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2
votes
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
4answers
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
0answers
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
0answers
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
1answer
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
1answer
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
3answers
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
1answer
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
0answers
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 ...

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