# All Questions

**3**

votes

**1**answer

126 views

### Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$
such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...

**-3**

votes

**0**answers

25 views

### Model of function of 2 random variables [on hold]

In my model W = f(E, K). f is a complex function (several operations on E and K).
for any W, infinity pairs of (E, K) exist that satisfy f.
E and K are between [0, +oo]
I have observations for W ...

**3**

votes

**0**answers

26 views

### On a continuous extension of a linear 2nd order PDE

Consider an elliptic (hyperbolic) equation
$A(x,y) u_{xx} + 2B(x,y) u_{xy} + C(x,y) u_{yy} = 0$
in a bounded open plane set $D$, with real-valued functions $A$, $B$, and $C$. Is it true that at least ...

**1**

vote

**0**answers

37 views

### How the modular theory of von Neumann algebras, deal with generating C*-algebras?

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Suppose the existence of a bicyclic ...

**1**

vote

**0**answers

47 views

### Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]:
Let $K$ be a real ...

**0**

votes

**1**answer

106 views

### Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$.
Let $p,q \in M_{\infty}(A)$ be ...

**3**

votes

**1**answer

376 views

### What's so special about $1$-categories?

I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms ...

**0**

votes

**0**answers

52 views

### Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?

If $\zeta(s)$ is the Riemann Zeta function, then $\zeta(n)^z$, with $z \in \mathbb{C}$, $\Re(s)>1$, can be represented as
$$\zeta(s)^z=\sum_{n=1}^\infty \frac{d_z(n)}{n^{-s}}$$
where $d_z(n)$ ...

**0**

votes

**0**answers

15 views

### What happens to the angles of an isosceles triangle if one vertex is at infinity? [migrated]

My son and I were trying to decide whether an isosceles triangle can ever have 90 degree base angles. I would argue that if the two equal length sides are both infinitely long, they must have 90 ...

**2**

votes

**0**answers

54 views

### variants of ramification groups - need terminology and sources

I've asked this question in several more elementary forums, and haven't get any answer. So I presume this is not so an elementary question.
Let $L/K$ be a Galois extension, and $w$ be a valuation of ...

**1**

vote

**1**answer

123 views

### Bounded dg algebra vs unbounded dg algebras

1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull ...

**7**

votes

**0**answers

309 views

### Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:
Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...

**1**

vote

**1**answer

110 views

### Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that
$$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ...

**0**

votes

**0**answers

80 views

### this sequence $A_{n}$ have recursive relations?

Let $$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$
I want find the $A_{n}$ recursive relations,such as following form
...

**0**

votes

**1**answer

140 views

### How to solve this differential equation with an infinite sum?

I would like to find solutions of the following differential equation:
$ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$
For example in space of function from $\mathbb R^*$ to $\mathbb ...

**2**

votes

**3**answers

675 views

### Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...

**5**

votes

**1**answer

208 views

### Is there a Riemann existence theorem for orbifolds?

For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified ...

**1**

vote

**0**answers

44 views

### Floquet-Bloch solutions of the quasiperiodic Schrödinger equation

I am concerned about the Schrödinger equation
$-x''(t)+q(t)x(t)=Ex(t).$
Here, the potential $q$ is real and quasiperiodic with frequency vector $\omega$. That is, we let $T^d$ be the $d$-dimensional ...

**0**

votes

**0**answers

15 views

### how to efficiently compute the mean function for non-homogeneous poisson process?

Suppose that I know all intensity functions lambda(t) during given period [0,t], how can I compute the mean function m(t) for non-homogeneous Poisson process?
Basically, m(t) in the integral of ...

**1**

vote

**0**answers

41 views

### Intersection of ordinary subspaces at different primes

Choose two distinct primes $\ell$ and $\ell'$, and embeddings $\iota_\ell : \overline{\mathbb Q} \to \overline{\mathbb Q_\ell}$, $\iota_{\ell'} : \overline{\mathbb Q} \to \overline{\mathbb ...

**3**

votes

**1**answer

169 views

### Strong Morita equivalence and representation theory

In the context of pure algebra we say that two algebras (in general: rings) $A,B$ are Morita equivalence when there are bimodules $_AP_B,_BQ_A$ such that $P \otimes_B Q \cong _AA_A$ and $Q \otimes_A P ...

**3**

votes

**1**answer

88 views

### Eigenvectors of a symmetric positive definite Toeplitz matrix

I wish to efficiently compute the eigenvectors of an n x n symmetric positive definite Toeplitz matrix K. A full eigendecomposition would be even better.
Although I assumed this would be a well ...

**1**

vote

**2**answers

232 views

### How many k-subsets of the integers {1,…,n} sum to N?

Given the set of integers $S = \{1,..n\}$, how many subsets of $S$ with $k$ elements sum to $N\in \mathbb Z$?

**11**

votes

**3**answers

558 views

### Complete resolutions of GCH

Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ...

**2**

votes

**1**answer

81 views

### Simple Isogeny Question

I'm looking for a reference of an isogeny fact that I've used many times but am having a hard time proving formally.
One can define the degree of an isogeny as the degree of extension fields of the ...

**-5**

votes

**0**answers

39 views

### transition matrix [on hold]

Gene mutation. Suppose a gene in a chromosome is of type $A$ or type $B$. Assume that the probability that a gene of type $A$ will mutate of type $B$ in one generation is $10-4$ and that a gene of ...

**11**

votes

**1**answer

162 views

### Nonperiodic points of homeomorphisms of a ball

Suppose $B$ is a $d$-dimensional ball (for some $d \geq 1$) and $T$ is a homeomorphism from $B$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that ...

**0**

votes

**0**answers

41 views

### Does an arbitrary product of $f$ and $f^\dagger$ belong to a universal enveloping algebra of the Heisenberg algebra? [migrated]

The Heisenberg algebra is essentially the canonical commutation relations (CCR) for bosons $[f,f^\dagger]=1$. $f$ is called an annihilation operator in physics ($f^\dagger$ creation operator). ...

**2**

votes

**0**answers

38 views

### Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...

**2**

votes

**1**answer

205 views

### Example of a Frobenius algebra that is not projective over a Frobenius subalgebra

I'd like to know an example of a Frobenius algebra $A$, with a subalgebra $B$ that is itself a Frobenius algebra, such that $A$ is not projective as a left $B$-module. I don't require any ...

**14**

votes

**2**answers

1k views

### The letters of the word “ART”

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is ...

**-3**

votes

**0**answers

61 views

### Linear algebra over principal rings 1 [on hold]

If N is a left-idea of ring R and R is a left R-module, then submodule N is a direct sum of R if and only if N has a right unit.

**14**

votes

**1**answer

790 views

### Joyal's letter to Grothendieck

Mostly out of curiosity: Where do I find Joyal's letter to Grothendieck in which he defines a model structure on simplicial sheaves?
The question was already asked in this MO post, but that ...

**-1**

votes

**1**answer

161 views

### journal to submit mathematic books' review

it has been asked to me to write a review on a book about the history of mathematics in Italy between the two world wars.
The book is a non-technical one. I would like to know which journal accepts ...

**0**

votes

**0**answers

127 views

### On matrix rank inequality

Let $A$ be a $\{0,1\}$ square matrix.
Let $J$ be all $1$ matrix.
Let $\bar{A}=J-A$.
Is it possible for $rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1$ and $rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1$ for some ...

**5**

votes

**1**answer

212 views

### Can ergodic theory help to prove ergodicity of general Markov chain?

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...

**2**

votes

**2**answers

201 views

### Real algebraic solution

Suppose a system of polynomial equations with rational coefficients has a real solution. Does necessarily there exists a real solution with algebraic coordinates? What about the simplest case of one ...

**3**

votes

**3**answers

197 views

### Injective map between two schemes

Assuem we have a finite surjective map between two irreducible, separated schemes, $f:X \rightarrow Y$, and for a dense open $U \subset Y$ and for any $y \in U$, $|X_y| =1$, then can we say $f$ is ...

**1**

vote

**2**answers

131 views

### Dual connections for Information Geometry

In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied ...

**-2**

votes

**1**answer

128 views

### Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [on hold]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that
$$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$
...

**4**

votes

**1**answer

274 views

### Results about moduli of surfaces

There are early successes of the moduli theory
- the construction and compactification of the moduli spaces of curves $\overline{\mathcal{M}}_g$ .
I want to study about the moduli of algebraic ...

**-2**

votes

**0**answers

59 views

### Finding an example for [on hold]

Let $\varphi$ be a periodic function s.t. at zero and every integer points it is equal to 1. Moreover it's equal to one in at least one point between each integer. Can we have two distinct density ...

**2**

votes

**0**answers

72 views

### invariant measures of the expanding maps on the circle

I would be very happy to know about original references for the following results;
For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1)
(1) There exist ...

**2**

votes

**1**answer

88 views

### How to construct a graph with arbitrarily large girth and large chromatic number? [on hold]

Erdos theorem says it is possible and it is not so easy. What is the general procedure to construct graphs like Grötzsch graph?

**0**

votes

**0**answers

13 views

### Looking to derive bound for modulus of harmonic eigenfunction on weighted graph

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus:
$\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$
and wish to lower ...

**-4**

votes

**0**answers

45 views

### Height of a tripod [on hold]

I am trying to determine the height of a tripod when the length on the tripod's legs (81") and the distance between the ends that touch the ground are 57" apart.
My thought process so far: When the ...

**4**

votes

**1**answer

401 views

### Beginners Guide to Cartan for Beginners [on hold]

I am working through parts of Cartan For Beginners by Ivey and Landsberg. Thankfully some exercises have solutions, but, we would benefit from some additional guidance.
Question: I am seeking ...

**-3**

votes

**0**answers

118 views

### Cylinder in a topological space? [on hold]

There is a notion of path in a topological space $X$, namely a continuous function f with domain X and codomain the interval $[0, 1]$.
Given that, in a Quillen model category, the dual of a path ...

**0**

votes

**0**answers

37 views

### Is the blowing up the rectifiable set cone?

Let $M$ be a rectifiable set in $\mathbb{R}^N$. For any point $p\in M$, is the following true?: $\lambda_i M$ subconverges to a cone in $p$ for $\lambda_i\to\infty$, i.e. $(\lambda_i M)\cap B(p,R)$ ...

**-2**

votes

**0**answers

48 views

### Proof of Kolomogrov-Sinai Theorem [on hold]

I've seen reference to the result, but have not been able to actually locate a proof of said theorem. If anybody here could point me that way, then I'd greatly appreciate it.