User per alexandersson - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T16:14:49Z http://mathoverflow.net/feeds/user/1056 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130920/what-mathematical-field-work-with-this-kind-of-concept/130930#130930 Answer by Per Alexandersson for What mathematical field work with this kind of concept? Per Alexandersson 2013-05-17T09:02:22Z 2013-05-17T09:02:22Z <p>It also sound a bit like neural networks; these are "trained" (adapted) to yield certain output for certain input.</p> <p>Maybe you will also be interested in Markov processes, (which can also be trained to give certain output for some given input).</p> <p>Both these concepts are related to something called genetic programming, where a large parameter space is sampled, evaluated, and mutated, so that the next iteration performs "better" than the previous.</p> http://mathoverflow.net/questions/130418/polynomial-zero-within-a-square/130425#130425 Answer by Per Alexandersson for polynomial zero within a square Per Alexandersson 2013-05-12T17:12:33Z 2013-05-12T17:12:33Z <p>Since $f$ is non-zero in a neighbourhood of the unit square, it follows that $1/f$ is holomorhic in the unit square. By the maximum modulus principle, $|1/f|$ attains its maximum on the boundary. If follows that the minimum of $|f|$ is attained at the boundary, of the unit square. This sort of hints that it should be very hard to find such a polynomial.</p> http://mathoverflow.net/questions/65034/useful-tricks-in-experimental-mathematics/129779#129779 Answer by Per Alexandersson for Useful tricks in experimental mathematics Per Alexandersson 2013-05-05T21:26:41Z 2013-05-06T03:30:17Z <p>The tricks I regularly use:</p> <ul> <li>Create more examples. Always.</li> <li>As a corollary of the above, time is well spent on making algorithms that presents examples nicely.</li> <li>The Online Encyclopedia of Integers (OEIS), is your friend.</li> <li>Or, if that does not work, put your sequence or constant into WolframAlpha.</li> <li>If the numerical data looks strange, redo! Some software do not warn when the precision is lost. Some software (Mahtmematica for example), do not consider $1/2$ and $0.5$ to be equal.</li> <li>Take time to learn your software! You are more tempted to try new stuff, if it is easy to code.</li> </ul> http://mathoverflow.net/questions/129735/elementary-proof-of-k-saturation-conjecture Elementary proof of K-saturation conjecture Per Alexandersson 2013-05-05T16:10:04Z 2013-05-05T16:10:04Z <p>One variant of Fulton's K-saturation conjecture is as follows:</p> <p>$K_{\lambda/\mu,w} > 0 \Leftrightarrow K_{n\lambda/n\mu,n w} > 0$ for any integer $n>0.$</p> <p>Here $K_{\lambda/\mu,w}$ denotes the Kostka numbers (number of skew SSYT of shape $\lambda/\mu$ and weight $w.$</p> <p>This has been proved in various ways, (Knutson, Tao), so it is no longe a conjecture, but to me the proofs are quite involved. The proof shows the similar statement for Littlewood-Richardson coefficients, using K-hives and the above follows as a corollary.</p> <p><strong>Question:</strong> Is there an elementary proof of the above statement? Could one expect a short proof of this?</p> http://mathoverflow.net/questions/129706/zeta3-in-terms-of-derivatives-of-zeta-at-1-2-and-pi/129721#129721 Answer by Per Alexandersson for zeta(3) in terms of derivatives of zeta at 1/2 and pi Per Alexandersson 2013-05-05T14:15:09Z 2013-05-05T15:02:04Z <p><s>This is just partial numerics, but the following Mathematica code strengthens the conjecture, with 500 decimal places:</s></p> <p>Here is code for those that want to perform numerics in Mathematica:</p> <pre><code>prec = 500; (* Precision of calculations. *) lhs = N[Zeta[3], prec] rhs = N[(-Zeta'''[1/2]/Abs[Zeta[1/2]] - 3 Zeta''[1/2] Zeta'[1/2]/Abs[Zeta[1/2]]^2 - 2 Zeta'[1/2]^3/Abs[Zeta[1/2]]^3 - Pi^3/4)/7, prec] lhs == rhs (* Gives true *) </code></pre> http://mathoverflow.net/questions/128961/probability-of-random-0-1-toeplitz-matrix-being-invertible/129189#129189 Answer by Per Alexandersson for Probability of random (0,1) Toeplitz matrix being invertible Per Alexandersson 2013-04-30T08:32:08Z 2013-04-30T08:32:08Z <p>This is just a partial answer:</p> <p>Let $A$ be a $N \times N$ Toeplitz matrix, and consider the sequence of matrices $A_n$ for $n>N$ where we increase the size of $A$ to $n\times n$ and the new diagonals are filled with 0. Then the sequence of determinants $|A_n|$ will satisfy a linear recurrence. The zeros in a linear recurrence appears either in arithmetic progressions, or are sporadic. There is an upper bound on the number of sporadic zeros in linear recurrences, and these are "quite rare" in some sense. To have an infinite number of zeros in this sequence of determinants, we require that the <em>symbol</em> of the matrix have roots of unity, (this is sort of the same as the characteristic equation for the linear recurrence). Now, since the matrix is binary, the coefficients of the symbol (a polynomial) are also either 0 or 1.</p> <p>So, in some sense, the question is related to the probability that a polynomial with coefficients either 0 or 1 has a root of unity.</p> http://mathoverflow.net/questions/128897/l-systems-and-sierpinski-triangle/128936#128936 Answer by Per Alexandersson for L-systems and Sierpinski Triangle Per Alexandersson 2013-04-27T17:12:47Z 2013-04-27T17:12:47Z <p>Once you get the grips around constructing the Hilbert curve: <a href="https://en.wikipedia.org/wiki/Hilbert_curve" rel="nofollow">https://en.wikipedia.org/wiki/Hilbert_curve</a> then this is not really all that hard to construct. </p> <p>Any fractal that consists of $n$ identical, smaller copies of itself, and where the pieces are connected "start to finish", could probably be generated by a suitable l-system.</p> <p>As an exercise: * The Koch kurve consists of 4 smaller copies of itself, connected in the end-points. * n-flakes: <a href="https://en.wikipedia.org/wiki/N-flake#Hexaflake" rel="nofollow">https://en.wikipedia.org/wiki/N-flake#Hexaflake</a> etc...</p> http://mathoverflow.net/questions/106040/sequence-of-semi-standard-young-tableaux-counting Sequence of semi-standard Young tableaux, counting Per Alexandersson 2012-08-31T14:37:31Z 2013-03-30T11:37:36Z <p>Let $\lambda$ be a partition and define $T_\lambda^n(k)$ to be the number of semi-standard young tableaux of shape $(k\lambda_1,k\lambda_2,\dots,k \lambda_n).$ Now, one can prove that $T_\lambda^n(k)$ is a polynomial in $k.$</p> <p><strong>Is this well-known? Is there a known formula for $T_\lambda^n(k)$? Would it be interesting to find a general formula for this polynomial?</strong></p> <p>(I know about the hook-length formula, but since it is a product with $|\lambda|$ factors, this is not really useful for studying asymptotics of $T_\lambda^n(k)$.)</p> <p>(The asymptotics for the corresponding Schur polynomials are also quite interesting. The roots of $S_{k \lambda}(t,1,1,\dots,1) = 0$ will for example accumulate close to the unit circle as $k$ grows.)</p> <p>EDIT: It seems that finding Kostka numbers, $K_{k\lambda, k\beta}$ is a refinement of the above problem, and it is known that these grows polynomially. An article from 2007, Degrees of Stretched Kostka coefficients, by B. McAllister finds the degree of these, but the exact formulas for the polynomials seems to be unknown.</p> http://mathoverflow.net/questions/124456/learning-stochastic-calculus-want-to-know-what-the-notation-of-this-function-mea/124457#124457 Answer by Per Alexandersson for Learning stochastic calculus, want to know what the notation of this function means Per Alexandersson 2013-03-13T21:20:45Z 2013-03-13T21:20:45Z <p>I would strongly suspect $1_{[a_i,b_i)}$ is the function which is one on the interval $[a_i,b_i)$ and zero elsewhere.</p> http://mathoverflow.net/questions/123670/all-possible-linear-combinations-of-positive-half-integers-with-coefficients/123673#123673 Answer by Per Alexandersson for All possible linear combinations of positive half-integers with coefficients +/- 1 Per Alexandersson 2013-03-05T22:07:17Z 2013-03-05T22:07:17Z <p>Hm, maybe rewriting this by adding $p_1+p_2+\dots$ on both sides, and dividing by two, then this is isomorphic to $P' = f_1 p_1 + \dotsc$ where $f_i$ is either 0 or 1. Thus, it is the number of ways to construct $P'$ as a sum using 0 or one $p_1.$ Thus, expand $\prod_i (1+x^{p_i})$ and look for the coefficient of $x^{P'}$ to find the number you are looking for.</p> <p>I don't know if this helps you, but maybe this interpretation helps a bit. What do you know of the numbers $p_i$?</p> http://mathoverflow.net/questions/123563/eigenvalues-of-principle-minors-vs-eigenvalues-of-the-matrix/123609#123609 Answer by Per Alexandersson for Eigenvalues of principle minors Vs. eigenvalues of the matrix Per Alexandersson 2013-03-05T10:48:41Z 2013-03-05T10:48:41Z <p>No, it is not true; Consider the matrix</p> <p><code>$$A= \begin{pmatrix} \frac{9}{2} &amp; \frac{9}{20} &amp; \frac{21}{20} &amp; -\frac{3}{2}\\ -\frac{79}{11} &amp; -\frac{3}{110} &amp; \frac{23}{110} &amp; \frac{31}{11}\\ \frac{6}{11} &amp; \frac{21}{55} &amp; \frac{114}{55} &amp; \frac{6}{11}\\ \frac{16}{11} &amp; \frac{12}{55} &amp; \frac{128}{55} &amp; \frac{5}{11} \end{pmatrix}$$</code> which has eigenvalues 0,1,2,3 (so it is positive semi-definite, but not definite.) The four principal minors are $$\frac{27}{22},\frac{189}{110},\frac{153}{55},\frac{36}{11}$$ sorted in increasing order. This should give a definite negative answer to your question.</p> http://mathoverflow.net/questions/122973/homogenous-polynomials-as-sum-or-differences-of-squares-and-symmetric-polynomials Homogenous polynomials as sum or differences of squares and symmetric polynomials Per Alexandersson 2013-02-26T12:33:55Z 2013-02-27T20:39:21Z <p>I seem to recall that a general <em>homogenous</em> real polynomial $P$ of even degree in $n$ variables, $n\geq 3,$ cannot always be expressed as $P(x_1,\dotsc,x_n)=\sum_j a_j Q_j^2(x_1,\dotsc,x_n)$ where $a_j \in \mathbb{R},$ and the $Q_j^2$ are <em>homogenous of the same degree</em> as $P.$ (Please, correct me if I am wrong).</p> <p>Now, what if we know that $P$ is symmetric? Is it still true that a general polynomial cannot be expressed as above? And if it can, what if we require that the $Q_j$ themselves are symmetric? </p> <p><strong>Question:</strong> If $P$ is symmetric and homogenous of even degree, can it be expressed as a sum/difference of squares of homogenous and symmetric polynomials?</p> <p>I know that if $P$ is symmetric, and $P(x_1+t,x_2+t,\dotsc,x_n+t)=P(x_1,\dotsc,x_n),$ then there is always such a representation as the one above, but can one lose the translation-invariant condition and still have sum-and-difference of squares representation?</p> http://mathoverflow.net/questions/122973/homogenous-polynomials-as-sum-or-differences-of-squares-and-symmetric-polynomials/123143#123143 Answer by Per Alexandersson for Homogenous polynomials as sum or differences of squares and symmetric polynomials Per Alexandersson 2013-02-27T20:39:21Z 2013-02-27T20:39:21Z <p>My college provided me with a simple example: $$x^2 + xy + y^2$$ cannot be expressed as a sum/difference of symmetric, homogenous polynomials of degree 1, for obvious reasons.</p> <p>(Each homogenous, symmetric polynomial of degree 1 in two variables are of the form a(x+y). Thus, all possible $Q_j$ are of the form $a'(x+y)^2$ which, of course, cannot give the polynomial above.)</p> http://mathoverflow.net/questions/123025/the-right-conformal-map-to-make-a-certain-picture/123039#123039 Answer by Per Alexandersson for The right conformal map to make a certain picture Per Alexandersson 2013-02-26T22:03:35Z 2013-02-26T22:03:35Z <p>Well, if $z \mapsto z^2$ gives circles, then $(x,y) \mapsto (x^2-y^2,xy)$ should give you ellipses with the horizontal axis twice as large as the vertical. Note, this map is not conformal.</p> <p>It therefore "feels" like you cannot find such a map that you are looking for, ellipses are not really conformal objects in some sense, maybe someone else can give a rigorous proof on why there cannot be such a map?</p> http://mathoverflow.net/questions/121205/periodicity-of-a-specific-non-linear-ode-of-second-order Periodicity of a specific non-linear ODE of second order Per Alexandersson 2013-02-08T15:40:00Z 2013-02-11T23:00:23Z <p>Consider the second-order ODE: $$\ddot{x} + x+x^2=0,$$ here $\ddot{x}$ is the second derivative w.r.t. $t$. Take initial values $x(0)=0.5$ and $\dot{x}(0)=0.$</p> <p><strong>Question:</strong> is the solution periodic or not?</p> <p>Comment: Numerical experiments seems to show that the solution is periodic when $x(0)&lt;0.5$ and if $x(0)>0.5$ then the solution fails to be periodic, in fact, $x(t)\rightarrow -\infty.$</p> <p>(Asked by Prof. J.E. Björk, Stockholm Univ.)</p> http://mathoverflow.net/questions/121116/to-what-extent-should-a-pure-mathematician-care-about-the-meaning-of-things/121118#121118 Answer by Per Alexandersson for To what extent should a pure mathematician care about the meaning of things? Per Alexandersson 2013-02-07T21:15:18Z 2013-02-07T21:15:18Z <p>To me, it is a bit like asking the meaning of music; it does not have to be "applied", but may exist and be beautiful in it self.</p> <p>Now, beautiful mathematics is usually simpler to understand, than ugly mathematics, so sometimes, it might be necessary to do beautiful, non-meaningful things, to simplify mathematics which has a meaning. This was sort of the story when the complex numbers were invented; these were strange, and did not have a meaning, but simplified the mathematics. Now, we cannot live without them.</p> http://mathoverflow.net/questions/120681/elementary-question-distinct-elements-in-a-set/120688#120688 Answer by Per Alexandersson for Elementary question: distinct elements in a set Per Alexandersson 2013-02-03T17:17:05Z 2013-02-03T17:17:05Z <p>"Let $x,y,z,$ be pairwise distinct", is perfectly fine.</p> http://mathoverflow.net/questions/81607/discriminant-on-boundary-of-semi-algebraic-surface Discriminant on boundary of semi-algebraic surface Per Alexandersson 2011-11-22T13:32:17Z 2013-02-02T16:06:33Z <p>Let $P(t)$ be a polynomial in $t$ of degree $n$, with some contiguous coefficients (not the first or last) being $x_1,\dots,x_k$ and the rest of the coefficients are fixed.</p> <p>(E.g. $p(t)=1+2t+x_1t^2+x_2t^3+(5i+7)t^6$ is ok).</p> <p>Consider the set $S$ defined as $(x_1,\dots,x_k) \in \mathbb{C}^k$ with the property that the roots $t_1,\dots,t_{n}$ of $P(t)=0$ may be ordered increasingly w.r.t modulus, and such that $$|t_j|=|t_{j+1}| = \dots = |t_{k+j}|,$$ for some fixed $j.$</p> <p>This set is real $k$-dimensional.</p> <p>Is it true, (in general), that the discriminant $D(x_1,\dots,x_n) = Discr_t(P)$ satisfy $D(x_1,\dots,x_n)=0$ on the $k-1$-dimensional boundary of $S$?</p> http://mathoverflow.net/questions/120298/picture-of-a-3-dimensional-amoeba/120310#120310 Answer by Per Alexandersson for Picture of a 3 dimensional amoeba. Per Alexandersson 2013-01-30T12:30:37Z 2013-01-30T13:09:29Z <p><a href="http://en.wikipedia.org/wiki/Amoeba_%28mathematics%29" rel="nofollow">There is now a 3-dimensional amoeba on wikipedia.</a></p> <p>You are welcome. The main reason for the difficulty of making such amoebas is that it is the projection of a 4-dimensional surface (the zero set of the polynomial). Finding a parametrization of the zeros set of an arbitrary polynomial in 2 variables is highly non-trivial. The case on wikipedia is linear, thus we may easily parametrize it. Now, this set is projected with $Log|\cdot|$, thus one has to be a bit careful which points to include to make the picture "pretty", since there may be zeros of the polynomial "far" away, and close to 0. </p> <p>I did not draw the projection of the parametrized 4-dimensional surface, but rather the projection of a lot of points on the surface, chosen in a manner that makes them somewhat evenly distributed.</p> <p>EDIT: Also, rumour has it that Frank Sottile has some 3-dimensional amoebas somewhere on his <a href="http://www.math.tamu.edu/~sottile/index.html" rel="nofollow">web page</a>, but I have not been able to find them.</p> <p>EDIT 2: Now, amoebas never look like soap bubbles, but the connected components of their complements do. (These components are always convex, aka. soap bubbles)</p> http://mathoverflow.net/questions/117805/asymptotic-behavior-of-non-analytic-function-of-the-eigenvalues/118192#118192 Answer by Per Alexandersson for Asymptotic Behavior of Non-Analytic Function of the Eigenvalues Per Alexandersson 2013-01-06T08:52:17Z 2013-01-06T08:59:38Z <p>To create a partial answer, eigenvalues of banded Toeplitz matrices accumulate on some real algebraic curves in $\mathbb{C}$, (this has been proved by Schmidt and Spitzer around 1970.). Now, if we also attach a point-mass at each eigenvalue, (for fixed matrix size) with equal mass at each point, and with total mass one, we get a probability measure. These measures converge in a certain sense to some limit measure, see for example the book by <a href="http://www.amazon.com/Spectral-Properties-Banded-Toeplitz-Matrices/dp/0898715997" rel="nofollow">Bender and Böttcher </a>.</p> <p>Now, the sum $c_n = \frac{1}{n} \sum_{k=1}^n |\lambda_{n,k}|$ can then be interpreted as the <s>center of mass</s> mean distance to 0 of all the eigenvalues from matrix of size $n \times n.$ Say that the associated point measure is called $\mu_n$ (mass $1/n$ at each eigenvalue), then we know that $\mu_n \to \mu$ for some $\mu$ in some sense.</p> <p>Note that $c_n = \int_{\mathbb{C}} |z| d\mu_n(z)$ and then what you are looking for is $c = \int_{\mathbb{C}} |z| d\mu(z)$.</p> <p>As an explicit example, taking your matrix to be tridiagonal, will give rise to characteristic polynomials which are also a family of orthogonal polynomials, w.r.t some measure (the limit of point measures, actually), see e.g. <a href="http://www.google.com/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=1&amp;ved=0CDkQFjAA&amp;url=http%3A%2F%2Feuler.us.es%2F~renato%2Fpapers%2Fchain-revised-1.pdf&amp;ei=iTrpUIi5MaOH4ASojYDADQ&amp;usg=AFQjCNGMKq-EYEq5rqZeXhVcv4eqrHj2Lw&amp;bvm=bv.1355534169,d.bGE&amp;cad=rja" rel="nofollow">this paper i just googled</a>.</p> http://mathoverflow.net/questions/117028/sequences-with-a-fractal-dimension/117042#117042 Answer by Per Alexandersson for sequences with a fractal dimension Per Alexandersson 2012-12-22T18:46:48Z 2012-12-22T18:46:48Z <p>Taking a truncated integer sequence is essentially the same as defining the "heights" of the line endpoints of a 2-dimensional curve, similar to <a href="http://www.gameprogrammer.com/fractal.html" rel="nofollow">http://www.gameprogrammer.com/fractal.html</a> my guess is that this is not really new, but rather a cumbersome way to define self-similar curves.</p> <p>Making a reasonable scaling will in some examples, as above, give a well-defined limit curve. Maybe it is possible to compute the Hausdorff dimension by doing a discrete Fourier transform of the sequence. The spectrum should be quite interesting, I think.</p> http://mathoverflow.net/questions/116171/littlewood-richardson-coefficients-from-kostka-coefficients Littlewood-Richardson Coefficients from Kostka coefficients Per Alexandersson 2012-12-12T13:46:13Z 2012-12-13T02:06:36Z <p>I read on page 4 <a href="http://lipn.univ-paris13.fr/~toumazet/biblio/ARTICLES/NTB.pdf" rel="nofollow">here</a> that the Kostka coefficients $K_{\lambda,\mu}$ are specializations of the Littlewood-Richardson coefficients $c^\tau_{\sigma,\lambda}$ by specializing $\sigma,\tau$ depending on $\mu$ in a simple manner (certain sums of parts of $\mu$).</p> <p>I have two questions: Is there a similar specialization/translation for Kostka coefficients obtained from skew shapes, $K_{\lambda,\mu}^\nu$ where $\lambda/\nu$ is a skew shape?</p> <p>Secondly: Is there a way to "go back", i.e. can I express $c^\tau_{\sigma,\lambda}$ as some linear combination of some skew Kostka coefficients $K_{\lambda,\mu}^\nu$?</p> <p>I would like to see if polynomiality of the map $n \mapsto K_{n \lambda, nw}^{n \nu}$ implies polynomiality for a similar map with LW-coefficients.</p> http://mathoverflow.net/questions/114745/monic-polynomial-with-integer-coefficients-with-roots-on-unit-circle-not-roots-r Monic polynomial with integer coefficients with roots on unit circle, not roots root of unity? Per Alexandersson 2012-11-28T09:07:31Z 2012-11-28T17:58:05Z <p>There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example.</p> <p>Now, for a monic polynomial of degree $n$, this is impossible (I think).</p> <p><strong>So, my question is,</strong> given a monic polynomial with integer coefficients of degree $n$, what is the maximal number of roots that can lie on the unit circle, and not be roots of unity?</p> <p>For example, $1 + 3 x + 3 x^2 + 3 x^3 + x^4$ has two roots on the unit circle, and two real roots.</p> http://mathoverflow.net/questions/114184/a-question-about-the-axiom-of-choice-and-straight-lines-in-the-euclidean-plane/114216#114216 Answer by Per Alexandersson for A question about the Axiom of Choice and straight lines in the Euclidean plane. Per Alexandersson 2012-11-23T06:54:12Z 2012-11-23T06:54:12Z <p>How about tangents to a deltoid, <a href="http://mathworld.wolfram.com/Deltoid.html" rel="nofollow">http://mathworld.wolfram.com/Deltoid.html</a> ?</p> <p>It do look like the tangents really do cover the entire plane, and all of those are non-parallel.</p> <p><a href="http://imageshack.us/photo/my-images/42/deltoidtangents.png" rel="nofollow">http://imageshack.us/photo/my-images/42/deltoidtangents.png</a></p> http://mathoverflow.net/questions/112074/how-many-matrices-are-possible-for-the-given-arrangement/112093#112093 Answer by Per Alexandersson for How many matrices are possible for the given arrangement? Per Alexandersson 2012-11-11T17:48:42Z 2012-11-11T17:54:29Z <p>Let $a_n$ be the number of $2 \times n$ -matrices avoiding constant 2*2-submatrices. Then </p> <p>$$a_n = \frac{2^{-n} \left(4 \left(17+4 \sqrt{17}\right) \left(3+\sqrt{17}\right)^n+\left(\sqrt{17}-17\right) \left(\sqrt{17}-3\right)^n e^{i \pi n}\right)}{17 \left(3+\sqrt{17}\right)}$$</p> <p>This should be fairly straightforward to prove, let $v(n)=(e_{01}(n),e_{10}(n),e_{00}(n),e_{11}(n))$ be the vector of number of $2\times n$-matrices ending with column 01, 10, 00 resp. 11.</p> <p>We then have the recursion <code>$$v(n+1)=\begin{pmatrix} 1 &amp; 1 &amp; 1 &amp; 1 \\ 1 &amp; 1 &amp; 1 &amp; 1 \\ 1 &amp; 1 &amp; 0 &amp; 1 \\ 1 &amp; 1 &amp; 1 &amp; 0 \\ \end{pmatrix} v(n)$$</code></p> <p>Since this is symmetric, we may diagonalize this and from here, it should be straightforward to find the formula above. (I cheated a bit in Mathematica).</p> <p>EDIT: Of course, $e_{01}(n)=e_{10}(n)$ and $e_{00}(n)=e_{11}(n)$ by symmetry, so one can of course reduce the above to a 2 by 2 matrix recursion instead, with entries 2,2 and 2,1. Eigenvalues of this matrix are $1/2 (3 + \sqrt{17}), 1/2 (3 - \sqrt{17})$ which explains the strange formula above.</p> http://mathoverflow.net/questions/110337/are-there-any-non-planar-graphs-containing-only-k3-3-as-a-subgraph-that-are-not/110865#110865 Answer by Per Alexandersson for Are there any non-planar graphs containing only K(3,3) as a subgraph that are not 4-colourable? Per Alexandersson 2012-10-27T22:24:51Z 2012-10-27T22:24:51Z <p>Wikipedia:</p> <p>As Wagner showed, every graph that has no K5 minor can be decomposed via clique-sums into pieces that are either planar or an 8-vertex Möbius ladder, and each of these pieces can be 4-colored independently of each other, so the 4-colorability of a K5-minor-free graph follows from the 4-colorability of each of the planar pieces.</p> <p>From <a href="http://en.wikipedia.org/wiki/Hadwiger_conjecture_%28graph_theory%29" rel="nofollow">http://en.wikipedia.org/wiki/Hadwiger_conjecture_%28graph_theory%29</a></p> http://mathoverflow.net/questions/109904/prove-that-a-particular-polynomial-sequence-is-a-sturm-sequence/109946#109946 Answer by Per Alexandersson for Prove that a particular polynomial sequence is a Sturm sequence Per Alexandersson 2012-10-17T20:14:53Z 2012-10-18T18:02:55Z <p>This is just a partial answer, but anyway: ( I use $s[n,k]$ for $s^n_k$ since I just copy-pasted from Mathematica:</p> <p>I do not know if this helps, or if you already know, but you have the recurrence $$s[n- 1,k + 1] = s'[n, k] - x \cdot s'[n - 1, k]$$</p> <p>Note, by induction, we assume $s[n,k]$ and $s[n-1,k]$ to have interlacing roots. By thm 1.47 in <a href="http://arxiv.org/abs/math/0612833" rel="nofollow">http://arxiv.org/abs/math/0612833</a> (look into this work), we know that the derivatives have interlacing roots as well.</p> <p>Now, the recursion is quite similar to many other that appear in the link above, so it might not be too hard to prove stability.</p> <p>EDIT: I show/sketch below that the operator $f \mapsto A f - x f'$ always produce roots interlaced (and to the right) of the roots of $f$, provided $A>\deg f$ and that all roots of $f$ are positive. (We may assume leading term of $f$ has positive coefficient).</p> <p>Clearly if f has a root of multiplicity m, then the result will have the same root with multiplicity m-1, so the problem is essentially to ensure that no new multiple roots may appear.</p> <p>Assume now we have two consecutive roots $0\le a \le b$ of $f$ and that $f$ is positive between these. The derivative of $f$ this first positive, and then negative in $[a,b]$, thus $-x f'$ is first negative and then positive. Hence, $A f - x f'$ is negative in $a$, and positive in $b.$ Thus, we have a root of $A f - x f'$ in the interval $[a,b]$. (Mutatis mutandis for the case when f is negative between $a$ and $b$).</p> <p>Now, if the degree of $f$ is even, $f$ is positive to the right of its largest root. The derivative is also positive here (since even degree poly), so $x f'$ is positive. Hence, $A f - x f'$ is negative in this point. However, notice this is a polynomial with even degree, and with a leading coefficient! Hence, it must eventually cross the real line and grow to infinity. (Mutatis mutandis for the case when f is odd).</p> <p>This proves that $f \mapsto A f - x f'$ produces interlacing roots, and eventual multiplicities are decreased, no new multiple roots can be introduced.</p> http://mathoverflow.net/questions/109567/eigenvalues-of-infinite-matrices/109602#109602 Answer by Per Alexandersson for Eigenvalues of infinite matrices Per Alexandersson 2012-10-14T11:32:59Z 2012-10-14T11:32:59Z <p>I have done some research on banded Toeplitz matrices, (where we consider a sequence of finite matrices, and show where the eigenvalues accumulate). There is also quite old literature on this subject, see references in this paper I shamelessly advertise: <a href="http://arxiv.org/abs/1208.5607" rel="nofollow">http://arxiv.org/abs/1208.5607</a></p> http://mathoverflow.net/questions/107915/collisions-between-rooks-taking-random-flights-on-an-n-by-m-chessboard/107923#107923 Answer by Per Alexandersson for Collisions between rooks taking random flights on an $N$ by $M$ chessboard Per Alexandersson 2012-09-23T19:46:58Z 2012-09-25T06:11:52Z <p>So, I did some numerical experiments on 4 rooks on a k times k board. Each data point is the mean of 500 runs.</p> <p><img src="http://imageshack.us/a/img831/2701/rookhunt.png" /></p> <p>The x axis is the width/height of the board, the y axis is the number of iterations needed for it to be only one rook left.</p> <p>EDITED: So, I did some changes and now my data conforms with the others:</p> <p>1) Choose rook. 2) Choose direction. 3) If direction does not allow moving in that direction, goto 2. 4) Move 1,2,.. or k steps in direction chosen, where k gives a boundary square.</p> <p>I.e. this does not count non-moves (which the image ABOVE do).</p> <p>The image below shows mean of 500 runs, k rooks on a k*k board, starting at k=1. <img src="http://imageshack.us/a/img577/2701/rookhunt.png" /></p> http://mathoverflow.net/questions/106606/new-formula-for-counting-ssyts New formula for counting SSYTs? Per Alexandersson 2012-09-07T13:54:01Z 2012-09-07T13:54:01Z <p>I have a proof that given a partition $\lambda=(\lambda_1,\dots,\lambda_l)$ then the number of semi-standard Young tableaux of shape $\lambda$ with entries in $1,2,\dots, n$ is given by</p> <p>$$\frac{1}{1!2!\cdots (n-1)!} \prod_{1\leq i\lt j\leq n} (\lambda_i-i)-(\lambda_j-j).$$ (We define $\lambda_j:=0$ if $j \gt l.$)</p> <p>The product is also recognized as a Vandermonde determinant.</p> <p>There are plenty of product formulas (over boxes in the tableau) and determinant formulas (but not in Vandermonde form, as far as I can tell) for the number of such SSYTs, but I have not seen a this particular one in the literature or in any article I've come across.</p> <p>Is this formula known? Is this formula of any interest?</p> http://mathoverflow.net/questions/130735/approximate-closed-form-solution-for-a-recurrence Comment by Per Alexandersson Per Alexandersson 2013-05-15T16:00:02Z 2013-05-15T16:00:02Z Is this maybe some type of Eulerian numbers? http://mathoverflow.net/questions/130520/how-to-use-a-numerical-method-similar-to-newtons-method-to-solve-multiple-roots Comment by Per Alexandersson Per Alexandersson 2013-05-13T21:21:11Z 2013-05-13T21:21:11Z This is a really messy question, and I think you better ask over at math.stackexchange or programmers.stackexchange since this is not research and really more about programming. http://mathoverflow.net/questions/41283/is-always-possible-to-slice-a-pizza-in-a-fair-way/41295#41295 Comment by Per Alexandersson Per Alexandersson 2013-05-13T21:19:32Z 2013-05-13T21:19:32Z There should be some generalization of the ham sandwich theorem that takes care of more than two people (I think). It is trivial to generalize to a power of two people, for example (by iterating ham sandwich thm). However, a related, and mostly unsolved problem, is to divide a polygon into n parts with equal area and perimeter (some special cases are known only). http://mathoverflow.net/questions/130418/polynomial-zero-within-a-square/130423#130423 Comment by Per Alexandersson Per Alexandersson 2013-05-12T17:14:29Z 2013-05-12T17:14:29Z haha, nice there! http://mathoverflow.net/questions/129735/elementary-proof-of-k-saturation-conjecture Comment by Per Alexandersson Per Alexandersson 2013-05-06T08:12:28Z 2013-05-06T08:12:28Z Correct me if I'm wrong, but for the non-skew version, $K_{\lambda,w}&gt;0$ iff $\lambda \geq_d w$ in dominance order. And I suppose that $\lambda \geq_d w \Leftrightarrow n\lambda \geq_d nw$ is easy to show... http://mathoverflow.net/questions/129706/zeta3-in-terms-of-derivatives-of-zeta-at-1-2-and-pi/129721#129721 Comment by Per Alexandersson Per Alexandersson 2013-05-05T15:01:22Z 2013-05-05T15:01:22Z Oh, read it as &quot;to $10^{-4}$ precision&quot;. Silly me... http://mathoverflow.net/questions/129719/the-prime-real-numbers-and-their-applications Comment by Per Alexandersson Per Alexandersson 2013-05-05T14:25:05Z 2013-05-05T14:25:05Z This is nonsense. Also, the product of two non-primes is not necessary composed, so I believe you example is faulty. Vote to close. http://mathoverflow.net/questions/128961/probability-of-random-0-1-toeplitz-matrix-being-invertible Comment by Per Alexandersson Per Alexandersson 2013-04-30T08:18:46Z 2013-04-30T08:18:46Z marshall: I think it is ok, I read the question too quickly. http://mathoverflow.net/questions/128961/probability-of-random-0-1-toeplitz-matrix-being-invertible Comment by Per Alexandersson Per Alexandersson 2013-04-30T06:46:13Z 2013-04-30T06:46:13Z You probably would like to be more specific about &quot;random&quot;. There is no canonical way to choose random elements from R. http://mathoverflow.net/questions/126614/solution-of-the-recurrence-relation Comment by Per Alexandersson Per Alexandersson 2013-04-05T14:47:49Z 2013-04-05T14:47:49Z So what is the question? http://mathoverflow.net/questions/105689/techniques-to-solve-equations-involving-a-definite-integral Comment by Per Alexandersson Per Alexandersson 2013-03-08T07:44:53Z 2013-03-08T07:44:53Z Solve numerically, with 10-12 digits, and see Plouffes inverter, or search in wolframalpha. If it is expressible in known constants, you might get a hit. Knowing the exact value may now indicate how to PROVE it is the exact value you got. http://mathoverflow.net/questions/123669/calculation-of-a-fourier-transform Comment by Per Alexandersson Per Alexandersson 2013-03-05T21:56:35Z 2013-03-05T21:56:35Z This is for research level questions. Your question is not on research level. Ask over at math.stackexchange http://mathoverflow.net/questions/123563/eigenvalues-of-principle-minors-vs-eigenvalues-of-the-matrix/123609#123609 Comment by Per Alexandersson Per Alexandersson 2013-03-05T17:26:21Z 2013-03-05T17:26:21Z Terry Tao: Right, that is a possibility, but I can see why the poster believed that the current version was true; I did some quick numerics (that is how I got the example), and the counterexamples are quite rare, (with appropriate definition on &quot;rare&quot;). http://mathoverflow.net/questions/123569/fractal-dimension-of-1d-set-what-if-logn-vs-loge-is-a-polygonal-chain Comment by Per Alexandersson Per Alexandersson 2013-03-05T10:56:24Z 2013-03-05T10:56:24Z It sounds a bit like a multi-fractal, (<a href="http://en.wikipedia.org/wiki/Multifractal_system" rel="nofollow">en.wikipedia.org/wiki/Multifractal_system</a>) which has &quot;mixed&quot; fractal dimensions. If your data do not have sufficient resolution, it might be an artifact that it eventually becomes zero. Now, some DLA-systems (<a href="http://en.wikipedia.org/wiki/Diffusion-limited_aggregation" rel="nofollow">en.wikipedia.org/wiki/&hellip;</a>) have something similar happening in them, if I recall correctly. http://mathoverflow.net/questions/123061/brute-force-lattice-problems Comment by Per Alexandersson Per Alexandersson 2013-02-27T06:38:18Z 2013-02-27T06:38:18Z I'd say this is almost better to ask over at mathematica.stackexchange, my bet is that if you ask nicely, you will even get working code for your problem.