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7 views

Operator norm for random vectors

I am looking for a bibliographic reference addressing the following norm for a random variable $X \sim \mathcal P$, where $\mathcal P$ is a distribution over $\mathbb R^d$: $$\Vert X\Vert_{p\to 1} := \...
-1
votes
0answers
17 views

Increase a number gradually and exponentially

I have 3 given values: xt, x0 and t. I want to gradually and exponentially increase x0 to xt in t number of days like in below graph (in below graph example; x0=1, xt=43000 and t=30 days). I want to ...
-2
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0answers
30 views

how to factorize x^p^2-1? [closed]

I know how to factorize x^n-1, but how to factorize this expression, x^p^2-1? I tried to factor out x-1 first. Thank you.
-1
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0answers
7 views

Probability density for the result of a random matrix based linear equation

Suppose $P_k$ and $Q_k$ are k-dimensional independent random vectors with normal distributions. $a$, $b$, $c$, $d$, $e$ and $f$ are scalar matrixes. What's the probability distribution of $V$ under ...
0
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0answers
18 views

Periodcs of Coxeter matrices for truncated Nakayama algebras

For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,...,i+r-1$ (we only do this until $i+r-1>n$). So for example for $n=7$ and $r=3$ we ...
0
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0answers
37 views

On Sums of powers 2

In my previous question, I asked about nontrivial sums of four powers $a^m+b^n+c^k=d^l$, and whether the nature of the solutions depend on whether $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{l}$ is ...
1
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0answers
19 views

Hyper-degree sequences: How to count them and how to construct hyper-graphs from them?

From an answer to this question I have learned how to ask this question properly. Consider a $k$-uniform hypergraph on $n$ nodes, i.e. a family of $k$-subsets of $[n]= \{1,2,\dots,n\}$ (the hyperedges)...
1
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0answers
21 views

Cone of morphism in families

I am working in derived category $D^b(X)$ of coherent sheaves on a smooth projective varitey. Let $E,F$ be two sheaves on $X$, with $\mathrm{R}Hom(E,F)=k\oplus k[-1]$, I consider the following ...
3
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0answers
62 views

Emergence of the Discrete from the Continuum

An almost eternal theme in Mathematics is the approximation of the Continuum by the Discrete. This core idea goes back at least to Archimedes, and remains active to these very days (and quite likely ...
0
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0answers
9 views

How do I fit flow values to connections in a known network?

this is not my area and I'm new to its terminology, and am posting my problem in the hope that someone will be able to direct me to where it has been solved, or who has written about it. I have a flow ...
1
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0answers
54 views

Is every countable discrete group a subgroup of a non discrete Lie group?

1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group? 2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word ...
-1
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0answers
34 views

$RC=CR $ $\forall R\in SO(n)$ then $C$ is diagonal [closed]

I read in a paper of Park, that if $RC=CR$ holds for every $R\in SO(n)$ then $C$ is a diagonal matrix with only the same elements on the diagonal. Can someone give me a hint how to prove this?
0
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0answers
19 views

Hausdorff dimension of cookie-cutter sets using topological pressure

I'm reading the proof of Theorem 5.3 from "Techniques in Fractal Geometry" by Kenneth Falconer - the theorem wich gives the Hausdorff dimension of a cookie-cutter set using topological ...
1
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0answers
31 views

Circle geometry

I met an interesting task, tried a lot of ways to prove but didn't found. Problem: Let KLM be a triangle, let I be an incircle's center. Let Q be the point on an extension of the segment KM beyond a ...
3
votes
1answer
35 views

Probability of complex eigenvalues

I find this is the best site to post this question, even though I considered cs. It is a Monte Carlo experiment over the set of 10.000 n×n matrices. If a single matrix eigenvalue is complex then ...
1
vote
2answers
89 views

Intrinsically defining smooth/continuous/analytic functions

In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing ...
0
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0answers
27 views

Percentage and ratio [closed]

Sam wants to convert a mixture of acid and water in the ratio 16 : 19 to 19 : 16. What is the amount of acid as a percentage of initial solution that should be added to achieve the required ratio?
1
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0answers
28 views

Is the maximum of derivatives of a function in (s,2)-Sobolev space (an RKHS) bounded by their norms?

Let $f(x) \in W^{s,2}(\Omega) \equiv H^s$, where $\Omega \subseteq \mathbb{R}^d$ and $W^{s,2}$ is a $(s,2)$-Sobolev space. Clearly, $W^{s,2}$ is an Reproducing Kernel Hilbert Space (RKHS) and ...
-1
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0answers
23 views

Joint distribution of distances between random points

Let $X$ be the 2-dimensional flat torus. That is, $X=[0,1]\times [0,1]$ with the "cyclic" euclidean metric. Let $x_1,x_2$ be two random points in $X$ and let $y^{(1)},y^{(2)},\dots,y^{(k)}$ ...
1
vote
0answers
54 views

Number of models vs. complexity for SOL theories

This was previously asked at MSE without success. Suppose $T$ is a complete first-order theory with continuum-many countable models up to isomorphism. We define two sets of Turing degrees associated ...
2
votes
0answers
58 views

Existence of twisted metaplectic categories

The paper Classification of metaplectic modular categories by Ardonne-Cheng-Rowell-Wang (2016) mentions (in Section 3) the Grothendieck ring for the metaplectic modular categories, i.e. $SO(N)_2$, $N&...
0
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0answers
49 views

Codifferential of a pullback

Given oriented Riemannian manifold $M$, a k-form $\alpha \in \Lambda^{k}(M)$, and an orientation preserving diffeomorphism $\eta:M \rightarrow M$: is there a "nice" coordinate invariant ...
3
votes
0answers
35 views

Mutually orthogonal Latin hypercubes

A $d$-dimensional Latin hypercube with side length $n$ is a $d$-dimensional array with $n$ symbols such that along any line parallel to an axis, each symbol appears exactly once. Let us call a $(n,d)$ ...
2
votes
0answers
18 views

Existence of inner twisting preserve the $*$ action on simple roots

I've been trying to learn the theory of inner and outer Galois twisting in algebraic groups. Let $k$ be a field of characteristic 0, and let $k^s$ be a separable closure. Let $G$ be a connected ...
3
votes
0answers
60 views

About Homotopy Transfer Lemma

If M, A are two differential graded complexes over a commutative ring R with the following data, $$(M,d_M) \overset{\Delta}{\longrightarrow} (A,d_A)$$ $$(A,d_A) \overset{f}{\longrightarrow} (M,d_M)$$ ...
0
votes
0answers
21 views

Large deviation for Brownian occupation time

I am asking for reference about the large deviation principle (LDP) for the occupation time of a Brownian motion/bridge. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth and compactly supported. My ...
1
vote
0answers
70 views

Nontrivial integer homology class implies orientability

I posted this question on MSE and I would like to see if my reasoning is correct. Let $M^3$ be a compact, connected and oriented $3$-manifold with nonempty boundary and let $\Sigma^2$ be a compact and ...
1
vote
1answer
146 views

Examples of conjectures whose direct falsity implies different consequences than indirect falsity

Mathematics several times has statements of form $$\mathsf{Statement A}\implies\mathsf{Statement B}$$ where $\mathsf{Statement A}$ and $\mathsf{Statement B}$ are conjectures while the implication is ...
6
votes
1answer
85 views

The cohomology of modular curves as a module over the Galois group

Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \...
4
votes
0answers
50 views

Information density of proofs?

I am a CS person so please excuse the hand-waving. Given a set of machine-represented proofs, each different (but not necessarily proving a different thing), what sort of information-theoretic ...
6
votes
1answer
87 views

Sets of residues with only a single intersection under translation

A combinatorial game I am studying has given rise to the following question. Consider the group $\Bbb Z/n\Bbb Z$. What is the largest $m$ such that there exist $k$ sets of $m$ residues such that the ...
2
votes
0answers
39 views

Reference Request: Is every interval-valued probability measure consistent?

Short version: Does every interval-valued probability measure contain a conventional probability measure? I have a sense that this is a basic result about an obscure topic but I am having trouble ...
6
votes
0answers
61 views

Chevalley-Eilenberg cohomology of polynomial vector fields on $\mathbb{A}^2$

I have a question similar to one given here. What is the cohomology of the Lie algebra of polynomial vector fields on an affine space $\mathbb{A}^2$ over a field of characteristic $0?$ (I'm interested ...
0
votes
1answer
60 views

Type Theory with no Base Types

Is there any work in type theory where no base types are assumed, e.g., that there are only function types in place ($t_1 \to t_2$ is a type whenever $t_1$ and $t_2$ are types)? If not, are there ...
1
vote
1answer
78 views

Presentations of $\mathbf{𝐏𝐆𝐋}_3(\mathbb{F}_2)$ by three involutions, 2

I am searching for a presentation of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
3
votes
0answers
15 views

Reference request: diffusion approximation to the radiative transport

I'm looking for a good, modern reference for the diffusion approximation to the radiative transport problem. I'm aware of the text of Dautray and Lions, as well as the monograph by Bensoussan, Lions, ...
-1
votes
0answers
52 views

Bound on minimum polynomial value coprime to given integer

Given a polynomial $p(x)$ of degree $d$ in $\mathbb Z[x]$ and an odd integer $n$ is there a technique to know if there is an integer $t\in\mathbb Z_{\geq0}$ such that $\mathsf{GCD}(p(t),n)=1$? If so ...
3
votes
0answers
27 views

Finding particular closed paths in geometric plane regions

Let $X_m$ denote a set of $m\geq 3$ lines in $\mathbb{R}^2$ that are not all parallel. Consider the problem of determining a closed path of $kn$ points in $X_m$ $k, n \in \mathbb{Z}^+$, such that the ...
1
vote
0answers
19 views

What relationship exists between samples of a function and samples of its vector gradient field?

A real function $f(x)$ is defined on $N$-dimensional real space where $N \ge 3$. $f(x)$ is differentiable and its gradient with respect to x is $g(x)$. So $g(x)$ is a vector field. Assume we do not ...
2
votes
0answers
35 views

Analysis of coefficients for quickly vanishing analytic vector field

Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
1
vote
1answer
35 views

Proof of “Prove that a sub-gaussian and isotropic random vector over a finite set T implies that the set is exponentially large”

Here the original question was asked and answered. However I have a question to the solution. If I get it right they try to show $\frac 12 I_n \leq \mathbf{E} YY^T \leq I_n$ by proving $$ \mathbf{E} \...
8
votes
0answers
121 views

Representation theory of $GL_2(\mathbb Z/n\mathbb Z)$

Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $GL_2(\mathbb Z/n\mathbb Z)$ and $PGL_2(\mathbb Z/nZ)$ for varying $n$ and in the limit, for $PGL_2(\...
5
votes
1answer
110 views

The variety induced by an extension of a field

If $K$ is a finitely generated field extension of $k$, then there exists an irreducible affine $k$-variety with function field $K$. The idea is that if $x_1, \dots, x_n$ are generators of $K$ under $k$...
13
votes
0answers
93 views

Graph embeddings in the projective plane: for the 35 forbidden minors, do we know their Colin de Verdière numbers?

The Graph Minor Theorem of Robertson and Seymour asserts that any minor-closed graph property is determined by a finite set of forbidden graph minors. It is a broad generalization e.g. of the ...
0
votes
0answers
27 views

Double commutant theorem when $C^*$-subalgebra does not contain identity operator $1$

Double commutant theorem: For a unital $C^*$-subalgebra $M \subset B(H)$, one has $$\smash{\overline M}^\text{SOT}=\smash{\overline M}^\text{WOT}=M''.$$ My question: For a $C^*$-subalgebra $M \subset ...
5
votes
1answer
83 views

$C^j$-topology considered by Greene and Krantz

My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
3
votes
0answers
49 views

Semisimple subgroup of Euclidean group

Let $G$ be a closed and connected semisimple subgroup of the Euclidean group $E(n)$ (the group of isometries of $\mathbb R^n$). Can we prove that $G$ is conjugate to a subgroup of $O(n)$?
25
votes
2answers
888 views

Intuitively, what does a graph Laplacian represent?

Recently I saw an MO post Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs? that got me interested. ...
-1
votes
0answers
32 views

What is the relationship between two polynomials with the same common factors on a polynomial ring ? [closed]

Let $f(x),g(x)\in\mathbf{Z}_{q^\alpha}[x]$, $q$ is a prime. If the polynomials $f(x)$ and $g(x)$ have the same factors, is there any relationship between $f(x),g(x)$ ? If $f(x),g(x)\in\mathbf{Z}_n[x]$,...
7
votes
1answer
114 views

discontinuous functions on the Sobolev borderline

The Sobolev embedding theorem implies that every function of class $W^{k,p}$ on a reasonable $n$-dimensional domain is continuous if $kp > n$. Cases with $kp=n$ are known as "borderline" ...

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