3
votes
0answers
62 views

Minimal algebraic degree of symmetric unit distance embedding of Heawood graph

I'm looking at embeddings of the Heawood graph in the plane as unit distance graph. Apparently the first such embedding was given by Gerbracht, 2009 and has algebraic (over the rationals) coordinates ...
0
votes
0answers
79 views

What is the sharpest bound of this sum?

Fix $y\geq 1$ and let $\delta$ be a small enough positive real number. Put $$\mathcal{D}^{+}=\left\{d=p_{1}...p_{l}: p_{l}<...<p_{1},\ p_{m} \leq y_{m} \ \textrm{for all odd} \ m \right\},$$ ...
0
votes
1answer
79 views

A uniform Lebesgue density theorem

The Lebesgue density theorem in $\mathbb{R}^n$ may be stated as follows. For a Lebesgue-measurable $A\subseteq\mathbb{R}$ and $r>0, x\in\mathbb{R}^n$, define $$ \chi_{A,r}(x)=\frac{\mu(A\cap B_r(x))...
-4
votes
0answers
71 views

How do I go about solving the following problem? [closed]

Given $(m_1,m_2, ...,m_r)\in Z^r_{\geq 0}$, and $a_1, a_2, · · · , a_r \in \mathbb{N}$ such that: $\sum_{i=1}^r a_im_i=qL$ where $L$ denotes the least common multiple of $a_1, a_2, · · · , a_r$ and $...
3
votes
1answer
129 views

Number of distinct variables used in axiomatizating (Classical) Propositional Logic

The first part of the present question is concerned with Classical Propositional Logic (CPL). The second part involves its fragments or alternative logical systems. There are in the literature many ...
3
votes
0answers
75 views

Random polyominoes containing $2\times2$ squares

The construction quoted in the question "How to sample a uniform random polyomino?" claims to produce a "uniform random polyomino". But apart from the mentioned possibility of getting stuck, it also ...
0
votes
0answers
55 views

An idea of two sided invertible matrix [closed]

Can you suggest me a reading material that expand on the idea of two matrices that are inverse to each other but which are not square matrices? Here's what I think of, take $A$ a matrix of order $n\...
0
votes
0answers
88 views

Quotient of Cohen-Macaulay ring

Let $A$ be a Cohen-Macaulay ring, and $I$ an ideal of $A$. What can we say about $\operatorname{depth}(A/I)$? I know that $\operatorname{depth}(A/I)\le \dim(A/I)$.
8
votes
0answers
228 views

De Rham Cohomology in positive characteristic

This question is about the possible existence of a "compactified" algebraic De Rham cohomology in positive characteristic. Namely, one knows that, for a smooth, but not proper, variety $U$ over a ...
7
votes
2answers
518 views

When is a functor a right derived functor?

Suppose we have Grothendieck abelian categories $\mathcal{A}, \mathcal{B}$. Suppose also we have given an exact functor of triangulated categories $$ F \colon D(\mathcal{A}) \to D(\mathcal{B}) $$ ...
-5
votes
1answer
88 views

What are the most important mathematical prerequisites for machine learning? [closed]

Next week I like to start the machine learning class with Andrew Ng and now I like to brush up on some mathematical topics. My inquires let me to some recommendations: Linear Algebra: matrices ...
1
vote
1answer
185 views

What is the structure group of the Hopf fibration $S_1\rightarrow S_3 \stackrel{p}\rightarrow S_2$?

I am studying fiber bundles and have thoroughly reviewed the famous example of the Möbius strip. In that example, I learned how to discover that the structure group of the Möbius strip fiber bundle ...
1
vote
0answers
44 views

BV functions with values in metric space

$ \newcommand{\IR}{\mathbb{R}} \newcommand{\IN}{\mathbb{N}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\divergence}{\operatorname{div}} \newcommand{\Lip}{\operatorname{Lip}} $ Let $E$ be a ...
0
votes
0answers
21 views

Existence of solution to first order pde [closed]

Let $U : = \big(-\frac 12, \frac 12 \big)^2 \setminus B_R(0)$ for some sufficiently small $R > 0$. I would like to prove the existence of a solution $\rho = \rho (x_1, x_2)\in C^1(U)$ to the ...
1
vote
0answers
64 views

Eigenvalues of the Laplacian modulo primes

Let $d\geq 1$ be an integer and consider $M = \mathbb{R}^d / \mathbb{Z}^d$. Let $\Delta$ be the Laplace operator on $M$. Then the eigenvalues are $$\frac{1}{4 \pi^2}\lambda_{\vec{k}} = |\vec{k}|^2.$$ ...
8
votes
1answer
112 views

Small set such that $\{1 , \ldots , n\} \cdot A = \mathbb{Z} / p \mathbb{Z}$

Let $p$ be a large prime and $n < p$. What is the smallest size of a set $A \subset \mathbb{Z} / p \mathbb{Z}$ such that $A \cdot \{1 , \ldots , n\} = \mathbb{Z} / p \mathbb{Z}$? Here $\cdot$ ...
5
votes
2answers
161 views

Some examples of $\mathbb Q$-Gorenstein smoothing

I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition. Definition. For a normal projective surface $X$ with quotient ...
0
votes
0answers
33 views

Controlling the size of the balls in Hausdorff dimension/measure

Let $X$ be a compact metric space, and let $$ \nu_s(X):=\sup\limits_{\varepsilon>0} \inf\limits_{\mathcal{E}} \sum\limits_{E \in \mathcal E} \mathrm{diam}(E)^s $$ be the $s$-dimensional Hausdorff ...
6
votes
3answers
360 views

linear independence of $\sin(k \pi / m)$

I have tried searching the literature for a result like the following, but have not found anything. For a positive integer $m$, is it known that $$\{ \sin (k \pi / m): 1 \leq k \leq m/2, (k,m)=1 \}$$ ...
0
votes
1answer
35 views

What is the shatter coefficient / VC - dimension of some hypothesis set?

Let $H:=\{h:\mathbb{N}_0^n \rightarrow \{0,1\}| h(x_1,\cdots,x_n) = \mathbb{1}_0(\sum_{i\in I}{x_i}-\sum_{j \notin I}{x_j}) \text{ for some } I \subset \{1,\cdots,n\}\}$ where $\mathbb{1}$ is the ...
1
vote
0answers
33 views

What mathematical background is preliminary for reading and understanding books/papers on wavelets? [migrated]

Please excuse my english. I have had the following math courses for mechatronics engineering education: Calculus (single and multivariable) Linear algebra (introductory) Differential equations (ode'...
0
votes
0answers
8 views

Representation Theorem for levy process

The answer given by The Bridge, Martingale representation theorem for Levy processes was useful for me. Thank you. But I have a question, can this theorem be given for $X_t$ being not just $R^n$ ...
5
votes
0answers
82 views

Constructible sheaves on general stratified spaces

I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...
0
votes
0answers
21 views

Interpolation inequality for fractional Sobolev spaces

In Theorem 5.2 of the book Robert A. Adams and John J. F. Fournier, MR 2424078 Sobolev spaces, ISBN: 0-12-044143-8. is stated that, for any integers $0 \leq j \leq m$ and $u \in W^{m,p}(\Omega)$ (...
1
vote
2answers
88 views

Factorial Series

Is there a closed form expression for $$ \sum_{k=n}^\infty\frac{k!^2}{(k+x)(k-n)!(k+n+1)!} $$ where $0<x<1$ ? (For $n=0$, I know that $$\sum_{k=0}^\infty\frac{1}{(k+x)(k+1)}=\frac{\psi(x)+\...
-2
votes
0answers
18 views

Drawing graph with some different metric values [closed]

I have some metrics but all of them have different values. Some are decimals, some are integers and some are large numbers. Assume the metrics are (average values): - metric1 - 1500 - metric2 - 0....
2
votes
1answer
65 views

Version of Donsker-Invariance-Principle

Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$...
-2
votes
1answer
120 views

Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]

I am reading from the book Topics in Galois theory by Serre. I have the following question , take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by $$\sigma x\;=\;1/(1-x)$$ where $\sigma$ ...
0
votes
1answer
154 views

Characters and Galois stability

Let $G$ be a finite abelian group and $\widehat{G}$ the character group. Let $S \subset \widehat{G}$ be a Galois-stable subset i.e. if $\chi \in S$, then the Galois conjugates $\chi^{\sigma} \in S$ ...
-1
votes
0answers
22 views

Equality of sum of fractions implies correspondence of terms [closed]

I am working in a theorem of Jhonson and Newman about cospectrality and got stucked un this claim. can you help me? $a_i$ and $b_i$ are non negative numbers, $z\in\mathbb{C}$ and $d_i \neq d_j$ for $...
0
votes
1answer
61 views

On some examples of critical families

I'm reading the book on Injective choice functions by Holz, Podewski and Steffens, and I find it to be at the same time well written and quite difficult. It has almost no examples - and in quite a few ...
4
votes
1answer
147 views

Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?

I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory. As I understand it, there is a very special class of ...
2
votes
1answer
74 views

Minimize matrix distance to tensor product

Minimize the following function: $ f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product. ...
0
votes
0answers
49 views

A generalization of the Tucker circle theorem and the Thomsen theorem associated with a conic

I gave a generalization of the Tucker circle theorem and the Thomsen theorem at here. Now, I give a more generalization of these theorems as following: Problem: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, ...
0
votes
0answers
35 views

Software for matching theorems to inputted conditions/hypotheses

Many times I find myself going through analysis books, wikipedia and papers, looking for what is known for my functions/objects at hand. So is there any software that at least tries to move in that ...
3
votes
1answer
114 views

Does this PDE have a name?

I'm looking for any and all information that might be known about the following second-order PDE for one function $u(x,y)$: $ u_{xy} = u_x e^u + u_y e^{-u} $ e.g., Does it have a name? Is it known ...
6
votes
2answers
97 views

Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables

Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...
7
votes
3answers
197 views

When can the Cayley graph of the symmetries of an object have those symmetries?

Let $P$ be an object in $\mathbb{R}^n$ with symmetry group $G$. Let $C$ be the a Cayley graph of $G$. When can $C$ be embedded in $\mathbb{R}^m$ so that the embedded graph has the same symmetry ...
2
votes
0answers
87 views

example of fuchsian groups acting on 2-sphere by G. Martin

Currently I am reading a paper "Infinite group actions on spheres" by Gaven Martin. I am a first year graduate students and I got lots of questions, so one of them is about the following example: (...
10
votes
3answers
282 views

How to sample a uniform random polyomino?

A polyomino is formed by joining finitely many unit squares edge to edge. It may be regarded as a finite subset of the regular square tiling with a connected interior. In particular, for us, ...
29
votes
2answers
624 views

$x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?

This question was asked on MathStackexchange here, but there was no answer, so I am asking it here. Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...
-4
votes
0answers
97 views

Finite groups are isomorphic [closed]

For two finite groups $G_1, G_2$ if for every integer $n\geq 0$, $|G_1^n| = |G_2^n|$, then is it true that $G_1\cong G_2$? By $G^k$ we mean set $\{g^k|g\in G\}$.
5
votes
2answers
193 views

Criterion for being reflexive via Ext

In this question it was claimed that if a module $M$ over a noetherian domain $R$ satisfies $\rm{Ext}^i(M,R)=0$ for $i=1,2$, then $M$ is reflexive. Is this true? Does someone know a reference or a ...
-3
votes
0answers
139 views

How to explain the subject Operator Algebra to dummies? [closed]

I have to appear for an interview to pass the requirements of AI instructor. I have to explain what is operator algebra which is my subject to Professors of Department of Second Language Studies. Any ...
-4
votes
0answers
27 views

summation of operators in Lp space [closed]

How can we find upper bounded for following term in L∞ space? the term is : ||u_1 C_(φ_1 )+ u_2 C_(φ_2 )|| according to this point that the terms u_1 C_(φ_1 ) and u_2 C_(φ_2 ) are not bounded. ...
1
vote
0answers
54 views

Reference on Probability theory on functional spaces (in special Hilbert spaces)

Currently, I am working on some sort of stochastic optimization problems defined over function spaces. I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")...
0
votes
0answers
67 views

(∀x.(p(x)⇒∀x.p(x)) )= ((∃x.p(x))⇒(∀y.p(y)))? [closed]

In dealing with my homeowrk, someone has told me ∀x.(p(x)⇒∀x.p(x)) could be transformed to (∃x.p(x))⇒(∀y.p(y)) However, intuitively, this doesn't make sense to me, could anyone give me a ...
1
vote
0answers
109 views

Vector bundles and equivariant vector spaces

It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles. In so far as I understand it, the reason for that is the ...
-1
votes
1answer
155 views

When the Ratio of Two Factors is a power of $2$ (i.e. $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$)

We can write, $n!= 2^s \times a \times b$ where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$. Problem: Is there infinite number of $n$ when $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$? Question: 1.How ...
0
votes
0answers
70 views

Variation on stones in buckets

This is a spinoff, see Collecting stones in n buckets. Frankly speaking my only motivation is that I became curious: what happens if one redistributes the stones into the same buckets? More ...

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