User igor rivin - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:47:20Z http://mathoverflow.net/feeds/user/11142 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93437/math-blog-directory Math blog directory Igor Rivin 2012-04-07T18:18:16Z 2013-05-16T08:11:57Z <p>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.</p> <p><strong>EDIT</strong> As I say in my comment, the key point is that I am looking for <em>high quality</em> 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.</p> http://mathoverflow.net/questions/110709/a-riemannian-manifold-with-finitely-many-closed-contractible-geodesics/110768#110768 Answer by Igor Rivin for A riemannian manifold with finitely many closed contractible geodesics Igor Rivin 2012-10-26T16:38:11Z 2013-05-12T08:17:31Z <p>Ellipsoids with almost but not quite equal axes have exactly three simple closed geodesics.</p> http://mathoverflow.net/questions/94682/history-surrounding-gauss-theorema-egregium-and-differential-geometry/94690#94690 Answer by Igor Rivin for History surrounding Gauss Theorema Egregium and differential geometry Igor Rivin 2012-04-20T21:03:42Z 2013-04-30T21:49:33Z <p>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.</p> http://mathoverflow.net/questions/94084/groups-where-every-two-generator-subgroup-is-free Groups where every two generator subgroup is free Igor Rivin 2012-04-15T01:45:09Z 2013-03-31T07:55:02Z <p>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...</p> http://mathoverflow.net/questions/74798/polytopes-with-few-vertices Polytopes with few vertices. Igor Rivin 2011-09-07T21:43:30Z 2013-03-27T18:23:02Z <p>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?</p> http://mathoverflow.net/questions/77537/legendre-and-sums-of-three-squares/77541#77541 Answer by Igor Rivin for Legendre and sums of three squares Igor Rivin 2011-10-08T15:23:08Z 2013-03-20T03:34:26Z <p>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: <a href="http://dl.dropbox.com/u/5188175/WeilNumbers.pdf" rel="nofollow">http://dl.dropbox.com/u/5188175/WeilNumbers.pdf</a></p> http://mathoverflow.net/questions/122056/on-average-length-of-sums-of-unit-vectors-in-rn/122251#122251 Answer by Igor Rivin for On average length of sums of unit vectors in R^n Igor Rivin 2013-02-19T02:04:46Z 2013-02-19T02:04:46Z <p>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 <em>is</em> 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:</p> <p>Random Unit Vectors II: Usefulness of Gram-Charlier and Related Series in Approximating Distributions</p> <p>David Durand and J. Arthur Greenwood Source: Ann. Math. Statist. Volume 28, Number 4 (1957), 978-986.</p> <p>(available for free on Project Euclid). </p> http://mathoverflow.net/questions/122124/closed-form-solution-of-an-odes/122132#122132 Answer by Igor Rivin for closed form solution of an ODEs Igor Rivin 2013-02-18T03:00:58Z 2013-02-18T14:07:02Z <p>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:</p> <p>Kovacic, Jerald J. "An algorithm for solving second order linear homogeneous differential equations." Journal of Symbolic Computation 2, no. 1 (1986): 3-43.</p> <p>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:</p> <p>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]</p> http://mathoverflow.net/questions/93879/riemann-zeta-at-even-integers Riemann zeta at even integers Igor Rivin 2012-04-12T16:14:26Z 2013-02-07T03:12:59Z <p>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 <a href="http://dl.dropbox.com/u/5188175/zagierzeta.pdf" rel="nofollow">Don Zagier's paper</a> (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?</p> <p><strong>EDIT</strong> 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 <a href="http://dl.dropbox.com/u/5188175/splinezeta.pdf" rel="nofollow">another proo</a>f, not contained in Robin's list, so maybe there are more yet out there...</p> http://mathoverflow.net/questions/119980/concise-model-of-modern-fiat-money-and-its-non-conservation/119983#119983 Answer by Igor Rivin for Concise model of modern fiat money and its non-conservation Igor Rivin 2013-01-27T02:07:35Z 2013-01-27T02:07:35Z <p>For a mathematical model see <a href="https://dl.dropbox.com/u/5188175/fiatmoney.pdf" rel="nofollow">Hayashi and Matsui, 1994.</a> For an in-depth discussion without too many (actually, any) equations, see many books by Murray Rothbard (all available on Amazon.com).</p> http://mathoverflow.net/questions/119918/the-knot-whose-complement-is-the-hantzsche-wendt-manifold/119979#119979 Answer by Igor Rivin for The knot whose complement is the Hantzsche-Wendt manifold Igor Rivin 2013-01-27T01:57:36Z 2013-01-27T01:57:36Z <p>Much enlightenment (though not an explicit answer to the question) can be gleaned from Bruno Zimmermann's paper <a href="https://dl.dropbox.com/u/5188175/ZimmermanHW.pdf" rel="nofollow">"On the Hantzsche-Wendt manifold".</a></p> http://mathoverflow.net/questions/118405/immersed-surface-with-circle-as-a-boundary/118412#118412 Answer by Igor Rivin for Immersed surface with circle as a boundary Igor Rivin 2013-01-09T00:57:15Z 2013-01-09T00:57:15Z <p>Check out <a href="http://www.ugr.es/~rcamino/publications/pdf/art14.pdf" rel="nofollow">this paper of Rafael Lopez</a> and references therein.</p> http://mathoverflow.net/questions/118141/does-every-polynomial-diophantine-equation-have-solutions-modulo-p/118167#118167 Answer by Igor Rivin for Does every polynomial diophantine equation have solutions modulo p? Igor Rivin 2013-01-06T01:28:34Z 2013-01-06T01:28:34Z <p>I believe the OP is asking about a version of Hilbert nullstellensatz, see for example <a href="http://atlas.mat.ub.es/personals/dandrea/atlanta.pdf" rel="nofollow">these nice notes of D'Andrea.</a></p> http://mathoverflow.net/questions/116219/rational-orthogonal-matrices Rational orthogonal matrices Igor Rivin 2012-12-13T00:54:39Z 2012-12-24T07:35:56Z <p>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).</p> http://mathoverflow.net/questions/108372/translation-distance-in-the-curve-complex Translation distance in the curve complex Igor Rivin 2012-09-28T19:06:47Z 2012-12-17T16:17:58Z <p>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...</p> <p>I know there are algorithms for computing distances IN the curve complex, but this is not quite the same...</p> http://mathoverflow.net/questions/115192/power-log-distance-between-matrices power log distance between matrices Igor Rivin 2012-12-02T18:35:32Z 2012-12-10T03:20:29Z <p>In <a href="http://ptmat.fc.ul.pt/~pedro/thesis.pdf" rel="nofollow">this thesis</a>, 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).</p> http://mathoverflow.net/questions/115566/int-infty-0xr-s-11-x-s1-x2-fracrm2dx/115568#115568 Answer by Igor Rivin for $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$ Igor Rivin 2012-12-06T03:00:51Z 2012-12-06T03:35:34Z <p>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 <code>$F(k)=\frac{2^{(k-1)/2} G_{3,3}^{3,3}\left(1\left| \begin{array}{c} -\frac{1}{2},-\frac{1}{2},0 \\ 0,\frac{1}{2},(k-3)/2 \\ \end{array} \right.\right)}{(k-2)!! \pi ^{3/2}}$</code></p> <p>Where the $G$ is the Meijer G function, and $n!!$ is the product of all odd numbers not exceeding $n.$</p> <p>For <em>even</em> $k$ the answer is always a linear function of $\pi$ with rational coefficients, e.g. $F(4) = \frac12 - \frac{\pi}8$</p> <p>$F(6)= \frac14 - \frac{\pi}{16}$</p> <p>$F(8) = \frac1{12} - \frac{\pi}{64}$</p> <p>$F(10)= \frac{\pi}{128}$</p> <p>$F(12) =-\frac{1}{30}+ \frac{17\pi}{1024}$</p> <p>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).</p> <p>I did not try the other $r, s$ but this already seems pretty interesting.</p> http://mathoverflow.net/questions/115541/period-of-linear-congruential-generator/115546#115546 Answer by Igor Rivin for Period of linear congruential generator Igor Rivin 2012-12-05T20:51:30Z 2012-12-05T20:51:30Z <p>Kurlberg, Pär, and Carl Pomerance. "On the period of the linear congruential and power generators." arXiv preprint math/0405120 (2004).</p> http://mathoverflow.net/questions/115110/eversion-of-the-6-sphere-in-7-space/115130#115130 Answer by Igor Rivin for Eversion of the 6-sphere in 7-space Igor Rivin 2012-12-02T01:07:02Z 2012-12-02T01:07:02Z <p>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.</p> <p>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").</p> <p>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...</p> http://mathoverflow.net/questions/115125/iterating-random-matrix-operations/115129#115129 Answer by Igor Rivin for Iterating Random Matrix Operations Igor Rivin 2012-12-02T00:57:54Z 2012-12-02T00:57:54Z <p>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:</p> <p>Bougerol, Philippe, and Jean Lacroix. Products of random matrices with applications to Schrödinger operators. Birkhäuser, 1985.</p> http://mathoverflow.net/questions/114977/spectrum-of-transition-matrix-for-symmetric-random-walk/115018#115018 Answer by Igor Rivin for Spectrum of transition matrix for symmetric random walk Igor Rivin 2012-11-30T19:42:09Z 2012-11-30T19:42:09Z <p>The spectrum of the matrix is computed in the beginning of my preprint: </p> <p>Rivin, Igor. "Growth in free groups (and other stories)." arXiv preprint math/9911076 (1999).</p> <p>(there is a published version, too).</p> http://mathoverflow.net/questions/114911/abstract-characterization-of-polygonizations/114947#114947 Answer by Igor Rivin for Abstract characterization of polygonizations Igor Rivin 2012-11-30T01:16:01Z 2012-11-30T01:16:01Z <p>For the OP's claim re *infinite*polyhedral graphs, the answer is yes, this is true, and a proof is in my paper:</p> <p>Rivin, Igor. "Combinatorial optimization in geometry." Advances in Applied Mathematics 31.1 (2003): 242-271.</p> <p>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 <em>any</em> Jordan domain.</p> http://mathoverflow.net/questions/114824/how-to-solve-a-specific-multivariate-recurrence-relation-or-general-ones/114833#114833 Answer by Igor Rivin for How to solve a specific multivariate recurrence relation (or general ones) Igor Rivin 2012-11-29T01:55:00Z 2012-11-29T01:55:00Z <p>Read Bender and Orszag (Advanced Mathematical Methods for Scientists and Engineers), and you will be enlightened.</p> http://mathoverflow.net/questions/114194/how-to-divide-a-n-dimensional-simplex-in-n1-equal-parts/114197#114197 Answer by Igor Rivin for How to divide a n dimensional simplex in n+1 equal parts Igor Rivin 2012-11-22T23:40:17Z 2012-11-22T23:48:20Z <p>The magic word is "barycenter". The convex hull of the barycenter and any one face of the simplex has the right volume.</p> http://mathoverflow.net/questions/114018/fastest-way-to-factor-integers-260/114135#114135 Answer by Igor Rivin for Fastest way to factor integers < 2^60 Igor Rivin 2012-11-22T04:09:19Z 2012-11-22T04:09:19Z <p>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 $300 to double your RAM than spending an unbounded amount of your time on a wild goose chase.</p> http://mathoverflow.net/questions/114084/absolute-sum-of-coefficient-of-1-xb-1xn-b/114134#114134 Answer by Igor Rivin for Absolute sum of coefficient of (1-x)^b (1+x)^{(n-b)} Igor Rivin 2012-11-22T04:03:50Z 2012-11-22T04:03:50Z <p>This is analyzed exhaustively (also exhaustingly) by Domenici in <a href="http://arxiv.org/abs/math/0501042" rel="nofollow">this 2005 preprint.</a></p> http://mathoverflow.net/questions/114120/continuous-pointwise-ergodic-theorem/114123#114123 Answer by Igor Rivin for Continuous pointwise ergodic theorem? Igor Rivin 2012-11-22T01:36:12Z 2012-11-22T01:36:12Z <p>There is a difference between ergodicity and unique ergodicity. Notice that for the geodesic flow space averages equal time averages for almost all, but definitely not for all orbits (as shown by the existence of closed geodesics). On the other hand the horocycle flow IS uniquely ergodic (on a <em>compact</em> surface). That doesn't quite answer your Q2 (the average may exist, just not equal the space average). </p> <p>In any case, the best (in my opinion) exposition of the ergodicity of the geodesic flow is given by Curt McMullen in his <a href="http://www.math.harvard.edu/~ctm/home/text/class/berkeley/277/96/course/course.pdf" rel="nofollow">dynamics course notes.</a></p> http://mathoverflow.net/questions/114052/primitive-subwords-in-a-free-group-of-rank-2/114110#114110 Answer by Igor Rivin for Primitive subwords in a free group of rank 2 Igor Rivin 2012-11-21T22:26:44Z 2012-11-21T22:26:44Z <p>To answer your last question, check out <a href="https://dl.dropbox.com/u/5188175/CohenMetzlerZimmerman.pdf" rel="nofollow">Cohen-Metzler-Zimmerman (1981)</a> and references there in.</p> http://mathoverflow.net/questions/114107/what-are-first-eigenfunctions-of-laplacian-for-cpn-with-fubini-study-metric/114109#114109 Answer by Igor Rivin for What are first eigenfunctions of Laplacian for$CP^n\$ with Fubini-Study metric? Igor Rivin 2012-11-21T22:13:53Z 2012-11-21T22:13:53Z <p>See <a href="http://mathoverflow.net/questions/85481/the-first-eigenvalue-of-the-laplacian-for-complex-projective-space" rel="nofollow">this question</a>, and the answers to it.</p> http://mathoverflow.net/questions/114056/growth-of-energy-of-eigenfunctions-on-hyperbolic-surface/114061#114061 Answer by Igor Rivin for growth of energy of eigenfunctions on hyperbolic surface Igor Rivin 2012-11-21T14:32:34Z 2012-11-21T14:32:34Z <p>You might want to look at Chris Judge's papers, particularly:</p> <p>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. </p> http://mathoverflow.net/questions/122422/please-write-me-links-to-pdf-articles-or-books Comment by Igor Rivin Igor Rivin 2013-02-20T21:56:42Z 2013-02-20T21:56:42Z Not appropriate (in particular, since there is no question). Voting to close. http://mathoverflow.net/questions/122306/how-many-unit-balls-can-be-put-into-a-unit-cube Comment by Igor Rivin Igor Rivin 2013-02-19T16:42:49Z 2013-02-19T16:42:49Z This is related to this question: <a href="http://mathoverflow.net/questions/98007/covering-a-unit-ball-with-balls-half-the-radius" rel="nofollow" title="covering a unit ball with balls half the radius">mathoverflow.net/questions/98007/&hellip;</a> http://mathoverflow.net/questions/122056/on-average-length-of-sums-of-unit-vectors-in-rn Comment by Igor Rivin Igor Rivin 2013-02-19T01:56:30Z 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. http://mathoverflow.net/questions/120380/bivariate-polynomial Comment by Igor Rivin Igor Rivin 2013-01-31T05:03:15Z 2013-01-31T05:03:15Z This has a certain homework flavor. Voting to close. http://mathoverflow.net/questions/119980/concise-model-of-modern-fiat-money-and-its-non-conservation Comment by Igor Rivin Igor Rivin 2013-01-28T16:32:05Z 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 &quot;can be found in standard undergraduate textbooks&quot; is a completely worthless statement. Give us a book and a page number. http://mathoverflow.net/questions/119980/concise-model-of-modern-fiat-money-and-its-non-conservation/119983#119983 Comment by Igor Rivin Igor Rivin 2013-01-28T16:29:51Z 2013-01-28T16:29:51Z @Greg: certainly a fair point. http://mathoverflow.net/questions/119980/concise-model-of-modern-fiat-money-and-its-non-conservation/119983#119983 Comment by Igor Rivin Igor Rivin 2013-01-28T00:36:39Z 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 &quot;Austrian&quot; (by which I very much DO NOT mean &quot;like that of Michael Greinecker&quot;), and has a certain amount of predictive power (which is what makes it of any interest). http://mathoverflow.net/questions/119980/concise-model-of-modern-fiat-money-and-its-non-conservation/119983#119983 Comment by Igor Rivin Igor Rivin 2013-01-27T19:20:49Z 2013-01-27T19:20:49Z @Michael: (a) I explicitly said that Rothbard made no attempt to provide a &quot;mathematical&quot; model. (b) but why don't you let whoever here cares read his work for themselves. (c) &quot;I don't care what you think&quot; is very open-minded and polite of you. http://mathoverflow.net/questions/119980/concise-model-of-modern-fiat-money-and-its-non-conservation/119983#119983 Comment by Igor Rivin Igor Rivin 2013-01-27T15:03:34Z 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 &quot;most economists&quot; 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. http://mathoverflow.net/questions/119918/the-knot-whose-complement-is-the-hantzsche-wendt-manifold/119982#119982 Comment by Igor Rivin Igor Rivin 2013-01-27T02:08:08Z 2013-01-27T02:08:08Z As shown in the reference I gave. http://mathoverflow.net/questions/118431/representation-of-teichmuller-space-teichmuller-space Comment by Igor Rivin Igor Rivin 2013-01-09T15:40:05Z 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. http://mathoverflow.net/questions/112198/solvable-groups Comment by Igor Rivin Igor Rivin 2013-01-07T00:04:20Z 2013-01-07T00:04:20Z Why is this not getting closed? http://mathoverflow.net/questions/117899/probability-density-of-vector-sum Comment by Igor Rivin Igor Rivin 2013-01-05T01:20:54Z 2013-01-05T01:20:54Z The correct verb is &quot;convolving&quot;. @Uwe Franz has a way to convolve correctly in his answer. http://mathoverflow.net/questions/118098/boundary-problem-with-an-area-constraint Comment by Igor Rivin Igor Rivin 2013-01-05T01:06:58Z 2013-01-05T01:06:58Z Homework, voting to close. http://mathoverflow.net/questions/117558/are-there-heronian-triangles-that-can-be-decomposed-into-three-smaller-ones Comment by Igor Rivin Igor Rivin 2012-12-30T00:48:08Z 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.