# All Questions

116,369
questions

**0**

votes

**0**answers

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

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**0**answers

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

**1**answer

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

**2**answers

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**

votes

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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" ...