User igor rivin - MathOverflow

Math blog directory

2012-04-07T18:18:16Z

Does anyone have a list of high quality mathematics (or related) blogs. I am of course aware of Terry Tao's most excellent blog, and also of ldtopology.wordpress.com, but I am sure the complete list is far longer.

EDIT As I say in my comment, the key point is that I am looking for high quality blogs. nLab and mathblogging both give a VERY long list, and while they are both useful resources (neither of which I was aware of before asking the question) neither is sufficiently selective to be really useful.

Answer by Igor Rivin for A riemannian manifold with finitely many closed contractible geodesics

2012-10-26T16:38:11Z

Ellipsoids with almost but not quite equal axes have exactly three simple closed geodesics.

Answer by Igor Rivin for History surrounding Gauss Theorema Egregium and differential geometry

2012-04-20T21:03:42Z

You should read Gauss' book (General theory of curves and surfaces). It is written in a fairly unenlightening manner (back then, people did not trust pictures, so Gauss just does pages and pages of horrible computations), but if you read it carefully, you will see that he had a very geometric view of things. Secondary sources (e.g., Spivak) are based entirely on Gauss' book and since Gauss did not write an autobiography (to the best of my knowledge) you should go to the said book of his.

Groups where every two generator subgroup is free

2012-04-15T01:45:09Z

Is there a name for the class of groups in the title, and any sort of characterization? Free groups and surface groups are in the class, but presumably there are many more...

Polytopes with few vertices.

2011-09-07T21:43:30Z

Suppose I have a convex polytope in $\mathbb{R}^d$ which I know has few vertices (in the case which prompted this question, I seem to have a polytope in $\mathbb{R}^9$ which has sixteen vertices). Is there some constructive way to enumerate the possibilities? If the polytope has $k$ vertices, is there some not-too-horrible upper bound on the number of possibilities?

Answer by Igor Rivin for Legendre and sums of three squares

2011-10-08T15:23:08Z

There is a discussion of this by Andre Weil in "Number theory from Hammurapi to Legendre", where seems to imply that Legendre's proof might have been a bit problematic (p. 332). Weil also gives some nice proofs of the $n$-squares theorems (appendix 2 to the Euler chapter). You can find a link to the Weil book here: http://dl.dropbox.com/u/5188175/WeilNumbers.pdf

Answer by Igor Rivin for On average length of sums of unit vectors in R^n

2013-02-19T02:04:46Z

I am assuming that the OP means that the initial $k$ vectors are themselves a sample of the uniform distribution on the sphere, so the question is both about the limiting distribution of the length (which is normal) and the speed of convergence to this distribution. This is a much studied (and highly nontrivial) question: for a nice analysis of the two dimensional case (and extensive references), see:

Random Unit Vectors II: Usefulness of Gram-Charlier and Related Series in Approximating Distributions

David Durand and J. Arthur Greenwood Source: Ann. Math. Statist. Volume 28, Number 4 (1957), 978-986.

(available for free on Project Euclid).

Answer by Igor Rivin for closed form solution of an ODEs

2013-02-18T03:00:58Z

This is a second degree linear ODE with rational function coefficients, and this problem has been completely solved algorithmically by Kovacic in the mid-eighties:

Kovacic, Jerald J. "An algorithm for solving second order linear homogeneous differential equations." Journal of Symbolic Computation 2, no. 1 (1986): 3-43.

All the major computer algebra systems implement this, and (not surprisingly) Mathematica does not find a solution in closed form in the generality you give. However, since your $c_i$ are ``known real numbers'', perhaps you are lucky, and a closed for solution is known in your special case. The relevant Mathematica incantataion is "DSolve", as in:

DSolve[y''[x] + (c1/x^2 + c2/(1 - x)^2 + c3 (1/x + 1/(1 - x))) y'[ x] + (c4 (1/x + 1/(1 - x)) + c5/x^2 + c6/(1 - x)^2) y[x] == 0, y[x], x]

Riemann zeta at even integers

2012-04-12T16:14:26Z

I am talking about this in a course I am teaching, and hence am wondering: what are the various derivations of the values of Riemann zeta function at even integers? There are two incredibly cool proofs in Don Zagier's paper (section 1), but there must several other proofs floating around. Also, I recall reading that Euler originally proved the formula for $\zeta(2)$ by thinking of $\sin(x)$ as a polynomial -- has this argument been made rigorous since?

EDIT I did not realize that this was known as the "Basel Problem", so did not find @Yemon's answer myself. I conjecture, however, that the Robin Chapman list is incomplete, since I have found yet another proof, not contained in Robin's list, so maybe there are more yet out there...

Answer by Igor Rivin for Concise model of modern fiat money and its non-conservation

2013-01-27T02:07:35Z

For a mathematical model see Hayashi and Matsui, 1994. For an in-depth discussion without too many (actually, any) equations, see many books by Murray Rothbard (all available on Amazon.com).

Answer by Igor Rivin for The knot whose complement is the Hantzsche-Wendt manifold

2013-01-27T01:57:36Z

Much enlightenment (though not an explicit answer to the question) can be gleaned from Bruno Zimmermann's paper "On the Hantzsche-Wendt manifold".

Answer by Igor Rivin for Immersed surface with circle as a boundary

2013-01-09T00:57:15Z

Check out this paper of Rafael Lopez and references therein.

Answer by Igor Rivin for Does every polynomial diophantine equation have solutions modulo p?

2013-01-06T01:28:34Z

I believe the OP is asking about a version of Hilbert nullstellensatz, see for example these nice notes of D'Andrea.

Rational orthogonal matrices

2012-12-13T00:54:39Z

``everybody knows'' that an integral orthogonal matrix is a signed permutation matrix, so there are exactly $2^n n!$ such matrices in $O(n).$ Now, what if we ask for the enumeration of elements of $O(n$ with rational elements, where all the denominators are equal to $q$ (where $q$ is an arbitrary integer, though if the question is much easier for $q$ prime, that's fine too...)? I would suspect this has been studied... (one can ask the same question for arbitrary algebraic groups/number fields, of course).

Translation distance in the curve complex

2012-09-28T19:06:47Z

Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the translation distance is/is not very small. That is, if the algorithm can pick among the possibilities: translation distance is 0, 1, 2, 3, many, then I am happy...

I know there are algorithms for computing distances IN the curve complex, but this is not quite the same...

power log distance between matrices

2012-12-02T18:35:32Z

In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of the argument. This is presumably standard, and goes back to Minkowski, but I am having trouble finding references or any kind of comprehensive treatment. Any suggestions for where this might be discussed? (number theory tag because this is closely related to the Siegel half space, etc).

Answer by Igor Rivin for $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

2012-12-06T03:00:51Z

Not quite an answer, but: If you call the integrand $f(x, r, s, m),$ and evaluate for explicit values of $r, s, m$ you get a clue. For example, if you define $g(x, k) = f(x,1, 2, k),$ then $F(k)=\int_0^\infty g(x, k) dx,$ for odd $k$ seems to equal $F(k)=\frac{2^{(k-1)/2} G_{3,3}^{3,3}\left(1\left| \begin{array}{c} -\frac{1}{2},-\frac{1}{2},0 \\ 0,\frac{1}{2},(k-3)/2 \\ \end{array} \right.\right)}{(k-2)!! \pi ^{3/2}}$

Where the $G$ is the Meijer G function, and $n!!$ is the product of all odd numbers not exceeding $n.$

For even $k$ the answer is always a linear function of $\pi$ with rational coefficients, e.g. $F(4) = \frac12 - \frac{\pi}8$

$F(6)= \frac14 - \frac{\pi}{16}$

$F(8) = \frac1{12} - \frac{\pi}{64}$

$F(10)= \frac{\pi}{128}$

$F(12) =-\frac{1}{30}+ \frac{17\pi}{1024}$

And so on. I am not sure I am quite catching the pattern of the coefficients (the powers of two in the denominator are not quite regular, even).

I did not try the other $r, s$ but this already seems pretty interesting.

Answer by Igor Rivin for Period of linear congruential generator

2012-12-05T20:51:30Z

Kurlberg, Pär, and Carl Pomerance. "On the period of the linear congruential and power generators." arXiv preprint math/0405120 (2004).

Answer by Igor Rivin for Eversion of the 6-sphere in 7-space

2012-12-02T01:07:02Z

The comments give a rather bogus version of history. It is true that a movie ("Outside in") was made at the geometry center, but the explicit eversion precedes the movie by three decades, and is due to Arnold Shapiro (1960), simplified by Bernard Morin in 1967. A good reference is an Intelligencer article by Morin and George Francis in 1980.

The Thurston "crinkling" technique is not due to Thurston, but rather to Nico Kuiper, who used it in the sixties to prove the amazing result that EVERY Riemannian manifolds admits a $C^1$ isometric embedding into its topological embedding dimension, and not only that, the image of the embedding can be constrained to lie in an arbitrarily small ball. This circle of ideas was later made into a science by Gromov ("the h-principle").

As for writing something explicit for $S^6,$ maybe, but where does this method get stuck for $S^4,$ e.g.? No amount of crinkling can overcome the obstruction...

Answer by Igor Rivin for Iterating Random Matrix Operations

2012-12-02T00:57:54Z

You are taking a random product of transvections corresponding to row and column operations, and then multiplying your initial matrix $M$ by this product. Obviously, $M$ is a bit of a red herring, it is the random product you care about, and this has been studied extensively. The canonical reference is:

Bougerol, Philippe, and Jean Lacroix. Products of random matrices with applications to Schrödinger operators. Birkhäuser, 1985.

Answer by Igor Rivin for Spectrum of transition matrix for symmetric random walk

2012-11-30T19:42:09Z

The spectrum of the matrix is computed in the beginning of my preprint:

Rivin, Igor. "Growth in free groups (and other stories)." arXiv preprint math/9911076 (1999).

(there is a published version, too).

Answer by Igor Rivin for Abstract characterization of polygonizations

2012-11-30T01:16:01Z

For the OP's claim re *infinite*polyhedral graphs, the answer is yes, this is true, and a proof is in my paper:

Rivin, Igor. "Combinatorial optimization in geometry." Advances in Applied Mathematics 31.1 (2003): 242-271.

Basically, you can construct a circle packing with any prescribed (three-connected) combinatorics. What you lose when you go from finite to infinite is uniqueness, in a spectacular way: it should be true that one can get the carrier of the packing to be any Jordan domain.

Answer by Igor Rivin for How to solve a specific multivariate recurrence relation (or general ones)

2012-11-29T01:55:00Z

Read Bender and Orszag (Advanced Mathematical Methods for Scientists and Engineers), and you will be enlightened.

Answer by Igor Rivin for How to divide a n dimensional simplex in n+1 equal parts

2012-11-22T23:40:17Z

The magic word is "barycenter". The convex hull of the barycenter and any one face of the simplex has the right volume.

Answer by Igor Rivin for Fastest way to factor integers < 2^60

2012-11-22T04:09:19Z

This is really a comment, but what the heck. I would strongly advise against implementing your own factoring. Systems such as Pari/GP are very carefully optimized, and it would take you months, if not years, to get as efficient. 8GB of ram is nothing these days (the laptop I am typing this on has 16GB), so you are better off spending $300 to double your RAM than spending an unbounded amount of your time on a wild goose chase.

Answer by Igor Rivin for Absolute sum of coefficient of (1-x)^b (1+x)^{(n-b)}

2012-11-22T04:03:50Z

This is analyzed exhaustively (also exhaustingly) by Domenici in this 2005 preprint.

Answer by Igor Rivin for Continuous pointwise ergodic theorem?

2012-11-22T01:36:12Z

There is a difference between ergodicity and unique ergodicity. Notice that for the geodesic flow space averages equal time averages for almost all, but definitely not for all orbits (as shown by the existence of closed geodesics). On the other hand the horocycle flow IS uniquely ergodic (on a compact surface). That doesn't quite answer your Q2 (the average may exist, just not equal the space average).

In any case, the best (in my opinion) exposition of the ergodicity of the geodesic flow is given by Curt McMullen in his dynamics course notes.

Answer by Igor Rivin for Primitive subwords in a free group of rank 2

2012-11-21T22:26:44Z

To answer your last question, check out Cohen-Metzler-Zimmerman (1981) and references there in.

Answer by Igor Rivin for What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?

2012-11-21T22:13:53Z

See this question, and the answers to it.

Answer by Igor Rivin for growth of energy of eigenfunctions on hyperbolic surface

2012-11-21T14:32:34Z

You might want to look at Chris Judge's papers, particularly:

Judge, Christopher M.(1-IN) Tracking eigenvalues to the frontier of moduli space. I. Convergence and spectral accumulation. (English summary) J. Funct. Anal. 184 (2001), no. 2, 273–290.

Comment by Igor Rivin

2013-02-20T21:56:42Z

Not appropriate (in particular, since there is no question). Voting to close.

Comment by Igor Rivin

2013-02-19T16:42:49Z

This is related to this question: mathoverflow.net/questions/98007/…

Comment by Igor Rivin

2013-02-19T01:56:30Z

@Per: I am not sure I understand your question: the mean value of the length of the sum, is not the same as the length of the mean of the sum.

Comment by Igor Rivin

2013-01-31T05:03:15Z

This has a certain homework flavor. Voting to close.

Comment by Igor Rivin

2013-01-28T16:32:05Z

@Jyotirnoy: While it is obviously debatable whether the question is mathematical, everything else you say is unreasonable: The OP clearly spent a lot of effort formulating the question, and "can be found in standard undergraduate textbooks" is a completely worthless statement. Give us a book and a page number.

Comment by Igor Rivin

2013-01-28T16:29:51Z

@Greg: certainly a fair point.

Comment by Igor Rivin

2013-01-28T00:36:39Z

I think I misunderstood the question -- with no offence intended, the mechanics of monetary policy is fairly trivial, but the longterm consequences are interesting, and it is, in particular, interesting why (speaking slightly figuratively) printing a bunch of paper tickets should make any real difference to the economy, and what the difference actually is. I have a very strong opinion, which is strongly "Austrian" (by which I very much DO NOT mean "like that of Michael Greinecker"), and has a certain amount of predictive power (which is what makes it of any interest).

Comment by Igor Rivin

2013-01-27T19:20:49Z

@Michael: (a) I explicitly said that Rothbard made no attempt to provide a "mathematical" model. (b) but why don't you let whoever here cares read his work for themselves. (c) "I don't care what you think" is very open-minded and polite of you.

Comment by Igor Rivin

2013-01-27T15:03:34Z

@Michael: Economics is not really a science, and its social dynamics are tribal. Rothbard is a leading of proponent of (ironically) the Austrian School of Economics (in honor of von Mises and Hajek), which is not the politically dominant school. I personally find the Austrian school extremely compelling, and find what "most economists" think (or claim to think) of little interest. I could expound at length on this, but at this point we are veering quite far from mathematics.

Comment by Igor Rivin

2013-01-27T02:08:08Z

As shown in the reference I gave.

Comment by Igor Rivin

2013-01-09T15:40:05Z

@Misha's guess is probably right, but the OP should pose a well-formed question before s/he expects to get an answer.

Comment by Igor Rivin

2013-01-07T00:04:20Z

Why is this not getting closed?

Comment by Igor Rivin

2013-01-05T01:20:54Z

The correct verb is "convolving". @Uwe Franz has a way to convolve correctly in his answer.

Comment by Igor Rivin

2013-01-05T01:06:58Z

Homework, voting to close.

Comment by Igor Rivin

2012-12-30T00:48:08Z

Also, the OP's alternative formulation seems to be wrong, since it is not clear how this specifies that the area is rational.