# Steve Huntsman

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## Registered User

 Name Steve Huntsman Member for 3 years Seen 4 hours ago Website Location Inside the Beltway Age 34
My mathematically-oriented work generally explores discrete geometric and probabilistic themes in physics, computation, and communication. Jack of most trades, master of none. I can be reached at sh#eqnets#com (make the obvious substitutions for #).
 May14 awarded ● Popular Question May8 comment Minimal number of colours for colouring Voronoi-cells of a $d-$dimensional latticeRelated: mathoverflow.net/questions/11453 May7 comment Probability that one RV will exceed many othersen.wikipedia.org/wiki/… May3 comment Where do the product expansions of modular forms come from?The Mellin transform provides a bridge between modular forms and Dirichlet series, which in turn can be expressed as Euler products. It may help to view the pair (Mellin transform, multiplicative convolution algebra) as analogous to the pair (Fourier transform, additive convolution algebra). Apr27 comment The discrete theory of compressible fluids dynamicspra.aps.org/abstract/PRA/v46/i4/p1967_1 Apr25 comment Langevin equation with position-dependant damping: existence of an invariant measure?Perhaps look at search results for "nonlinear Langevin equation" Apr23 comment Constraint optimization problem for any dimensionality $n>1$.@Mark: Your matrix is circulant. The DFT diagonalizes circulant matrices. Apr17 comment Double series solution of wave equationWhat is $f$?... Apr14 comment when are two Markov chains same distributionsmathoverflow.net/questions/14729 Apr12 awarded ● Popular Question Apr11 comment Efficient computation of Markov chain transition probability matrixIf $Q$ is diagonalizable, say $Q = V\Lambda V^{-1}$, then using $e^{tQ} = V e^{t\Lambda} V^{-1}$ seems likely to be faster than expm. You can also approximate the exponential using Krylov subspace (Lanczos or Arnoldi) techniques. Apr9 comment Iterated Ito Integral, Gaussian Volterra ProcessYour $I_n$ is basically a Hermite function, by Ito's classical results on the "multiple Wiener integral". See, e.g. section 5.4.1 of books.google.com/books?id=V2BS_Dmp0XoC Apr9 comment Expected value with a kronecker product and Gaussian distributional assumption$I \otimes B$ is just a block diagonal matrix with diagonal blocks equal to $B$. So $\mathbb{E}[I \otimes B] = I \otimes \mathbb{E}[B]$. Apr8 comment Reference for ultrametric spacesYou might also be interested in the use of ultrametric spaces in the functorial approach to hierarchical clustering algorithms: see, e.g., Carlsson and Memoli's papers. Apr8 comment Reference for ultrametric spacesrmp.aps.org/abstract/RMP/v58/i3/p765_1 Apr3 revised Using Fourier Transform to speed up calculation of forces following an inverse square lawtags Apr3 comment Using Fourier Transform to speed up calculation of forces following an inverse square lawNB. arxiv.org/abs/0911.4114 appears to describe a more manifestly Fourierish FMM. Apr3 answered Using Fourier Transform to speed up calculation of forces following an inverse square law Mar28 comment How to compute difference between 2 similarity matrices?en.wikipedia.org/wiki/… Mar28 comment Adding a damping term to a dynamical system or Markov process: what happens to invariant measures?I wouldn't be surprised if things could break. Imagine a random walk on a graph induced by nearest integer neighbors in a dumbbell-shaped region of the plane. Suppose that the walk is uniform except on the bar, where it's heavily biased in one direction. Convergence to the invariant measure will be very slow because one of the ends of the dumbbell will take a long time to reach. With damping, we might be able to break ergodicity. Mar27 comment A short question about the DFT matrixI didn't know that I noted the character tables, but thanks for spelling it out! Now it's clear to me what direction I was headed... Mar26 comment A short question about the DFT matrixI should have said "$N$th roots of unity" above. Mar26 comment A short question about the DFT matrixLet $M$ a matrix of the form you describe, with $M_{jk} = C e^{i\omega_{jk}}$ and $C > 0$. Unitarity implies that $C^2 \sum_\ell e^{i(\omega_{\ell k} - \omega_{\ell j})} = \delta_{j k} = C^2 \sum_\ell e^{i(\omega_{j \ell} - \omega_{k \ell})}$. Taking $j = k$ gives that $C = N^{-1/2}$, where $N = \dim M$. For $j \ne k$, $\sum_\ell e^{i(\omega_{j \ell} - \omega_{k \ell})} = 0$. The only way this can happen is if the angles $\omega_{j \ell} - \omega_{k \ell}$ balance out''. Roots of unity are a particularly nice way for this to happen, but as Mark Meckes points out, not the only one. Mar26 comment A short question about the DFT matrixThe identity matrix has lots of zeros. Mar24 comment Cube roots in $C^*$-algerbaFunctional calculus? Mar24 comment Self-containing structuresThis strikes me as a kind of opposite of Cantor diagonalization, BTW. Mar24 comment Self-containing structuresCommunity wiki? Mar23 comment What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?I was a student of McKean, perhaps that explains my take. Mar19 comment Rigorous numerics for maxima and minima (one variable)I can't say off the top of my head. Certainly there are free/public C++ libraries for interval methods. It also seems likely that INTLAB can be made to work with Octave. Mar19 answered Rigorous numerics for maxima and minima (one variable) Mar16 comment The “interplay” between additive and multiplicative structure in a fieldMy personal feeling is that the polynomial description of fields is very unintuitive: I have learned and forgotten many details more than once over the years. To my mind, a much more intuitive (but not computationally better) approach is to treat elements of $\mathbb{F}_{p^r}$ as matrices over $\mathbb{F}_p$: the additive and multiplicative structures coexist nicely then. This approach reduces much of the theory of finite fields concerned with the interplay of which you speak to representation theory. It is detailed in a cute article by Wardlaw: jstor.org/discover/10.2307/2690850 Mar15 comment Time varying random walk on number line Time-inhomogeneous Markov processes are dicussed in Stroock's Springer GTM book with simulated annealing as the motivating example. Your phraseology is in fact the usage. Mar13 comment Rigorous numerical integrationI'll add the trivial note that a change of variables for improper integrals will be helpful from the POV of implementation in silico. Mar13 comment Rigorous numerical integrationWon't the following work for all of these? Use automatic differentiation (en.wikipedia.org/wiki/Automatic_differentiation, covered at a basic level in Tucker's book) to get expressions for any derivatives appearing in integrands, and then apply integration with IA (or Taylor forms) to the results. Mar6 comment Rigorous numerical integrationOne could combine IA and automatic differentiation to address derivative bounds rigorously. In fact so-called Taylor forms generalize IA and can provide this capability. Mar5 answered Rigorous numerical integration Jan22 awarded ● Popular Question Dec30 awarded ● Nice Answer Dec22 revised The human body’s random number generatoradded reference w/ description Dec22 answered The human body’s random number generator Dec6 answered Is Fourier analysis a special case of representation theory or an analogue?