-1
votes
1answer
71 views

Variety of commutative semi group [on hold]

V is a variety of commutative semi group satisfying the identity $x^2 = x^3$. I need to prove that: $|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$. Any hints on this ? $F_V$ is V-free algebra.
5
votes
2answers
184 views

How to define the square root of $1-\Delta $?

If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$? The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential ...
-3
votes
1answer
77 views

SHPS and SPHS inequality using monounary algebra

Let $A_n = \{(1,\ldots,n) , f \}$ where $f(i) = (i+1)$ if $i \neq n $ otherwise $f(n) = 1$. This describes a mono unary algebra. The proof for $HPS \neq SPHS$ I know uses metabelian groups and was ...
2
votes
1answer
85 views

Is every sufficiently dense well mixed set an additive basis?

Let $B \subset \mathbb{N}$ be a set of natural numbers such that $|B \cap [1,N]| \sim N^\gamma$, for some $\gamma > 0$ with the following property: For any pair of positive integers $k,n$ we ...
13
votes
0answers
156 views

A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$? This question is motivated ...
3
votes
0answers
83 views

“Almost” prime k-tuples in intervals

In this paper, Heath-Brown proves that there are $ \gg x(\log{x})^{-k}$ integers $n \leq x$ such that $$ \prod_{i=1}^{k} (a_{i}x + b_{i}) $$ is squarefree and that each term has a "small" number of ...
4
votes
2answers
498 views

Why Cech cohomology does not compute sheaf cohomology on an open annulus

Let $A=\{z\in\mathbf{C}:1/2<|z|<1\}$ be an open annulus. Let us cover $A$ by 3 open sets: $U_0,U_1$ and $U_2$ which we assume to be all homeomorphic to a 2 dimensional open disc. Moreover, we ...
0
votes
1answer
72 views

Normal coordinates near the boundary

Let $M$ be an Riemannian manifold with boundary $\partial M$ and $e_n$ be a unit normal vector on $\partial M$. With respect to $e_n$, around a point $p$ on boundary, we have the usual normal ...
0
votes
1answer
60 views

Equivalent of Stirling-like numbers

let $b_{n,k}$ be the numbers defined formally by $$X^n=\sum_{k=0}^n b_{n,k}\binom{X}{k}$$ where $\binom{X}{n}=\frac{1}{n!}\prod_{k=0}^{n-1}(X-k)$. I am looking for an equivalent of $b_{n,k}$ when $k$ ...
-1
votes
0answers
42 views

Is an even-dimensional real projective space (RP^2 or RP^4) a spin(or spin^C) manifold or not? [on hold]

I have a dumb question. Let us consider an even-dimensional real projective space (for instance, RP^2 or RP^4). I wonder if those spaces allow spin structure. In other words, is the real projective ...
1
vote
0answers
30 views

Pseudofunctors of 2-variables and Gray tensor product of bicategories

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be bicategories. We say a 'pseudofunctors of 2-variables' consists of two families of pseufunctors Fix $A\in obj\mathcal{A}$, we have a pseudofunctor ...
2
votes
1answer
79 views

M/M/1 Queue with probability of new customer leaving

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...
4
votes
2answers
145 views

Generic Ultrapower as a Class

If $X$ is a set and $I$ is an ideal on $X$. Let $\mathbb{P}$ be the forcing poset consisting of $I^+$ subsets of $X$ with the subset partial ordering. Let $G$ be $\mathbb{P}$-generic filter over $M$, ...
1
vote
0answers
57 views

Finite Volume 1D Anderson Tight Binding Model

My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the ...
6
votes
1answer
140 views

Limit of distance between two random points in a unit-radius $n$-sphere

This is a companion contrast to the earlier analogous question for unit $n$-cubes, where the answer (provided by several respondents) is $\infty$ . What is the limit, as $n \to \infty$, of the ...
2
votes
3answers
170 views

is the tensor product of projective modules again projective?

Let $R$ be a commutative ring and let $A_1$ and $A_2$ be (not necessarily commutative) $R$-algebras. Under which conditions on $A_1$ and $A_2$ is the following true: For every projective $A_1$-module ...
2
votes
0answers
81 views

Is this a valid Hadamard product for $\frac{2\,\xi(s)-1}{s\,(s-1)}$?

This question builds on this MSE question: Take the well known equation: $$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}-1} + ...
0
votes
0answers
122 views

Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
1
vote
1answer
42 views

The class of bounded uniformly continuous functions in viscosity solution theory for Hamilton-Jacobi equations

Dumb question: Usually in viscosity solution theory for Hamilton Jacobi equations (with convex, coercive Hamiltonians), solutions are said to be in the class $BUC(\mathbb{R}^n)$ or ...
-1
votes
0answers
112 views

What are the areas of modern math? [on hold]

question: In undergraduate mathematics there are very clearly defined areas (Calculus, Linear Algebra, Analysis, et cetera), however these are very well developed ares of mathematics that seem to not ...
2
votes
1answer
272 views

What is the “complex third derivative”?

Background I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian. If $f:\mathbb{R}^n ...
0
votes
0answers
42 views

Is there an example where we cannot lift an analytic arc of irreducible $SL_2(\mathbb{C})$-character to an analytic arc of irreducble representation

Is there an example of an irreducible and boundary irreducible 3-manifold $M$ with torus boundary and a non-abelian representation $\rho: \pi_1(M) \to SL_2(\mathbb{C})$, a non-constant analytic arc ...
1
vote
4answers
170 views

Bruhat order and Schubert cycles

I am looking for a good (textbook) reference for the basic fact (due to Chevalley) that for every semisimple Lie group $G$ (without compact factors) with Weyl group $W$, the Bruhat order on $W$ ...
3
votes
0answers
71 views

understanding the definition of $\infty$-operad of module objects

I'm just trying to understand the following definition: Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following: Let $O^\otimes$ be ...
3
votes
0answers
93 views

(Non trivial) coidempotents

I was interested to know about coalgebraic version of "Idempotents". So I seached the web and I found the following interesting post : ...
5
votes
1answer
219 views

Finite group acting on sphere

Let $G$ be a finite abelian group (of odd order if it's significant) acting on sphere $S^2\subset\mathbb{R}^3$. So my question: is it true that $G$ has a fixed point?
5
votes
0answers
88 views

Unique maximal ideal in group C*-algebras

Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$ ...
3
votes
1answer
60 views

H S class operator and its equality

$A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some member of $K$ It is trivial to see the containment $SH \leq HS$. Taking a simple ...
6
votes
2answers
122 views

eisenstein part of the theta function

If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ ...
3
votes
1answer
69 views

A problem related to routing in a graph

I have come across a new problem - I want to know whether this problem is similar to some existing problem or not. The new problem is this. There is a tourist who has a having the following ...
-1
votes
0answers
58 views

showing that a set of linear forms is closed (Bruns and Herzog, Theorem 4.2.12)

Let $k$ be an infinite field and $R$ a homogeneous $k$-algebra, i.e. a $k$-algebra that is generated by linear forms. Let $s = \sup\left\{\dim_k h R_{n-1} : h \in R_1\right\}$, where $R_i$ denotes the ...
4
votes
1answer
338 views

Did Nash prove that every game or every symmetric game has a symmetric equilibrium?

Most references seem to state that Nash showed every symmetric game has a symmetric equilibrium point, but as far as I can tell from Nash's paper, he actually showed the much more general statement ...
8
votes
0answers
187 views

Lift chain complex from $F_2$ to $Z$

We start with a finite dimensional chain complex over $F_2$, equipped with a basis. That is, we have finitely many finite dimensional $F_2$-vector spaces $C_0,\dots,C_k$ with bases $B_0,\dots,B_k$, ...
0
votes
0answers
53 views

Reference request: Heat kernel regularity near the boundary

Let $D$ be a domain in $\mathbb{R^d}$ and $p(t,x,y)$ be the heat kernel of $D$ (for the Dirichlet problem). I was told that if the boundary of $D$ is real-analytic, then the function $y\mapsto ...
2
votes
1answer
202 views

How to minimize $-\sum p_b \ln{p_b}$?

Consider multisets of the form $A = \{a_1,\dots,a_n\}$ of integers. Let $q = P(a_i = a_j)$ when $i$ and $j$ are chosen independently and uniformly from $\{1,\dots, n\}$. Let $B$ be the set of ...
1
vote
0answers
59 views

On Flajolet's analytic urn model: a unified approach or just an interesting trick?

Recently I'm reading Flajolet's work on analytic urn models. In around 2006 He introduced a new analytical method that can give exact solutions to many classical urn models in a unified way. For a ...
2
votes
0answers
67 views

Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?

Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...
5
votes
1answer
203 views

Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?

Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points? Given a convex body K,, such that t K=-K, is there a point; such that $diam(\partial K)=d(x,-x)$ I am ...
2
votes
0answers
57 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

I duplicate here a question I asked on math.stackexchange. Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some ...
13
votes
3answers
1k views

Algebra and Cancer Research

Let me start by acknowledging the existence of this thread: Mathematics and cancer research ? It is well-known that mathematical modeling and computational biology are effective tools in cancer ...
2
votes
0answers
100 views

Heat kernel and Wiener measure

A theorem by Barry Simon says that for arbitrary open sets $\Omega\subset \mathbb{R}^n$, we have $$[\exp(t\Delta_{\Omega}^D)](x,y) = \mu_{x,y,t}\lbrace \omega \text{ } \vert \text{ } \omega(s) \in ...
0
votes
1answer
71 views

Singular distributions: Applications and Instances

Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...
3
votes
0answers
124 views

Dead Flies Problem [duplicate]

If a set of points in the plane contains one point in each convex region of area 1, then can it have finite density? what is the density of the points? In my understanding, it means the average ...
-1
votes
0answers
43 views

vector-matrix notation and expectation of matrix and Hermitian product [on hold]

Let $\textbf{h} \in \mathbb{C}^{N\times 1}$, $\textbf{a} \in \mathbb{C}^{N\times 1}$, $\textbf{b} \in \mathbb{C}^{N\times 1}$ and $\textbf{c} \in \mathbb{C}^{N\times 1}$. The variable $h_i$ is defined ...
1
vote
0answers
25 views

abstract simplicity results for anisotropic quasi-simple algebraic groups defined over a non-archimedean local field

I was wondering if anyone has some references I can look at that deal with results that identify some abstractly quasi-simple normal subgroup of the group of rational points of an anistropic ...
4
votes
2answers
224 views

Simple groups and words

Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting ...
3
votes
2answers
172 views

Schreier's index formula

A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies ...
1
vote
0answers
70 views

number of times Brownian motion hits boundaries

Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...
0
votes
0answers
84 views

Find two triangles of longest side length 25? [on hold]

I'm using the quadratic Diophantine equations to solve for two integer triangles of longest side 25. It's been shown that for $a^2 + b^2 = c^2$, which goes to $x^2 + y^2 = 1$ where $x = a/c$, $y = ...
0
votes
0answers
37 views

Canonical relations and phase functions of a Fourier Integral Operator

I'm thinking about the (semiclassical) Fourier Integral Operator $T$ given by $T=h^{-n}\int{e^{i\phi(x,y,\theta)/h}a(x,y,\theta,h)d\theta}$ (that is, $T$ has phase $\phi$ and amplitude $a$). ...

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