# Tagged Questions

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I would like a better understanding of the set of pairs $(f_1,f_2)$ of functions $[0,1] \times [0,1] \to [0,1]$ which satisfy the following conditions: For all $y \in [0,1]$, $f_1(x,y) \geq ... 2answers 334 views ### How to integrate an exponential function of an exponential function? Does any one know how to calculate the following integration? $$\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.$$ This post is related to my previous question here , ... 0answers 59 views ### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II This is a modification of a previous question. Question: Suppose$a(s)\in C^\infty([0,1])$and$H(s,x)\in C^\infty([0,1]\times [0,1])$with$H(s,x)>0$,$\forall s,x\in [0,1]$. Suppose, ... 0answers 266 views ### Uniqueness for a non-local differential equation Question:Fix$\epsilon>0$. Consider the differential equation, defined for functions$f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$defined by $$\frac{\partial}{\partial t} ... 2answers 225 views ### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform Question: Suppose a(x,y)\in C^\infty([0,1]\times [0,1]) and suppose$$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$Is a(x,0)=0, \forall x\in[0,1]? ... 2answers 266 views ### how to solve a singular integral equation involving the kernel 1/x Dear all, Suppose we know that f(x) is nonnegative and Hölder continuous at zero with exponents 1/2. We also know that$$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$... 0answers 839 views ### An inverse Laplace transform involving Error function Dear MOs, I need to calculate the inverse Laplace transform of the following function$$ g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$I have checked, among many ... 0answers 193 views ### Regularity class of certain diffeomorphisms of the real line. I care about the following class of homeomorphisms of \mathbb R, which I'll call \mathcal C^?. For simplicity, let us restrict attention to compactly supported homeomorphisms (a homeomorphism ... 1answer 396 views ### Properties of a matrix-valued generalization of the \Gamma function I am interested in the following function (Mellin transform of matrix exponential):$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$Where x and s are scalars, but A and B are matrices with B\succ ... 3answers 735 views ### A definite integral Hello, I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might ... 2answers 463 views ### Fonction spéciale Est-ce qu'il y a une fonction spéciale sous la forme suivante:$$\int e^{-st}e^{-(\alpha/(1+t))}t^{\beta-1}(1+t)^{-(\beta-1)-\gamma}dt$$ou$$\int ... 2answers 909 views ### Complex structure on$L^2(\mathbb R)$generalizing the Hilbert transform. The Hilbert transform on the real Hilbert space$L^2(\mathbb R)$is the singular integral operator $$\mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy.$$ It satisfies ... 1answer 371 views ### Integral kernel of form$e^{-<x,y>^2}$Let$K(x,y): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$given by$K(x,y) = e^{-< x,y>^2}$where$<\cdot,\cdot>$denote the canonical inner product. Define integral operator ... 3answers 1k views ### When I can safely assume that a function is a Laplace transform of other function? If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is$\hat{f}(s)$such that my function$f(x)$can be written as:$f(x) ...
Introduction Let $C$ be a code of block length $n$ having $A_i^C$ words of Hamming weight $i$, for $i\in [n]$, where $[n]:=\{0,\ldots,n\}$. Then, the sequence $\{ A_i^C \}_{i \in [n]}$ is called the ...
### inverse Laplace transform of $\delta_1(\cdot)$
Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify ...