# All Questions

**0**

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6 views

### 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 ...

**2**

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**0**answers

15 views

### 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 ...

**0**

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**0**answers

6 views

### 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 ...

**1**

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**0**answers

14 views

### 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 ...

**0**

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20 views

### 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 ...

**1**

vote

**0**answers

14 views

### 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 ...

**0**

votes

**0**answers

25 views

### 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 $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$

**0**

votes

**1**answer

24 views

### 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 ...

**0**

votes

**0**answers

7 views

### 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 ...

**1**

vote

**0**answers

36 views

### 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 ...

**2**

votes

**1**answer

20 views

### 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 ...

**-4**

votes

**0**answers

20 views

### 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:
...

**16**

votes

**1**answer

546 views

### 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?

**0**

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**0**answers

44 views

### Symplectic forms and Chern classes

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

**7**

votes

**1**answer

105 views

### 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, ...

**2**

votes

**1**answer

62 views

### p-adic L-function of curves

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 ...

**1**

vote

**2**answers

107 views

### 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 ...

**3**

votes

**0**answers

29 views

### Dimension of the sum of images of transpose

$\newcommand{\rank}{\operatorname{rank}}\newcommand{\im}{\operatorname{im}}$
Given $A,B\in M_{n\times n}(k)$, define $\rank(A,B):=\dim(\im A+\im B)$. I'm looking for results regarding relationships ...

**-4**

votes

**0**answers

23 views

### 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 ...

**1**

vote

**0**answers

113 views

### 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?

**0**

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**0**answers

17 views

### 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 ...

**6**

votes

**1**answer

151 views

### 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 ...

**0**

votes

**0**answers

57 views

### 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 ...

**2**

votes

**2**answers

237 views

### 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: ...

**1**

vote

**1**answer

77 views

### Compute higher direct image for Gm under open embedding

Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j_*\mathbb ...

**0**

votes

**0**answers

72 views

### A question about “Bounded analytic functions” by J.B.Garnett, 2007 [on hold]

I am reading the book "Bounded analytic functions" by J.B.Garnett, 2007. On page 134, I don't understand why G is in fact a rational function? Why f has at most n-1 zeros?

**-1**

votes

**0**answers

49 views

### About the Lorentz space ∧(ω,1)?

I ask if the Lorentz space ∧(ω,1) ( as a Banach lattice) has order continuous norm? Is it discrete? As it is known, in this space the lattice operations are not sequentially weakly continuous ( i.e ...

**1**

vote

**0**answers

109 views

### Finite Cohomology and free groups

Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H ...

**28**

votes

**3**answers

676 views

### 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?

**0**

votes

**1**answer

32 views

### 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
...

**7**

votes

**0**answers

80 views

### 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 ...

**1**

vote

**0**answers

33 views

### How to find a Lax Pair for the modified KdV equation

Question
I am having trouble trying to find a matrix $T$ so that with $X$, they form a Lax pair for the modified KdV equation $u_t - 6 u^2 u_x + u_{xxx} = 0$. Where $X$ is defined as:
$
X = ...

**8**

votes

**0**answers

71 views

### Commuting nets for commuting projections

I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange.
Let $A$ be a $C$*-algebra and let $p,q\in A^{**}$ be two commuting projections. Then there ...

**2**

votes

**1**answer

88 views

### How do small changes in a filtered complex affect the associated spectral sequence

I have recently been learning about spectral sequences, not in the context of any problem, but mostly out of curiosity. There are two questions that have occurred to me which I've been unable to ...

**1**

vote

**0**answers

30 views

### Bounding expected value of maximum of dot product with random chirp

Let $\mathbf{x}\in\mathbb{C}^n$ with $\|\mathbf{x}\|=1$ with $n<\frac{N}{2}$. I am interested in a bound of the form
\begin{equation*}
...

**0**

votes

**1**answer

53 views

### If a (linear) relation maps Lagrangian subspaces to Lagrangian subspaces, is it a Lagrangian relation?

Let $(U,\omega),(V,\rho)$ be symplectic vector spaces. Call a relation $U \to V$ a (linear) Lagrangian relation (also Lagrangian correspondence) if it is a Lagrangian subspace of $\overline U \oplus ...

**2**

votes

**1**answer

53 views

### 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 $$\int_D|f''(z)|^2 dxdy =\infty.$$

**4**

votes

**1**answer

152 views

### 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 ...

**-1**

votes

**1**answer

52 views

### Machine learning and Linear programming [on hold]

Her's the problem I'm trying to solve :
I want to find $n$ function $f_i : \mathbb{R} \rightarrow\mathbb{R}$ and the vector $X=(x_1,...x_n)$ that minimizes :
$\displaystyle{\sum_{i=1}^{n} ...

**5**

votes

**1**answer

172 views

### Immersion of $S^1$ in $\mathbb{R}^2$ that can be extended to $\mathbb{D}$

I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow ...

**3**

votes

**0**answers

93 views

### 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 ...

**0**

votes

**0**answers

22 views

### 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 ...

**-3**

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**0**answers

66 views

### 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 ...

**-4**

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**0**answers

76 views

### 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: ...

**0**

votes

**1**answer

68 views

### 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 $Z(k)$ ...

**0**

votes

**2**answers

246 views

### 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 ...

**-1**

votes

**0**answers

21 views

### 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 ...

**0**

votes

**1**answer

68 views

### 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, ...

**3**

votes

**0**answers

100 views

### different proofs of the fact that compact riemann surface has a non-trivial meromorphic function

I would have like to have a list of different proofs of the fact that compact riemann surface has a non-trivial meromorphic function.This is certainly one of the main results of compact riemann ...

**0**

votes

**0**answers

56 views

### plane curves with two points of high multiplicity

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 ...