Questions tagged [specific-calculation]
Every question that deals with calculating the value of a specific integral, derivative, summation etc.
21 questions
3
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0
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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 ...
- 18.7k
1
vote
0
answers
41
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 ...
- 69
0
votes
0
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76
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 ...
- 71
2
votes
0
answers
206
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,...
- 29
2
votes
0
answers
103
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 ...
- 1,566
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, ...
- 21
0
votes
1
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295
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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 ...
1
vote
1
answer
194
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 $\...
- 75
1
vote
1
answer
632
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/...
- 1,466
3
votes
1
answer
354
views
Evaluation of an interesting Integral [duplicate]
Supposedly the answer is 1 but I have no idea how to evaluate this thing analytically.
$$f(n) = \frac{2}{\pi} \int_{0}^{\infty} 2\cos(x) \cdot \frac{\sin(x)}{x} \cdot \frac{\sin(x/3)}{x/3} \cdot \...
- 49
3
votes
1
answer
186
views
packing with special sets in high dimensional Euclidean space
Let $\lambda$ be Lebesgue measure on $[0,1]$. For $\mathbf{x}=(x_1,x_2,..,x_k)\in[0,1]^k$, define $$A(\mathbf{x}):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[...
- 75
4
votes
2
answers
228
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 $$A(x_1,..,x_k):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[0,1]$$ ...
- 75
0
votes
1
answer
142
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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 ...
user95470
3
votes
1
answer
942
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.
$$F(\beta)=...
- 51
0
votes
1
answer
1k
views
Integrals involving associated legendre polynomials
Do the following integrals have a closed-form solution for any integer value of $m,l,k$ and $n$?
$\int^{\pi}_{0} P^{m}_{l}\left(\cos\theta\right)P^{n}_{k}\left(\cos\theta\right)\cot\theta d\theta$
$\...
2
votes
1
answer
224
views
How to compute the following integral $I_{\alpha,\beta}$
We have the following identity (see Bateman, H. (1953). Higher Transcendental Functions [Volumes I], p. 25.)
$$(*)\quad \Gamma(\mu)\, \zeta(\mu,\nu) = \int_{0}^{1} x^{\nu-1} \,(1-x)^{-1} \Bigr(\log ...
- 650
8
votes
1
answer
607
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 ...
- 21.8k
3
votes
2
answers
263
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 ...
- 21.8k
4
votes
1
answer
347
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(...
- 1,997
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:
$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$
...
- 453
2
votes
0
answers
443
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(\...
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