October 2008 - mathfun@mailman.xmission.com

Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle)
by Eugene Salamin 11 Oct '08

11 Oct '08

----- Original Message ---- From: Christian G. Bower <bowerc(a)usa.net> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Saturday, October 11, 2008 3:24:43 PM Subject: Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle) ------ Original Message ------ Received: Fri, 10 Oct 2008 10:27:44 AM PDT From: Eugene Salamin <gene_salamin(a)yahoo.com> To: math-fun <math-fun(a)mailman.xmission.com> Subject: Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle) > > Here's a nice challenge problem. Let f(z) = sum( z^(2^n) / n!, n=0..infinity). We know that f(z) is analytic on the open unit disk and that f and all its derivatives are continuous and bounded on the closed unit disk. Find a function g(z) analytic on the open complement of the unit disk (possibly including infinity), with g and its derivatives continuous and bounded on the closed complement, and that agrees with f and its derivatives on the unit circle. The obvious extension method, Schwartz reflection, does not work, since f does not map the unit circle into itself. > > Gene Can't we just use f(1/z)? Christian ________________________________ Won't work. At z = exp(i t), 1/z = exp(- i t), so we have g(z) = f(1/z) = f(exp(- i t)) /= f(exp(i t)) = f(z). Gene

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Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle)
by Christian G. Bower 11 Oct '08

11 Oct '08

------ Original Message ------ Received: Fri, 10 Oct 2008 10:27:44 AM PDT From: Eugene Salamin <gene_salamin(a)yahoo.com> To: math-fun <math-fun(a)mailman.xmission.com> Subject: Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle) > > Here's a nice challenge problem. Let f(z) = sum( z^(2^n) / n!, n=0..infinity). We know that f(z) is analytic on the open unit disk and that f and all its derivatives are continuous and bounded on the closed unit disk. Find a function g(z) analytic on the open complement of the unit disk (possibly including infinity), with g and its derivatives continuous and bounded on the closed complement, and that agrees with f and its derivatives on the unit circle. The obvious extension method, Schwartz reflection, does not work, since f does not map the unit circle into itself. > > Gene > > Can't we just use f(1/z)? Christian

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Re: [math-fun] Financial Crisis = Mathematical Malpractise ?
by Dan Asimov 11 Oct '08

11 Oct '08

Robert wrote: << . . . . . . George Soros was talking about how economists assume that things will average out and self-correct. This is, of course, false. Instead, there seems to be a tendency for self-reinforcing feedback loops to occur (in both positive and negative directions), which cause exponentially fast changes to occur. Unforunately, that's often too fast for society or the political system to react. >> I would agree only that self-correction doesn't happen in the short term. In the long term, however, I believe self-correction does occur: A commodity is ultimately worth its intrinsic value, rather than what investors speculate people might be willing to pay for it in the future. The meanings of short and long term are hard to pin down. But also, my claim is almost tautological, since "intrinsic" worth is essentially defined by what people are willing to pay for it. Yet I think there's also a grain of non-tautological truth there, too. --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] 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] CORRECTED Geometry puzzle
by Dan Asimov 11 Oct '08

11 Oct '08

Michael Reid wrote: << . . . by the way, this is reminiscent of a problem posed a while back by dan, concerning a sequence of unit spheres such that each was tangent to the previous three. dan, did you ever send solution(s) to the list? i remember working on it and sending my solution off-list as requested >> I *thought* I posted an answer -- I recall composing one -- but maybe I never sent it to math-fun. Repeat of (equivalent) question, in case anyone else wants to claim an answer: ------------------------- Given mutually tangent unit balls B_k, 1 <= k <= 4, in 3-space that are mutually tangent, inductively define B_(n+4), n >= 1, (uniquely) by the condition that it is tangent to B_(n+k), 1 <= k <= 3, and that it is not B_n. QUESTION: Do there exist distinct positive j, k, such that the interiors of B_j and B_k intersect ? --------------------------------------------- --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] CORRECTED Geometry puzzle
by Michael Reid 11 Oct '08

11 Oct '08

> CORRECTED Geometry puzzle: > -------------------------- > > (Posted to math-fun on Oct. 10, 2008) > > Subject: Geometry puzzle > > Let B = B_0 be a unit ball in 3-space that's > tangent to the z-axis. > > Let K > 0 be an unknown but fixed constant. > > Now inductively, for all n > 0, place B_(n+1) so that > it's tangent to both B_n and the z-axis, so that its > center's z-coordinate exceeds that of B_n by K. > > Assume this is done so there exists a rigid motion of > 3-space that takes each B_n into B_(n+1). > > QUESTION: > --------- > Find K_min := the least K such that all nonempty > intersections between two B_n's are tangencies. the smallest such value of K is sqrt(3) - 1 . let P_n be the center of the n-th unit sphere, and suppose that P_0 = (1, 0, 0) . it is easy to see that the rigid motion must preserve the orientation of the z-axis, and thus is a rotation about the z-axis, followed by a translation along the z-axis. this transformation can be expressed as (x, y, z) |---> (cos(2 alpha) x - sin(2 alpha) y , sin(2 alpha) x + cos(2 alpha) y , z + K) , where we may assume that 0 <= 2 alpha <= pi . then P_1 = (cos(2 alpha), sin(2 alpha), K) , and since its distance to P_0 is 2 , we find that K^2 = 4 cos^2(alpha) , so K = 2 cos(alpha) . we then get the formula P_n = (cos(2 n alpha), sin(2 n alpha), 2 n cos(alpha)) . the square of the distance between P_0 and P_2 is 32 cos^2(alpha) - 16 cos^4(alpha) = 8 K^2 - K^4 . for K in [0, sqrt(3) - 1) , this distance squared is < 4 , so the 0-th sphere and the 2nd sphere intersect transversally. for K = sqrt(3) - 1 , the 0-th sphere and the 2nd are tangent, while for n >= 3 , the z-coordinate of P_n is >= n ((sqrt(3) - 1)) > 2 , so the 0-th sphere does not intersect the n-th sphere for n >= 3 . by the way, this is reminiscent of a problem posed a while back by dan, concerning a sequence of unit spheres such that each was tangent to the previous three. dan, did you ever send solution(s) to the list? i remember working on it and sending my solution off-list as requested. mike

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[math-fun] Geometry puzzle
by Dan Asimov 10 Oct '08

10 Oct '08

Let B = B_0 be a unit ball in 3-space that's tangent to the z-axis. Let K > 0 be an unknown but fixed constant. Now inductively, for all n > 0, place B_(n+1) so that it's tangent to both B_n and the z-axis, so that its center's z-coordinate exceeds that of B_n by K. QUESTION: --------- Find K_min := the least K such that all nonempty intersections between two B_n's are tangencies. --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] CORRECTED Geometry puzzle
by Dan Asimov 10 Oct '08

10 Oct '08

(I seem to be hopeless for getting things right the first time.) CORRECTED Geometry puzzle: -------------------------- (Posted to math-fun on Oct. 10, 2008) Subject: Geometry puzzle Let B = B_0 be a unit ball in 3-space that's tangent to the z-axis. Let K > 0 be an unknown but fixed constant. Now inductively, for all n > 0, place B_(n+1) so that it's tangent to both B_n and the z-axis, so that its center's z-coordinate exceeds that of B_n by K. Assume this is done so there exists a rigid motion of 3-space that takes each B_n into B_(n+1). QUESTION: --------- Find K_min := the least K such that all nonempty intersections between two B_n's are tangencies. _____________________________________________________________________ "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] Sum z^2^n (Was: Theta_3(0,q) near the unit circle)
by Eugene Salamin 10 Oct '08

10 Oct '08

----- Original Message ---- From: Dan Asimov <dasimov(a)earthlink.net> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Thursday, October 9, 2008 6:11:46 PM Subject: Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle) Gene wrote: << I wrote: << Is [Sum_n=1^oo z^(2^n)/n!] that function Hille shows converges both inside and outside the unit circle? And, IIRC, that at least radially is continuous at the points of convergence *on* the circle? (If not, there is such an example in that book, and it's very cool.) >> I found the example you were looking for. The discussion begins on p. 21 of vol. 2. >> Dumb of me to even wonder, since this weird example obviously can't be a power series in z. --Dan _____________________________________________________________________ Here's a nice challenge problem. Let f(z) = sum( z^(2^n) / n!, n=0..infinity). We know that f(z) is analytic on the open unit disk and that f and all its derivatives are continuous and bounded on the closed unit disk. Find a function g(z) analytic on the open complement of the unit disk (possibly including infinity), with g and its derivatives continuous and bounded on the closed complement, and that agrees with f and its derivatives on the unit circle. The obvious extension method, Schwartz reflection, does not work, since f does not map the unit circle into itself. Gene

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Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle)
by Dan Asimov 10 Oct '08

10 Oct '08

Gene wrote: << I wrote: << Is [Sum_n=1^oo z^(2^n)/n!] that function Hille shows converges both inside and outside the unit circle? And, IIRC, that at least radially is continuous at the points of convergence *on* the circle? (If not, there is such an example in that book, and it's very cool.) >> I found the example you were looking for. The discussion begins on p. 21 of vol. 2. >> Dumb of me to even wonder, since this weird example obviously can't be a power series in z. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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