# All Questions

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### Why the Szpiro conjecture over number fields doesn't depend on the discriminant of the number field?

Szpiro's conjecture states that the Szpiro ratio is: $$\sigma_{E/K}=\frac{\log{|N_{K/Q}\Delta_{E/K}|}}{\log{|N_{K/Q} f_{E/K}}|}$$ Given $\varepsilon >0$ there are only finitely many $E/K$ with ...
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### Is this differential equation on the sphere known?

Just right now, I met the PDE $$\left(-\Delta_{\theta,\phi} - a \cos(2\theta) - b \cos(4 \theta) \right) \psi (\theta,\phi) = \lambda \psi(\theta,\phi).$$ where $\Delta_{\theta,\phi}$ is just the ...
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### Rank one (phi,Gamma)-modules

Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Consider an unramified representation $\rho : Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p) \to \mathbb{F}_p^{\times}$ which sends the arithmetic ...
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### Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
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### Diagonally change the matrix [on hold]

if we have a matrix 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 then we have to change the elements diagonally from top left to bottom right ? what it ...
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### Finding a smooth 6-manifold with a closed 2-form which is degenerate only along some embedded 2-spheres

Given a symplectic 6-manifold $(M,\omega)$ and an embedded symplectic 2-sphere $C\subset M$ whose normal bundle has the first Chern class -2. How to find on $M$ another closed 2-form $\eta$ which only ...
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### A bound on the size of the center

Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
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### Reference for finite number of Weyl groups of reductive groups of rank $r$

I'm posting this question on behalf of someone without access to mathoverflow: Can anybody give me a reliable reference (not a proof) to the following statement? Up to isomorphism, there are only ...
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### Liouville type theorems; linear PDE with decaying potential

Dear Mathoverflowers, I am interested in the following pde: $$-\Delta u(x) + C(x) u(x) = 0$$ in $R^N$. Lets assume that $C(x)$ is bounded and (smooth if you like) and satisfies the ...
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### a colouring / matching problem

While trying to find a bijection which preserves various combinatorial statistics, I was led to the problem below. Very much to my surprise, a closely related question, Coloring summands of given ...
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### Which is the right way to compute the Approximate Entropy (ApEn)?

My problem is the inconsistency between the definition and the computation of the Approximate entropy (ApEn). Suppose $u = (u_i:1\leq i \leq N)$ is a sequence of ...
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### Success Ranking Methodology [on hold]

I'm trying to calculate the success of a couple of students but they aren't all in sync with their attended exam count. These are the data I have right now: ...
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### Hilbert's Hotel

Hilbert's Hotel is a famous story on infinty attributed to David Hilbert (1862-1943). Is it doumented that Hilbert's Hotel is in fact due to Hilbert and if yes: where?
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### Symplectic forms and Chern classes

Given a symplectic manifold, is there any explicit formula showing the relation between its symplectic form and Chern classes?
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### Isotropic Riemannian manifolds

Let $M$ be a Riemannian manifold and $G$ a closed connected subgroup of isometries of $M$. Call the pair $(M,G)$ an isotropic pair if $G$ acts transitively on the sphere bundle $SM$. As an example, ...
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Given a smooth projective curve $C$ over $\mathbb{Q}$ one has the $L$-function $L(C, s)$ and the Beilinson conjectures predict its values at integers $s=n$ in terms of regulators. Is there a p-adic ...
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### Sheaf of isogenies representable?

It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be ...
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### Dimension of the sum of images of transpose

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### Probability question (win three games in a row = win or 4 wins total = win) [on hold]

Two teams play each other repeatedly until either one of them wins three games in a row or one of them wins a total of four games. What are all the ways in which the tournament can be played? What is ...
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### Davenport's proof that almost all integers are the sum of 4 cubes

Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?
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### Mathematics of simple performance testing [migrated]

I have a set of sorted tables T that have known but different dimensions. There are two types of functions in this system: f(T) g(T, n), where n is an integer parameter. ... and two types of costs ...
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### History of powers beyond squares and cubes

The ancient Babylonians understood squares:       Plimpton 322 The ancient Athenians understood cubes, if we can take doubling the cube, i.e., the Delian problem, as evidence. My ...
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### Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?

In this MSE question/thread, I have been discussing the equation $$(x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{\star}$$ where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to ...
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### Is there a von Koch-type theorem for the generalized Riemann hypothesis?

Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound $$\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).$$ Q1: ...
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### If any open set is a countable union of balls, does it imply separability?

If a metric space is separable, then any open set is a countable union of balls. Is the converse statement true?
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### Generalized assignment problem with no integrality gap

Suppose I am solving the generalized assignment problem, so that I am given matrices $U$ and $W$ and a vector $c$ (all three of which have, say, positive entries), and I want to solve ...
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### Computing exact or asymptotics for number of strings over an alphabet of size $n$ that have no non-trivial substrings that appear more than once

I ran across a seemingly relatively simple combinatorics problem that appears open. For an alphabet of size $n$, let $A(n)$ be the number of strings over the alphabet that have no substring of length ...
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### Integrability of second derivative of conformal mappings

How to construct a conformal mapping $f$ of the unit disk $D$ onto a Jordan domain with $C^1$ boundary such that $$\int_D|f''(z)|^2 dxdy =\infty.$$
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### transcendence of beta values

(1) Can anybody suggest a readable reference for Schneider's theorem that the number $$\beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...
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### Partial differential Equation over characteristic p

I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
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### ILP for minimum edge coloring problem

We know that for a Graph G=(V,E) ,minimum edge coloring is a coloring of E, i.e., a partition of E into disjoint sets E1,E2...,Ek such that, for 1<=i<=k, no two edges in Ei share a common ...
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### EXAMPLES OF SMOOTH FUNCTION IN L^2(R) [on hold]

Is there any function $f$ which lies in $L^2(R)$ that belongs to $C^n$ space and whose $nth$ derivative is bounded. Is the example $f(t)= t^2$, for $0\leq t< 1$ and 0 otherwise which satisfies the ...
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### A big list of non-trivial examples of functions from outside mathematics [on hold]

I asked this question on Mathematics SE, but got disappointingly little interest. Therefor I repeat the question here. Link to MSE question is: ...
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### Does the homogeneous spaces $K^{\mathbb{C}}/{{Z(k)}^{\mathbb{C}}}$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, $K^{\mathbb{C}}$ be complexified Lie group of $K$. Denote $Z(k)$ by the centralizer of k∈K and $Z^{\mathbb{C}}(k)$ be the complexified Lie group of $Z(k)$ ...
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### What are all positive integers n for which the congruence $a^{n+1} \equiv a (mod n)$ holds? [on hold]

Fermat's little theorem says that the congruence $a^p \equiv a (mod p)$ if $p$ is a prime number. $a^{n+1} \equiv a (mod n)$ works for all integers $a$ and some positive integers $n$, how can we ...
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### Finding the source for minimizer of a functional for all $C^2$-curve $x(t_0)=x_0$ and $x(t_1)=x_1$

I am trying to find where this problem comes from and its corresponding proof for my students, but I cannot find the source anywhere. If anyone can find the source of this, or has any ideas where I ...
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### Existence of bounded $n-$th derivative of the solution of differential equation

This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...
Let $\mathcal{C}$ be an irreducible plane curve in $\mathbb{P}^2_\mathbb{C}$ of degree $d$. Let $D$ be a quartic with three irreducible components with normal crossing singularities, i.e. a conic and ...