# All Questions

**3**

votes

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**3**answers

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

**0**answers

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

**3**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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 ...