Carlo Beenakker
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Registered User
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physicist at Leiden University
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1d |
comment |
How can I randomly draw an ensemble of unit vectors that sum to zero? @Allen, wouldn't the rescaling to unit length spoil the sum to zero? |
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2d |
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Derivation of Bessel functions yes, $u\ll 1$ means that the quadratic (convective) terms are neglected; and indeed, the Bessel equation, which has Bessel functions as solutions, is a linear equation, so it cannot include the convective term; see my full answer below. |
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2d |
revised |
The relations between the Perelman’s entropy functional and notions of entropy from statistical mechanics Perelman quote |
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2d |
revised |
The relations between the Perelman’s entropy functional and notions of entropy from statistical mechanics rewording |
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2d |
answered | The relations between the Perelman’s entropy functional and notions of entropy from statistical mechanics |
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2d |
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The relations between the Perelman’s entropy functional and notions of entropy from statistical mechanics it seems to be little more than a formal correspondence, judging from page 11 of arxiv.org/abs/math.DG/0211159 |
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2d |
revised |
Derivation of Bessel functions p_0 |
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2d |
answered | Derivation of Bessel functions |
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2d |
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Derivation of Bessel functions as far as I can see, the solution you have written down for the density $\rho$ neglects the nonlinear term $u\cdot\nabla u$, so any effect of convection is neglected and it only applies in the limit of small velocities $u$; in that limit, indeed, the differential equation for $\rho$ is just the Poisson equation, which in cylindrical coordinates has the Bessel function solution; all the complications of hydrodynamics (shock waves, turbulence, etc.) come from the nonlinear convective term $u\cdot\nabla u$ which is neglected; once you add that term, there is no closed form solution. |
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May 17 |
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how to proof this Stirling related equation sorry, I misunderstood the left and right hand side both as fractions, my mistake |
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May 17 |
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Series expansion with remaining $log n$ the series expansion is easiest if you first take the logarithm, and then you find directly a powerseries in $n^{-1}\ln n$, $$-\frac{k(k+1)}{2n}\ln n+2\ln(2kn)\left[\sum_{p=0}^{\infty}\frac{1}{p}k^p(1+k)^p(2kn)^{-p}(\ln n)^p \right]$$ |
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May 17 |
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How to maximize the determinant of a matrix of the form VDV^H thank you, Federico, I stand corrected; the correct formula is $${\rm Det}A=\sum_{S}|{\rm Det}V_{S}|^2\prod_{k\in S}D_{kk}$$ where $S$ is a subset of $M$ indices out of $1,2,...2M$ and $V_S$ is an $M\times M$ matrix constructed from $V$ by deleting the $M$ columns that are not in $S$. it would seem that to maximize this is in general not trivial. |
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May 17 |
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how to proof this Stirling related equation the left hand side is infinite, so the inequality is obviously false. |
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May 17 |
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How to maximize the determinant of a matrix of the form VDV^H since $Det A = (Det V^H V) \prod_n D_n$, this is trivial |
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May 16 |
revised |
computing Bernoulli numbers zeta function |
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May 16 |
answered | computing Bernoulli numbers |
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May 15 |
answered | help on antiderivative of a vector function |
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May 15 |
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help on antiderivative of a vector function hmm, votes to close .... is it trivial? |
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May 15 |
accepted | Principal value of integral |
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May 15 |
revised |
Eigenvalues of random Hamiltonian matrices added reference |
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May 14 |
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Principal value of integral I added several more intermediate steps; is it clear now? |
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May 14 |
revised |
Principal value of integral math corrected |
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May 14 |
revised |
Principal value of integral explanation |
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May 14 |
revised |
Principal value of integral derivation |
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May 14 |
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Principal value of integral the integral can be done as a contour integral in the complex plane, using the definition of principal value which I gave in a comment to your question above (so as the average of a contour closed in the upper and lower half of the complex plane, once picking up the pole at $x=i\epsilon$ and once at $x=-i\epsilon$; I will try to add this calculation to my answer later today. |
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May 14 |
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Principal value of integral @Gerald: it is equivalent to the textbook definition ${\cal P}\int_a^b dx f(x)=\frac{1}{2} \lim_{\epsilon\rightarrow 0}\left(\int_{a-i\epsilon}^{b-i\epsilon} dx f(x)+ \int_{a+i\epsilon}^{b+i\epsilon} dx f(x)\right)$ |
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May 14 |
revised |
Principal value of integral typo |
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May 14 |
answered | Principal value of integral |
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May 14 |
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Principal value of integral I would presume each pole at $x_n=n\pi$ is excluded in an interval $(x_n-\epsilon,x_n+\epsilon)$, and then the limit $\epsilon\rightarrow 0$ is taken. |
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May 13 |
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Eigenvalues of random Hamiltonian matrices no, we checked numerically that the eigenvalue density vanishes linearly with the distance from either real or imaginary axis; moreover, there is a linear repulsion of pairs of eigenvalues on the real axis, as well as on the imaginary axis; in these respects the Hamiltonian ensemble behaves just like the Ginibre ensemble. |
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May 13 |
accepted | Eigenvalues of random Hamiltonian matrices |
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May 13 |
comment |
Analytical continuation of electrostatic potentials since your charge density solves the Poisson equation, your potential has the Fourier representation $V(\vec{r})\propto\int d\vec{k} \exp(i\vec{k}\cdot\vec{r})k^{-2}\rho_{C}(\vec{k})$. So you see that if the Fourier transformed charge density $\rho_C(\vec{k})$ decays only as a power law in $k$, then the potential must diverge exponentially for any imaginary $\vec{r}$. This seems unavoidable. |
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May 13 |
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Eigenvalues of random Hamiltonian matrices @Federico: you're completely correct, I did not properly define the limit in which the density becomes uniform; I've corrected that now; the uniformity applies to the rescaled density; the nonuniformities in the rescaled density are of order $n^{1/2}$, so the factor $n^{-1}$ removes them in the large−$n$ limit. |
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May 13 |
revised |
Eigenvalues of random Hamiltonian matrices rescaling |
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May 13 |
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Eigenvalues of random Hamiltonian matrices thank you for asking: this is certainly not an artifact; in the Ginibre ensemble the eigenvalue density is known to vanish linearly on approaching the real axis; basically the depletion zone signals the accumulation of eigenvalues on the real axis, as if the eigenvalues are "attracted" towards the real axis, where they condense. It seems that the repulsion from the imaginary axis is stronger than linear, but this is something we have not yet checked. |
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May 13 |
revised |
Eigenvalues of random Hamiltonian matrices typo |
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May 13 |
answered | Eigenvalues of random Hamiltonian matrices |
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May 11 |
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A problem about Joint sine and cosine fourier transform hmm, there is no square root in the denominator? if there was a square root, then we could define $\alpha=\arccos(\lambda/\sqrt{\lambda^2+h^2})$ and your kernel would simply be $K(x,\lambda)=\sqrt{2/\pi}\cos(\lambda x-\alpha)$ |
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May 11 |
revised |
Lower bounds on derivative around zero set of a positive smooth function. small edit |
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May 11 |
revised |
Lower bounds on derivative around zero set of a positive smooth function. simpler |
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May 11 |
revised |
Lower bounds on derivative around zero set of a positive smooth function. one more step |
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May 11 |
answered | Lower bounds on derivative around zero set of a positive smooth function. |
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May 9 |
comment |
Prove that the sum of a certain infinite series is 1 can you give some background how this identity to unity arises? does that suggest a way to a proof? |
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May 8 |
accepted | Probability distribution for two-state system that depends on residence time |
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May 8 |
revised |
Probability distribution for two-state system that depends on residence time remove header |
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May 8 |
comment |
Probability distribution for two-state system that depends on residence time @ionlet: I expanded the answer, hopefully answering all your queries; is it clear now? |
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May 8 |
revised |
Probability distribution for two-state system that depends on residence time expanded |
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May 8 |
revised |
Probability distribution for two-state system that depends on residence time corrected |
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May 7 |
answered | Probability distribution for two-state system that depends on residence time |
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May 5 |
accepted | Non-asymptotic results for bulk of random Wishart matrix |
