April 2005 - mathfun@mailman.xmission.com

RE: [math-fun] history of the mathematical story problem
by Daniel Asimov 19 Apr '05

19 Apr '05

This makes it sound as though the mathematical story problem is a phenomenon, rather than just a collection of examples. But unlike "the chess problem", I've never heard anyone refer to "the mathematical story problem" before, even using a different phrase for it. (Has any math-funner?) Of course we all know about so-called "word problems" esp. in H.S. algebra, but this doesn't seem quite the same. So, is "the mathematical story problem" an actual phenomenon? --Dan ------------------------------------------------------------------------------ << Margaret Cohen of the Stanford French dept is writing a book on literature of the sea and asks: does anyone know a good history of the mathematical story problem? >>

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[math-fun] history of the mathematical story problem
by Thane Plambeck 19 Apr '05

19 Apr '05

Margaret Cohen of the Stanford French dept is writing a book on literature of the sea and asks: does anyone know a good history of the mathematical story problem? She offers the following example from a sixteenth century navigation manual (I think she said that's where it comes from) * * * * Suppose a Merchant-Ship in 45d 30m North Latitude, falls into the Hands of Pyrates, who take away his Sea-Compass; after which he saileth as directly as he can 67 leagues between the S. and W. and at the end of two days meets a Ship of War, who also had been the Day before in 45d.30m. North Latitude, and had sailed thence S.E. by S. 39 Leagues; now the Merchant-ship left those Pyrates lying to and fro where they robbed him, and the Man of War being desirous to find them: I demand what Course he must shape to speak with them?Â” (73). * * * * I thought of Archimedes' Problema Bovinum but she's not really satisfied with that, she wants the history of the math story problem as 'dramatic literature' (my term, not hers) I think; ie she wants to know the history of problems that really go to the effort of coming to meet the possibly unmotivated reader with some exciting stuff surrounding the problem. Any thoughts? -- Thane Plambeck http://www.plambeck.org/ehome.htm

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Re: [math-fun] Density question
by Daniel Asimov 19 Apr '05

19 Apr '05

Bill Thurston wrote: << 1. A point of density of a set is a point such that for small enough neighborhoods, the density of the set in that neighborhood is arbitrarily close to 1. I.e. every neighborhood smaller than delta has 99.99% of points in the set, etc. The Lebesgue density theorem asserts that almost every element of any measurable set is a point of density. I.e. for any measurable subset of R (or any other space) almost every point has neighborhoods with density approaching 1 or density approaching 0. This is a fundamental principle of measure theory, I think it is (or at least it should be) in the basic texts. It really comes from the way measure is defined. >> I think the following is right, though I haven't done measure theory in a while: If H is a measurable subset of [0,1] such that (*) meas(H \int [a,b]0i) = (b-a)/2 for all 0 <= a <= b <= 1 then of course meas(H) = meas(H \int [0,1]) = (1-0)/2 = 1/2. But this means that the infimum of |open cover| of H by intervals is 1/2, where |{o_i : i in Z+}| is defined as (sum over i in Z+ of meas(o_i)). So let O = {o_i} be an open cover of H with (**) 1/2 <= |O| < 1/2 + eps for any fixed eps > 0. But by (*) we also have meas(H \int o_i) = meas(o_i)/2. So for each o_i, the set (H \int o_i) has its own open cover C_i = {c_ik : k in Z+}, such that (***) meas(o_i)/2 <= |C_i| < meas(o_i)/2 + eps/(2^i) But the union the c_ik (over all i and k) contains H. Hence (R to L, then top to bottom) 1/2 <= sum over i in Z+ of (sum over k in Z+ of meas(c_ik)) = =sum over i of |C_i| < sum over i of (meas(o_i)/2 + eps/(2^i) (by ***) =(1/2) |O| + eps < (1/2)(1/2 + eps) + eps (by **) = 1/4 + (3/2)eps. By choosing eps = 1/6, this long inequality becomes 1/2 < 1/2 => <=. QED. << 2. You can take an irrational rotation of the circle R/Z like x -> x + sqrt(2). The orbits of this rotation are all isomorphic to Z --- i.e. points never return to themselves. Pick a point from each orbit, using the axiom of choice. Partition the orbit into the even images and the odd images of this point. The union of all odd points in all orbits is "half" the circle in the sense that its congruent by a translation to its complement. In fact you can take odd powers of the rotation arbitrarily close to the identity, so you can make the set match its complement by an arbitrarily small rotation. This example is covered by an example in R --- you could think of this as taking any action of Z^2 (or Z^k would also work) by translations on R with dense orbits, then partitioning the orbits into cosets of a subgroup of index 2 acting on a selected point It's known that there's no definition or rule or humanly comprehensible procedure to actually make such a selection. Bill >> This is the same example in R/Z that I had in mind. While the two pieces of R/Z are dense, and isometric by a translation, I don't think it's clear they have a third quality I claimed: that each is homogeneous. (But I believe I've seen a construction that causes this, too, to be true.) Also: When I first learned that the Axiom of Choice is equivalent to the statement that the cartesian product of nonempty sets is nonmpty . . . I became a believer. --Dan -------------------------------------------------------------------------------- On Apr 18, 2005, at 9:24 PM, Michael Kleber wrote: > Dan Asimov wrote: > >> << >> Can someone give me an example of a set of density 1/2 on every >> interval of the real numbers? >>>> >> >> If such a set must be measurable, then no such set exists. >> >> On the other hand, there do exist partitions of the reals into >> two dense homogeneous subsets that are related by a translation. > > Probably I should know why both of these are true, but right now > it seems that I don't. Can you explain?

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RE: [math-fun] Density question
by Andy Latto 18 Apr '05

18 Apr '05

> -----Original Message----- > From: math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On > Behalf Of Thomas Colthurst > Sent: Sunday, April 17, 2005 11:16 AM > To: dasimov(a)earthlink.net; math-fun > Subject: Re: [math-fun] Density question > > > > David Wilson wrote: > > Can someone give me an example of a set of density 1/2 on every > > interval of the real numbers? > > Take [0,1/2), and then for every point x in it, flip a coin. > If the coin is heads, keep x as is. If the coin is tails, replace > x with x+1/2. Then shift the resulting set by all the integers. That doesn't work. A set has S density 1/2 on [0,1] if the measure of S intersect [0,1] is 1/2. The set you describe is nonmeasurable (with probability 1). Andy Latto andy.latto(a)pobox.com

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[math-fun] need density definition
by Chris Landauer 17 Apr '05

17 Apr '05

hihi, all - David Wilson is looking for a set of reals that has density 1/2 in every real interval - would it not suffice for a set that is uniform in some useful way and also density 1/2 in any single interval since the rationals are dense in the reals, isn't the density of the set of rationals equal to 1? so we should probly clarify what kind of density is expected, whether using topology or just cardinality more soon, cal

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[math-fun] Density question
by David Wilson 17 Apr '05

17 Apr '05

Can someone give me an example of a set of density 1/2 on every interval of the real numbers? - David W. Wilson "Truth is just truth -- You can't have opinions about the truth." - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"

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Re: [math-fun] Density question
by lkmitch＠att.net 17 Apr '05

17 Apr '05

Allow me to jump out here and clearly display my ignorance: how about something like the set of rational numbers in lowest terms whose numerators (or denominators) are odd (or even)? Kerry Mitchell -------------- Original message from "David Wilson" <davidwwilson(a)comcast.net>: -------------- > Can someone give me an example of a set of density 1/2 on every > interval of the real numbers? > > - David W. Wilson

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[math-fun] Polygon problem
by David Wilson 16 Apr '05

16 Apr '05

In a self-crossing polygon of n vertices, what is the maximum number of possible crossings? - David W. Wilson "Truth is just truth -- You can't have opinions about the truth." - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"

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Re: [math-fun] coprime problem
by David Wilson 16 Apr '05

16 Apr '05

The question I asked arises in one attack of the following problem: Let S be a set of real numbers. Integer k >= 1 is a divider of S if a rational of reduced form j/k lies strictly between every two elements of S. The question is, is a divider of {0, 1/n, 1/(n-1), ..., 1}also a divider of the set { j/k | j, k in Z, 1 <= k <= n }. ----- Original Message ----- From: perry(a)globalnet.co.uk To: ham ; math-fun ; David Wilson Sent: Saturday, April 16, 2005 4:52 AM Subject: Re: [math-fun] coprime problem We could add to this: a,b,x,y are all pairwise coprime. Jon Perry perry(a)globalnet.co.uk On Sat, 16 Apr 2005 00:20 , David Wilson <davidwwilson(a)comcast.net> sent: Let a, b >= 1, gcd(a, b) = 1. It is not difficult to show that every sufficiently large integer is of the form ax+by where x, y >= 1. Is the same true with the added condition gcd(x, y) = 1? _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ------------------------------------------------------------------------------ Message sent via Global Webmail - http://www.globalnet.co.uk/ Global MAX Broadband 2Mb now just £19.99 a month

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[math-fun] coprime problem
by David Wilson 16 Apr '05

16 Apr '05

Let a, b >= 1, gcd(a, b) = 1. It is not difficult to show that every sufficiently large integer is of the form ax+by where x, y >= 1. Is the same true with the added condition gcd(x, y) = 1?

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