0
votes
0answers
9 views

Try to prove that a discrete distribution function is a singular distribution function

actually it is the 6th question in the section 3 of chapter 1 in the book named by The course in probability theory, author:Chun Kai Lai. someone asserts that the derivate of the discrete function on ...
0
votes
0answers
25 views

Compact elements of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field

Let $F$ be a nonarchimedean local field and let $\mathcal{O}$ be its ring of integers. An element $g$ of $\mathrm{GL}_n(F)$ is called compact if the cyclic subgroup that it generates has compact ...
3
votes
2answers
50 views

Contractibility of space of embeddings of a disc

I'm pretty sure that both of the following spaces are contractible. However, I can't seem to find a proof or a reference. Can anyone provide one? Let $D^2$ be the unit disc in $\mathbb{R}^2$. The ...
1
vote
0answers
64 views

Connes on Integers / Primes and Quantum Field Theory / Elementary Particles

I was browsing Matthew Watkins' Number Theory and Physics archive, and came across the following quote by Alain Connes (without context, aside from the fact that it was in a popular book called Dr. ...
8
votes
1answer
169 views

Did Leibniz really get the Leibniz rule wrong?

A couple of posts ([1], [2]) on matheducators.SE seem to suggest that Leibniz originally got the wrong form for the product rule, perhaps thinking that $(fg)'=f'g'$. Is there any actual historical ...
1
vote
2answers
48 views

curvature flow for loops in S^2

Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, ...
2
votes
0answers
19 views

co-dimension one minimizing verifolds

It is known that a minimizing co-dimension verifolds within a manifold may need to be singular. I think a famous example first partially analyzed by Jim Simons is the cone on in the 8-ball of the ...
0
votes
0answers
7 views

Online algorithm for nested optimization problem(with locally optimization)

How to construct a sequence ${x_t;\theta_t}$, which is online algorithm for following optimization problem: $\arg\min_\theta \sum_t\min_{x_t}\ell_t(x_t;\theta)$ For simply, we can assume ...
1
vote
0answers
16 views

Tori in Compact Riemannian Symmetric Spaces

The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic ...
7
votes
0answers
130 views

same paper was published in the same journal twice

I just realized that the paper "Hitchin's connection and differential operators with values in the determinant bundle" by Xiaotao Suna and I-Hsun Tsai and was published twice in the Journal of ...
0
votes
0answers
16 views

Question about B. Host paper 'Nombres, normaux entropie, translations'

I put this question on mathstack but it seems more suitable to put it here: I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = ...
0
votes
0answers
30 views

Graham's Number and Ramsey Theory [on hold]

I had a few questions regarding Graham's number and Ramsey theory. I understand what Graham's number is and what it is attempting to solve. My question is, is a hypercube with dimension equal to ...
-4
votes
0answers
22 views

What is the difference between Representation and Fibre Bundle? [on hold]

When a Group G have a homomorphism to General Liner Group GL(n, K), we call GL(n, K) Liner Representation. When a Space X have a map to another Space Y, We call the inverse image of y, or f~-1(y), ...
6
votes
1answer
291 views

Did differential geometry undergo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by ...
0
votes
0answers
21 views

Sumset of parallel arithmetic progressions in Euclidean space

Let $A$ be a finite subset of $\mathbb R^n$ such that dim $A=n$. It is a known result from Freiman that the size of the sumset $A+A$ is lower bounded as $|A+A|\geq (n+1)|A|-\frac{n(n+1)}{2}$ where ...
1
vote
0answers
54 views

Stalks of étale sheaves

I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at ...
0
votes
0answers
24 views

Asymptotic decay for the inhomogeneous Schrödinger equation

Let $H$ be an Schrödinger operator $(-\Delta+V(x))$ with a radially symmetric and smooth potential and $H$ is e.s.a. on $C^\infty_0(\mathbb{R}^n)$, furthermore let $\lambda$ be an element of the ...
0
votes
0answers
47 views

Generalization of the alternating sign test for convergence of a series?

I'm struggling with a series of the form $$\sum_n |a_n|\, s_n $$ where $s_n$ is the sign of a simple function of $n$. The $|a_n|$ monotonically decrease and are relatively simple functions of ...
1
vote
0answers
74 views

Hilbert triples

I apologize if this is a trivial matter for the analysts out there, but I looked this up in a number of books (e.g., H. Brezis, J. Wloka) and could not find a satisfying discussion, one that would ...
1
vote
0answers
122 views

Is there a relationship between the standard conjectures and Langlands program? [on hold]

I would like to know are there connections between Standard conjectures on algebraic cycles and Langlands program (in the light of Motives, I assume)? What implications would a development of the ...
2
votes
0answers
50 views

A perturbation question for the intersection of C*-subalgebras

This feels like something I may have asked before (in which case, apologies) and it also might be some kind of "standard counterexample in a book on C* algebras". Let M be a unital C*-algebra and let ...
1
vote
0answers
112 views

Old Math books, will research and sell most [on hold]

I saw an earlier thread on selling old math books. My dad was a professor at Manhattan College for many years, he passed away a couple of years ago and has tons of old math books. I promised him I ...
2
votes
0answers
50 views

$L^p$ estimates for Ornstein-Uhlenbeck: what is known beyond hypercontractivity?

Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho ...
1
vote
1answer
87 views

L-function of twist

I'd like to ask the following easy question, since I can't find a reference. Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in ...
0
votes
0answers
14 views

On the local L-parameter of U(3) obtained by the theta lift from U(1)

Let $E/F$ be a quadratic extension of number fields and $v$ a place of $F$. Let $\sigma$ be a automorphic character of $U(1)(\mathbb{A}_F)$ and $\Theta(\sigma)$ the theta lift of $\sigma$ to ...
1
vote
1answer
105 views

Characterization of a subset of $[0,1]$

Let $T\subseteq[0,1]$ be a subset containing $1$. Now we know that $T$ satisfies the following property: For every $t\in [0,1)$, if there exists a decreasing sequence $\{t_n\}_{n\ge 1}\subset T$ such ...
0
votes
0answers
31 views

Holder continuous analytic function

Assume that $0<\alpha<1/2$ and $f$ is analytic in the unit disk $D$ with $|f(z)-f(w)|\le M|z-w|^\alpha$. Can we state that in general $$\int_D |f'(z)|^2 dx dy <\infty?$$
3
votes
1answer
113 views

Ways to show a system of polynomial equations has no solution

I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$: $$ \begin{cases} 2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\ ...
-1
votes
0answers
17 views

average dirichlet distribution

say that A and B are 2 Dirichlet distributions. Is there a way to know if the average of the values is still a Dirichlet distribution? If not, how to merge 2 Dirichlet "similar" distributions?
0
votes
0answers
21 views

One dimensional foliation of surfaces with prescribed graph of foliation

According to the definition of the graph of a foliation by Winkelnkemper we ask the following questions: Let $G$ be one of the following non hausdorff 3 dim manifold 1) $G$ is a ...
-3
votes
0answers
41 views

How to calculcate frecency? [on hold]

I recently came across the concept of Frecency. I thought it was a typo, but apparently it's a combination of recency and frequency, used in Mozilla's URL bar. What are possible formulas for ...
12
votes
2answers
226 views

Minimal number of generators for $GL(n,\mathbb{Z})$

$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = ...
1
vote
1answer
93 views

Solution or Reference Request for a Closed Form of the Sum

I have been working for quite a while on finding a closed formula for the Legendre Symbol. Inspite of my best efforts I can't come anything better with a formula for the symbol ...
1
vote
0answers
28 views

How to distribute groups over activities in rounds [on hold]

This problem started with my sister asking me how she can distribute 12 groups over 6 activities in 6 rounds. She wants to organize a camp for for year students. There are 6 acitvities and in every ...
4
votes
1answer
49 views

Separability of the C*-algebra in the definition of K-homology

There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book ...
4
votes
1answer
60 views

Comodule analogue for statement that a faithful representation of an affine group scheme generates all

If $V$ is a faithful finite dimensional representation of an affine group scheme $G$ over a field $k$, then every finite dimensional representation of $G$ is isomorphic to a subquotient of $\otimes^n ...
8
votes
1answer
90 views

Fast computation of a Groebner basis - What is Possible

I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...
-1
votes
0answers
23 views

graphical estimate of convergence rates [looking for textbook reference] [on hold]

I asked this question already in mathematics, but got no sufficient answer. I am writing a paper for engineers. There, under other things, I compare convergence rates of sequences $x_n \to x^*$, ...
1
vote
0answers
37 views

Lower central series in a free pro-p group

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$. Is ...
2
votes
1answer
54 views

Intersection of closed geodesics in hyperbolic surface

This question may be easy but I could not come up with a proof. Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed ...
3
votes
1answer
160 views

Are the Szpiro ratios of 37b1 over certain number fields {33,39,42,48,51,66}?

Related to this question. Szpiro's conjecture over number fields states that the Szpiro ratio is: $$ \sigma_{E/K}=\frac{\log{|N_{K/Q}\Delta_{E/K}|}}{\log{|N_{K/Q} f_{E/K}}|}$$ Given $ \varepsilon ...
0
votes
1answer
35 views

edge transitivity and edge deletion

Let G be a graph which has the following properties: 1) For every $e_1,e_2 \notin E(G)$, $G \cup e_1 \cong G \cup e_2$ 2) For every $e_1,e_2 \in E(G)$, $G\setminus e_1 \cong G\setminus e_2$ i.e. ...
0
votes
0answers
56 views

Maximal subgroups of indefinite special orthogonal group SO(p,q)

Can someone answer the following question: Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions: ...
6
votes
2answers
189 views

Square filling self avoiding walk

I want to create an algorithm that fills a square grid with a random Hamiltonian path starting at a particular point. See this example. One approach is to try a free direction as a next step, and ...
1
vote
0answers
136 views

Learning roadmap in Algebra [on hold]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
1
vote
0answers
45 views

Expanding graphs from iterated zig-zag product

Let $\Gamma \, zz\, G$ denote the zig-zag product of graphs $\Gamma$ and $G$. Reingold, Vadhan, and Wigderson proved that the second largest eigenvalue of the normalized adjacency matrix of the ...
-4
votes
0answers
100 views

How can I prove: If A is a subset of C and B is a subset of D, then the union of A and B is a subset of the union of C and D? [on hold]

How could I write a proof for the above statement, considering I'm studyng the first Enginnering math's course? (in other words, my math level is pretty basic) Thanks in advance
5
votes
0answers
83 views

How many simultaneous polynomial equations of degree 2 can software solve today?

Consider the following problem: Input: $n$ polynomial equations of degree $2$ in approximately $n$ variables. Each equation contains about $\sqrt{n}$ monomials. We would like to find one ...
0
votes
1answer
89 views

Convergence of complex series that are not absolutely convergent?

Does anyone know of a convergence test for a complex series of the form $$\sum_n a_n \cdot \exp(i \cdot b_n)$$ ? The particular series I need to understand has a_n going to zero as n goes ...
1
vote
1answer
83 views

Expression and growth bound for $r_{p^m,k}(n)$

Let's define , $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$ what will be growth bound of $R_{p^m,k}(n)$? This can be thought as a ...

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