# All Questions

116,034
questions

**2**

votes

**0**answers

98 views

### Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that
Minkowski spacetime has
$$ISO(d-1,1)/SO(d-1,1),$$
de Sitter spacetime has
$$SO(d,1)/SO(d-1,1)$$ and
anti-de Sitter spacetime ...

**4**

votes

**1**answer

403 views

### Are there infinite many two sided prime numbers?

A prime number $p=\overline{a_na_{n-1}\ldots a_1a_0}$ is called a two sided prime number if its reverse representation $q=\overline{a_0a_1\ldots a_{n-1}a_n}$ is a prime number too.
Are there ...

**9**

votes

**0**answers

204 views

### On the status of some conjectures mentioned/used in Harish Chandra's 1970 lecture notes

In van Dijk's notes of Harish Chandra's lectures on harmonic analysis, several conjectures are mentioned throughout, such as in Part 1, section 4 of van Dijk's notes
Conjecture I : Let $\omega$ be ...

**8**

votes

**1**answer

391 views

### Set of points with a unique closest point in a compact set

Let $K\subset\mathbb{R}^n$ be any compact set. Let $\operatorname{Unp}(K)$ be the set of points in
$$
\operatorname{Unp}(K)=\{x\in\mathbb{R}^n\setminus K:\, \exists ! y\in K \ \ |x-y|=d(x,K)\}.
$$
...

**6**

votes

**0**answers

196 views

### Example of a tensor triangulated category with two different monoidal t-structures?

What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures?
While I'm at it: is there an example of a tensor ...

**4**

votes

**1**answer

99 views

### Expected supremum of normalised random walk

Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$.
Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix.
Define $S^k=...

**0**

votes

**0**answers

40 views

### Optimization of an integral functional when the multiplier rule yields no useful information

Let
$(E,\mathcal E,\lambda)$ be a measure space
$\mu\ll\lambda$ be a probability measure on $(E,\mathcal E)$
$p\in[1,\infty)$
$k\in\mathbb N$
$f:E\times\mathbb R^k\times\mathbb R^k\to[0,\infty)$ such ...

**6**

votes

**1**answer

218 views

### Anti-holomorphic involutions of a complex linear algebraic group

Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$.
Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$.
Let $\sigma$ be an anti-...

**2**

votes

**1**answer

75 views

### Index and congruence subgroup from scaling variables of Jacobi form

Let $J_{k,m}(N)$ be the space of Jacobi forms of weight $k$, index $m$, and congruence subgroup $\Gamma_{0}(N) \rtimes \mathbb{Z}^{2}$. I do not believe it is relevant here to specify what type of ...

**1**

vote

**1**answer

119 views

### On existence of certain operators in von Neumann algebra

Let $M\subset B(\mathcal{H})$ be an infinite dimensional vN algebra in standard form. Fix $\xi\neq 0 \in \mathcal{H}$, does there exist $M\ni x_{\xi}\neq I$ such that $x_{\xi}(\xi)=\xi$?

**6**

votes

**1**answer

484 views

### Are the coefficients of certain product of Rogers-Ramanujan Continued Fraction non-negative?

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
The following equality is famous:
$$\cfrac{q^{1/5}}{R(q)} = \prod_{k>0} \cfrac{(1-q^{5k-2})(1-q^{...

**4**

votes

**2**answers

120 views

### Understanding equiprobable trinomial identity

With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...

**3**

votes

**1**answer

176 views

### Sum-product estimate in finite fields

There is a paper by Bourgain, Katz and Tao
Bourgain, Jean; Katz, N.; Tao, Terence C., A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14, No. 1, 27-57 (2004). ZBL1145....

**4**

votes

**0**answers

217 views

### Conventions for Riemann curvature tensor

I am aware of two conventions for the Riemann curvature tensor, namely the expression
$$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$
is either declared to be $R(X,Y,Z,W)$ or $...

**0**

votes

**1**answer

121 views

### Notion of module over commutative post lie algebra

Let $ (S, \{. \}, [.]) $ be an algebra over a vector
space endowed with two bilinear maps $ \{. \}, [.] : S \times S \rightarrow S$ and satisfying some compatibility conditions.
In example, if S is ...

**2**

votes

**1**answer

121 views

### subelliptic Sobolev compact embedding theorem

Consider the smooth vector fields $X=(X_1,X_2,...,X_m)$ defined in a open bounded set $\Omega\in R^n$. And the non-isotropic dimension is $Q$ (see https://arxiv.org/pdf/1502.06332.pdf page 398)
In the ...

**7**

votes

**0**answers

163 views

### Meaning of Elliptic Irregular Primes

The Bernoulli numbers are defined by the equation
$$
\frac{t}{e^t-1}=\sum_k b_k \frac{t^k}{k!}.
$$
A prime number $p$ is irregular if it divides the numerator of one of the even Bernoulli numbers up ...

**7**

votes

**1**answer

249 views

### Kinematic formula for Euler characteristic

Is there a formula for $\int \chi(K \cap gL) \: dg$ (where $\chi$ is Euler characteristic) analogous to the kinematic formula for $\int \mu(K \cap gL) \: dg$ (where $\mu$ is Lebesgue measure)? In both ...

**0**

votes

**0**answers

36 views

### Can we have a stratified theory equivalent to NFU + Infinity + choice with atmost failure of Extensionality?

It is known in NFU + Infinity + Choice, that we can partition the set $U$ of all Ur-elements (empty objects other than the empty set) such that each piece is as big as the set $V$ of all objects, and ...

**1**

vote

**0**answers

74 views

### Probabilistic lower bound on largest singular value of matrices

I have a distribution $\mathcal{D}$ that spits out vectors in $\{-1, 1\}^N$. Suppose I have a sample of $H$ of these vectors which I arrange into a matrix $M$ of the form $H \times N$.
Consider the ...

**2**

votes

**0**answers

125 views

### View Dirichlet character as a character of Galois group

In Jaclyn Lang's article "On the image of the Galois representation associated to non-CM Hida family" section 2, the Dirichlet character $\chi$ module $N$ is also viewed as a character $\chi\colon\...

**-1**

votes

**1**answer

84 views

### Coupling argument involved in the contracting and mixing properties of the Glauber dynamics for an Ising model

While doing a research work, I had to read about the Glauber dynamics for an Ising model. A wonderful account on this is given in the book Markov Chains and Mixing Times by Levin, Peres and Wilmer.
...

**6**

votes

**1**answer

224 views

### Guessing the number of other $1$'s in a binary sequence

I have posed the following question on math.stackexchange.com but have not received an answer. So I would like to seek experts' opinion here.
Consider the set of all binary sequence of length $n+1$, $...

**0**

votes

**0**answers

17 views

### Quantiles of the Q values of an unknown MDP

Consider an MDP with $n$ states, $k$ actions, and discount factor $\gamma \in [0,1)$. We are uncertain of its reward function $R \in \mathbb{R}^{n \times k}$ and transition function $T \in \mathbb{R}^{...

**5**

votes

**1**answer

250 views

### Is it possible to use Feynman diagrams to represent a dot product $a \cdot b$?

Feynman diagrams are topological entities, but they describe linear
operators
It has been observed that Feynman diagrams are in particular string diagrams (morphisms in monoidal categories)) in ...

**1**

vote

**0**answers

69 views

### Existence of fundamental solution of fractional laplacian

Does the fundamental solution of $$(-\Delta)^sF(x, y)+ V(x) F(x, y)= \delta_{y}(x) \text{ in } \mathbb R^N $$ exist. Here $V(x)$ is a positive smooth non-constant bounded function which satisfy $V(x)\...

**2**

votes

**1**answer

177 views

### Representative in ideal class group coprime to the conductor

Working in an order $\mathcal{O}$ in an imaginary quadratic field $K = \mathbb{Q}(\sqrt{d})$ and given an invertible ideal $\mathfrak{a}\subseteq \mathcal{O}$, I would like to produce another integral ...

**1**

vote

**1**answer

118 views

### Exponential upper bounds for sums of martingale differences

Let $(X_{i})_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}_{i})_{i \geq 1}$. Define $S_{n} = \sum_{i=1}^{n}X_{i}$ and $\Sigma^{2}...

**4**

votes

**1**answer

135 views

### Commensurator of a subgroup of matrices

Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the ...

**5**

votes

**0**answers

126 views

### Is there an equivariant simplicial deformation retract of Teichmüller space?

Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...

**0**

votes

**0**answers

21 views

### Name for subset selecting matchings

Tutte and also Lovasz and Plummer reduce the calculation of (optimal) f-factors in graph to non-bipartite matching via replacing each vertex with a $K_{f,\delta}$, refered to as a 'gadget' whose ...

**0**

votes

**1**answer

137 views

### A question about entire functions of order 1

Suppose $f:\mathbb C \to \mathbb C$ is an entire function on the complex plane of order $1$.
Additionally, suppose that:
$$ \forall\, c \in \mathbb R, \quad \lim_{t \to \pm \infty} \, f(t+ic) =0.$$
...

**2**

votes

**0**answers

148 views

### Counting special metrics on finite fields

Define a Galois coding norm of degree n as a map $|\space| : \Bbb F_{2^n}\rightarrow {\Bbb Z}$ with the following properties :
(I) $(\Bbb F_{2^n},|\space|)$ is a self-orthogonal code ; i.e. $(x,y)\...

**4**

votes

**2**answers

417 views

### Convex hull in a discrete space [closed]

I know some algorithms which compute the convex hull in a continuous space.
Are there efficient algorithms to compute it in a discrete domain?
For example in 3D discrete space, given the blue points, ...

**0**

votes

**0**answers

59 views

### Sum of i.i.d discrete random vectors, restricted sum of multinomial

We are given a random vector of dimension $n$ which takes a value one with the probability $p_{1,1}$ in the first coordinate, value two with the probability $p_{1,2}$ in the first coordinate and so ...

**2**

votes

**1**answer

102 views

### Finding all unitary representations of the connected Poincaré group

I am studying representation theory of Lie groups and its combination to theoretical physics, and I am concerned about the following. Is there an exhaustive way to find all unitary representations of ...

**4**

votes

**2**answers

120 views

### 4-polytopes with only one kind of regular facet

Is there a neat way to show (or a reference that already proves) that
the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes?
the 24-cell is the only convex 4-polytope in ...

**4**

votes

**1**answer

333 views

### Does Spec functor sends pushouts of rings into pullbacks of sets?

This question was posted here on StackExchange.
Let $A$ be a commutative ring and $B,C$ be two commutative $A$-algebras.
Consider the pushout square of ring homomorphism
$\require{AMScd}$
\begin{CD}
...

**5**

votes

**1**answer

139 views

### Translated version of a Caratheodory article

This excellent introduction to Compressive Sensing cites a couple of (seemingly) interesting Caratheodory papers from 1907-1911.
These are:
[46] C. Caratheodory. Uber den Variabilitätsbereich der ...

**6**

votes

**1**answer

272 views

### Is min exponents of three positive integers $n$, $n+1$ and $n+2$ $=1$ true or false?

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$
...

**4**

votes

**0**answers

79 views

### Is every locally compact connected homogeneous metric space a manifold cross a continuum?

Suppose that $(X,d)$ is a locally compact connected homogeneous metric space, where by homogeneous I mean that for any $x_0,x_1 \in X$ there exists an isometry $f:X\rightarrow X$ such that $f(x_0)=x_1$...

**5**

votes

**0**answers

142 views

### What are the effective epimorphisms of presentable $\infty$-categories?

Let $\mathcal C$ be a sufficiently nice $\infty$-category, and let $f: U \to X$ be a morphism in $\mathcal C$. Recall that $f$ is said to be an effective epimorphism if the induced map $|U^{\times_X (\...

**5**

votes

**4**answers

458 views

### General topological space with closure operation as in Russian translation of Hausdorff's 1914 and 1927 Mengenlehre

In the Russian translation (by P. Alexandroff and A. Kolmogorov) of chapter "Point Sets in General Spaces" Hausdorff (1914), the notion of a general topological space is defined as set $R$ with ...

**28**

votes

**0**answers

729 views

### Is this representation of Go (game) irreducible?

This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph.
Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...

**6**

votes

**0**answers

142 views

### Is there an $\infty$-topos of monochromatic spaces?

Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain ...

**2**

votes

**1**answer

83 views

### Ramsey type properties of $F_\sigma$ ideals

Let $I \subseteq 2^\omega$ be any $F_\sigma$ ideal containing every finite sets :
$\forall X \in I\ \forall Y \subseteq X\ \text{ we have } Y \in I$
$\forall k\ \forall X_1,\dots,X_k \in I\ X_1 \cup \...

**0**

votes

**1**answer

222 views

### Weak convergence of Hilbert Schmidt operators

So I am stuck at this situation. Let $\{A_n\}$ be a weakly convergent sequence in $B_2(H)$ converging to $0$ in the weak topology on $B_2(H)$. Which means that $\left<A_n,D\right>=\operatorname{...

**5**

votes

**3**answers

253 views

### Tensoring irreducible $B$-series representations/ Type B Littlewood-Richardson

When tensoring finite dimensional representations of the Lie algebra ${\frak sl}_n$, we have an explicit algorithm given in terms of Young diagrams. See Section 4 of this paper.
Do there exist ...

**5**

votes

**1**answer

392 views

### Banach-Mazur game and infinite products

Studying the article "Games that involve set theory or topology" of Marion Scheepers, I found the following result
Theorem 46 Let $\{(X_{i}, \tau_{i}) : i\in I \}$ be a family of topological spaces. ...

**5**

votes

**0**answers

169 views

### Generating the monoid of injective endomorphisms of the free group

Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injective group endomorphisms $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set ...