# All Questions

**2**

votes

**1**answer

64 views

### Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes:
$$
\forall f,g,h\in G:hg(f)=h(g(f))
$$
Now suppose there is additional axiom, or constraint if you prefer, ...

**7**

votes

**1**answer

1k views

### prime numbers and Pythagorean triplets

What are the necessary conditions for two of the terms in the Pythagorean triplet $a^2 = b^2 + c^2$ to be prime numbers?

**0**

votes

**0**answers

14 views

### Hamiltonian potential invariant under lie group action?

Let $(M,g)$ be a riemannian manifold and $G$ a Lie group acting on $M$ by isometries.
Taking a $G$-invariant function $f\colon M \to \mathbb{R}$, we have the riemannian gradient $\operatorname{grad} ...

**0**

votes

**0**answers

8 views

### Can all Local Martingales Be Represented using Only Brownian Motion and Finite Variation Processes?

This is a cross-post of my unanswered (more than a week) question on Math.SE. Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to try to ...

**1**

vote

**1**answer

72 views

### If a Weyl element preserves a root, then it has a representative which preserves the root space?

Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\...

**6**

votes

**0**answers

363 views

+100

### Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...

**2**

votes

**0**answers

21 views

### Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$

Does anybody know an algorithm to solve the following matrix equation?
$$X^{-1}=\sum_{i=1}^n D_i X A_i$$
where $D_i$s are diagonal and $A_i$s are symmetric matrices.

**0**

votes

**0**answers

16 views

### A kind of saturation property related to forcing notions

Forcing is typically done over well-founded models. There are lots of good reasons for this, but it can feel confining at times. Fortunately, we can equally well force over non-well-founded models! It ...

**0**

votes

**0**answers

15 views

### Banach spaces: A ball being a subset of the interior of the union of two balls

Let $X$ be a separable reflexive Banach space and let $A$, $B_1$, and $B_2$ be three closed balls in $X$. Is there a `handy' necessary and sufficient condition for checking whether $A \subseteq (B_1 \...

**0**

votes

**0**answers

10 views

### Expected value of stochastic process

How can i calculate the expected value of $$S_t= S_0 \exp\left( mt-\frac12\int_{0}^{t}e^{2Y_s}ds+\int_{0}^{t}e^{Y_s}dB_s\right)\quad $$ where $${Y_t}$$ is the solution of a sde and follows tha normal ...

**0**

votes

**1**answer

490 views

### Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...

**1**

vote

**1**answer

39 views

+50

### Relaxed path decomposition of a graph

Definition
Given a directed connected graph $G$ without multiple edges or self loops. We call a final path of $G$ a path ending with a vertex with no successor (the path can not be extended anymore) ...

**3**

votes

**1**answer

193 views

### History of the notation for substitution

One of the very common notations for syntactic substitution is $[\ /\ ]$.
There seems to be a large inconsistency in the literature about its use.
Many write $[t/x]$ for substitute $t$ for $x$ (e.g. ...

**11**

votes

**1**answer

208 views

### Multiplicative infinitesimals in q-analogs?

Risking to be downvoted, here is a very lightweight question.
In various fields - say, algebraic geometry, nonstandard analysis, synthetic differential geometry - infinitely small quantities, i. e. ...

**1**

vote

**0**answers

54 views

### Dirac functional embedding

I got the following set up:
Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with the pointwise ...

**4**

votes

**0**answers

110 views

### Classification of representation-finite algebras up to stable equivalence of Morita type

Assume $K$ is an algebraically closed field.
I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...

**0**

votes

**1**answer

71 views

### When do two quasi-Banach spaces with identical dual spaces have equivalent norms?

Let $X$ and $Y$ be two quasi-Banach spaces such that the dual spaces satisfy $X^*=Y^*$.
I want to know if there are some conditions that imply $X=Y$ (in the sense of equivalent norms).

**3**

votes

**0**answers

47 views

### Bernstein sets of large cardinality

A metrizable space $X$ will be called a generalized Bernstein set is every closed completely metrizable subspace $C$ of $X$ has cardinality $|C|<|X|$.
It is well-known that the real line contains ...

**1**

vote

**1**answer

110 views

### Evaluation of sum of factorials

Is there an evaluation of this sum (possibly involving gamma functions)? $k$ and $n$ are natural numbers and $x$ is real with $0<x<1$.
$$ \sum_{\substack{k=0\\n-k\text{ even}}}^n \frac{(-1)^{(n-...

**7**

votes

**0**answers

98 views

### What is the smallest density of a metrizable space without countable separation?

A Tychonoff space $X$ is defined to have countable separation if some (equivalently, any) compactification $bX$ of $X$ contains a countable family $\mathcal U$ of open sets such that for any points $x\...

**1**

vote

**1**answer

52 views

### Concentration of matrix norms under random projection.

Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables.
Are there cases where one can been able to quantify $...

**4**

votes

**1**answer

126 views

### Bijection modeling isomorphism of infinite-dimensional vector spaces

Let $T : V \to W$ be an isomorphism of vector spaces with bases $B_V$ and $B_W$, which may be of any cardinality.
Does there exist a bijection $f : B_V \to B_W$ such that, for each
$b_V \in B_V$,...

**1**

vote

**0**answers

20 views

### $TSU(n)$ completely integrable with 3 $SU(2)$ invariant functions?

Consider the Lie group $SU(n)$ endowed with the standard bi-invariant metric. Then $SU(n)$ can be viewed as a symmetric space of $K_{n,n} := SU(n) \times SU(n)$.
Define $M := K_{n,n} /SU(n)$. Using ...

**11**

votes

**4**answers

1k views

### Is there any generalization of the Dold-Kan correspondence?

The Dold-Kan correspondence gives an equivalence between simplicial abelian groups and chain complexes of abelian groups supported on negative degrees. It actually works for any abelian category. I'm ...

**4**

votes

**0**answers

182 views

### Description of connecting maps of Derived functors

Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\...

**0**

votes

**1**answer

69 views

### the relation between projective and quasi-projective modules

An $R$-module $M$ is called quasi-projective if $\text{Hom}_R(M,M)\to\text{Hom}_R(M,N)$ is surjective for every surjective homomorphism $M\twoheadrightarrow N$.
What are the rings $R$ for which every ...

**3**

votes

**2**answers

162 views

### Maximal Cohen-Macaulay modules of type one

Does any body know an example of a Noetherian local ring $(R,m)$ which admits a maximal cohen-macaulay module of type one, but the ring $R$ itself is not CM.
If $C$ is the maximal CM module then the ...

**1**

vote

**0**answers

62 views

### Tensor and symmetric invariants of Symmetric group

For the action of $S_n$ on $\mathbb C^n$ the elementary symmetric polynomials generate the ring of polynomial invariants. What are the generators for the action of $S_n$ on $\mathbb C^n \otimes \...

**0**

votes

**1**answer

62 views

### Exact formula for computing n-step transition probability of random walks with self-transitions

Consider a semi-infinite random walks $X_n$, $n=0,1,2,\ldots$, whose state space is a set of consecutive integers and whose one-step transition probabilities are $P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}$...

**0**

votes

**0**answers

52 views

### Find equidistant points on surface of sphere [on hold]

Given a sphere, find the maximum number of points can be placed on the surface of the sphere such that all are equidistant from each other.
I am not at all good in maths. Please let me know what can ...

**4**

votes

**1**answer

192 views

### Hilbert modular forms twist-equivalent to their conjugates

Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$.
Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate ...

**1**

vote

**1**answer

128 views

### Left adjoint to Double Nerve?

The well known nerve functor from small categories to simplicial sets has a left adjoint, namely the fundamental category functor. Does the double nerve functor $N^2:2Cat\rightarrow sSSet$ from 2-...

**6**

votes

**4**answers

436 views

### Speed of convergence in Lebesgue's density theorem

Let $\lambda=\text{unif}([0,1])$ be uniform distribution on $[0,1]$ and $B$ be any Borel set. Lebesgue's density theorem states that for $\lambda$-almost all $x\in[0,1]$ the limit
$$\lim_{\epsilon\...

**1**

vote

**1**answer

206 views

### $G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? $G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? [on hold]

As summarized in the title, suppose there is an isomorphism between $G_1 \times G_2$ and $G_3$, is it always true that $BG_1 \times BG_2$ is homotopy equivalent to $BG_3$? If it is not always true ...

**0**

votes

**0**answers

25 views

### Basic Definition and Notations in RWRE [on hold]

From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ ...

**5**

votes

**1**answer

207 views

### Intuitive descriptions of some large cardinals

I was trying to formulate intuitive descriptions of some large cardinals.
Roughly something equivalent to "A manifold is an object which looks like patches of $R^n$ glued together". Not perfectly ...

**2**

votes

**0**answers

35 views

### Approximate unit in C*-algebra with additional properties

In the book about K-theory (a friendly approach) by Wegge and Olsen I came across the following notion: in Lemma 16.4 authors assume that the $C^*$-algebra $A$ possess "(commuting) approximate unit $(...

**4**

votes

**2**answers

685 views

### Unreasonable application of mathematics to the other areas [on hold]

What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science?
I found ...

**6**

votes

**1**answer

133 views

+150

### Lexicographic distribution of irreducible polynomials

Let $A = {\mathbb F}_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its ...

**1**

vote

**0**answers

59 views

### Relative Leopoldt defect

Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$.
Is there a bound of the Leopoldt defect of $M$ ...

**15**

votes

**5**answers

828 views

### Simplicial Model of Hopf Map?

The Hopf fibration is a famous map S3 --> S2 with fiber S1, which is the generator in pi_3(S2). We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces ...

**-5**

votes

**0**answers

56 views

### A property of minimal prime ideals [on hold]

Let $R$ be a commutative ring with $1$, and let $p$ be a minimal prime ideal of $R$. If $p\subseteq I_1+ I_2$, where $I_1$ and $I_2$ are two ideals of $R$, can we deduce that $p\subseteq I_1 $ or $...

**3**

votes

**0**answers

46 views

### Do the normal forms for braid groups really conceal information about better than randomly applying the braid relations?

In braid based cryptography, one typically wants to conceal the way a certain braid $b$ has been obtained. One therefore puts $b$ into some normal form. Since every braid has a unique normal form, the ...

**2**

votes

**1**answer

153 views

### Cambridge Mathematical Tripos papers from late 19th century

Are the scanned images of Cambridge Mathematical Tripos papers from late 19th century available anywhere on Internet?