**0**

votes

**0**answers

1 view

### All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid?

Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open:
Given ...

**31**

votes

**3**answers

1k views

### Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula.
I. Given the fundamental unit ...

**4**

votes

**2**answers

156 views

### Extending inequality for $\ell^p$ with integer $p$ to real $p$

Suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are two decreasing sequences of positive numbers such that
$a_1<b_1$ and
$$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k<\infty$$ ...

**-1**

votes

**0**answers

15 views

### Exact statistics in the Frobenius coin problem

The Frobenius coin problem guarantees that if $(a,b)=1$, then
$$ax+by$$ does not represent exactly $(a-1)(b-1)/2$ numbers all below $ab-a-b$ if $x,y\geq0$ holds.
I am confused by following argument.
...

**-1**

votes

**0**answers

16 views

### Self avoiding walk problem?

As in the image we can see that there are black spots and moving from spot to another is 1 move.
Can we create a function which will tell us the position after say 119 moves, 143 moves etc without ...

**0**

votes

**1**answer

329 views

### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...

**6**

votes

**0**answers

71 views

### Upper bound for a sum including Andre polynomial coefficients

Let $ c_{n,k} $ be a sequence defined by the following relations: $\displaystyle c_{n,0} = 1, \hspace{0.1cm} (n \ge 1)$;
$$ c_{n,k} = (k+1) c_{n-1,k} +(n-2k+1) c_{n-1,k-1}, \hspace{0.5cm} (1 \leq k ...

**8**

votes

**3**answers

369 views

### Distance between two knots

Are there some well-studied functions defining natural distance measures between two knots? One can imagine a function that counts, say, the minimum number of
moves, each of which passes one strand of ...

**-4**

votes

**0**answers

23 views

### Counting subset of repeated permutations [on hold]

I'm trying to find a way to find to count specific ordered arrangements of a set with k elements, where each arrangement has n elements.
For example for k=2 and n=3
aaa
aab
aba
abb
baa
bab
bba
bbb
...

**-4**

votes

**0**answers

28 views

### how to represent a sorting function versus absolute value function

we can represent $max\left(x_1,x_2\right),min\left(x_1,x_2\right)$ as follow:
$$
max\left(x_1,x_2\right)=\frac{1}{2}\left(x_1+x_2 + \left|x_1-x_2\right| \right)
$$
$$
...

**0**

votes

**0**answers

17 views

### Reference request: Linear evolution equations of “hyperbolic type”

Does anyone have any accessable link to the following paper by Kato?
Linear evolution equations of “hyperbolic type”
Note: It is the first paper, not the sequel numbered by II. After several ...

**-2**

votes

**0**answers

61 views

### spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane.
The question is ...

**5**

votes

**3**answers

175 views

### Smoothness of the fourth power of the geodesic distance in a Finsler geometry

The simplest form of Finsler metric is:
$ds (X, dx)=\sqrt[4]{g_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g_{\alpha, \beta, \gamma, \delta}$ is a fourth ...

**3**

votes

**1**answer

55 views

### Genericity of irreducible automorphisms of free groups

I have seen in the literature that Irreducible outer automorphisms of a free group $F_n$ are "generic".
I would like to ask that if for example : it is true that for any generating set $X$ of ...

**1**

vote

**0**answers

48 views

### Weighted maximal number of disjoint chains in the integer divisibility poset for $\{1,2,\ldots,n\}$

In the mathoverflow question here the asymptotic growth of antichains in the divisibility poset ${\cal P}_n$ of the set of natural numbers $\{1,\ldots,n\}$ is considered. I have a somewhat dual ...

**3**

votes

**1**answer

383 views

### Size of KL-divergence neighbourhoods

I am new here. I was reading another
post
here and this got me wondering what can be said about the size of the following kl divergence neighborhoods.
Consider these two kl-divergence neighbourhood ...

**-1**

votes

**0**answers

27 views

### Distribution of $e^*f$, if $e$ is a complex Gaussian vector and $f$ is a unit norm complex vector

Let $e$ be a complex Gaussian vector where its elements are of zero mean and variance equals to $\sigma^2$. In addition, we define $f$ as a complex random unit norm vector uniformly distributed. Note ...

**1**

vote

**1**answer

79 views

### General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot ...

**2**

votes

**1**answer

37 views

### Reference: First publishing of Mallivain Derivative as First Variation

Who first showed the Malliavin derivative to be expressible in terms of the first Variation of the process it was deriving?

**1**

vote

**0**answers

19 views

### Constructing symmetric invariants of the exceptional simple Lie algebra as restrictions

Let $\mathfrak{g}$ be a complex simple Lie algebra with maximal torus $\mathfrak{h}$, Weyl group $W$. The adjoint representation $\operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)}$ extends ...

**33**

votes

**11**answers

24k views

### Sum of 'the first k' binomial coefficients for fixed n

I am interested in the function $\sum_{i=0}^{k} {N \choose i}$ for fixed $N$ and $0 \leq k \leq N $. Obviously it equals 1 for $k = 0$ and $2^{N}$ for $k = N$, but are there any other notable ...

**4**

votes

**1**answer

158 views

### realizing uniform boundedness of Galois representations associated to elliptic curves

This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear).
Suppose I have an ...

**2**

votes

**1**answer

41 views

### Supercommutator of exterior multiplication operators and their adjoints

Let $\mathfrak{h}$ be a complex Hilbert space and consider Grassmann algebra $\mathcal{F}=\bigwedge\mathfrak{h}$ with its induced inner product. For $\omega\in\mathcal{F}$ we also consider the ...

**2**

votes

**3**answers

205 views

### Classification of open subset of $\mathbb{R}^{3}$ [on hold]

There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this
Theorem ? Let ...

**2**

votes

**1**answer

338 views

### optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...

**5**

votes

**1**answer

448 views

### Grothendieck's paper on principal bundles, reduction to a torus step

In Grothendieck's paper "Sur la Classification des Fibres Holomorphes sur la Sphere de Riemann", there is a step I don't understand in section 4, where he proves reduction to a torus. He states (lemma ...

**25**

votes

**5**answers

3k views

### Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA?

After having read Gunnar Carlsson's Topology and Data I feel enthusiastic to use some topological data analysis (TDA) methods in my current research, mostly in social sciences. We often handle huge ...

**2**

votes

**1**answer

93 views

### How many points are in such set with the same norm-2

Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking ...

**8**

votes

**1**answer

216 views

### Nice sign-expansions of special surreal numbers

What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic?
I can think of more than one natural way to ...

**0**

votes

**0**answers

16 views

### Calculation of minimal right $\operatorname{add}(M)$-approximations

given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 ...

**0**

votes

**0**answers

73 views

### A family of maximal ideals

Let $m_i $, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} ...

**3**

votes

**0**answers

176 views

### How close to being well-orderable does this make my powerset?

Let's work in a set theory without assuming AC (for instance, but not necessarily, ZF). Fix a set $k$ satisfying $k\times k \simeq k$, and consider its powerset $X = 2^k$. I have a technical condition ...

**8**

votes

**1**answer

92 views

### On linear integer inequalities with infinitely many solutions

Suppose that a linear system of inequalities $Ax \le b$, where $A\in Z^{m\times n}$ and $b\in Z^m$ have integral coefficients, has an infinite number of integral solutions $x$.
Can one conclude that ...

**7**

votes

**2**answers

185 views

### $p$-torsion of an abelian variety of $p$-rank $0$

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...

**1**

vote

**1**answer

95 views

### Schauder estimate for the heat equation on compact manifolds

I asked this question on math.stackexchange.com, however I didn't get any answers so I'll try it here.
Let $M$ be a compact manifold without boundary. Consider $Lu:=\partial_tu-\Delta u$. Let $f\in ...

**3**

votes

**0**answers

83 views

### Is stable map space $\overline{M_{0,n}}(\mathbb{P}^n,d)$ is irreducible for all $n,d$?

I read a paper "Notes On Stable Maps And Quantum Cohomology, W.Fulton and R.Pandharipande". And I think that $\overline{M_{0,n}}(\mathbb{P}^n,d)$ is irreducible. But I cannot find an exact statement ...

**6**

votes

**1**answer

103 views

### Inequality for the maximum of Gaussian variables

Let $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_n)$ be centered Gaussian vectors
with variance matrix $\Gamma_X$ and $\Gamma_Y$. We assume that the matrix
$\Gamma_Y-\Gamma_X$ is positive definite. Is it ...

**2**

votes

**0**answers

52 views

### Universal Witt vectors in full complete closed p-adic space omega?

Is there a p-adic mathematical structure that incorporates the advantages of both universal Witt vectors (not p-typical-limited; implementing Frobenius and Verschiebung operations) and permitting ...

**-4**

votes

**0**answers

39 views

### How to calculate product of the coefficients of a polynomial? [on hold]

I have a recurrence equation that results in a polynomial. The quantity I am interested in is the product of the coefficients of the terms in the polynomial.
For example, if f_2(x)=a0+a1x+a2x^2, I ...

**-2**

votes

**0**answers

43 views

### Composition of derivations is zero on commutative ring? [on hold]

Let $Z$ be a commutative domain char $Z>n>1$.If $d_{1},..,d_{n}$ are such derivations of $Z$ that the composition $d_{1}...d_{n}$ is a derivation,then $d_{i}=0$ for some $1\leq i \leq n$. Can ...

**-1**

votes

**0**answers

24 views

### For a $\sigma$-finite measure is $u_*(E)=lim_iu_*(E_i)$? [on hold]

let $u$ be a $\sigma$-finite measure on a $\sigma$-ring $S$, let $u_*$ be the inner measure induced by $u$ and denote $H(S)$ as hereditary $\sigma$-ring generated by $S$.
{$E_i$} is an increasing ...

**6**

votes

**0**answers

244 views

+100

### “The” natural double complex associated to a principal $G$-bundle?

Let $\pi: P \to M$ be a principal $G$-bundle. We have the associated adjoint bundle $ad(P)= P \times_{ad} \mathfrak g$ whose sections correspond to infinitesimal guage trasformations.
Consider the ...

**0**

votes

**0**answers

41 views

### What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$?

What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$ in terms of $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$? Any help will be greatly appreciated!

**13**

votes

**3**answers

491 views

### An inequality improvement on AMM 11145

I have asked the same question in math.stackexchange, I am reposting it here, looking for answers:
How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$:
...

**4**

votes

**1**answer

247 views

### Improper integral $\int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx$ with $-a$ and $b$ positive

Is the following function real analytic in $t>0$:
$$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$
where $-a$ and $b$ are positive, and $c\not=a$?
...

**22**

votes

**2**answers

437 views

### Uniform proof that a finite (irreducible real) reflection group is determined by its degrees?

Given a finite (irreducible real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated ...

**0**

votes

**0**answers

128 views

### Time Hierarchy Theorem and P vs NP

One obvious strategy for proving P not equal to NP would be to show that there is some problem in NP which is hard for a time class strictly containing P (the origin of this question is the recent ...

**5**

votes

**1**answer

312 views

### Homogeneous metrics of given volume element, on the n-sphere.

Let $\omega$ be an $n$-form on $S^n$, nowhere vanishing.
Is there a Riemannian metric $g$ on $S^n$, so that its volume form is $\omega$, and $(S^n,g)$ is homogeneous? Is it unique, and if not, what ...

**-4**

votes

**0**answers

83 views

### A tricky 2d integral [on hold]

I tried to calculate such integral:
$$
\int d^2q \frac{\bf{q+q_2}}{(\mathbf{q}^2+m^2)((\mathbf{q-q_1})^2)^{1-i\eta}(\mathbf{q+q_2})^2}
$$
where $q$,$q_1$ and $q_2$ are two dimensional vectors. Can ...

**5**

votes

**2**answers

241 views

### A result of Sierpiński on non-atomic measures

There is a classical result commonly attributed to W. Sierpiński that reads as follows:
Theorem 1. If $f: \Sigma \to \bf R$ is a non-atomic (*) measure on a set $S$, then for every $X \in \Sigma$ ...