# All Questions

**3**

votes

**2**answers

479 views

### A double centralizing theorem for finite groups

I have a proof for the following assertion which employs Model Theory. It has certainly a pure group theoretic proof, but what is such a proof? Is the assertion trivial?
Theorem Let $G$ be a finite ...

**5**

votes

**2**answers

172 views

### Identity for Power Series and Binomial Coefficients

This question concerns a combinatorial identity obeyed by power series coefficients. Throughout we let $[x^{M}]\{\phi(x)\}$ denote the coefficient of $x^{M}$ in a power series $\phi(x)$.
Let $k$ be ...

**2**

votes

**1**answer

85 views

### Legendre transform and Lipschitz approximation

Let $(X,d)$ be a compact metric space and $f:X \to \mathbb{R}$ a real valued continuous function. Let us agree that a modulus of continuity means concave, nondecreasing, uniformly continuous function ...

**2**

votes

**0**answers

47 views

### Analysing functions on zero-length intervals and super-small values

Suppose a function that has a pole in $x=0$:
Here we see the graphic of the real part of the Gamma function.
As we can see on it, there is a vertical line at $x=0$ that comes from $-\infty$ to ...

**4**

votes

**0**answers

145 views

+50

### On the boundary of the twindragon

Let $\mathcal T$ be the famous twindragon, i.e.,
$$
\mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}.
$$
Then, as is well known, $\mathcal T$ has a ...

**3**

votes

**0**answers

61 views

+100

### Induced subgraphs of small strongly regular graphs

Consider a strongly regular graph $G$ with parameters $(76,30,8,14).$ Hoffman's bound tells us that $\overline{G}$ has an independent set of size at most $4$ and its not hard to see there are indeed ...

**3**

votes

**1**answer

31 views

### Can we extend a multiplicative linear functional of a closed left ideal on whole of the algebra?

Let B be a closed left ideal of a Banach algebra A. Also, B has a right approximate identity (in B).
If g is a nonzero multiplicative linear functional on B, can we always extend g to a ...

**0**

votes

**1**answer

25 views

### holomorphic continuation

consider the function given by $f(t):=\sum\limits_{n=0}^{\infty}e^{-\left(n+\frac{1}{2}\right)^2t}$ for $t\in (0,\infty)$.
This function can be continued holomorphically for all complex numbers with ...

**16**

votes

**6**answers

580 views

### Under what conditions $\|x-y\|=n\iff\|f(x)-f(y)\|=n.$ for $n\in\mathbf{N}$ implies isometry?

Let $X, Y$ be normed space and $f:X\to Y$ be a mapping. Assume that for all $n\in\mathbf{N}$, $$\|x-y\|=n\iff\|f(x)-f(y)\|=n.$$
Under what conditions this map will be an isometry?
Thanks

**4**

votes

**1**answer

144 views

### Irreducibility of a polynomial

For $n\ge 1$, let $g(x_1,x_2,\ldots,x_n)$ be an irreducible homogeneous polynomial in $n$ variables over a field $k$ and $f(x)$ an irreducible polynomial of $k[x]$. Is $f(g(x_1,x_2,\ldots,x_n))$ ...

**11**

votes

**0**answers

227 views

+50

### Monotone embedding of complete binary tree in hypercube

Embedding different graphs, especially binary trees, in the hypercube has a huge literature. However, I could not find anything if we restrict the embedding to be monotone. So I would like to ...

**0**

votes

**0**answers

11 views

### Solve PDE system almost equal to the Ernst equation of general relativity

I am trying to find a solution to the following elliptic quasilinear system of PDEs that arises from a 1962 mathematical physics theory for which no solutions have ever been found:$$\Delta G+\nabla ...

**1**

vote

**1**answer

177 views

### Divisibility of divisors in some tori and lattices

Let $E$ and $E'$ be two general elliptic curves. We consider the $2$-dimensional torus $A:=\frac{E\times E'}{(u\times u')\left((\mathbb{Z}/2\mathbb{Z})^2\right)}$, where ...

**0**

votes

**0**answers

31 views

### Properties of “incomplete finite simplicial complexes”

Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K.
...

**7**

votes

**1**answer

103 views

### A positivity problem involving the number of ways of expressing $n$ as a product of $k$ factors

Let $d_k(n)$ denote the number of ways of expressing $n$ as a product of $k$ factors, and let $$D_k(x)=\sum_{n\leq x}d_k(n)$$ be the summatory function. During a study of Mertens' function I was lead ...

**0**

votes

**0**answers

13 views

### Tiling with restricted overlap

Non-overlapping tilings of regions is a well-studied topic.
I wonder if the following variant has been considered:
A tile can be partitioned into several regions, where such regions from different ...

**13**

votes

**2**answers

215 views

### Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can ...

**0**

votes

**0**answers

48 views

### How do you differentiate $x_i^j\sum_i\sum_jx_i^j$? [on hold]

I'm trying to find the jacobian of of a function that contains matrices, e.g.
$$\dot{x_i^j} = \ \ x_i^j\sum_i\sum_jx_i^j + \sum_i\sum_jx_i^jy \ \ \ + \ ... \ = \ \ f(x_i^j,y) $$
Where $x_i^j$ is ...

**16**

votes

**2**answers

910 views

### Two functors from Grp to Grp?

It has been many years since I first read Categories for the Working Mathematician, but I still have a question about one of the first exercises. Question 5 in section 1.3 asks you to find two ...

**0**

votes

**1**answer

57 views

### Uniqueness of a Integro-parabolic differential equation?

Let $r, q,\lambda,\sigma,\kappa,\mu$ are positive real numbers and let $c(t)$ is a differential function of $t$. $\Gamma(\eta)$ is a probability density function.
When I consider price of American ...

**8**

votes

**0**answers

74 views

### How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...

**8**

votes

**1**answer

257 views

### Descending chain condition in noncommutative rings

By Hopkins Theorem it is well-known that every right (resp. left) artinian unitary ring is right (left) noetherian. Suppose that a noncommutative unitary ring R satisfies the descending chain ...

**3**

votes

**0**answers

23 views

### Identities for ${~}_3\phi_1$?

I am looking for some source of summation formulas for the $q$-hypergeometric function ${~}_3\phi_1$ in the sense of Gasper-Rahman book. For some reason, the book focuses heavily on ${~}_{r+1}\phi_r$ ...

**3**

votes

**1**answer

92 views

### Can every finite abelian $p$-group with duality pairing be written as cokernel of a symmetric matrix over the $p$-adic integers?

Let $G$ be a finite abelian $p$-group (where $p$ is a prime). Suppose there exists a symmetric bilinear map $\delta\colon G\times G\to \mathbb{Q}/\mathbb{Z}$ such that the induced map $g\to\langle ...

**4**

votes

**0**answers

69 views

### Continuous-piecewise-linear versus piecewise-linear

Some authors use the term "continuous piecewise-linear" where other authors use the shorter term "piecewise-linear" (with continuity tacit).
I'd be interested in people's thoughts about this ...

**1**

vote

**0**answers

80 views

### Reference request for the relation of Ext groups and bar construction

I need a reference for the description of Ext groups in mixed categories (i.e. abelian categories with a weight filtration and semisimple graded quotients) by using the bar complex, as mentioned in ...

**2**

votes

**0**answers

46 views

### Multiplicativity of combinatorial l classes

For closed smooth manifolds $M$ and $N$, the Hirzebruch $L$ class is multiplicative, i.e. $L(M\times N)=L(M)L(N)$. Is this property still true if $M$ and $N$ are assumed to be closed topological ...

**4**

votes

**2**answers

126 views

### Largest eigenvalue adjacency matrix-link deletion

Let G be a connected undirected graph and G\e be a graph obtained by removing a random link e from the graph G. Let $\lambda_1(A(G))$ be the largest eigenvalue of the adjacency matrix of graph G. Is ...

**6**

votes

**2**answers

281 views

### Surjectivity of curl

Let: $\mathbb R^3\ni x\mapsto v(x)\in\mathbb R^3$ be a vector field with null divergence belonging to the Schwartz class such that
$$
\int_{\mathbb R^3} v(x) dx=0.
$$
Is it true that there exists a ...

**4**

votes

**2**answers

342 views

### Artin Jacobson-semisimple rings are semisimple. Constructively, too?

Notation. When I say "ring", I mean "ring with unity" (not necessarily commutative).
Definition. A ring $R$ is said to be left-Artinian if for every sequence $I_0\supseteq I_1\supseteq I_2\supseteq ...

**-4**

votes

**0**answers

46 views

### the triangle inequality for shortest paths of graphs [on hold]

In why-the-triangle-inequality
I found the statement:
for example if $d(a,b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite ...

**1**

vote

**0**answers

113 views

### Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?

Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective?
The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments ...

**10**

votes

**1**answer

172 views

### Parametric solutions of Pell's equation

Given a positive integer $n$ which is not a perfect square, it is well-known that
Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$.
Question: Let $n$ be a ...

**10**

votes

**1**answer

111 views

### Choosing a metric in which homeomorphism is Holder continuous

Let $X$ be a compact metrizable space, and let $f:X \to X$ be a homeomorphism. Is it always possible to choose a compatible metric on $X$ in which $f$ is Holder continuous? I've tried some simple ...

**0**

votes

**0**answers

133 views

### Is this topological looking space metrizable?

First of all I am sorry if something is missing or if I am making a mistake. I am a bit new in this field. The question is as follows:
The topology that is considered is called gross error model, or ...

**2**

votes

**0**answers

44 views

### Convergence in distribution of stochastic equation solutions

I post this post en MSE (link) but I think that is more suitable for this site.
I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the ...

**0**

votes

**0**answers

158 views

### Lower bound for constructing $n!$ with additions, subtractions, and multiplications starting from $1$

Found this on Complexity Zoo warning expired certificate
check NP Over The Complex Numbers.
[BCS+97] show the following striking result. For a positive integer $n$, let $t(n)$ denote the minimum ...

**5**

votes

**1**answer

251 views

### Adding a real with infinite conditions

Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion.
Does $\Bbb P$ ...

**0**

votes

**1**answer

55 views

### Practical Algorithm for Comparing the Discrepancy of Point Sets (on Unit Hyper Spheres)

I have devised a simple geometric algorithm for generating a sequence of points on unit hyper spheres; that algorithm depends on a single real parameter, which I would like to optimize in order to get ...

**5**

votes

**1**answer

207 views

### Some calculus in $O(n)$

How can one compute each of the following matrices, explicitly:
$$\int_{O(n)} e^{g}dg$$ or
$$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$
What is the explicite entries of the resulting ...

**1**

vote

**0**answers

59 views

### Probabilistic statement on matrix ranks

Given $A\in\{0,1\}^{n\times n}$ with $rank(A)=r=2^{O(\sqrt{\log_2n})}$.
Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s.
Does
$$\lim_{n\rightarrow\infty}\mathsf{P_{A\in\{0,1\}^{n\times ...

**-3**

votes

**0**answers

162 views

### Why is SO(3) not $S^1 \times S^2$? (Where is the mistake?) [migrated]

I was trying to calculate the fundamental group of SO(3). In order to represent the group I reasoned the following way:
In order to build the 3X3 orthogonal matrix I need an orthonormal positive ...

**11**

votes

**0**answers

148 views

### Elements of order 3 normalizing no non-identity 2-subgroups in Almost Simple Groups

This question is partly motivated by a situation which arises in modular representation theory. A finite group $G$ is said to be almost simple if $G$ has a unique minimal normal subgroup which is a ...

**33**

votes

**4**answers

8k views

### Non Borel sets without axiom of choice

This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of ...

**22**

votes

**6**answers

3k views

### Why the triangle inequality?

[Maybe this is asking to be closed; but I thought I'd risk it.]
A metric satisfies the axioms:
$d(x,y)=0$ if and only if $x=y$.
$d(x,y) = d(y,x)$.
$d(x,y) \leq d(x,z) + d(z,y)$.
Similarly (and ...

**5**

votes

**1**answer

325 views

### Faithful transitive actions by large groups on small sets

How large is the largest transitive subgroup of $S_n$ other than itself and $A_n$? In particular, does its size grow at least exponentially in $n$?

**5**

votes

**1**answer

137 views

### Generic filters of inverse limits

Maybe this doubt is silly, but I do not understand the final step of the proof of Lemma 5.2 in Hamkins' paper Fragile measurability, Journal of Symbolic Logic 59 (1994) 262-282.
There, $\mathbb ...

**6**

votes

**2**answers

194 views

### Decidability of diophantine equation in a theory

Given a theory $T \subseteq \operatorname{Th}(\mathbb{N})$, define the decision problem $D_T$ as follows:
Given a polynomial $p$ with integer coefficients and variables $\bar{x}$, decide whether
...

**2**

votes

**0**answers

42 views

### Compiling self-referential forms

Fix $1\leq d\in\mathbb{N}$ and set $D:=\{0,1,\ldots,d-1\}$.
Consider the system of equations
\begin{equation}
x_i=c_i + \sum_{j\in D}\delta_{x_j,i}
\end{equation}
with $c_i\in D$ given and $x_i\in D$ ...

**7**

votes

**1**answer

187 views

### When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For ...