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Exponential decay of kernel

Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by \begin{equation} (Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta) \end{equation} where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
Marcel's user avatar
  • 11
-4 votes
2 answers
228 views

An elementary-looking integral inequality

This might seem a bit easy but I still like to ask it for pedagogical reasons. QUESTION. Is this inequality true for non-negative integers $n$? $$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
T. Amdeberhan's user avatar
-4 votes
1 answer
200 views

How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?

Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily ...
Safwane's user avatar
  • 1,197
-4 votes
1 answer
387 views

Eigenvalues of real symmetric matrix [closed]

Suppose $A$ is a $n \times n$ real symmetric matrix with entries $a_{ij}\geq 1 $ and $a_{ii} = 0 $. Is it possible to have sum of the absolute eigenvalues of $A < 2 (n - 1).$
L S B. user255259's user avatar
-4 votes
1 answer
302 views

A Question in Fourier Analysis proposing a conjecture

Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\...
Rajesh D's user avatar
  • 698
-5 votes
1 answer
753 views

Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]

The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
Anixx's user avatar
  • 10.1k
-5 votes
1 answer
86 views

Why is the second order correction to energy zero for a fully degenerate eigensystem? [closed]

Consider the system given by, $$ H|n\rangle = E|n\rangle$$ where: $H$ is the hamiltonian. $|n\rangle$ is the eigenstate. $E$ is the energy of the eigenstate. Using degenerate perturbation theory and ...
user544899's user avatar
-5 votes
1 answer
270 views

Calculus based on pdf [closed]

Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
Marius S.L.'s user avatar
-5 votes
1 answer
184 views

Two inequalities in $\mathbb{R}$ [closed]

How to prove that for real numbers $a$ and $b$, the following inequalities hold? $(a|a|^{p-2}-b|b|^{p-2})(a-b)\geq 2^{2-p}|a-b|^p$,if $p\geq 2$ $(a|a|^{p-2}-b|b|^{p-2})(a-b)\geq (p-1)\frac{|a-b|^2}{(|...
sidi mohamd deval's user avatar
-5 votes
1 answer
184 views

a question of definite integral [closed]

1.$$\int_{0}^{1} \frac{1}{1+e^{-(x+\ln(u/(1-u)))/\tau}}\, du$$ 2.$$\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{+\infty}\frac{e^{-u^{2}/2}}{1+e^{-(x-u)/\tau}}\,du$$ please help me. I tried to use MATLAB but ...
Haoyu Wang's user avatar
-6 votes
2 answers
2k views

Is there a transformation or a proof for these integrals?

Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality. Question. Is this true? If so, is there an underlying transformation or just a ...
T. Amdeberhan's user avatar
-6 votes
1 answer
614 views

Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
Mikhail Katz's user avatar
  • 16.6k
-6 votes
1 answer
175 views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ with fixed size finite fibers [closed]

During a business meeting, I was trying to find a continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $|f^{-1}(\{y\})| = 2$ for all $y\in \mathbb{R}$, and after some experimentation I found $$f:\...
Dominic van der Zypen's user avatar
-6 votes
1 answer
141 views

Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ [closed]

Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$ $$ f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right), $$ ...
James's user avatar
  • 105
-8 votes
2 answers
1k views

why do we need algorithms, and why is non-convex optimization difficult? [closed]

A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global ...
Casella's user avatar
-9 votes
1 answer
338 views

Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?

Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ hold true? $\|A\|_{2}$ denotes ...
Rohit Shukla's user avatar

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