# Tagged Questions

Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics.

59 views

### uniform one-sided van der Corput inequality

Is the following true (and if yes, where the best proof is written?)? For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$? Hm,...
12 views

### Conditions for convergence to non-isolated fixed points

Consider a dynamical system of the form $$\dot{x}=f(x), \quad x\in X,$$ and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ ...
95 views

72 views

76 views

I also put this question on MSE here Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable). Let $\... 1answer 210 views ### The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed] In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ... 1answer 68 views ### Optimal constant for a Sobolev-type inequality Let$\overset{\circ}{H^s}(\mathbb T)$, where$s\ge 0$, be the space zero average of$2\pi$-periodic functions$u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$such that $$\lvert u\rvert_s = \... 0answers 122 views ### Parametric normalized Fermat curves (Fermat functions) The normalized Fermat curve is X^n+Y^n=1. We have of course infinite possibilities for parametrisation, but the periodicity is a special characteristic here. E.g. \cos^2x+\sin^2x=1 has ... 1answer 433 views ### Why sum of three squares of real polynomials is a sum of two squares? If f(x),g(x) are real polynomials, then f^2+g^2+1 is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are ... 1answer 56 views ### characterization of normality by selection theorem The Urysohn's extension theorem states that a space X is normal iff every continuous function f:A \rightarrow \mathbb{R}, with A a closed subset of X, can be extended to a continuous function ... 1answer 126 views ### The “Peano phenomenon” for differential equations Consider the following statement: If f:\mathbb{R} \rightarrow \mathbb{R} is a continuous function, for the autonomous equation$$x' = f (x)$$the "Peano phenomenon" can arise only at those values ... 3answers 283 views ### Nonlinear ODE: y'=(1+axy)/(1+bxy) Consider the first order nonlinear ODE problem:$$ y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0 $$where a, b>0 are some constants. I would like to know if these kind of equations were ... 2answers 358 views ### An integral identity evaluating the gamma function While reading a number theory paper I encountered the identity$$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$... 0answers 86 views ### Optimal Kakeya Maximal Bound for Bushes Let \{T_{\alpha}\} be a collection of 1\times\cdots\times 1\times N tubes, where N\gg 1, with maximal 1/N-separated directions, which all are centered at the origin (i.e. they form a bush). In ... 1answer 76 views ### Behavior of a Solution of a Nonlinear ODE In my work, I encountered the following equation:$$ (a'(x)+1)^2+k^2(x) a^2(x)=1,\;\;k(x)=2 {\mbox{sech}}(x). $$I would like to know as much as possible about the solution. More particularly, I would ... 2answers 312 views ### Needing proof of convergence for a sequence Let \left\{u_i\right\}_{i=1}^\infty be a sequence of real vectors (i.e. u_i\in R^n, i=1,2,... ) and m an integer large enough such that \sum_{i=1}^m u_i u_i^T is a positive definite matrix. ... 1answer 153 views ### Domain of Laplacian Let L be an operator on C^2(\mathbb R), defined by$$L \phi (x) = \int_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \text{ for all } x\in \mathbb R$$for a measure \nu(dy) = |y|^{-2} ... 1answer 99 views ### Is there a way to solve this integral equation? I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem. For \xi = (\alpha\theta)^{1/\alpha} and for ... 0answers 24 views ### diffusion and potentials in several dimensions In a Lagrangian field theory there are conserved quantities, like total energy. This allows geometric arguments to be made about the behaviour of a system with known potential. (E.g. on asymptotic ... 1answer 93 views ### Smooth dependence on the initial condition of the integral of an ODE I am considering an ODE \dot{x}=f(x), with x\in\mathbb{R}^d and d<\infty. f is a C^k function and I denote by \Phi_t x be the value of the solution at time t. I assume that my ODE ... 2answers 382 views ### Generalizations of the Euler-Maclaurin Summation Formula I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely Specifically, I would like to ... 1answer 179 views ### Is there a matrix that converts the gradient of every possible function to gradient of other function? I have already asked this question on math.stackexchange.com http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe Now I ... 1answer 214 views ### Is the exponent 2 sharp in the Balog-Szemerédi-Gowers Theorem? The Balog-Szemerédi-Gowers theorem can be stated in the following form: let A,B be subsets of \mathbb{Z}/n\mathbb{Z} (say) with equal cardinality, such that$$ \|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 \|... 1answer 153 views ### Ramanujan-type sum Could you show that $$\sum _{k=0}^{\infty } \frac{k}{e^{\frac{\pi k}{2}}+1}=\frac{7 \pi ^2+6 \left(\psi _{e^{2 \pi }}^{(1)}(1)+\psi _{e^{2 \pi }}^{(1)}\left(1-\frac{i}{2}\right)-\psi _{e^{2 \pi }}^{(... 0answers 104 views ### Ramanujan sum type I try to show$$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }... 0answers 48 views ### Dirichlet series decomposition of arbitrary function Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694 Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ... 0answers 59 views ### History of Cauchy-Euler Equations As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of -$\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\...
I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...