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Questions tagged [specific-calculation]

Every question that deals with calculating the value of a specific integral, derivative, summation etc.

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Fast checking that a system of polynomial equations is satisfiable over $\mathbb{F}_2$

I have a (fairly large) system of polynomial equations, of the form $$ c_1d_1=0,\ c_1d_2+c_2d_1=0,\ldots $$ (In case it is relevant, all the polynomials are homogeneous of degree 2, except for exactly ...
Pace Nielsen's user avatar
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1 vote
0 answers
40 views

Computational comparison in solving two optimization problems

Can I get some inputs on whether the following two optimization problems are computationally the same, or one of the problems is easier to solve computationally than the other, such as, finding their ...
muddy's user avatar
  • 69
0 votes
0 answers
74 views

Closed form of integral $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$

How could one calculate the closed form solution of this integral: $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$ Here the integral is ...
NuKuYul's user avatar
  • 71
2 votes
0 answers
199 views

How to calculate Gauss Manin connection?

If $f:X\rightarrow B$ is a holomorphic family of compact complex manifold. Fix a $k$, then all the $H^k(X_t,\mathbb{C})$ is the same with respect to $t$. Say take a $d$-closed form $\alpha\in H^k(X_t,...
user426106's user avatar
2 votes
0 answers
102 views

Projectors and idempotents

An $n\times n$ matrix $P$ (over a commutative ring with identity $R$) is called a projector if $P^2=P$. Let $X$ denote the $\mathbb{Z}$-affine subscheme of $\mathbb{A}^{n^2}$ that is defined by the ...
Kapil's user avatar
  • 1,546
2 votes
1 answer
73 views

Solving equation for higher degree of composition

Given this function $f(x) = x - 1/x$, the equation $f(f(x)) = x$ has two solutions: $\frac{1}{\sqrt{2}}$, $\frac{-1}{\sqrt{2}}$. But how about solving this equation for a higher degree of composition, ...
meh98's user avatar
  • 21
0 votes
1 answer
243 views

Showing equality of Eberlein polynomials

I have thought about the following question a long time and still got no progress. Currently I am writing my master thesis about association schemes in combinatorics and need an equality which seems ...
McRatchet's user avatar
1 vote
1 answer
193 views

Is there a 1/poly(n) or 1/polylogn upper-bound for this tail bound?

Is there a good tail bound for $\operatorname{P}\!\Bigg[\bigg\vert\dfrac{\sum_{j=1}^n(\sum_{i=1}^n a_{i,j})^2}{n^2} -1\bigg\vert > \epsilon\Bigg]\,,$ where all $a_{i,j}$'s are iid, with $\...
Pascalprimer's user avatar
1 vote
1 answer
596 views

Time ordered integral involving beta function:

Any help on unpacking integrals of the following type, would be helpful: $$ \int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt $$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$. Edit/...
cheyne's user avatar
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3 votes
1 answer
353 views

Evaluation of an interesting Integral [duplicate]

Supposedly the answer is 1 but I have no idea how to evaluate this thing analytically. $$f(n) = \frac{2}{\pi} \int_{0}^{\infty} 2\cos(x) \cdot \frac{\sin(x)}{x} \cdot \frac{\sin(x/3)}{x/3} \cdot \...
Igor Lehne's user avatar
3 votes
1 answer
182 views

packing with special sets in high dimensional Euclidean space

Let $\lambda$ be Lebesgue measure on $[0,1]$. For $\mathbf{x}=(x_1,x_2,..,x_k)\in[0,1]^k$, define $$A(\mathbf{x}):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[...
Cuize Han's user avatar
4 votes
2 answers
225 views

lower bound volume of a set

Let $\lambda$ be Lebesgue measure on [0,1]. For any $x_{1},\dots,x_{k}$ in $[0,1]$, define $$A(x_1,..,x_k):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[0,1]$$ ...
Cuize Han's user avatar
0 votes
1 answer
141 views

Numerical Calculations

i have this numerical calculation problem : $$\prod \limits_{i=121443371}^{455052511} 1+\frac{1}{p(i)} \leq 1.06406506887043952285362856325019948 $$ such that $p(i)$ is the $i$-th prime number i ...
Ahmad's user avatar
  • 595
3 votes
1 answer
886 views

Cumulative integral of the Marchenko-Pastur density for Wishart eigenvalues

I can't find any work on the cumulative density function of the well-known Marchenko-Pastur density for the eigenvalues of a standard Wishart Matrix as its dimension goes to $\infty$, i.e. $$F(\beta)=...
blacklist's user avatar
0 votes
1 answer
1k views

Integrals involving associated legendre polynomials

Do the following integrals have a closed-form solution for any integer value of $m,l,k$ and $n$? $\int^{\pi}_{0} P^{m}_{l}\left(\cos\theta\right)P^{n}_{k}\left(\cos\theta\right)\cot\theta d\theta$ $\...
Gilberto Zapata's user avatar
2 votes
1 answer
222 views

How to compute the following integral $I_{\alpha,\beta}$

We have the following identity (see Bateman, H. (1953). Higher Transcendental Functions [Volumes I], p. 25.) $$(*)\quad \Gamma(\mu)\, \zeta(\mu,\nu) = \int_{0}^{1} x^{\nu-1} \,(1-x)^{-1} \Bigr(\log ...
Z. Alfata's user avatar
  • 598
8 votes
1 answer
576 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
asv's user avatar
  • 21.2k
3 votes
2 answers
256 views

Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain: $$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$ where $m>0$ is fixed. Question. To ...
asv's user avatar
  • 21.2k
4 votes
1 answer
340 views

Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} \int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(...
user173856's user avatar
  • 1,987
23 votes
2 answers
1k views

Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues: $$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ ...
ocg's user avatar
  • 453
2 votes
0 answers
441 views

What is the Fourier transform of this function?

Consider the function $$ f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du. $$ It is known that $f(x_1,x_2)\in L^2(\...
Uchiha's user avatar
  • 87