churches in the land teach things that are in opposition to Scripture, and I don't care if 100% of their [CapitalThorn]lthy members laugh at me, what I care about is doing the things that God desires for me. And He wants me to post this material in mathematical, political and religious forums. ZAPHOD: > and what does satan care about a coin toss result? if you > have a problem with probablility theory, then why not > tell us which parts exactly, and we can discuss it... DARYL: Satanic forces can and do make coins fall into positions that practitioners of I Ching then interpret and tell fortunes. Some laws allow for tied elections to be determined my coin tosses, but this is not a good idea for the candidate that Satan and his dark angels most highly esteems will be the inevitable winner. ZAPHOD: > your posts here are completely devoid of any kind of > mathematics. disagree? then describe your methods > and show your proofs. DARYL: I am giving example after example of mathematical harmony that ties together people's names, birthdays and the numbers of the Bible. The 29x29th chapter of the Bible contains 29 verses, why doesn't your analytical tools note this and similar patterns? ZAPHOD: > until then, you're simply a numerologist in denial who > uses his faith as a shield DARYL: I have faith that God provides us with our names and lives, and that this God is the God of the Bible rather than the god of the Koran or some other religious book. Note that the 4th Book of the Bible is called Numbers, it seems that God has an interest in numbers and that discussion of such should be permitted in these religious and mathematical forums. -Daryl S. Kabatoff === Subject: : Re: Hypocritical Fortune Tellers (What are the odds?) > DARYL: > When you accused my work of being a glori[CapitalThorn]ed horoscope, > was there not an implicit underlying message that my material > was cultish? And I don't really care if 100% of the churches > in the land teach things that are in opposition to Scripture, > and I don't care if 100% of their [CapitalThorn]lthy members laugh at me, > what I care about is doing the things that God desires for me. > And He wants me to post this material in mathematical, > political and religious forums. your material IS cultish!! you're afraid to go to www.chrisitianty.com because you KNOW that its cultish. and if He WANTED you to post in religious forums, then why dont you go there and show them how smart you are. where else do you post other than here? give me some links. > DARYL: > Satanic forces can and do make coins fall into positions that > practitioners of I Ching then interpret and tell fortunes. > Some laws allow for tied elections to be determined my coin > tosses, but this is not a good idea for the candidate that Satan > and his dark angels most highly esteems will be the inevitable > winner. yikes. so does God have any say in this? i mean, doesnt God care that Satan is taking control of all these coin tosses? if so, maybe we can conjecture that whenever the coin ends up being tails its Satan, and whenever it ends up being heads, its God. and my theory is that as the coin toss is repeated, God will win the cointoss about 50% of the time (if the coin isnt Satanic). can you not see how psychotic your theory about satan and his dark angels is about a cointoss? (how old are you, btw?) > DARYL: > I am giving example after example of mathematical harmony > that ties together people's names, birthdays and the numbers > of the Bible. The 29x29th chapter of the Bible contains 29 > verses, why doesn't your analytical tools note this and > similar patterns? only because i dont care about the patterns of the bible. did you know that if you look hard enough, you'll be able to [CapitalThorn]nd patterns anywhere you look? i'm sure if you study the other Koran, i'm sure you'll [CapitalThorn]nd similar patterns if you take a good look. > DARYL: > I have faith that God provides us with our names and lives, > and that this God is the God of the Bible rather than the > god of the Koran or some other religious book. Note that > the 4th Book of the Bible is called Numbers, it seems that > God has an interest in numbers and that discussion of such > should be permitted in these religious and mathematical > forums. -Daryl S. Kabatoff religion & mathematics = oil & water if you use faith to try to mix the two, we have a word for that. its called NUMEROLOGY. deal with it === Subject: : Re: Dice Probabilities by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i29KDiX02788; What is the probability of rolling yahtzee of 1's === Subject: : Re: Dice Probabilities Diane wibbled: > What is the probability of rolling yahtzee of 1's 1 in 6*6*6*6*6 You can probably work it out from there === Subject: : Re: Dice Probabilities > What is the probability of rolling yahtzee of 1's One in six to the [CapitalThorn]fth, or 1/7776. === Subject: : Re: 1st-order Pigeonhole Challenge > (i) There are at least m objects > (ii) There are exactly n A's > (iii) The f of each object is an A > Therefore, > (iv) There are two distinct objects a, b such that f(a) = f(b). There is a premise missing from this (maybe it was present in the original); > m has to be GREATER THAN n for this to go through. > Well, actually *he* added (in the very same post): If m > n, this argument is valid (Jeffrey Ketland) Well, actually, he was full of . Valid arguments in FOL are valid in virtue of their syntax. The interpretations of the terms occurring in them do NOT matter. In other words, that argument doesn't become VALID UNTIL it is ACCOMPANIED BY SOME AXIOMS DEFINING >. This whole thing has to be done in SOME axiomatic background that must be CHOSEN AND PROMULGATED by the advocate! In other words, in addition to getting the order of the relevant logic wrong, Ketland also failed to specify a framework. If the axiomatic framework is ZFC then there Is NOTHING To Prove, for the sole and simple reason that > is NOT a primitive symbol of ZFC and the usual reasonable de[CapitalThorn]nition of it winds up BEING exactly this. === Subject: : Re: 1st-order Pigeonhole Challenge X-ID: XprwAUZ-oecDUUATaQWe0HqzigqXm3Q2fdBniXDTxYk0G8K4ZpzO0N Well, actually, he was full of . *plonk* === Subject: : Re: Why are there 63360 inches in a mile? John Baez wibbled: But why didn't they just de[CapitalThorn]ne a rod to be 16 feet, instead of 16.5??? I imagine the chain came [CapitalThorn]rst. Why it's 66feet (22 yards) I don't know. It's entirely possible that it results from a convergence of two different measuring systems: there used to be a lot of different inches etc in Europe. === Subject: : Re: Why are there 63360 inches in a mile? In message , >John Baez wibbled: >> But why didn't they just de[CapitalThorn]ne a rod to be 16 feet, instead of 16.5??? >I imagine the chain came [CapitalThorn]rst. >Why it's 66feet (22 yards) I don't know. >It's entirely possible that it results from a convergence of two >different measuring systems: there used to be a lot of different inches >etc in Europe. I would think that the fundamental unit is the furlong (furrow-long), a length which was supposed to be the average distance which could be ploughed in a day in the Middle Ages. There was a strong desire for decimalisation, so the Chain was invented as being 1/10 of a furlong. But since the mile had already been de[CapitalThorn]ned, they were forced to only allow 8 furlongs in a mile. -- Jeremy Boden === Subject: : Re: Why are there 63360 inches in a mile? >>Brießy: Kummer discovered to his dismay that the [CapitalThorn]eld Q(sqrt(-5)) >>has two ideal classes, one principal and one not. These correspond >>to two points in the moduli space of elliptic curves, and if compute >>the modular j-function at both these points you get two numbers which - >>according to Kronecker's dream, now known to be true - are both roots >>of a quadratic polynomial with integer coef[CapitalThorn]cients. It's a pretty >>bizarre polynomial, and if you write it as simply as possible, the >>constant term is >>880^3 >>which you will recognize as precisely the number of cubic yards in a >>cubic half-mile! >>It's all the more weird because the de[CapitalThorn]nition of the j-function >>involves the number 1728 [...] Which is one less than Ramanujan's smallest number expressible as the >sum of two cubes in two different ways... This is *one* reason why Ramanujan didn't need to be so brilliant > as some people claim, in order to notice that 1729 = 1728 + 1 = 1000 + 729 is the smallest number expressible as a cube in two ways. > He knew about the j function. He had also studied problems of expressing numbers as a sum of > powers. And, it's just *not that hard* to notice that 1728 + 1 = 1000 + 729 > and get interested in whether there are any smaller numbers like > this - at least for someone interested in numbers. hmmm...this conspiracy is deeper than it [CapitalThorn]rst appeared. Yes. But what I found out last weekend took it to a whole new level! If you take the j function and evaluate it at (sqrt(-67)+1)/2, > you get minus the number of cubic feet in a cubic mile, which is -5280^3 = -147,197,952,000 And, this number (sqrt(-67)+1)/2 is not just any old weird > number. It's the algebraic integer generating the second largest > imaginary quadratic [CapitalThorn]eld with class number one! One consequence > is that exp(pi sqrt(67)) = 147,197,952,743.999997.... is very, very close to an integer. And this integer is (5280)^3 + 744 In fact, if you type the number 147197952000 between what I just said and another famous curiosity, namely that exp(pi sqrt(163)) = 262537412640768743.9999999999992... is very close to the integer (640320)^3 + 744 the point being that sqrt(-163) is the algebraic integer generating > the *largest* imaginary quadratic [CapitalThorn]eld with class number one. 147197952000 cubic feet in a cubic mile. So, the conspiracy is obvious: the druid priests who put 5280 feet > in a mile, 12 inches in a foot, and therefore 63360 inches in a mile, > must secretly have been experts on the j function. No? No. There are 5280 ft in a *statute* mile. Hence, it immediately that the Druids where not Druids. They were British *Lawyers*. AKA Newton & Galileo-Wannabee LTD. that was written by my evil twin, who is trying to give me a bad name.] === Subject: : Re: Why are there 63360 inches in a mile? > .... > Of course it is trivial to remember there are 320 rods per mile, > so no right-thinking person has any excuse whatsoever to not > know the actual length of a rod.... So there are no right-thinking persons outside the U.S.A.? Most of us don't even know the actual length of a mile (although I do: 1.609 Km). Ken Pledger. === Subject: : Re: Why are there 63360 inches in a mile? > .... > Of course it is trivial to remember there are 320 rods per mile, > so no right-thinking person has any excuse whatsoever to not > know the actual length of a rod.... So there are no right-thinking persons outside the U.S.A.? Most > of us don't even know the actual length of a mile (although I do: 1.609 > Km). No, no, of course I didn't claim there are no right-thinking persons outside the USA. Just that they have no excuse not to know the length of a rod. But it occurs to me now that not knowing the length of a mile in sensible units *is* an excuse of sorts for not knowing the length of 1/320 mile. I guess we will just have to go with about 199 rods per kilometer. Ok? Now you agree? -jiw === Subject: : Re: Why are there 63360 inches in a mile? >No, no, of course I didn't claim there are no right-thinking persons >outside the USA. Just that they have no excuse not to know the >length of a rod. But it occurs to me now that not knowing the >length of a mile in sensible units *is* an excuse of sorts for >not knowing the length of 1/320 mile. I guess we will just have >to go with about 199 rods per kilometer. Ok? Now you agree? Whoa! You mean all those spammers are covertly trying to destroy the metric system when they keep sending me offers to increase the length of my rod? Lee Rudolph === Subject: : Re: Why are there 63360 inches in a mile? Ken Pledger wibbled: > Most > of us don't even know the actual length of a mile Depends if you're on the way to the pub, or on the way back. === Subject: : Re: Why are there 63360 inches in a mile? > I tried to determine the source of the factor of 11 a few months ago. I'm surprised no-one has referred to the length of a cricket pitch as a serious argument for retaining the traditional linear units. Or has the EU required the MCC to reduce (?) the length to 20 metres? It can't be a coincidence that 22 yards (1 chain) is exactly the right length for the trajectory of a cricket ball. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: : Re: Why are there 63360 inches in a mile? >Brießy: Kummer discovered to his dismay that the [CapitalThorn]eld Q(sqrt(-5)) >has two ideal classes, one principal and one not. These correspond >to two points in the moduli space of elliptic curves, and if compute >the modular j-function at both these points you get two numbers which - >according to Kronecker's dream, now known to be true - are both roots >of a quadratic polynomial with integer coef[CapitalThorn]cients. It's a pretty >bizarre polynomial, and if you write it as simply as possible, the >constant term is >880^3 >which you will recognize as precisely the number of cubic yards in a >cubic half-mile! >It's all the more weird because the de[CapitalThorn]nition of the j-function >involves the number 1728 [...] >>Which is one less than Ramanujan's smallest number expressible as the >>sum of two cubes in two different ways... >This is *one* reason why Ramanujan didn't need to be so brilliant >as some people claim, in order to notice that >1729 = 1728 + 1 = 1000 + 729 >is the smallest number expressible as a cube in two ways. >He knew about the j function. >He had also studied problems of expressing numbers as a sum of >powers. >And, it's just *not that hard* to notice that 1728 + 1 = 1000 + 729 >and get interested in whether there are any smaller numbers like >this - at least for someone interested in numbers. >>hmmm...this conspiracy is deeper than it [CapitalThorn]rst appeared. >Yes. But what I found out last weekend took it to a whole new level! >If you take the j function and evaluate it at (sqrt(-67)+1)/2, >you get minus the number of cubic feet in a cubic mile, which is >-5280^3 = -147,197,952,000 >And, this number (sqrt(-67)+1)/2 is not just any old weird >number. It's the algebraic integer generating the second largest >imaginary quadratic [CapitalThorn]eld with class number one! One consequence >is that >exp(pi sqrt(67)) = 147,197,952,743.999997.... >is very, very close to an integer. And this integer is >(5280)^3 + 744 >In fact, if you type the number >147197952000 >between what I just said and another famous curiosity, namely that >exp(pi sqrt(163)) = 262537412640768743.9999999999992... >is very close to the integer >(640320)^3 + 744 >the point being that sqrt(-163) is the algebraic integer generating >the *largest* imaginary quadratic [CapitalThorn]eld with class number one. >147197952000 cubic feet in a cubic mile. >So, the conspiracy is obvious: the druid priests who put 5280 feet >in a mile, 12 inches in a foot, and therefore 63360 inches in a foot, >must secretly have been experts on the j function. >No? My guess would be yes. It says that there is something physically practical with the measurement. My deepest assumption is that nature is very ef[CapitalThorn]cient. As for the [CapitalThorn]elds plowed and everything...My dad uses a hooked stick for a cane. It used to be used as a measurement for spacing rows when planting. I don't know the length but I betcha its a fraction of a rod. I can think of two reasons for the spacing: 1. It provides the most ef[CapitalThorn]cient growing space for plants (IIRC, he spaced [CapitalThorn]eld corn); and, 2. It provides spacing for the width needed for cultivators (weeding) and the tractor pulling them. The second point has a history that could be probably be traced back to yokes and chariots etc. This can be traced to the physical properties that made axles and wheels, etc. most ef[CapitalThorn]cient w.r.t. weight bearing and maneuverability(sp?). So, you'ld be right; it's a conspiracy of nature. /BAH Subtract a hundred and four for e-mail. === Subject: : Re: Why are there 63360 inches in a mile? > As for the [CapitalThorn]elds plowed and everything...My dad uses a hooked > stick for a cane. It used to be used as a measurement for > spacing rows when planting. I don't know the length but I betcha > its a fraction of a rod. > I can think of two reasons for the spacing: 1. It provides > the most ef[CapitalThorn]cient growing space for plants (IIRC, he spaced > [CapitalThorn]eld corn); and, 2. It provides spacing for the width needed > for cultivators (weeding) and the tractor pulling them. > The second point has a history that could be probably be traced > back to yokes and chariots etc. This can be traced to the > physical properties that made axles and wheels, etc. most > ef[CapitalThorn]cient w.r.t. weight bearing and maneuverability(sp?). > So, you'ld be right; it's a conspiracy of nature. Standard American railroad gauge supposedly traces back to the distance between thw wheels on Roman wagons. === Subject: : Re: Why are there 63360 inches in a mile? |>It's all the more weird because the de[CapitalThorn]nition of the j-function |>involves the number 1728, It has to do with the discriminant of a plane cubic. The factors of 2 and 3 come basically from derivatives of square and cube terms. I used to know more of the details, about ten years ago. |Which is one less than Ramanujan's smallest number expressible as the |sum of two cubes in two different ways...hmmm...this conspiracy is |deeper than it [CapitalThorn]rst appeared. If I remember correctly, in one of Feynman's stories he describes being challenged by a fellow with an abacus. The last calculation suggested by the guy with the abacus is extracting the cube root of 1729. According to his story, Feyman recognizes this right away as one more than the number of cubic inches in a cubic foot, getting the integer part right away, and then he gets the next digit with an approximation. I think in the movie In[CapitalThorn]nity (where he's played by Matthew Broderick) they have him getting more digits. That particular choice of number to take the cube root of seems almost too convenient, if you ask me. Keith Ramsay === Subject: : Re: New Randomness Test Random means passing each and all of his stated list of randomness > tests. It doesn't require a rocket scientist to [CapitalThorn]gure that out. >> 2. Can you explain why the digits of pi don't meet that de[CapitalThorn]nition? > Because they are the digits of pi, the ratio of a circle's > circumference to its diameter in zero curvature 2D space. That isn't > random. It doesn't require a rocket scientist to [CapitalThorn]gure that out. >> If we accept your de[CapitalThorn]nition of random above, and if the digits of >> pi do pass his stated list of tests, then how can these digits not >> be random, according to your de[CapitalThorn]nition? > Because if you know how to measure it anywhere, you get the same > answer everywhere. To be more explicit, if it were random, you'd > never get the same answer twice in one place much less anywhere else. >> Why did you paste stuff into my post? > What stuff? I just hit Reply, which brought up your post as a quote > (indicated by the indenting and stacked carets/vertical lines, depending > on how your newsreader does such things, on the left margin), deleted > your sig, and typed in my response. The quoted material I left > added automatically by my newsreader to indicate which quoted material > goes with which poster. I notice there's some extra spaces above between 2. Can you > explain... and Because they... but I didn't deliberately add them. I > suspect browser clash. The only deliberate changes I made were removing your sig (because you What are you seeing that I'm not? Sorry, a brain fart. What threw me off was that my point was that _whatever_ was given as the de[CapitalThorn]nition of random, anything that passes the tests must be random. In other words, the details of how the numbers were generated were irrelevant, unless addressed by the de[CapitalThorn]nition (with its tests). Anyway, my mistake. Gib === Subject: : Re: New Randomness Test X-SessionID: lPm3c-7360-45-23776@news.uchicago.edu X-Hash-Info: post-[CapitalThorn]lter,v:1.4 X-Hash: 183a19aa bcda5c3a 04d1b738 a903dced b7ef16fb >In message , >In message , > Mati Meron | When you argue with a fool, >> meron@cars.uchicago.edu | chances are he is doing just the same Keep meaning to ask you, Mati... your responses to my posts do not >appear on an indented branch under mine, but appear on the same >branch. Any idea why? >I've no idea. This is being posted from an old machine (which I >>*only* use for posting, nothing else) so perhaps there are some >>peculiarities in its interactions with the net. >><4047AA20.B79AE01D@hate.spam.net> <4FT1c.37$T4.19306@news.uchicago.edu> <4GX1c.38$T4.21222@news.uchicago.edu><5Gh2c.45$T4.31736@news.uchicago.edu> from the list. It looks as if you're >following up your own posts, not his. >>Well, as I said, it is an old machine. Still, useful since it is >>immune to viruses and spam atacks do not interfere with anything I do. >>So, if it has some peculiarities, so be it. >And there's con[CapitalThorn]rmation. My last message-ID doesn't appear in the >References, as it should. Oh, well. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same === Subject: : Re: New Randomness Test <4047AA20.B79AE01D@hate.spam.net> <4FT1c.37$T4.19306@news.uchicago.edu> <4GX1c.38$T4.21222@news.uchicago.edu> <5Gh2c.45$T4.31736@news.uchicago.edu> In message , >>In message , > Mati Meron | When you argue with a fool, > meron@cars.uchicago.edu | chances are he is doing just the same >>Keep meaning to ask you, Mati... your responses to my posts do not >>appear on an indented branch under mine, but appear on the same >>branch. Any idea why? >I've no idea. This is being posted from an old machine (which I >*only* use for posting, nothing else) so perhaps there are some >peculiarities in its interactions with the net. ><4047AA20.B79AE01D@hate.spam.net> <4FT1c.37$T4.19306@news.uchicago.edu <4GX1c.38$T4.21222@news.uchicago.edu from the list. It looks as if you're >>following up your own posts, not his. >Well, as I said, it is an old machine. Still, useful since it is >immune to viruses and spam atacks do not interfere with anything I do. >So, if it has some peculiarities, so be it. And there's con[CapitalThorn]rmation. My last message-ID doesn't appear in the References, as it should. -- Richard Herring === Subject: : Re: New Randomness Test > 2. Can you explain why the digits of pi don't meet that de[CapitalThorn]nition? The sequence of decimal digits of pi is highly compressible; > a random sequence in general won't. L.Rodriguez response: > You can have a short program for constructing the sequence of pi, but > that don't means that you can compress any string of digits of pi, > more that its original length. Right. Suppose you receive a string of 100 digits of pi situated after the K > th digit of pi. If K is a number of 200 digits, the total length of > the minimum possible program is: Length of program of pi + 200 > 100 Unless, of course, the 200 digit number is expressible as > the result of a shorter calculation of some form... But you are right that there must be breakeven points > where it is just as ef[CapitalThorn]cient to transmit the original > sequence as the formulaic version. This does not alter the fact that the sequence is not > truly random, since it is entirely predictable given > the knowledge of the appropriate formula. That fact is zero though. Since a formula is not a prediction. Hence pseudo-random numbers are used because to compute the 2^3^5^7^11^13^17^19 th digit of pi takes longer than the universe has been around. Nevermind how long pi formulas have been around. Perhaps a given sequence might be thought of as > relatively random with respect to a given compression > method, sort of like numbers can be relatively prime? But, perhaps not. Since you [CapitalThorn]rst have to convince the mathema-philosophers to develop a theory of relative that works for longer than the expected lifetime of tritium, which is not that long. === Subject: : Re: New Randomness Test But you are right that there must be breakeven points > where it is just as ef[CapitalThorn]cient to transmit the original > sequence as the formulaic version. This does not alter the fact that the sequence is not > truly random, since it is entirely predictable given > the knowledge of the appropriate formula. > That fact is zero though. Since a formula is not > a prediction. Really? Amazing. Do you expect to get different digits everytime you use the same formula to generate the digits of pi? Do you expect different formulas that give the digits of pi to give different results? > Hence pseudo-random numbers > are used because to compute the 2^3^5^7^11^13^17^19 th > digit of pi takes longer than the universe > has been around. Nevermind how long pi > formulas have been around. Nevertheless, the 2^3^5^7^11^13^17^19th digit is given by a formula, regardless of the computing time that might be involved. Accessiblity is not a prerequisite for existence in mathematics. Also, what would one do with the 2^3^5^7^11^13^17^19th digit of pi that one couldn't do with the 2^3^5^7th? Perhaps a given sequence might be thought of as > relatively random with respect to a given compression > method, sort of like numbers can be relatively prime? > But, perhaps not. Since you [CapitalThorn]rst > have to convince the mathema-philosophers > to develop a theory of relative that works > for longer than the expected lifetime of > tritium, which is not that long. I have no idea what you're trying to say there. === Subject: : Re: No such thing as a random number? > OK, let talk about a sequence of random numbers. > Use base 2. e.g. 0,1,0,1,1,1,0,1,1,0 > This may also be considered a random sequence of numbers? === Subject: : Re: A Paradox of The Central Limit Theorem eta(sigma/sqrt(n))+mu)= lim_[n approaches in[CapitalThorn]nity]Pr{_n = mu}, I'd already come close to deciding that you were a fool or > a troll yesterday - this morning I got an email stating > exactly that. I hope you enjoy living in your little world > of paradoxes - if you're ever interested in learning some > actual math you need to pay more attention. To write such things reveals that you have failed to solve the paradox, and you are at your wit's end, aren't you? You mentioned mistakes, but evey paradox is a kind of logical mistakes. Don't you understand this truth? Anyway, grapes always look sour to a skunk like you. === Subject: : Re: A Paradox of The Central Limit Theorem >eta(sigma/sqrt(n))+mu)= lim_[n approaches in[CapitalThorn]nity]Pr{_n = mu}, >> I'd already come close to deciding that you were a fool or >> a troll yesterday - this morning I got an email stating >> exactly that. I hope you enjoy living in your little world >> of paradoxes - if you're ever interested in learning some >> actual math you need to pay more attention. >To write such things reveals that you have failed to solve the >paradox, and you are at your wit's end, aren't you? No, it reveals that sci.math is not the only place you've been making a persistent fool of yourself. I solved the paradox in my [CapitalThorn]rst post on the topic. There are three n's in a certain expression, and you take a limit letting two of them tend to in[CapitalThorn]nity, leaving the third just sit there - of course that's going to lead to nonsense. >You mentioned mistakes, but evey paradox is a kind of logical >mistakes. Don't you understand this truth? Right. Like if I said 2 + 2 = 5 was a paradox - if you pointed out it was simply an _error_ you'd be showing you didn't understand this truth. >Anyway, grapes always look sour to a skunk like you. ************************ David C. Ullrich === Subject: : Re: A Paradox of The Central Limit Theorem > lim_{n-> in[CapitalThorn]nity} Pr( = mu) = 1, which is nothing but the Strong Law > of Large Numbers. Oh, No! I mean Pr{lim_{n-> in[CapitalThorn]nity}}( = mu) = 1 is nothing but the Strong Law of Large Numbers. === Subject: : Re: A Paradox of The Central Limit Theorem No. To quote from your message Pr{lim{n->in[CapitalThorn]nity}f(n)} > which appears to be the probability of the event lim{n->in[CapitalThorn]nity}f(n) > n'est-ce pas? Later you con[CapitalThorn]rmed that f(n) could be the event _n = mu. > Hence lim_{n-> in[CapitalThorn]nity}f(n) = lim_{n-> in[CapitalThorn]nity} ( = mu) > was exactly one of the things you were talking about. So, what is it? I can't clearly understand what you want to say. lim_{n-> in[CapitalThorn]nity} Pr( = mu) = 1, which is nothing but the Strong Law of Large Numbers. But waht does lim_{n->in[CapitalThorn]nity}( = mu) mean. if any? === Subject: : Re: A Paradox of The Central Limit Theorem >> No. To quote from your message Pr{lim{n->in[CapitalThorn]nity}f(n)} >> which appears to be the probability of the event lim{n->in[CapitalThorn]nity}f(n) >> n'est-ce pas? Later you con[CapitalThorn]rmed that f(n) could be the event _n = >> mu. Hence lim_{n-> in[CapitalThorn]nity}f(n) = lim_{n-> in[CapitalThorn]nity} ( = mu) >> was exactly one of the things you were talking about. >> So, what is it? I can't clearly understand what you want to say. > lim_{n-> in[CapitalThorn]nity} Pr( = mu) = 1, which is nothing but the Strong > Law of Large Numbers. But waht does lim_{n->in[CapitalThorn]nity}( = mu) mean. if any? That's what I am asking you --- it's *your* phrase: *you* used it. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: : Re: Die Cantor Die If mathematics, like religion, lived in the private domain, > then the mathematicians, like the religionists, would be fully > justi[CapitalThorn]ed in closing their ears to objections from outsiders. > But, of course, mathematics lives in the public domain. And therefore mathematicians are obliged to listen to unmathematical > twaddle about mathematics? I usually imagine that you're playing games and you know it, but maybe you're genuinely lost and confused. Here in the United States the intrusion of religion into the public domain is a serious issue. The religionists would love to have you on their side, at least if they were ever to get into positions of power. To think that they could dismiss any objections to their agenda as unreligious twaddle about religion, which they're not obliged to listen to. === Subject: : Re: Die Cantor Die > To be sure, formal systems, and hence set theory, are themselves > objects that live in the world of computation. And hence they > are objects which a mathematician might study. But, classical > set theory is not a faithful representation of the universe > of mathematics (it contains ghosts), and is totally unworthy > of its current status as a foundation of mathematics. > So is this a suggestion that mathematics would be better off with no > foundation at all? I can't believe I haven't made myself clear on this. I'm advocating that a reality check (what I'm calling observability) be incorporated into the foundations of mathematics. For lack of such a reality check, mathematics has drifted off into a fantasy world. === Subject: : Re: Die Cantor Die > here about 5 years ago] The objection to Cantor's Theory (i.e. classical set > theory) comes from the constructivist view that mathematics > necessarily has something to do with computation. This > point of view is especially appealing to the younger > generation. I tend to think of myself as being in the younger generation, but don't have the constructivist view. What has changed in the last thirty years is the ubiquity > of computers. Today's children are growing up surrounded > by computers, and they're growing up in families where > their parents are also comfortable with computers. They're > living in a computer culture. So how does this affect the perception of Cantor's Theory? To someone who has grown up with computers, the world > of computation [i.e. the virtual world, or the abstract world > living inside the mind of the computer] must seem like a > very real world. It's not > a physical world, to be sure, but it is concrete and has an > objective existence nonetheless. And anyone who plays computer > games surely believes that everything in the real world can be > modelled in the virtual world (the world of computation). Furthermore, unless you postulate the existence of some > non-computational ingredient in the human brain which no > arti[CapitalThorn]cial intelligence running on a digital computer has > access to, you are compelled to admit that intelligence > itself is a phenomenon that lives in the world of > computation. To deny that is to believe that humans have > access to something which cannot be examined or formalized, > but which nevertheless is a reliable guide to the truth. > And that's best described as mysticism, or blind faith. Many of these same people are aware of the concept of random and pseudo-random numbers. Would these be considered more computational or non-computational? If the behavior is random from the user's perspective, will it feel non-computational? So what's the point? The point is that the world of computation is a maximally > rich world. We don't lose any of our capacity to understand > the reality we live in if we believe that the purpose of all > abstract thinking is to relate the phenomena we observe in > the real world to phenomena we observe in the world of > computation. Why would I think the purpose of all abstract thinking is to relate observable phenomena to computational phenomena? One method of abstraction is to move away from numeric data to categorical data (such as thinking in terms of short or tall rather than numeric height). Other abstractions are even more extreme, such as those in the realm of art. To someone who is comfortable with thinking about the > world of computation as a real world, mathematics is > most usefully de[CapitalThorn]ned as the study of phenomena that are > observable in the world of computation. As a conceptual aid, > we can think of the computer as a microscope which helps us > peer into the world of computation, and mathematics as the > science that studies the phenomena observed through > that microscope. I would argue that this view of mathematics is extremely narrow, and fails to encompass the full breadth of the topic. Computation is one aspect of mathematics, but you seem to be arguing that it is the sole purpose. So where does Cantor's theory [CapitalThorn]t in to this scenario? It doesn't. And that's the problem. To someone who is comfortable with the notion of a world > of computation, it is extremely plausible to de[CapitalThorn]ne the > universe of mathematics to be the universe of objects that can > be seen through the microscope (i.e. objects that live in the > world of computation). Cantor's theory formally implies > the existence of objects that cannot be seen through that > microscope, and hence Cantor's theory is a mythology. What's being objected to here is Cantor's interpretation of > his diagonal argument, and de[CapitalThorn]nitely not the diagonalization > method itself. Informally, if we have a well de[CapitalThorn]ned list of > well de[CapitalThorn]ned real numbers (i.e. we can compute every digit of > every number on the list), the diaganolization argument gives > us a way to construct a real number not on that list. That > argument surely lives in the world of computation. But if > we accept Cantor's interpretation of the diagonalization > argument, as codi[CapitalThorn]ed in classical set theory, we are led to > believe in the existence of a super-in[CapitalThorn]nite world that has > no connection to the world of computation, other than the > borrowing of terminology. His argument simply states that if you construct a function f:N->[0,1], then he can construct a number in [0,1] that is not in the codomain of f. He concludes that there is no f that is an onto map. Conclusion: [0,1] has more elements than N, based on a formalized notion of comparing the number of elements in two sets. I'm not clear on why you feel this causes a problem. To be sure, formal systems, and hence set theory, are themselves > objects that live in the world of computation. And hence they > are objects which a mathematician might study. But, classical > set theory is not a faithful representation of the universe > of mathematics (it contains ghosts), and is totally unworthy > of its current status as a foundation of mathematics. What do you mean by ghosts? How is it not a faithful representation? When the classical mathematicians hear about all this, they > generally go into a tizzy and refuse to think straight about it. > They go into a defense mode. They seem to believe that > they are about to lose everything they've worked for. But in > fact, very little is lost. Ok, I'm now of[CapitalThorn]cially in a tizzy. Now, could you please be a little more precise in what you mean? So far you have made claims, but you have not stated clearly (to my mind at least) what your assumptions are. Perhaps de[CapitalThorn]ning what is or is not a computation would be a good start. Note: our notions of what it means to compute something are likely to be different, since I think of it as Turing Machine Computable, yet a Turing Machine does not and cannot exist in reality. For one thing, nothing of applied mathematics is lost. The > idea of applying mathematics means to create computational > models of real world phenomena in which the phenomena > observed in the model correspond to the phenomena observed > in the real world. So this notion of mathematics being the > study of phenomena observable through computation [CapitalThorn]ts > applied mathematics like a glove, so to speak. But we also don't lose most of the abstractions that the pure > mathematicians are fond of, though we are forced to understand > them from a different perspective. For starters, we can build up a theory of the in[CapitalThorn]nite, so long as > we don't allow anything into the universe of the in[CapitalThorn]nite that does > not have corresponding approximations that can be seen > through our microscope. Thus, for example, we can say that > perfect circles exist in the sense that we can see arbitrarily > close approximations to circles. Likewise, pi exists in that > we can compute arbitrarily close approximations. Functions > like exp(x) exist likewise. Fourier Transforms exist. > Manifolds exist. Solutions to differential equations exist. You seem to be saying that pi does not exist in the sense that most mathematicians think of it, the ratio of circumference to diameter in a perfect circle, because perfect circles do not exist. Yet the point of abstraction would be to go from what does NOT exist in reality to what can exist in theory as a model. The modelling process can then be reversed to that I can model the concept of a circle with a round [CapitalThorn]gure on paper. Is my notion of abstraction not corresponding to yours? What's more, in[CapitalThorn]nite sets exist, as long as we keep in mind > what it means to say that they exist. It means that we can > see arbitrarily close approximations to them. For example, for > the set of positive integers, we can see sets of the form {1..N} > for arbitrarily large N, and such sets can be taken to be > approximations of the set of all positive integers. For the set of > real numbers (a continuum), an approximation must be a [CapitalThorn]nite > set of [CapitalThorn]nite approximations of real numbers. So you are saying that the natural numbers are the limit of the sequence of [CapitalThorn]nite sets {1}, {1,2} , {1,2,3}, ... ,{1,2,...N}, ... ? So if in[CapitalThorn]nite sets exist, and power sets exist, what's wrong > with Cantor's theory? Besides the fact that the word exist is not clear to me any more? What I'm proposing is that mathematics should be built upon > a notion of observability. We can say that a mathematical > object is observable if we can observe arbitrarily close > approximations to it in our microscope. We can say that > the object has certain properties, if and only if we can see that > approximations to the property hold for approximations to > the object. We can say that mathematical assertions have > observable content if they make predictions that can be > observed in our microscope. We must recognize that what we > observe in our microscope are approximations, and we must > adopt a logic that is capable of handling imprecision and > uncertainty. This sounds like you want to apply the scienti[CapitalThorn]c method to mathematics. If your observations do not correspond to theory, does that represent a limitation of the theory, or of your computational device? I have a calculator that displays the 4194303th root of 2 as 1. Is that root 1, or is my calculator too limited? This notion of observability leads to a unique and minimalist > theory of the in[CapitalThorn]nite. And that theory is not Cantor's Theory. > Cantor's Theory implies that there exist things that are not > observable. And your theory implies, to my mind, that the in[CapitalThorn]nite does not actually exist. So what was Cantor's mistake? Cantor started off asserting that in[CapitalThorn]nite sets exist, without > consideration of what it means to observe those in[CapitalThorn]nite sets. He wasn't concerned with observing them. You are starting with a different set of axioms from Cantor, and drawing different conclusions. This is the fundamental conßict. If you state your axioms and de[CapitalThorn]nitions, you are free to use them as you see [CapitalThorn]t, but they are not a basis for stating that a different system of axioms is invalid. It appears to me that you want to change the rules, and based on your new rules, criticize what Cantor did with his. > He told us that power sets exist, without telling us what it > means to say that they exist. He went on talking about > properties of the objects, without telling us what it means > to say the objects have those properties (other than the > super[CapitalThorn]cial notion that what it means for an object to have a > property is that a sentence in the formal language can be > interpreted to say that the object has a property). For him, and many others, what you consider a super[CapitalThorn]cial notion is actually the heart of what it means to be doing mathematics. You appear to want math to be an experimental science, rather than a logical structure. There may be a place for your view within mathematics, but it appears that you wish to focus on something other than mainstream mathematics. > And then he > came to the conclusion that there must exist more objects > than can be observed. Cantor chose axioms which would > formally lead to the conclusion he wanted to reach, without > regard to the observable universe. (note that Cantor himself was > almost certainly not fully aware of the unrealistic implications > of his interpretations of the diagonalization method, and > judging by his religious convictions, was actually incapable > of distinguishing reality from fantasy) Are you sure about this? Is it not possible that he chose axioms that described his expectations, and some of the conclusions were a surprise to him? Is it not possible that his axioms were designed to describe [CapitalThorn]nite sets, and the idea of an in[CapitalThorn]nite set lead to interesting and non-intuitive conclusions? Emphatically, Cantor has not shown that there is any logical > inconsistency in assuming that the universe of observable > objects is the whole mathematical universe. He has not > given us a compelling reason to accept his fantasy world, > which goes far beyond the observable universe. Nor have you shown that Cantor's work is inconsistent or that the whole mathematical universe should be restricted to your notion of the observable. Cantor did, however, formalize his work. You have not shown in this paper that you have. As I have already pointed out, by accepting these ideas, > we do not even lose Cantor's theory (as a formal theory). That > is, formal theories themselves are objects that exist in the > observable universe. They are objects which mathematicians > might want to study. It's just that Cantor's theory has no right > to the claim of being the foundation of mathematics. It depends on how broad a view of mathematics you hold. > I actually believe the issue here is quite important. What's going on here can be understood as (quasi-) religious > fanaticism. The mathematicians are fanatically devoted > to beliefs that have no connection to reality, and they are > using their positions of authority to impose those beliefs on > others. They are quick to call those who question their > authority crackpots, idiots, and imbeciles. They exclude such > people from their community. This is not much different from the > Nazis calling their enemies untermenschen. Or, if you looked a little closer, you would [CapitalThorn]nd that different mathematicians prefer different axioms for their work, and while acknowledging other axioms systems as being self-consistent, consider them to be in violation of their intuition. A classic example from logic is whether or not to accept the Axiom of Choice as being true or false. You appear to be making a similar distinction, in which you have chosen not to accept the existence of in[CapitalThorn]nite sets. That is your choice, and as long as you can show that your Axioms are consistent, you can do useful and interesting work. Railing against a different set of Axioms is more in line with a religious belief system. Why do you wish to overthrow Cantor's set theory? Why not just work on developing your own, and then show how it is useful for doing something which Cantor's does not, or appeals to intuition in a way that Cantor's does not? Religious fanaticism is very serious threat to peace in this > world. If we could [CapitalThorn]nd a way to break through the barriers of > fanaticism within the mathematics community, we could likely > learn valuable lessons in dealing with fanaticism in the world > at large. Physician, heal thyself. You are expressing an anti-Cantor attitude, and seem to assume that all others have a pro-Cantor attitude. State your axioms and let's discuss what can be done with them. There is no need to kill a dead man. That being said, it's also true that [CapitalThorn]ghting fanaticism can > sometimes be more disruptive than the fanaticism itself. Caveat emptor. Formally state what you are working with, then. You have not offered an alternative to Cantor, just the desire for there to be one. -- Will Twentyman email: wtwentyman at copper dot net === Subject: : Re: Die Cantor Die > But if > we accept Cantor's interpretation of the diagonalization > argument, as codi[CapitalThorn]ed in classical set theory, we are led to > believe in the existence of a super-in[CapitalThorn]nite world that has > no connection to the world of computation, other than the > borrowing of terminology. His argument simply states that if you construct a function f:N->[0,1], > then he can construct a number in [0,1] that is not in the codomain of > f. He concludes that there is no f that is an onto map. Conclusion: > [0,1] has more elements than N, based on a formalized notion of > comparing the number of elements in two sets. I'm not clear on why you > feel this causes a problem. Small nit to be picked: I hope you mean that there is a number not in the RANGE, or IMAGE of f, since the Ôcodomain' of any f:A -> B is almost universally taken to be B. === Subject: : Re: Die Cantor Die Just to help me understand where you're coming from, would > you criticize an atheist for his ideas about religion, if > his ideas had no religious content? It is of course open to you to reject mathematics, as an atheist > rejects theistic religion. But why should such a rejection be of > interest to mathematicians? If mathematics, like religion, lived in the private domain, > then the mathematicians, like the religionists, would be fully > justi[CapitalThorn]ed in closing their ears to objections from outsiders. > But, of course, mathematics lives in the public domain. Nobody has that though. Mathematicians who think that imaginary numbers have something to do with doors are not only oblidged, but mandated to be dead before they can even contact their [CapitalThorn]rst set of moron Lawyers at AT&T and PBS to check their Licensing fee accounts with the idiots at Harvard to see if they're lastest unbootable UNIX REV 4.12.23.455. has been impounded by the FBI yet. === Subject: : Re: Die Cantor Die > If mathematics, like religion, lived in the private domain, > then the mathematicians, like the religionists, would be fully > justi[CapitalThorn]ed in closing their ears to objections from outsiders. > But, of course, mathematics lives in the public domain. > And therefore mathematicians are obliged to listen to unmathematical > twaddle about mathematics? Nice response. Now de[CapitalThorn]ne mathematics. That's the problem Torkel. Mathematics is not the endless generation of theorems from axioms and de[CapitalThorn]nitions. People outside of mathematics actually use the stuff to justify and explain. That is why I have turned to making rude remarks about faculty teas. Bureaucratic university administrations have so segregated thinkers that the presentation of knowledge is effectively incoherent. And it resolves to mathematics because everyone *believes* mathematics is logical and uses it justi[CapitalThorn]cationally. Very little presentation now, even with Ômotivation', explains the concepts. This is not at all surprising; few have seen a clear use of the concepts, and most mathematicians only have a working knowledge of them. One can prove theorems formally without understanding the underlying concepts. The statement was made after a mention of Wiles' proof, but, the paragraph that followed indicated that the remark was made with general applicability in mind. You may not like it. But it is reasonable to conclude that the concept of theoretical coherence is meaningless if there is no theory--and, I am not talking about nonsensical syntactic manipulation. Now go bury your head again like a good little ostrich. :-) mitch === Subject: : Re: Die Cantor Die > Mathematics is not the endless generation of > theorems from axioms and de[CapitalThorn]nitions. People outside of mathematics > actually use the stuff to justify and explain. Anybody is of course welcome to take an interest in Petry's remarks. My question was, are mathematicians obliged to take an interest in his tirades? I would say that of course they are not. Another question is, would you expect them to be interested in his remarks? There is no reason to expect any such interest. Mathematicians tend to be interested in mathematics - e.g. that of Kronecker, Brouwer, or Bishop - not in shrill tirades. === Subject: : Re: Die Cantor Die > The objection to Cantor's Theory (i.e. classical set > theory) comes from the constructivist view that mathematics > necessarily has something to do with computation. This > point of view is especially appealing to the younger > generation. Not _all_ of the younger generation I might note. (Though I suppose > that means I've been brainwashed by the older generation. :) Did you grow up with an interest in computers? Very much so! I was fascinated by computers, and machines in general, from an early age. I was an avid BBC Micro programmer from age 7 (long before I had any real interest in maths). Unfortunately I kind of lost interest in programming around age 15. Six years later, I still program a little bit, though out of necessity rather than as a hobby. But that doesn't really matter. The fact is, I know people my age who are _really_ into computers and programming, and they've never expressed this opinion that mathematics should be all about computation... Michael === Subject: : Re: Die Cantor Die > The fact is, I know people my age who > are _really_ into computers and programming, and they've never > expressed this opinion that mathematics should be all about > computation... Try a little experiment. Suggest to them a possible de[CapitalThorn]nition of mathematics as the study of phenomena observable through computation. Point out, by way of example, that all of the axioms of high school algebra make predictions about the results of computation. (For example, the distributive law tells us that if we compute a*(b+c) for arbitrarily chosen a,b,c, we will get the same result as when we compute a*b + a*c.) Then ask them if they see any possible de[CapitalThorn]ciency in that de[CapitalThorn]nition of mathematics. === Subject: : Re: Die Cantor Die md285@cam.ac.uk says... >But that doesn't really matter. The fact is, I know people my age who >are _really_ into computers and programming, and they've never >expressed this opinion that mathematics should be all about >computation... Well, I'm too old to be the younger generation, although I've used computers for more than half my life, and I think David's thesis is completely wrong. Backwards, even. It isn't that mathematics is about computation---it is that computation is a tool in understanding mathematics. Mathematics itself can describe things having nothing to do with computation: motion, population growth, gravity, quantum mechanics, etc. To make computation both a tool of mathematics and the *subject* matter of mathematics is much too limiting. Mathematics is about the world, not about our computing machines. David's calling classical mathematics religion is like the solipsist saying that belief in the existence of other people is a religion. It has no content, but expresses a very closed, insular mind. I don't know where David's bugaboo about classical mathematics comes from; maybe as a child he was frightened by a math professor. I especially don't understand why he isn't content to just say that certain branches of mathematics have no interest for him. Instead, he says that it should be of no interest to *anyone* other than wild-eyed fanatics. What is the point of that? === Subject: : Re: Die Cantor Die > Mathematics itself can describe things having nothing to > do with computation: motion, population growth, gravity, quantum > mechanics, etc. To make computation both a tool of mathematics and the *subject* matter > of mathematics is much too limiting. To say that I'm trying to limit the subject matter of mathematics to computation is akin to saying that set theory limits the subject matter of mathematics to sets. It's a despicable lie. I have no reason to value your opinion on anything related to this topic, but you might change my mind if you could coherently explain why you believe that set theory, with its wild and speculative implications about the existence of a super-in[CapitalThorn]nite world having nothing to do the world we live in, is a better foundation for mathematics than computation, especially in light of your observation that mathematics can describe motion, population growth, gravity, quantum mechanics, etc. === Subject: : SCHOENFELDS RANDOM THEOREM SCHOENFELDS RANDOM THEOREM: With probability 1, all [CapitalThorn]nite sequences of integers contained by [n,m] occur in[CapitalThorn]nitely in an in[CapitalThorn]nite sequence of random integers bound by [n,m]. P(S occurs in R) = 1 - lim j->+inf (1 - 1 / [(|m|-|n|) + 1]^|S|)^j = 1 - 0 = 1 where R : random sequence of integers bound by [n,m] |R| = +inf S : arbitrary sequence of integers |S| < +inf I also put forth that an in[CapitalThorn]nite sequence S occurs within an in[CapitalThorn]nite random sequence R given |S| < |R|. Example: given an uncountably in[CapitalThorn]nite sequence of random reals, all countably in[CapitalThorn]nite sequences of integers occur with probability 1, countably in[CapitalThorn]nite times. JS === Subject: : Re: SCHOENFELDS RANDOM THEOREM > As Uncle Al said: Trivially disproven by example in Hofstadter's Godel, Escher, Bach. It's only 777 pages long. If you look at each page for one second you can [CapitalThorn]nd the table within 12 minutes. === Subject: : Re: Spectral sequences in knot theory Originator: israel@math.ubc.ca (Robert Israel) > used to solve problems in knot theory. Any references ? > McCleary _A User's Guide to Spectral Sequences_ references Vassiliev, VA, Complements of Discriminants of Smooth Maps; Topology and Applications. Translated from the Russian by B. Goldfarb. Translations I know absolutely nothing about this, though -- just passing along the ref. Aaron === Subject: : Re: Die Petry, die >> So what is Petry's mistake? >> Petry started off asserting that observability matters, without >> consideration of what it means to observe. Actually, the question of what it means to observe is > exactly the issue I was addressing. Was it? But the point is that you failed to demonstrate the relevance > of your peculiar notions of seeing and observation > to mathematics. Silly circular discussion. If you de[CapitalThorn]ne mathematics to be the implications of set theory, then indeed the notion of observability has no relevance to mathematics. But if we agree from the outset that mathematics is intended to be a tool we can use to help us understand the world around us - the world we observe - then the notion of observability that I advocate is precisely the notion that will keep mathematics from drifting off into the world of make-believe. === Subject: : Re: Die Petry, die |> Classical [CapitalThorn]rst-order logic is the same as intuitionist logic with the |> law of excluded middle added. The law of excluded middle says that each |> proposition P is either true or false. Using classical logic usually is |> nonconstructive, because usually in a system there's no algorithm for |> determining in general whether any given proposition is true. | |? So classical logic is nonconstructive since it's undecidable. ? | |I thought most formalizations of intuitionist/constructive logics |were undecidable too --- does that make constructive logic |nonconstructive? The undecidable system I was referring to was the axiom system, not just the logic in it. The so-called disjunction property says that for every statements A and B, if A or B is a theorem, then either A is a theorem or B is a theorem. The disjunction property is sometimes considered a kind of litmus test for constructivity. A system that fails to satisfy the disjunction property normally is not considered constructive. (Another one is numerical existence property, which says that if there exists an n such that P(n) is a theorem, then so is P(N) for some numeral N. That's a decent approximation to what it means for an axiom system to count as constructive. It often implies the disjunction property since A or B often is equivalent to some rephrasing like for some n, if n=0 then A and if n<>0 then B.) If an axiom system is undecidable and has the law of excluded middle, it fails the disjunction property, because for each sentence P, there is a theorem P or not P, but it's not always true that either P is a theorem or not P is a theorem. A typical constructive system satis[CapitalThorn]es the disjunction property and is undecidable, and consequently must not have the law of excluded middle. For completeness sake, I was alluding to the possibility of being constructive AND using classical logic. By the above, it has to be a decidable system. A decidable system like Presburger arithmetic (which has addition as an unde[CapitalThorn]ned term, but not multiplication) with classical logic could still be considered constructive in a sense. In such a case, saying P or not P happens to be okay for each P expressible in the system, even on a constructive interpretation of all the terms, because you can always in principle prove P or prove not P. Keith Ramsay === Subject: : How are these minimal paths called? Let G=(V,E) be a connected graph without loops or parallel edges and W some subset of V. Consider all [CapitalThorn]nite paths through W, (which should not necessairly be closed, may involve some nodes from VW and which may go through the same node more than once.) Among all these paths, take those with a minimal length. How are they called? How is this minimal length called? Is anything done in this direction, does any bibliography exist? This looks a bit like a real travelling salesman through the cities from the set W, where the salesman is also allowed to use the cities from VW, and where each connection has weight 1. -- Alex. PS. My real email-address is formed by deleteing the letter combination loeschedies from the email address given. === Subject: : Randomness is a notion. Hi Servo, You asked what the word Ô random Ô means. First and foremost, randomness is a notion. And the notion is this: Insuf[CapitalThorn]cient information. For example: Otherwise predicable weather is labeled as random when one's information about it is insuf[CapitalThorn]cient. Likewise, an otherwise predicable coin toss is labeled random when one's information be insuf[CapitalThorn]cient. ( As it usually is ) Similarly, singularities are random, not physical. ( e.g. The start of the big bang is not perfectly known ) Because science is forever discovering more regularities in nature, it's best to assume that the future is perfectly [CapitalThorn]xed. ( Even though the future is imperfectly known ) This sticks in people's craws for the same reason that evolution is so repulsive. i.e. People would rather exploit unknowns to confabulate delusion of grandeur. === Subject: : Proof that Multiplication Modulo n is Associative? I'm trying to [CapitalThorn]nd an example of a proof that multiplication modulo n is associative. Every reference I can dig up on the web either deals with some other problem and tells the reader that the associativity of modular multiplication can be assumed, or else claims it trivially follows from the associativity of regular multiplication. It's been (too many) years since I took any courses in abstract algebra, and I don't currently have access to any textbooks that might supply the proof. This was actually a problem on a friend's quiz (he's taking an abstract algebra course at night school for fun... the weirdo :) and ever since he brought it up I've been driving myself crazy trying to prove this seemingly simple fact. I've come up with several candidate proofs myself... but all of them seem to depend upon other facts that I'm not sure I can assume. -Bill === Subject: : Re: Proof that Multiplication Modulo n is Associative? Adjunct Assistant Professor at the University of Montana. >I'm trying to [CapitalThorn]nd an example of a proof that multiplication modulo n >is associative. >Every reference I can dig up on the web either deals with some other >problem and tells the reader that the associativity of modular >multiplication can be assumed, or else claims it trivially follows >from the associativity of regular multiplication. Yes: you are looking at the quotient ring Z/nZ, so associativity follows. >It's been (too many) years since I took any courses in abstract >algebra, and I don't currently have access to any textbooks that might >supply the proof. Well, you can try proving it from the de[CapitalThorn]nition of congruent modulo n: a = b (mod n) if and only if n|(a-b). Now you want to prove that (ab)c = a(bc) (mod n); which means that you need to show that n|[(ab)c - a(bc)], which is trivial since the multiplication here is the usual multiplication for the integers. Since you say this is driving you crazy, I assume that what you really want to do is de[CapitalThorn]ne the multiplication just by remainders. That is, replace a with the remainder of dividing a by n, b with the remainder of dividing b by n; ab by the remainder of dividing ab by n, etc, and check it directly on the remainders somehow.... If so, I can see how it would drive you crazy! Take a, b, c, and we may assume that a, b, and c satisfy 0<= a,b,c < n. We want to show that remainder(remainder(ab)*c) = remainder(a*remainder(bc)), where remainder(x) means the remainder of dividing x by n. Write ab = p1*n + r1, 0<= r1 < n r1*c = q1*n + R1, 0<= R1 < n bc = p2*n + r2, 0<= r2 < n a*r2 = q2*n + R2, 0<= R2 < n and you want to show that R1 = R2. Consider R1-R2 = (r1*c - q1*n) - (a*r2 - q2*n) = (r1*c - a*r2) + (q2-q1)*n = (ab-p1*n)*c - a*(bc-p2*n) + (q2-q1)n = (ab)*c - a*(bc) - p1*c*n + a*p2*n + (q2-q1)n = n*(q2-q1 - p1*c + a*p2). That means that |R1-R2| is a multiple of n, and therefore is either equal to 0, or has absolute value greater than n. If R2<= R1, then 0<=|R1-R2| = R1 - R2 <= R1 < n, so R1-R2 = 0, so R1=R2. If R1<= R2, then 0<=|R1-R2| = R2-R1 <= R2 < n, so R1-R2 = 0, so R1=R2. Since either R2<=R1 or R1<=R2 (or both), we conclude that in either case, R1=R2. -- = = 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: : periodic wavelets Hi! I was wondering if there is anybody out there who has a matlab code,or anyother code, to produce periodic wavelets. I'm trying to approximate a function with help of periodic wavelets, but for some reason I cannot [CapitalThorn]nd enough information on how I can get the coe[CapitalThorn]cients for it. It don't really matter what kind of wavelet it is ( daubechies would be good). Also, I'm open to any help or suggestion you might have for me. Cheers. === Subject: : Random pi digits and Mahler's theorem charset=Windows-1252 People like to speculate about whether the digits of pi are random in some (or all) integer bases b >= 2. One way of posing the question is to ask whether the digits behave statistically like a realization of a sequence of iid rv's uniformly distributed on {0, 1, ..., b-1}. A fact sometimes used to argue in the negative is a theorem of Mahler stating that for all integers p,q > 1, |pi - p/q| > 1/q^42. For example, a consequence of the theorem is that given any n, the next 41*n digits after the nth digit cannot all be zero -- which seems to contradict statistical independence. But just how unlikely is this very same behavior with a corresponding sequence of iid random variables? In other words ... If X is uniform on (0,1), then what is the value of P= pr( for all integers p,q > 1, |3+X - p/q| > 1/q^42 )? Until P is shown to be small in some sense (which it may turn out to be), it seems to me that the argument based on Mahler's theorem is moot. --r.e.s. === Subject: : Re: Random pi digits and Mahler's theorem ................... >But just how unlikely is this very same behavior with >a corresponding sequence of iid random variables? >In other words ... >If X is uniform on (0,1), then what is the value of P= >pr( for all integers p,q > 1, |3+X - p/q| > 1/q^42 )? P < sum 2/q^42 < .5/10^12 -- 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: : Point in the right direction Hello I was wondering if someone could point me in the right direction when it comes to proving this proof. Thank you very much. Question: Let G be a group, and for any a E G let Ya : G -> G be the isomorphism given by conjugation; i.e., Ya(x) = ax(a^−1). If g, h E G prove that Yg = Yh if and only if g = hz for some element z E Z(G). Again Thanks for the help === Subject: : Re: Point in the right direction Assume [CapitalThorn]rst that Yg = Yh. Then Yg(x) = Yh(x), for every x in G. Since g is in G, why not look at Yg(g) and see where that leads? For the other direction, assuming that g = hz, I found to be pretty easy. Let x be any element of G and look at Yg(x) = Yhz(x), and manipulate. Hope this helps, Brian >Hello I was wondering if someone could point me in the right direction >when it comes to proving this proof. Thank you very much. >Question: >Let G be a group, and for any a E G let Ya : G -> G be the isomorphism >given by conjugation; i.e., Ya(x) = ax(a^−1). If g, h E G prove >that Yg = Yh if and only if g = hz for some element z E Z(G). >Again Thanks for the help === Subject: : Re: Sum, Over Some Prime Multiples, Is Always m (I am cross-posting this reply to rec.puzzles because the theorem can be used as the basis of one of those math-tricks which to amaze your friends with {your friends who know a little about math themselves} -- see below for details in reply.) > Let P be any subset of the primes. > (example: P contains all primes of even-index, > or those congruent to 1 (mod 6).) Let A be the set of positive integers: including 1 and every positive > integer which is a multiple of only the primes in P, and A contains > no member which is divisible by a prime not in P. Let B be the set of positive integers: including 1 and every positive > integer which is a multiple of only the primes *not* in P, and B > contains no member which is divisible by a prime *in* P. (Yes, there are positive integers in neither set, as long as P does > not contain every prime.) > Let g(x) be the number of distinct elements of A which are <= x, > for x = a positive real. > So then, for m = any positive integer: --- / g(m/k) > --- > k=elements of B, k <= m always = m. > In linear-mode: sum{k=elements of B, k<=m} g(m/k) always equals m. > (Right?) Leroy Quet So, as to make this a bit more accessible, you can amaze your semi-math-literate friends with this trick. First, tell them to come up with a positive integer m, where I would suggest that m be in the range of between 15 and 30, but much higher if they are using a computer. Have them keep this number secret. Next, have them write down all of the primes between 2 and m. Have them circle some of the primes, while leaving some uncircled. Which primes they circle is their choice, and they keep which primes are circled and which are not a secret as well. Next, form 2 lists of positive integers, list A and list B. Have them start list A with a 1. Next, write down in list A every integer from 2 to m which is divisible ONLY be primes which are circled. List A should contain EVERY integer from 2 to m which is divisible by the any of the circled primes exclusively, and contain 1, and contain no integers which are divisible by any other primes. And we do the same with the uncircled primes and list B: Have them start list B with a 1. Next, write down in list B every integer from 2 to m which is divisible ONLY be primes which are UNcircled. List B should contain EVERY integer from 2 to m which is divisible by the any of the UNcircled primes exclusively, and contain 1, and contain no integers which are divisible by any other primes. (And explain to your friends, that, yes, there are positive integers in neither set{as long as P does not contain every prime}.) Next, (hang in there, almost done here...) we de[CapitalThorn]ne a function g(x) which gives the number of elements of A which are <= x, for x = positive real. (In actuality, we need only to worry about the integer-part of x.) So, [CapitalThorn]nally we take this sum over all elements of B: g(m/k(1)) + g(m/k(2)) + g(m/k(3)) +...+g(m/k(n)), where k(j) is the j_th distinct element of B, and n is the number of distinct elements of B. And, what (miraculously!) is this sum of g's ALWAYS (if your friends have done their math correctly)????.... m, of course!!... Leroy Quet === Subject: : Re: zeta(r)^zeta(r) > Let a(1) = 1; Let, for m >= 2, a(m) = (1/ln(m)) sum{p=primes} sum{k|m,k>=2} a(m/k) ln(k) H(c(p,k)), where H(n) = 1+1/2+1/3+...+1/n, the n_th harmonic number, and c(p,k) is a nonnegative integer where > p^c(p,k) is the highest power of the prime p which divides k. (And, oh yeah, H(0) =0.) > So, we have then: sum{m=1 to oo} a(m)/m^r = zeta(r)^zeta(r), unless I made a mistake. > (My math was not rigorous.) But what I am wondering is, > what is a closed-form (non-recursive de[CapitalThorn]nition) for > a(m) ?? > I might as well give the [CapitalThorn]rst few terms of this sequence: a(m): 1, 1, 1, 2, 1, 3, 1, 7/2, 2,... Is suspect that each term is a rational which depends only on the exponents in the prime-factorization of that term's index, but I am not absolutely certain. If, for example, b(k) = a(p^k), p = prime, then: sum{k=0 to oo} x^k b(k) = (1/(1-x))^(1/(1-x)), I believe. And b(k)(k-1)! forms the sequence: 1, 2, 7, 35,... (Is this sequence, or anything related to the a-sequence, in the EIS??) Leroy Quet === Subject: : Re: New Knowledge > How, when and why does *New Knowledge* come to humankind? New knowledge must come after a new problem occcurs. New problems can be hmmm, og must eat fud but og have no fud or it can be our physical theories for the forces of nature are incompatible. Once we have a new problem, some human walks up to the problem and then the best part happens: Someone pulls an idea out of their ass. We don't have new knowledge yet of course, we merely have a conjecture, but a conjecture is possible knowledge! All we need to do is examine the conjecture. We ask: 1. does this conjecture really solve our problem? 2. does this conjecture cause other severe problems? 3. are there conjectures that solve our problem more elegantly than this conjecture? Thats just a sample of course, but after we sit around and talk this through for long enough (in other words, bull until we have a winner) it is then and how we get new knowledge. The bitch is this knowledge can be replaced in 5 minutes. I would look into stuff by Karl Popper if you want to learn more about what is called conjectural knowledge. Mike Helland === Subject: : Re: New Knowledge > How, when and why does *New Knowledge* come to humankind? Going for a slightly different approach: How: by the very act of thinking. When: constantly Why: because someone somewhere is always putting together ideas in a way they were never assembled before. The kicker is, not much of this new knowledge is revolutionary or memorable. That which is likely to have an impact on technology, for example, tends to happen in labs in incremental degrees, so as to not be unexpected when acquired. -- Will Twentyman email: wtwentyman at copper dot net === Subject: : Re: New Knowledge > How, when and why does *New Knowledge* come to humankind? The Ôfree-Mickey' lawyer (is his name Lessig?) said that all creativity is derivative. Darren. === Subject: : Re: New Knowledge charset=Windows-1252 > How, when and why does *New Knowledge* come to humankind? The Ôfree-Mickey' lawyer (is his name Lessig?) said that all creativity is > derivative. Darren. > ......ahahaha......but derivate is retro active, hence does not point to anything that is really novel. No surprise that this notion comes from a lawyer whose name is Essig = vinegar and who defends copyrights to last an average of 95 years. So, much for an impetus to foster creativity, looking and searching for the NEW..........ahahahaha.... OTOH green s would love to have their permit charges and user fees last for 95 years. But it ain't gonna happen. The creativity by others that the green turds feed off will be --- OUT SOURCED ---. ** Outsourcing the only real legacy of environmentalism *** hanson === Subject: : Re: New Knowledge charset=Windows-1252 N:dlzc D:aol T:com (dlzc) What am *I* driving at..... well, like I said: > How, when and why does *New Knowledge* come to humankind > In all the answers you have seen posted thus far, you can detect > a certain lost unease, a groping in the dark....after their laughs > have died down..... > Steve, if I'd knew the real, manifest and practical > answer to this, then I would not be posting here.....ahahahahaha... > The frontier of new thought is even scarier and closer then one > normally does admit to: Really new thoughts/ideas do come very > seldom. 99.9999999....% of what we think and say is parroting > of/about old and ancient stuff........ Now, let's see whether anybody > will come up with a *new* thought or idea that is truly original. Just a little niggle... > New Thought =/= New Knowledge > New Idea =/= New Knowledge > So I guess the question is, how would you de[CapitalThorn]ne knowledge, > new, and the domain for whom this might be new knowledge. > Tesla walked some strange ground, and even though his > philosophy/belief/reasoning were apparently two steps >from removed reality,they seemed to parallel reality pretty well. > I guess I'll see where your de[CapitalThorn]nitions line up, before I add more. ahahahaha........Smitty, make your own, be my guest > or be independent. I don't care, Smitty, except that > you should produce/express a novel unheard thought, > and instead of parakeeting precambrian arguments for > argument's sake.........ahahahaha......take care Smitty, A thought or idea that can be expressed in words, is > already known. It is constructed from known building blocks. > Even the language of music is written in building blocks we > already know. Knowledge is new to each person. > A baby gains the knowledge of how to pee, how to walk, > how to talk, how to manipulate his/her environment. > New knowledge comes from chaos. New knowledge comes > from walking new ground. New knowledge is more than > thought or idea, as it entails *testing*. > I suggest you study the works of Buddha (pre-cambrian indeed!) > if you want to know more about new knowledge. > David A. Smith > Smitty, stop belaboring old, well known crap. You are a mammal not an avian. Stop parakeeting. The fat dude your recommend, my friend Gautamo, said ~2600 years ago: ~ Do not believe that it (knowledge) is true because scriptures or authorities do say so. Believe only what you judge to true. Smitty, I don't know whether this is true, myself, but why are you doing exactly what he told you not to do? So, where is YOUR new thought, something of lasting & permanent value that humankind has NOT heard before? ............ahahahahaha........ahahahahanson === Subject: : Re: New Knowledge Dear hanson: ... > Smitty, stop belaboring old, well known crap. You are a mammal > not an avian. Stop parakeeting. Not parakeeting. I de[CapitalThorn]ned knowledge, and differentiated it from thought or idea. > The fat dude your recommend, > my friend Gautamo, said ~2600 years ago: ~ Do not believe that > it (knowledge) is true because scriptures or authorities do say so. > Believe only what you judge to true. Smitty, I don't know whether > this is true, myself, but why are you doing exactly what he told you > not to do? You asked for suggestions/responses to your question. I provided the response that new knowledge was constantly being gained. By the young. By the explorer. If you don't like it, quit yer bitchin'. > So, where is YOUR new thought, something of lasting & permanent > value that humankind has NOT heard before? > ............ahahahahaha........ahahahahanson Certainly not that. Okay, how about this: Despite my personal abhorrence, Bush's machine is more than Kerry's can overcome. Another narrow loss will make Bush President for another 4 years. Does that count as new knowledge? And you are welcome to rub my nose in it, if Kerry does manage to pull it off. I'll lose, but I'll win too. David A. Smith === Subject: : Re: Integral of cyclotomic polynomial > If I enter the function f(x) = (x^101 - 1)/(x-1) into Wolfram's Integrator, I get x^101/101 + x^100/100 + X^99/99 + ... + x^2/2 + x which is correct, being the term by term integration of x^100 + x^99 + ... + x^2 + x + 1 but this is quite unwieldly. Is there a simpler expression? Thanks! Barnaby Speci[CapitalThorn]cally, the integral between 0 and 1 of (x^n -1)/(x-1) is H(n), the n_th harmonic number, H(n) = sum{k=1 to n} 1/k. H(n) has a simpler representation as (ln(Gamma(n+1))' + c, where c is Euler's constant. But I do not know off-hand about x not 1. And I am pretty certain that the integral of (x^n -1) /(x-1) is most simply de[CapitalThorn]ned as constant + sum{k=1 to n} x^k/k. (If n = in[CapitalThorn]nity, then the integral is more simply -ln(1-x), of course, if -1 <= x < 1 anyway.) Leroy Quet