September 2008 - mathfun@mailman.xmission.com

Re: [math-fun] Equation mixing integers and reals
by Dan Asimov 11 Oct '08

11 Oct '08

Christian writes: << Let f(x , x , ..., x ) = k x + k x + ... + k x 1 2 n 1 1 2 2 n n where k , k , ..., k are n REAL constants 1 2 n and x , x , ..., x are n INTEGERS (positive or negative) 1 2 n If we try to have the best possible approximations of f(x , x , ... x ) = 0 1 2 n do you know good algorithms giving best values of (x , x , ..., x )? >> Assume x_j = 0 for all j is not true. Let K and X be the n-dim vectors of real constants and unknown integers, respectively. Then we want to find X so that <K,X> = 0 as near as possible. The set of all values G := {<K,X> : X in Z^n} is a subgroup of the reals, so it's either discrete or dense. The discrete case is easy to handle, and only occurs for measure 0 among possible vectors K in R^n. Otherwise the set G* := {<K,X> : X in Z^n - {0}} contains elements arbitrarily close to 0, so there is no "best" value of X. I'll leave finding "good" values of X to the number theorists, once "good" is defined. --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] Simultaneous solutions of 2 eqns., 2 unknowns
by Eugene Salamin 03 Oct '08

03 Oct '08

----- Original Message ---- From: Dan Asimov <dasimov(a)earthlink.net> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Friday, September 26, 2008 1:55:02 PM Subject: [math-fun] Simultaneous solutions of 2 eqns., 2 unknowns I have 2 equations in x & y, each of form trig(x) + P(x,y) = 0 where trig is sin or cos, and P is a real polynomial of degree 3 or 4. ________________________________ In Maple 12: > fsolve({cos(y) = x^4+y^4, sin(x) = x^3+y^3}, {x, y}); {x = 0.9667490675, y = -0.4317536212} Gene

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Re: [math-fun] Circulant determinants
by Dan Asimov 28 Sep '08

28 Sep '08

<< Should I think have read . . . I think this is my first sighting of the future conditional pluperfect. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Circulant determinants
by Fred lunnon 27 Sep '08

27 Sep '08

A n x n "circulant" determinant A = |A_ij| is constructed from n-vector S = [S_j] by setting A_{i,j} = S_{(i+j) mod n}. Let S be a random n-vector with elements in {-1,+1}. What is the probability that A = 0 ? Example: when n = 32, A has a maximum about 23 decimal digits; yet about 30% of the time it turns out to be zero! Remark: A well-known classical theorem states that A = \product_i \sum_j S_j w^{i j} where w is a primitive n-th root of unity. Fred Lunnon

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[math-fun] Can someone identify this number?
by Dan Asimov 27 Sep '08

27 Sep '08

If you have a number-identifying program, and the time, can someone please tell me if this number, to 20 places, seems to be expressible in closed form* ? 0.33650841691839529162 (I have no strong reason to believe it can be, or can't be.) Many thanks -- and I greatly appreciate all the advice I got for numerically solving two eqns., two unknowns. --Dan _______________________ * You know what I mean. _____________________________________________________________________ "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] Simultaneous solutions of 2 eqns., 2 unknowns
by Dan Asimov 26 Sep '08

26 Sep '08

Thanks, Edwin! I will try Maple -- it looks easier to get a high-accuracy solution than with Mma (which did work, but only to 6 decimal places). --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Simultaneous solutions of 2 eqns., 2 unknowns
by Dan Asimov 26 Sep '08

26 Sep '08

I have 2 equations in x & y, each of form trig(x) + P(x,y) = 0 where trig is sin or cos, and P is a real polynomial of degree 3 or 4. I just found out that I have no idea how to ask Mathematica to solve this numerically. (I tried Roots, Solve, NSolve, N[Solve[...],10], and few other things, but it always came back with a No Can Do message.) On the other hand, with the Mac "Grapher" utility, I can plot each locus and see where they intersect. Biggest problem with Grapher is, when I enter 100% logically equivalent forms of the same equation (like just putting one term on the other side of the equation!) (and yes, changing its sign), it sometimes plots entirely different curves, which have entirely different intersections. Can someone please advise me how to get either of these programs to succeed? (By the way, I'm almost sure my equations have infinitely many simultaneous solutions, but I'm only interested in the unique one that's in the first quadrant nearest the origin.) Thanks in advance. --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] ceiling function
by James Propp 25 Sep '08

25 Sep '08

>I'm really trying to sweep out the cobwebs in my memory here... And I'm trying to make sure I don't misuse words like "model" and "true". (If any of you who know this stuff better than I do think I'm misusing terminology or making some other sort of mistake, please let me know!) >The hardest thing to characterize in a theory of the integers is the "finite ># of steps between any two integers". I.e., "finiteness" is the hard property, >not infiniteness. This well-ordering property requires a _second order_ axiom, >and hence integers require a "second order" theory. I think I see what you mean. If one is going to quantify over objects called "subsets of Z", one will need a bunch of set-theoretic axioms to capture or partially capture what this means, and the answer to my question might depend on what bunch of axioms one chooses. I'm not sure what bunch of axioms corresponds most closely to what I meant to ask when I posed the question, but I will say that I think there might be some fun models in which, for every real x, the set of ALL integers bounded below by x has a least element, but in which there exists a set S of integers such that S is bounded below by some real number x but S doesn't have a least element. Can anyone come up with a model like this? >I think you may have to formally axiomatize what the properties of "ceil(x)" >are. I use just one axiom for this purpose: "for every real number x there exists an integer m such that for all integers n that are greater than or equal to x, n is greater than or equal to m." It is easy to prove that this m is unique, and we call it ceil(x). >Presumably the integers are the range of ceil() That follows as an easy consequence: for every integer n, ceil(n) = n. (Henry's email suggests an alternative way to try to ask my question: Axiomatize the ceiling function directly as a function on the reals, and then *define* Z as the image of this function. That's not what I had in mind, though it's interesting too.) Jim

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[math-fun] coffee-ordering logicians
by Steve Witham 24 Sep '08

24 Sep '08

(Sorry if this is old. Via Craig Swanson.) Six logicians have finished their dinner together. The waitress comes and asks, "Do you all want coffee?" The first logician answers, "I don't know." 2nd: "I don't know." 3rd: "I don't know." 4th: "I don't know." 5th: "I don't know." 6th: "Yes." --Steve

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[math-fun] epsilon-delta proofs
by James Propp 24 Sep '08

24 Sep '08

In trying to teach my honors calculus course, it's occurred to me that part of the problem is that there are two directions of time at work in most simple epsilon-delta proofs: the time in which the choices are made and logical implications take their inexorable course, and the order in which the steps are most helpfully explained. It seems to me that a two-dimensional style of proof might make this clearer to the students, and keep them from making certain sorts of mistakes. Has anyone seen anything like this, as a form of training-wheels for the mind? (I could create some for my students, but I'd rather not create teaching materials that already exist.) As background for my question, I should mention that there *is* a way to present proofs that sweeps the two-different-timelines issue under the rug: namely, omit all motivation, and simply say things like "Given epsilon > 0, we may take delta = epsilon / 2, and then 0 < |x-1| < delta implies |2x - 2| < 2 delta = epsilon, as required." But proofs in this style, especially when the details are complicated, may leave some students wondering "How could an ordinary human come up with such a proof?" What is more common, in books like Stewart's calculus, is to embed within the proof, starting from the beginning, helpful words about where we're trying to go, and what choices will enable us to get there. But the students who read such "motivated proofs" then sometimes write proofs that say things like "|2x-2| = 2|x-1| < epsilon, therefore delta = epsilon / 2," which confuses motivation with implication. I'd welcome thoughts about how to make things clearer to students on such matters, be they two-dimensional diagrams of proofs, or helpful analogies between proofs and other sorts of texts, or catchy slogans about the nature of proof, or whatever. I should mention that I already make extensive use of the whole quantified-proposition-as-two-player-game gimmick in my teaching; my students are probably really sick of (universal) Adam and (existential) Eve by now... Jim Propp

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