Igor Khavkine
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Registered User
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May 13 |
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Analytical continuation of electrostatic potentials It should be remarked that your integral kernel is not actually analytic in the neighborhood of the integration contour (the singularity at r=0 and the branch cuts spoil that). So, the expectation that $V(r)$ be analytic for arbitrary (under your restrictions) $\rho_C(r)$ is too optimistic. The reason $V_{\mathrm{gauss}}$ is actually analytic is because the corresponding $\rho_C$ is. |
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May 13 |
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Triangle area on surfaces of constant curvature For both cases, there are visual proofs, which can be considered elementary: mathoverflow.net/questions/8846/… and mathoverflow.net/questions/8846/… |
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May 12 |
awarded | ● Enlightened |
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May 11 |
awarded | ● Nice Answer |
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May 10 |
accepted | Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism |
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May 9 |
answered | higher order Noether identities |
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May 8 |
accepted | “Cohomology at the infinity”: what does one call it |
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May 7 |
answered | “Cohomology at the infinity”: what does one call it |
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May 5 |
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null controllability of linear wave equation @researcher: Deane's point stands independent of the sign of $k(x)$. So, unique initial data $(z,z_t)$ at $t=0$ is guaranteed for any $k(x)$ and $h(t)$. What is suspicious about your claim in reply to Deane's comment is the existence of such $h(t)$ for any given initial data set at $t=0$, provided that your notation means that $h(t)$ depends only on $t$. For example, what if $(z,z_t)$ have compact support at $t=0$? Then $h(t)$ will produce ripples arbitrarily far away that could not be canceled by the propagation of any part of the initial data. |
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May 4 |
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null controllability of linear wave equation Are you supposing by any chance that $h(t)$ vanishes in a neighborhood of $t=0$ or $T$? |
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May 4 |
accepted | naked singularity and null coordinates |
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May 3 |
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naked singularity and null coordinates Geodesics are defined by... the geodesic equation. Sounds like you need practice working with and solving the geodesic equation in coordinates in some simple examples. Wald is probably not the best book for that. On the other hand, Carroll, Landau & Lifshitz (vol 2), Misner & Thorn & Wheeler, Schutz, d'Inverno (roughly in decreasing order of sophistication) offer more worked examples, which should help you build up the experience you need. |
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May 1 |
revised |
Norms on tensor products TeXified notation. |
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May 1 |
answered | naked singularity and null coordinates |
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Apr 30 |
answered | Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism |
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Apr 29 |
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Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism Unfortunately, it's a bit unclear to me what aspects of the "Hamiltonian viewpoint" you think are missing. You might be interested in the answer I gave here, where a lot of these details are filled in, though not explicitly including the Hamilton-Jacobi equation: mathoverflow.net/questions/81800/81857#81857 |
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Apr 29 |
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Exact sampling from 2D Ising model where coupling is constant? The title of the question is confusing. Is your question about sampling algorithms or about an Onsager-like exact evaluation of the partition function? There's some detail about the latter on the Wikipedia page en.wikipedia.org/wiki/… |
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Apr 29 |
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Small Inhomogeneity of Differential Equation If you know $f_1(x)$ and $f_2(x)$, then you can easily construct a Green function for this linear ODE, with your favorite boundary condition. Solving an inhomogeneous system then comes down to integrating the inhomogeneity against the Green function, as usual. |
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Apr 26 |
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An integral with Gamma functions @Anirbit, probably the best thing for you to do is get a copy of Gelfand and Shilov and read carefully through their treatment of these distributions. |
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Apr 24 |
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Green’s fuction Yes and no, though mostly yes. Your question is too vague at the moment to get a useful answer. You may want to read the FAQ about how to ask and flesh out the question. |
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Apr 23 |
awarded | ● Enlightened |
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Apr 23 |
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Asymptotic decay for the wave equation Of course. I intuitively considered $\epsilon$ to be "sufficiently small". |
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Apr 23 |
awarded | ● Nice Answer |
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Apr 23 |
answered | Asymptotic decay for the wave equation |
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Apr 22 |
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An integral with Gamma functions @Anirbit: If that is what you are worried about, you should perhaps try an example, say, compute $k^3$ and $(k^2)^{1.5}$ for $k=(1,2,3)$. Think carefully about what the notation means. |
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Apr 21 |
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An integral with Gamma functions @Anirbit: $(k+q)^2=[(-k)-q]^2$. The $\nu$s could be any complex number, excluding the poles of the final answer. |
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Apr 15 |
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Does the derivative of log have a Dirac delta term? @Tom, in the text surrounding that formula, Dirac states precisely that (though less explicitly). |
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Apr 15 |
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Nonintegrable inverse powers as distributions @Mesoscopic_P, be careful with that "clearly". Your question as stated doesn't exclude derivatives of delta functions, but requiring homogeneity does. |
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Apr 15 |
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Nonintegrable inverse powers as distributions @Mesoscopic_P, you are almost there. If $T_f$ and $T'_f$ are two candidates, what is the support of $T_f-T'_f$? |
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Apr 14 |
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How unique is a conformal compactification? @Edward, for that you probably need to check out some early papers by Roger Penrose. |
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Apr 14 |
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How to solve this kinds of equations Counting in the most straight forward way, you have one equation and one unknown (which is $t$). Generically, unless some coincidence occurs in your choice of $A$, $B$ and $H$, for $t\in [-T,T]$ you will have a finite, discrete set of solutions (which could be empty). However, since $e^{iHt}Ae^{-iHt}$ potentially describes a complicated spiral in the space of $d\times d$ matrices, as $T\to\infty$, the solution set could become very complicated, including the extremes of remaining empty or becoming dense somewhere. Little can be said without more specific information about $A$, $B$ and $H$. |
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Apr 9 |
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Cauchy Data Problems. Homework? Please read the FAQ. |
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Apr 5 |
awarded | ● Citizen Patrol |
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Apr 2 |
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Would a closed universe with special relativity violate causality? Does the universe have to be simply connected? A very old question. The answer, as Aaron suggests, is standard. See, for instance, dx.doi.org/10.1119/1.16373. |
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Apr 1 |
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Does this matrix shape have a name? In numerical analysis, the addition of a $v v^T$ term is often called a rank-1 update. Thus, $A_n$ is a rank-1 update of a multiple of the identity. |
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Mar 30 |
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Uniqueness of compactification of an end of a manifold Having occasionally thought about this question some more, I think this answer is quite close to what I was looking for. In particular, if the hypotheses are strengthened such that the two possible boundaries are $s$-cobordant (rather than just $h$-cobordant), then the two boundaries are not just diffeomorphic, but one can be transformed into the other by a sequence of blow-up and collapse simple homotopy moves. en.wikipedia.org/wiki/… |
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Mar 27 |
revised |
An integral with Gamma functions Fixed typo in final exponent. |
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Mar 26 |
answered | Spectral theorem for self-adjoint differential operator on Hilbert space |
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Mar 26 |
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Replacing large-dimensional ODE systems with one PDE The answer obviously depends on the details of a specific case. Would you care to elaborate on yours? Note that the reverse operation is a routine aspect of numerical analysis of PDEs. However the resulting ODEs have special "sparse" structure. |
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Mar 25 |
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An integral with Gamma functions Yes, it's legitimate; no subtleties beyond consistently carrying out multiplication of series to a given order. Moreover, the $\Gamma$ function never vanishes on the real line, so the denominator does not contribute any poles. |
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Mar 22 |
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An integral with Gamma functions @curious,Anirbit: It's very simple to work through the details yourself, which is what I recommend. The steps are straightforward as long as you know some basic properties of Fourier transforms and convolutions. These you can look up on Wikipedia or in an elementary textbook. Same advice for figuring out the poles. The poles of a product are easy to get once you know the poles of the factors: just take products of the Laurent expansions. Even if you don't have access to Gelfand & Shilov, you can take my Fourier transform formulas for granted and check the rest of the arithmetic. |
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Mar 22 |
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An integral with Gamma functions Hmm, any comments to go along with the downvote? |
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Mar 22 |
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An integral with Gamma functions Alas, not my notation. |
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Mar 22 |
answered | An integral with Gamma functions |
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Mar 21 |
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Functional Analysis and Differential Manifold incompatibility The kind of incompatibility that you seem to have in mind is an urban myth. It is often mentioned, but rarely explained accurately outside specialized literature. The question of combining quantum mechanics with general relativity has been discussed here before. Best to approach it only after you know what a quantum field theory is. See for instance my answer here: mathoverflow.net/questions/13205/13261#13261 |
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Mar 20 |
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Open problems in PDEs, dynamical systems, mathematical physics added 1 characters in body |
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Mar 20 |
answered | Open problems in PDEs, dynamical systems, mathematical physics |
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Mar 18 |
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abstract algebra for component wise operations on “vectors” or what it might be called An extremely similar question was asked here before: mathoverflow.net/questions/9166. My preferred to think of $n$-vectors in this context as real-valued functions on a discrete space with $n$ points. |
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Mar 10 |
accepted | finding effective 2-form corresponding to an equation |
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Mar 10 |
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finding effective 2-form corresponding to an equation On a rereading, I noticed that this is exactly what Dean Yang already said in his comment. |
