0
votes
0answers
1 views

What is the difference between Representation and Fibre Bundle?

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), ...
3
votes
1answer
51 views

Did differential geometry undertgo 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
10 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
24 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
16 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
37 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 ...
0
votes
0answers
51 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 ...
0
votes
0answers
90 views

Is there a relationship between the standard conjectures and Langlands program?

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
27 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
votes
0answers
92 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
35 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
77 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
11 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
96 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
27 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
0answers
65 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_{2t}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\ ...
-1
votes
0answers
13 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
13 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 compact 3 dim manifold 1) $G$ ...
-3
votes
0answers
39 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 ...
10
votes
2answers
190 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
71 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 ...
0
votes
0answers
17 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
44 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
57 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 ...
7
votes
1answer
82 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
22 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
36 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
53 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 ...
2
votes
1answer
119 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
33 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
51 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: ...
5
votes
1answer
139 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
131 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
44 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
99 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
80 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
79 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 ...
4
votes
1answer
97 views

Inequality due to Siegel (assumptions) and upper bounds on number field discriminants

In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality $$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$ and compares with the bound due to Minkowski that ...
0
votes
0answers
40 views

Bounding the absolute value of a polynomial involving a Diophantine equation

Let $\mathbf{z}\in\mathbb{C}^n$ with entries $z_1,z_2,\ldots,z_n$. I would like to bound the following quantity \begin{equation*} ...
-7
votes
0answers
87 views

Proof that $\sqrt2$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{2}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
0
votes
1answer
87 views

Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of \begin{equation*} x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2, \end{equation*} where for all ...
0
votes
0answers
39 views

u-Invariants of p-adic function fields

In his Paper "Fields of u-invariant 9" Oleg Izhboldin points out that for a algebraic closed, finitely generated field $k$ we have $u(k)= 2^{cd(k)}$. In particular we have ...
-2
votes
0answers
38 views

Studying Signal Processing [on hold]

I'd Like to ask two questions : What is the difference between studying Signal processing (both Deterministic and statistical) in Department of Electrical Engineering versus Department of Mathematics ...
-1
votes
0answers
62 views

Proof that at least one of the nontrivial zeta zeroes has an irrational height (assuming RH) [on hold]

This seems quite simple so its likely someone has done this before (a few Google searches returned empty and I would be really grateful for a relevant link), but in case it's new, I wanted to check if ...
1
vote
0answers
100 views

Strong divisibility of Lucas sequences

Let $a$ and $b$ be relatively prime integers and let $u_n$ be their associate Lucas sequence, i.e., the second order linear recurrence sequence satisfying $u_0 = 0$, $u_1 = 1$ and $u_{n+2} = au_{n+1} ...
1
vote
2answers
81 views

The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...
5
votes
1answer
57 views

Trees with a maximal convex hull: are the only optimal solutions Steiner trees?

For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take ...
0
votes
0answers
51 views

Bounding Random Quadratic Gauss sums

I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$ and $|\epsilon_k|=1$ for all $k=1,2,\ldots,n$. We have \begin{align*} ...
0
votes
0answers
24 views

Period doubling bifurcations [on hold]

In the bifurcation diagram, is it true that if the function $f_r(x)$ at $x^*$ (where $x^*$ is a fixed point where a bifurcation occurs) can be locally written as a smooth function then ...

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