# All Questions

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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### “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 ...
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### 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 ...
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### 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 ...
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### 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 $k$ ...
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### 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 ...
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### 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 ...
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 $\lambda_i = ... 2answers 145 views ### Generic Ultrapower as a Class If$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$, ... 0answers 57 views ### Finite Volume 1D Anderson Tight Binding Model My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on$\mathbb{Z}_n$for$n \in \mathbb{N}$. For my question, these are the ... 1answer 140 views ### Limit of distance between two random points in a unit-radius$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$, of the ... 3answers 170 views ### is the tensor product of projective modules again projective? Let$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$-module ... 0answers 81 views ### Is this a valid Hadamard product for$\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} + ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 : ... 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? 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. ... 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 ... 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} ... 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 ... 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 ... 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 ... 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, ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 = ...
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 $a$). ...