December 2009 - mathfun@mailman.xmission.com

Re: [math-fun] Solving polynomial equations with roots, etc.
by rcs＠xmission.com 04 Dec '09

04 Dec '09

Date: Thu, 03 Dec 2009 22:18:51 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg(a)roadrunner.com> Dan Asimov wrote: > As interesting as this discussion could be, for the record > I assert that anything said about this issue is > total B.S. unless there is a precise definition made > of the term "God" and of the term "knowing". > > --Dan > > << > Date: Mon, 30 Nov 2009 19:44:17 -0800 > Subject: Re: [math-fun] Solving polynomial equations with roots, etc. > From: "Stephen B. Gray" <stevebg(a)roadrunner.com> > > On a different subject, for anyone to comment on. > > Everyone knows that sqrt(2), pi, and all irrational numbers have no > decimal strings that repeat an infinite number of times. That's not > necessarily the same as the decimal expansion being totally without > any pattern. That is, is there any way to predict the next digits of > say sqrt(2) or pi without doing one of the usual computations? In > other words, is it possible to "know" the entire decimal expansion > of any "ordinary" irrational? Is anything known about patterns in > "regular" irrational expansions? > > I'm excluding numbers invented for the sole purpose of being > irrational or transcendental and with an obvious pattern like > .101001000100001.... or .123456789101112..... ), etc.) I know about > the question of "normal" expansions but that has little to do with > my question. > Any info will be appreciated. > > _____________________________________________________________________ > "It don't mean a thing if it ain't got that certain je ne sais > quoi." --Peter Schickele > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > Dan - By "God" I mean the traditional Christian Omnigod, which is omnipotent, omniscient, omnibenevolent, omnipresent, infinite, eternal, personal, etc., etc., (all of which properties need to be defined and proved separately because they are independent of each other). By "knowing," I'm not currently able to define it because among other things I'm not sure what part of the common meaning you're not satisfied with. Steve Gray

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[math-fun] Tiger Woods drives sales of physics book sky-high
by Henry Baker 04 Dec '09

04 Dec '09

FYI -- Too bad it wasn't Serge Lang's "Algebra"... Tiger Woods drives sales of physics book sky-high A photograph showing a copy of Get A Grip On Physics by John Gribbin on the floor of Tiger Woods's wrecked SUV has seen the book rocket up Amazon's bestseller chart http://www.guardian.co.uk/books/2009/dec/04/tiger-woods-get-a-grip-on-physi…

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Re: [math-fun] I just reordered checks.
by Dan Asimov 04 Dec '09

04 Dec '09

<< Bank Lady: Where would you like the numbering to start? rwg: What was my last one? Bank Lady: 1093 rwg: Well then obviously 3511. Bank Lady: <Opens mouth. Decides not to ask. Resumes typing.> >> Ha! Clearly she didn't see the pattern. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Expansion of charpoly's in terms of traces ?
by Dan Asimov 04 Dec '09

04 Dec '09

<< > I'm not aware of mathematicians who define polynomials over > finite fields with respect to Fermat's Little Theorem. Indeed they don't --- that's exactly the problem! >> The polynomial ring over any field F is denoted by F[X]. It is a mathematical object that one may or may not have any interest in. If one is more interested in what becomes of F[X] after X^p is identified with X, then one need only consider the quotient ring F[X] / (X^p - X), where (X^p - X) denotes the ideal generated by X^p - X. I don't see any problem. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] I just reordered checks.
by Dan Asimov 04 Dec '09

04 Dec '09

<< On Thu, Dec 3, 2009 at 4:40 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > << > Bank Lady: Where would you like the numbering to start? > rwg: What was my last one? > Bank Lady: 1093 > rwg: Well then obviously 3511. > Bank Lady: <Opens mouth. Decides not to ask. Resumes typing.> >>> > > Ha! Clearly she didn't see the pattern. Well, she's not alone--is there a pattern to see? >> I have no idea. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Solving polynomial equations with roots, etc.
by Dan Asimov 04 Dec '09

04 Dec '09

<< . . . especially because I don't believe. . . . >> Even to not believe, it's necessary to define what it is one does not believe in, in order that it be meaningful. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Solving polynomial equations with roots, etc.
by rcs＠xmission.com 03 Dec '09

03 Dec '09

Date: Thu, 03 Dec 2009 09:16:47 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg(a)roadrunner.com> To: math-fun <math-fun(a)mailman.xmission.com> > > >>> If the digits have no pattern then God can't know all the digits >>> because there's no such thing as "all the digits," >>> > Why would this be true? Are you assuming that God has a bounded memory? > > That's one of the questions that makes this issue peculiar and probably meaningless, especially because I don't believe. If God has an infinite memory and the number of random-looking digits is infinite, who wins? Can God's memory hold only countable infinities? Can he know the full expansion of all transcendentals (whatever that might mean)? I'm trying to give an argument to theologians in their own language but my real feeling is that all this stuff doesn't matter.

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[math-fun] Expansion of charpoly's in terms of traces ?
by Henry Baker 03 Dec '09

03 Dec '09

I noticed in Wikipedia's description of eigenvalues, an expansion of the characteristic polynomial of a 3x3 matrix in terms of traces, so I wanted to see if this worked more generally. For a 2x2 matrix, the charpoly is: x^2-x*tr(M)+det(A) = x^2-x*tr(M)+ (tr(M)^2-tr(M^2))/2 For a 3x3 matrix, the charpoly is: x^3-x^2*tr(M)+x*(tr(M)^2-tr(M^2))/2-det(M) = x^3-x^2*tr(M)+x*(tr(M)^2-tr(M^2))/2-(tr(M)^3+2*tr(M^3)-3*tr(M)*tr(M^2))/6 For a 4x4 matrix, the charpoly is: x^4-x^3*tr(M)+x^2*(tr(M)^2-tr(M^2))/2-x*(tr(M)^3+2*tr(M^3)-3*tr(M)*tr(M^2))/6+det(M) Notice that the higher order terms are the same form even though the dimension is higher. Can we continue this indefinitely? What is the general term?

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Re: [math-fun] Expansion of charpoly's in terms of traces ?
by Henry Baker 03 Dec '09

03 Dec '09

[In the case of characteristic zero,] These Newton's Identity formulae seem to provide formulae for the sequence of coefficients of the characteristic polynomial which are completely "oblivious" to the dimension of the matrix, in the sense that the coefficients beyond the one that is equivalent to the determinant (i.e., what would be the constant term) all seem to become conveniently zero. Thus, it appears that you can provide a single infinite sequence A_i (A_0=1) of formulae for the polynomial coefficients, knowing that the terms beyond the n'th term are identically zero, no matter what the elements of the matrix are. This is very cool. At 09:16 AM 12/3/2009, victor miller wrote: >What I said is "almost" true. What one really needs to look at is the ring >of functions which are generated by all power sums as a subring of the ring >of all symmetric functions. As was pointed out, in characteristic 0 (or, >more generally anything larger than the degree) they are the same. In small >characteristic they are not. There are formulas for the index of the rings >(they're related to the theory of modular representations of the symmetric >group). > >Victor > >On Thu, Dec 3, 2009 at 12:13 PM, victor miller <victorsmiller(a)gmail.com>wrote: > >> There are actually formulas which work in the "small" characteristic cases. >> In your case you'd need to look at tr(M^3). This is related to the >> decoding of BCH codes (where they work out the formulas in characteristic >> 2). >> >> Victor >> >> On Thu, Dec 3, 2009 at 12:09 PM, Eugene Salamin <gene_salamin(a)yahoo.com>wrote: >> >>> Because of the appearance of n! denominators, all this is true only in >>> characteristic zero. As a counterexample, let M be the 2x2 identity matrix >>> in characteristic 2. The charpoly is (x - 1)^2 = x^2 + 1. But tr(M) = >>> tr(M^2) = 0. >>> >>> -- Gene >>> >>> ________________________________ >>> From: victor miller <victorsmiller(a)gmail.com> >>> To: math-fun <math-fun(a)mailman.xmission.com> >>> Sent: Thu, December 3, 2009 8:39:55 AM >>> Subject: Re: [math-fun] Expansion of charpoly's in terms of traces ? >>> >>> Yes, The key idea are the Newton Identities ( >>> http://en.wikipedia.org/wiki/Newton's_identities ), which give a way of >>> going from the power sums of the roots of a polynomial to the symmetric >>> functions of the roots. For matrices, the power sums of the eigenvalues are >>> the the traces of the powers of the matrix, and the coefficients of the >>> characteristic polynomial are the symmetric functions of the roots (up to >>> sign). Note that it is only necessary to prove this formula when the matrix >>> has all distinct eigenvalues because this set of matrices is Zariski dense >>> in the set of all matrices (i.e. if there is a polynomial equation giving >>> the characteristic polynomial in terms of the traces of the powers of the >>> matrix which holds for matrices with distinct eigenvalues, it must hold for >>> all matrices). >>> >>> Victor >>> >>> On Thu, Dec 3, 2009 at 11:18 AM, Henry Baker <hbaker1(a)pipeline.com> >>> wrote: >>> >>> > I noticed in Wikipedia's description of eigenvalues, an expansion of the >>> > characteristic polynomial of a 3x3 matrix in terms of traces, so I wanted to >>> > see if this worked more generally. >>> > >>> > For a 2x2 matrix, the charpoly is: >>> > >>> > x^2-x*tr(M)+det(A) = x^2-x*tr(M)+(tr(M)^2-tr(M^2))/2 >>> > >>> > For a 3x3 matrix, the charpoly is: >>> > >>> > x^3-x^2*tr(M)+x*(tr(M)^2-tr(M^2))/2-det(M) = >>> > >>> > x^3-x^2*tr(M)+x*(tr(M)^2-tr(M^2))/2-(tr(M)^3+2*tr(M^3)-3*tr(M)*tr(M^2))/6 >>> > >>> > For a 4x4 matrix, the charpoly is: >>> > >>> > x^4-x^3*tr(M)+x^2*(tr(M)^2-tr(M^2))/2-x*(tr(M)^3+2*tr(M^3)-3*tr(M)*tr(M^2))/6+det(M) >>> > >>> > Notice that the higher order terms are the same form even though the >>> > dimension is higher. >>> > >>> > Can we continue this indefinitely? >>> > >>> > What is the general term?

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Re: [math-fun] Solving polynomial equations with roots, etc.
by rcs＠xmission.com 03 Dec '09

03 Dec '09

Date: Wed, 02 Dec 2009 10:55:57 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg(a)roadrunner.com> Here's the real issue. I'm trying to make an argument against the supposedly omniscient God by raising the philosophical question of whether "God," if any, instantaneously and simultaneously knows "all" the digits of Pi or sqrt(2), for example. If the digits have no pattern then God can't know all the digits because there's no such thing as "all the digits," presumably disproving omniscience. But if there were some pattern to them, a theist could argue that knowing the pattern is equivalent to knowing all the digits. If you say that this whole issue is meaningless mystical mush, I agree, but I'm trying to show by some relatively elementary mathematical- philosophical argument that a theologian could not easily dispute, that omniscience is incoherent. (I'm writing a book about Christianity.) Omniscience is usually defined by theologians as knowing all the facts that it is possible to know. It's not clear whether if God is infinite (whatever that means) he can know an infinite string of digits or even the infinite digits in "each" of the uncountable number of algebraics or transcendentals. There are other issues related to this, for example in what sense do numbers "exist." One might argue that they're purely human constructs, but if humans know about numbers, "God" must, also. Steve Gray Mike Stay wrote: > On Tue, Dec 1, 2009 at 8:59 PM, <rcs(a)xmission.com> wrote: > >> Date: Tue, 01 Dec 2009 20:49:57 -0800 >> Subject: Re: [math-fun] Solving polynomial equations with roots, etc. >> From: "Stephen B. Gray" <stevebg(a)roadrunner.com> >> >> Henry Baker wrote: >> >> >>> Isn't this what the Borwein expansions are all about? >>> >>> At 06:07 PM 12/1/2009, you wrote: >>> >>>> from SBG ... >>>> Date: Mon, 30 Nov 2009 19:44:17 -0800 >>>> Subject: Re: [math-fun] Solving polynomial equations with roots, etc. >>>> From: "Stephen B. Gray" <stevebg(a)roadrunner.com> >>>> >>>> On a different subject, for anyone to comment on. >>>> >>>> Everyone knows that sqrt(2), pi, and all irrational numbers have no >>>> decimal strings that repeat an infinite number of times. That's not >>>> necessarily the same as the decimal expansion being totally without any >>>> pattern. That is, is there any way to predict the next digits of say >>>> sqrt(2) or pi without doing one of the usual computations? In >>>> other words, >>>> is it possible to "know" the entire decimal expansion of any "ordinary" >>>> irrational? Is anything known about patterns in "regular" irrational >>>> expansions? >>>> >>>> I'm excluding numbers invented for the sole purpose of being irrational >>>> or transcendental and with an obvious pattern like >>>> .101001000100001.... or >>>> .123456789101112..... ), etc.) I know about the question of "normal" >>>> expansions but that has little to do with my question. >>>> >>>> Any info will be appreciated. >>>> >>>> Steve Gray >>>> >>>> >> If by the Borwein expansions you mean the formula for computing hex digits >> without knowing the preceding ones, yes, I know about that. I also know >> about the contined fractions having simple patterns. Assuming that the >> "value" of the irrational is given only by the decimal expansion, is it >> possible to "know" the exact value by knowing the pattern? This is part of a >> theological issue which actually means very little to most people. (And it >> may be meaningless.). (I'm not a believer.) >> > What do you mean by "pattern"? Do you mean an algorithm for producing > each digit given only its index? If so, then any computable number > works, whereas uncomputable numbers (e.g. the complexity of the > rational approximations is unbounded) don't. > To: math-fun <math-fun(a)mailman.xmission.com> 2.1 unnamed [text/html] 4.48 KB Download

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