0
votes
0answers
36 views

Calculating the the ratio of two Dirac delta functions as the limit of the ratio of nasent delta functions?

I am in a situation where I find myself with the ratio of Dirac delta functions. Specifically, I find myself with the ratio of the nascent deltas: $\frac{\lim_{\varepsilon \rightarrow ...
5
votes
0answers
93 views

Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...
-2
votes
1answer
63 views

What does “group size” mean in the -G option of directg in nauty? [on hold]

To be sure I understand the definitions used in the nauty user manual: An automorphism group size (for a digraph) is the number of re-labelings (including the trivial original digraph) of the ...
14
votes
0answers
198 views

function field analogy and global/absolute geometry

The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...
3
votes
0answers
45 views

Symplectic sum and Symplectic cut

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...
0
votes
0answers
22 views

mixed semi definite and second order programming complexity order

Consider the following mixed semi definite and second order programming: $\begin{array}{l} \mathop {{\rm{min}}}\limits_{\bf{X}} \,{\rm{Tr}}\left( {{\bf{XA}}} \right)\\ {\rm{s}}{\rm{.t:}}\, & ...
1
vote
1answer
184 views

Computing the Chern class of $S^6$

I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...
-4
votes
0answers
35 views

Integer solutions for multiple variable equations [on hold]

Obviously it will take some brute-force. But how do I minimize the brute-force needed (optimize)? I know one can solve Diophantine equations and quadratic Diophantine equations. But what if I have ...
0
votes
0answers
60 views

Examples of manifolds whose second Stiefel-Whitney satisfies a nontriviality condition

I'm looking for examples of pairs $(M,L)$ where $M$ is a symplectic manifold, $L$ a (closed, connected) Lagrangian submanifold, such that the second Stiefel-Whitney of $L$, $w_2(TL)$, evaluates ...
5
votes
1answer
210 views

Detection of stable homotopy by K-theory spectra

This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from ...
-1
votes
0answers
33 views

Closed sets on the product space and operators [on hold]

$H$ is an hilbert space and $C$ is a closed subset of $H\times H$ with the product topology. If $P$ is the projection $P: (x,y) \in F\times F \to y \in F$ do we have that the set $$P(C)= \{ P((x,y)), ...
1
vote
2answers
110 views

Identity involving shifted Legendre coefficients

For small values of $n$ ($2\leqslant n\leqslant 5$), the coefficients $a_k = (-1)^k{n\choose k}{n+k\choose k}$ of the shifted Legendre polynomial $\tilde{P}_n(x)$ satisfy the identity ...
-2
votes
1answer
96 views

AI / Machine Learning related to high/modern/front mathematics [on hold]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
3
votes
1answer
63 views

Boolean completion (of a forcing notion) isomorphic to each of its cones

Suppose $ \mathbb{P} := (P, {\leq_P}, 1_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion. Fix some dense embedding $ i \colon P ...
1
vote
0answers
83 views

Collection of pointset topological spaces which form the terminal Grothendeick infinity topos

In Lurie's "Higher Topos Theory", he mentions two models of the terminal (Grothendieck) $(\infty,1)$-topos of "spaces". Firstly, the collection of Kan complexes and simplicial maps. Secondly, the ...
0
votes
1answer
45 views

Invariance of mutual information

Let $I(X,Y):=H(X)+H(Y)-H(X,Y)$ be the mutual information of the joint probability distribution $p_{XY}$ (here $H(\cdot)$ is the Shannon entropy of its argument). I know that the mutual information is ...
1
vote
1answer
42 views

Uniqueness of solution of a nonconvex optimization problem

What conditions need to be hold for a nonconvex optimization problem to have a unique solution? Specifically, I have the following minimization problem that I'd like to know whether it has a unique ...
0
votes
0answers
56 views

Can we deduce that all the real zeros of those $k^{th}$ derivatives are also simple?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...
-1
votes
0answers
45 views

Approximation of bounded continuous functions by Lispschitz bounded functions

Let $H$ be an Hilbert space and $f : H \rightarrow \mathbb{R}$ a continuous and bounded by $M>0$ function. Is it possible to construct a sequence of functions $f_n$ Lipschitz uniformly bounded by ...
2
votes
0answers
30 views

Approximating a superharmonic function, by smooth superharmonic functions

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$, $u\ge 0$ and $-\Delta u\ge 0$ in the sense of distributions ($u$ is superharmonic). The standard ...
-4
votes
0answers
38 views

non-degenerated vector spaces and Lie algebras [on hold]

A symplectic space is a finite dimensional vector space V over GF(2) equiped with an alternating bilinear form and if the form is non-degenerated then V is called a non-degenerated symplectic space. ...
-1
votes
1answer
172 views

How to prove this equality in $\ell_1$ [on hold]

Let $w$ be any element in $\ell_1$, and $(w_n)$, $(z_n)$ two bounded sequence in $\ell_1$ that converge to 0 pointwise. Then we have ...
2
votes
0answers
34 views

König-Wittstock norm on Banach space

Let $(E,\Vert.\Vert)$ be a real Banach space and $\ell\ne 0$ an non-continuous linear form on $E$. Let $a\in E$ be such that $\ell(a)=1$. König-Wittstock [Non-equivalent complete norms and would-be ...
3
votes
2answers
160 views

Is the hypersurface satisfying $\langle x-x_0,\nu\rangle>0$ diffeomorphic to sphere?

Let $p:M\to \mathbb{R}^{n+1}$ be the closed immersed hypersurface. Is the following thing right? If there exists a point $x_0$ in $\mathbb{R}^{n+1}$ such that $\langle x(p)-x_0,\nu(p)\rangle>0$ ...
-2
votes
0answers
65 views

Any other operators that may convert agebraic function into transcendental ones [on hold]

As we know,integral may convert or map a rational function or algebraic function into transcendental one,are there any other operators that may convert a rational function or algebraic function into ...
1
vote
0answers
37 views

d-refining covering of normal space

If $X$ is normal, it is well known that for any open-covering $(U_i)$ of $X$, there exist closed subspaces $F_i$ and $G_i$ and an open subspaces $O_i$ such that $$F_i\subset O_i\subset G_i\subset ...
1
vote
0answers
51 views

How to test whether a distribution follows a power law? [on hold]

I have the data of how many users post how many questions. For example, [UserCount, QuestionCount] [2, 100] [9, 10] [3, 80] ... ... it means each of the 2 users posts 100 questions, each of the 9 ...
3
votes
0answers
65 views

Lagrangian submanifolds in $T^\ast S^n$

Let $L\subset T^\ast S^n$ be a properly embedded Lagrangian submanifold homeomorphic to $\mathbb{R}^n$ with respect to the canonical symplectic structure on $T^\ast S^n$, and suppose $L$ intersects ...
2
votes
2answers
113 views

Product of Lebesgue and counting measures

Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...
3
votes
2answers
60 views

Is this generating family of a measurable space of point measures a pi-system?

I'm learning some probability and measure theory and working my way through the first few paragraphs of [1]. My question is perhaps too basic for Math Overflow, but I hope it is welcome here. Point ...
2
votes
0answers
62 views

Symplectic isotopies between small balls?

Let $(X,\omega)$ be a connected symplectic manifold, possibly with boundary. Let $g_1, g_2: B(1) \to X$ be two balls in $X$. Is it true that if $\delta$ is sufficiently small, then there is an ...
-4
votes
0answers
41 views

let A be an n*n matrix with real entries which of the following is coorect? [on hold]

let A be an n*n matrix with real entries which of the following is coorect? (a) if A^2 =0 then A diagonalisable over complex numbers (b) if A^2= I then A diagonalisable over real numbers (c) if A^2 ...
-4
votes
0answers
43 views

Which of the following conversion is wrong in the number system? [on hold]

Here is a diagram. which of the 2 boxes' conversion is wrong and please provide me a reason for it. I am doing the same conversion in 2 different procedure. where am i making the mistake.Please help ...
0
votes
0answers
64 views

Bases and transversals

Let $F$ be a free finitely generated group, $L \leq H \leq F$ subgroups of finite index. Given bases $B$ of $F$ and $C$ of $H$, must there be a Schreier transversal $T$ for $H$ in $F$ such that the ...
-4
votes
0answers
27 views

real analysis series and sequence [on hold]

If ∑(n=1)^∞▒an is absolutely convergent , then which of the following is not true? (a)∑(m=n)^∞▒am →0 as n→∞ (b)∑(n=1)^∞▒ansin⁡n is convergent (c)∑(n=1)^∞▒e^an is divergent (d∑_(n=1)^∞▒a_2^n ) is ...
4
votes
0answers
70 views

When is a Newton basin fractal continuously determined by the roots of its polynomial?

Newton basin fractals are visualizations of the Julia sets of functions of the form: $$f_p(z) = z - p(z)/p'(z)$$ where $p$ is a complex polynomial. My question is: When is the Julia set, ...
1
vote
0answers
42 views

Cheeger inequality for the maximal eigenvalue

Let $G = (V,E)$ be an undirected graph and let $L = I - D^{-1/2} A D^{-1/2}$ be its normalized Laplacian matrix. The Cheeger inequality asserts that: $$\frac{\Phi_G^2}{2} \leq \lambda_2 \leq 2 ...
7
votes
3answers
487 views

nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
4
votes
1answer
208 views

Set-theoretic tautologies

Let us consider unquantifed formulas of a set theory (for example, NBG), more precisely, the formulas, constructed from variables and the constants $\emptyset, V$ (the empty set and the class of all ...
1
vote
0answers
78 views

References for crystal bases and Demazure modules in representation theory

I was wondering what are some standard general references/books/survey articles about: (1) crystal bases, and string parameterizations and (2) Demazure modules, and Schubert varieties (containing ...
-3
votes
0answers
252 views

27 years old. Working in silicon industry. Considering PhD in Math [on hold]

All, I am 27 years old. I am working in the silicon industry. Did my Masters in Electrical Engineering. My personal life has impeded me and got in the way of my decision making in academics. ...
1
vote
0answers
79 views

Interpretation of the Gross-Zagier formula for Green function

I am reading the paper of Gross and Zagier on heights of Heegner points and would like to check with the experts whether the following (meta?)mathematical statement makes sense. In the calculation of ...
2
votes
0answers
109 views

Noncommutative HKR theorem

What is the analog of HKR theorem in the noncommutative world? Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a smooth commutative algebra $A$ of finite type ...
2
votes
1answer
98 views

BMO spaces on the torus

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is ...
0
votes
0answers
15 views

$L^\infty$ estimate for a fourth order (hyperbolic) equation

Consider the following fourth order equation $$u_{tt}+u_t= d\Delta u-\Delta^2u+f,$$ with Dirichlet or Navier boundary conditions, that is on $\partial\Omega$, we assume that ...
1
vote
2answers
90 views

Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions

For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, ...
-2
votes
2answers
77 views

Hexagon Formed by connecting Trisections of triangle sides [on hold]

Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...
2
votes
2answers
154 views

Example of linearization for GIT

Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle ...
1
vote
0answers
91 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
7
votes
0answers
74 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...

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