Steve Huntsman
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Registered User
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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 #).
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May 14 |
awarded | ● Popular Question |
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May 8 |
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Minimal number of colours for colouring Voronoi-cells of a $d-$dimensional lattice Related: mathoverflow.net/questions/11453 |
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May 7 |
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Probability that one RV will exceed many others en.wikipedia.org/wiki/… |
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May 3 |
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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). |
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Apr 27 |
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The discrete theory of compressible fluids dynamics pra.aps.org/abstract/PRA/v46/i4/p1967_1 |
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Apr 25 |
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Langevin equation with position-dependant damping: existence of an invariant measure? Perhaps look at search results for "nonlinear Langevin equation" |
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Apr 23 |
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Constraint optimization problem for any dimensionality $n>1$. @Mark: Your matrix is circulant. The DFT diagonalizes circulant matrices. |
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Apr 17 |
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Double series solution of wave equation What is $f$?... |
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Apr 14 |
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when are two Markov chains same distributions mathoverflow.net/questions/14729 |
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Apr 12 |
awarded | ● Popular Question |
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Apr 11 |
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Efficient computation of Markov chain transition probability matrix If $Q$ 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. |
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Apr 9 |
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Iterated Ito Integral, Gaussian Volterra Process Your $I_n$ 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 |
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Apr 9 |
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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 $\mathbb{E}[I \otimes B] = I \otimes \mathbb{E}[B]$. |
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Apr 8 |
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Reference for ultrametric spaces You 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. |
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Apr 8 |
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Reference for ultrametric spaces rmp.aps.org/abstract/RMP/v58/i3/p765_1 |
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Apr 3 |
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Using Fourier Transform to speed up calculation of forces following an inverse square law tags |
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Apr 3 |
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Using Fourier Transform to speed up calculation of forces following an inverse square law NB. arxiv.org/abs/0911.4114 appears to describe a more manifestly Fourierish FMM. |
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Apr 3 |
answered | Using Fourier Transform to speed up calculation of forces following an inverse square law |
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Mar 28 |
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How to compute difference between 2 similarity matrices? en.wikipedia.org/wiki/… |
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Mar 28 |
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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. |
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Mar 27 |
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A short question about the DFT matrix I 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... |
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Mar 26 |
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A short question about the DFT matrix I should have said "$N$th roots of unity" above. |
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Mar 26 |
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A short question about the DFT matrix Let $M$ 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. |
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Mar 26 |
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A short question about the DFT matrix The identity matrix has lots of zeros. |
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Mar 24 |
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Cube roots in $C^*$-algerba Functional calculus? |
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Mar 24 |
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Self-containing structures This strikes me as a kind of opposite of Cantor diagonalization, BTW. |
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Mar 24 |
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Self-containing structures Community wiki? |
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Mar 23 |
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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. |
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Mar 19 |
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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. |
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Mar 19 |
answered | Rigorous numerics for maxima and minima (one variable) |
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Mar 16 |
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The “interplay” between additive and multiplicative structure in a field My 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 |
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Mar 15 |
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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. |
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Mar 13 |
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Rigorous numerical integration I'll add the trivial note that a change of variables for improper integrals will be helpful from the POV of implementation in silico. |
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Mar 13 |
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Rigorous numerical integration Won'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. |
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Mar 6 |
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Rigorous numerical integration One could combine IA and automatic differentiation to address derivative bounds rigorously. In fact so-called Taylor forms generalize IA and can provide this capability. |
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Mar 5 |
answered | Rigorous numerical integration |
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Jan 22 |
awarded | ● Popular Question |
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Dec 30 |
awarded | ● Nice Answer |
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Dec 22 |
revised |
The human body’s random number generator added reference w/ description |
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Dec 22 |
answered | The human body’s random number generator |
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Dec 6 |
answered | Is Fourier analysis a special case of representation theory or an analogue? |
