# All Questions

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### Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx$

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
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### Decomposing polynomials with integer coefficients

Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
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### Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern Analysis") paper "The Final Problem: An Account of the Mock Theta Functions" the following formula of ...
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### Question about boundedness of a sequence in $W^{3,q}$ for any $1\leq q < \frac{N}{N-1}$

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can ...
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### Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence: Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ ...
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### Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
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### Jacobi field geodesic and calculus of variations.

How can we show that the second order variation to a geodesic is given by the Jacobi differential equation? In essence, \frac{D^2}{dt^2}J(t)+R(J(t),\dot \gamma (t))\dot \gamma ...
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### An interesting wire-tying problem to match wire ends in as few trips as possible

You have $N$ wires that all extend from one location to a second distant location. The wire ends at both locations are unlabeled, and the goal is to label them all (on both ends) with distinct labels ...
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### New Idea to prove $1+2x+3x^2+\cdots=(1-x)^{-2}$

If $|x|<1$ prove that $\\1+2x+3x^2+4x^3+5x^4+...=\frac{1}{(1-x)^2}$ 1st proof:suppose ...
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### Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
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### show this sequence inequality $x_{2^n}$

Define the sequence $\{x_{n}\}$ recursively by $x_{1}=1$ and $$\begin{cases} x_{2k+1}=x_{2k}\\ x_{2k}=x_{2k-1}+x_{k} \end{cases}$$ Prove that $$x_{2^n}>2^{\frac{n^2}{4}}$$ I have ...
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### Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
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### What's wrong in this dual derivation?

I have a function in the form \begin{align} f(q,M)=\sup_{0\leq \alpha \leq 1} -\alpha^T (R\odot M)\alpha+\alpha^Tq \end{align} which is a dual of a minimization problem, where $R$ and $M$ are ...
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### Is there a profitable way to read mathematical proofs

Mathematical proofs are often presented in a sequential way, i.e., presenting definitions, building lemmas based on these definitions, building further results on these lemmas and finally invoking a ...
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### Vertex Reconstruction Conjecture For Asymmetric Graphs

Simple question: (a) Is it known whether all graphs G having trivial aut(G) are vertex reconstructible, and (b) what is the proof if it exists?
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### Given a transformation, find the generating function

There's a mapping $(x,y) \mapsto(u,v)$ given by $u= x\cos\theta-y\sin\theta$ $v =x\sin\theta + y\cos\theta$ I'd like to find a generating function $G(x,y)$ for this mapping, which I understand to ...