5
votes
0answers
338 views

Number Theory over $\mathbb{F}_q [t]$, why is it important/interesting?

This is something I was curious about, but not really sure why. I was hoping maybe someone could give me a good explanation. I understand that there are similarities between $\mathbb{Z}$ and ...
-5
votes
0answers
63 views

Stronger than Gerretsen [on hold]

For all triangle prove that: $p\leq2r-R+\sqrt{9R^2-6Rr+3r^2}$, where $p$ is a semiperimeter, $R$ is a radius of circumcircle and $r$ is a radius of incircle of the triangle. I found a proof of this ...
3
votes
1answer
135 views

Normal subgroup lattice of the group $U_{6n}$

I need to find the normal subgroup lattice of the group $U_{6n} = \langle a,b | a^{2n} = b^ 3= 1, a^{-1}ba = b^{-1}\rangle$. To the best of my knowledge this group was introduced at first by GORDON ...
0
votes
0answers
27 views

evolution PDE in comlex domaim

i have got (an example for some other theorem) an existence theorem for such a type problem $$u_t(t,z)=-u(t,z)+a(t)z^m\frac{\partial^N u(t,z)}{\partial z^N},\quad u(0,z)=\hat u(z)\in \mathcal ...
-3
votes
0answers
36 views

Topological conjugacy [on hold]

Prove that any two linear systems with the same eigenvalues +/-ibeta, beta not equal to 0 are conjugate. What happens if the systems have eigenvalues +/- ibeta and +/- i*gamma with beta not equal to ...
2
votes
1answer
64 views

Unbounded convex not containing a ray - example without using a basis

I prove here that an unbounded convex in a finite dimensional space contains a ray. At the same place, I give an example of an unbounded convex not containing a ray in the case of an infinite ...
5
votes
1answer
283 views

Can an odd map be null homotopic?

Let $G$ be a compact Lie group with invariant measure $\mu$. An odd function is a continuous function, $\phi:G\to \mathbb{C}$, such that $\int_{G} \phi d\mu=0$. An odd map is a continuous map, $f:G\to ...
0
votes
1answer
59 views

About freeness of modules over the coordinate ring of an affine variety

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite ...
1
vote
0answers
46 views

If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable? Pietro Majer ...
3
votes
2answers
165 views

Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $

Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold? $$ \int_{k + 1/2}^{k + 3/2} \frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}} ...
-1
votes
0answers
24 views

Get fractional LP into canonical form [on hold]

I have the following LP problem and I would like get it into canonical form so that I can solve it for the x vector. min(y) = sum yi/(ci-yi) , where yi = sum over j Lij * xj st. H x = f ...
0
votes
1answer
186 views

A functorial isomorphism in derived category

This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...
3
votes
0answers
156 views

A good book on adeles and ideles

Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...
5
votes
1answer
361 views

Is fourier analysis necessary to prove this?

I have a couple of inequalities that I want to prove. The proof is easy using fourier analysis but I am wondering whether there is a proof that does not use fourier analysis. 1) For any $c, s > ...
-3
votes
0answers
72 views

Inequality with five variables [on hold]

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$ Easy to show ...
-2
votes
0answers
30 views

triangular inequality [on hold]

If $|a_n-L|\leq \epsilon$ and $|a_n-L'|\leq \epsilon$, then by the triangular inequality $|L-L'|\leq 2\epsilon$ I know that the triangular inequality says if $|a-b| \leq |a|+|b|$. However, I could ...
-2
votes
1answer
139 views

A calculus question [on hold]

Fix $q>1$. Define the function $$ f_q(c):=\int_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r. $$ The problem is whether the following is true, $$ \lim_{c\rightarrow 0} c \log(1/c)^q f_q(c) = C \in ...
3
votes
1answer
145 views

Is $G \rightarrow G/P$ surjective on $K$-points over a local field?

Let $K$ be a local field, $G$ a (connected) reductive $K$-group, and $P \le G$ a parabolic subgroup. Is the map $G(K) \rightarrow (G/P)(K)$ necessarily surjective, and, if so, then why?
0
votes
0answers
76 views

Big polynomial similar to a small rational function on a subset of points

Consider a real polynomial $p(x_1,x_2,\dots,x_n)$ that when evaluated on $x_i\in \{0,1\}$ takes values only in $\{0,1\}$. It is clear that $p(x_1,x_2,\dots,x_n)$ can be multilinear (multiaffine) ...
-5
votes
0answers
54 views

Fermat's little theorem with smaller powers [on hold]

I am having some trouble using Fermat's Little Theorem: (a^p-1) = 1 (mod p) I am able to solve something where the power is larger than what we are dividing by, but how would I go about solving ...
1
vote
0answers
182 views

Is the upper half plane an algebraic stack?

Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$. So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli ...
0
votes
0answers
46 views

Guess A Property Of The Integral Average Value Function [on hold]

Let $f$ be a function that is defined on $[a,b]$ and Riemann integrable on $[a,b]$. Def 1. $$\hat f(x)=\begin{cases} f(x),& \text{if }x\in[a,b], \\ f(a),& \text{if }x<a, \\ ...
5
votes
0answers
111 views

Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$. Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$? ...
-2
votes
0answers
53 views

Stable curves and degenerations of smooth ones [on hold]

i'm approaching the study of Deligne-Mumford compactification for the moduli space of smooth curves of genus $g$. I'm trying to understand the geometrial meaning of stable curves: i know they have a ...
0
votes
0answers
27 views

How is constrained optimization done? [on hold]

I am trying to implement an optimizer described in http://arxiv.org/pdf/1406.2572v1.pdf I have an objective function, gradient and hessian. The algorithm for unconstrained optimization is described ...
0
votes
1answer
52 views

About the regularity of the boundary of a set

Consider $\Omega_1, \Omega_2$ two convex, bounded domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ and suppose that $\operatorname{dist}(x, \partial \Omega_2) =\min_{y \in \partial ...
1
vote
0answers
32 views

Is there an efficient way to compute the “complete subset regression”?

Background: Let $X \in \mathbb{R}^{N\times K}$ and $y \in \mathbb{R}^{N\times 1}$ be data for a regression problem. The aim is to find $\beta \in \mathbb{R}^{K\times 1}$ such that $X\beta \approx y$ ...
5
votes
1answer
185 views

Associative Ring Spectra and Derived Completion

So, I was thinking before that this might have some nice, simple topos theoretic explanation, but Jacob disabused me of that notion. However, I'm still very interested in the following question: Is ...
3
votes
1answer
68 views

A lower bound on $\|\cdot\|_{p_{\ast}}$ image of $\ell^{q_{\ast}}$ vectors

The motivation for this question comes from this paper where we study lower bounds on the complexity of convex optimization algorithms (BTW, if you find a better title for my question, please let me ...
10
votes
3answers
314 views

Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?

I apologize in advance if this question is too vague for mathoverflow. My main aim is to get some references for a concept. First, we make the following observation: let $X: M \rightarrow TM $ be a ...
0
votes
0answers
41 views

Convergence of Selberg type Zeta function

Let $X=G/K$ be a Riemannian symmetric space without compact factor. We can assume $G$ is connected and real reductive. $K$ is a maximal subgroup of $G$. Let $\Gamma$ be a discrete torsion free ...
0
votes
2answers
144 views

Equidistribution of rational points on an algebraic variety

Suppose that we have a variety $X \subset \mathbb{P}^{n}$ defined over $\mathbb{Q}$. Suppose we have $S$ many rational points on $X$ inside the box defined by $|x_i| \leq B_i$ for $i = 0, \cdots, n$, ...
0
votes
0answers
41 views

quadratic knacksack problem, state of art [on hold]

What is the current status to quadratic knacksack problem? Say, how many variables can the state of art solver handle? Thank you.
0
votes
0answers
13 views

Pierce's Law in HOL Light [migrated]

Im fairly new to HOL light and ocaml. Could someone please explain to me how Pierce's Law can be written in HOL Light. Thanks in advance.
0
votes
1answer
122 views

Discriminant of a polynomial in two variables

I want to compute the discriminant of the following polynomial $$ F(X,Y)=X^mY^n+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}c_{ij}X^iY^j. $$ Here the discriminate means the equation $D(c_{i,j})$ in the variables ...
-2
votes
0answers
33 views

Demonstrate by extension with 3 changing value [on hold]

I want to demonstrate by extension this : Z x Z U Z It should give some stuff like , {x} element of Z I don't know how to demonstrate by extension because there will be 3 changing values, i, j and ...
6
votes
1answer
137 views

Center of one-point stabilizer in 2-transitive groups

In this MO question it was mentioned that the following fact seems to be true: If $G$ is doubly transitive on $X$ and the one-point stabilizer $G_x$ has a non-trivial center, then $G$ is of ...
1
vote
1answer
79 views

How to formulate approximation from above?

(This is perhaps a stupid question. If so, please give me a hint and a down vote.) I have a sequence of Banach spaces $X_{1}\supset X_2\supset ...$ and a sequence of elements $x_j\in X_j$ ...
2
votes
0answers
122 views

Tate's conjecture and symmetry of Hodge-Tate weights

I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf At page 10 he claims that an indirect ...
0
votes
0answers
49 views

Hill's discriminant and spectral properties of Schrödinger operator

I am currently reading this paper on Schrödinger operators see here. On page 6 and 7 they talk about Hill's discriminant and how this is connected with the spectral properties. They also show some ...
0
votes
0answers
23 views

Reference request for stably free modules

I am looking for some references on the theory of stably free modules. I will call (F) the following property for a ring $R$: every f.g. stably free module over $R$ is free. 1) Is there a standard ...
3
votes
1answer
89 views

Operator norm versus Hlawka inequality

Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality ...
7
votes
2answers
294 views

Rep-tiles of order 2

A 2-rep-tile is a geometric shape that can be partitioned into exactly 2 smaller (dilated) copies of itself. Although there are many rep-tiles of higher orders, the only 2-rep-tiles I could find ...
2
votes
2answers
101 views

Witt index of the sum of 24 squares

Consider the quadratic forms $q(x)=x_1^2+\dots +x_8^2$ in 8 variables and $p(x)=x_1^2+\dots+x_{24}^2$ in 24 variables over the field of rational numbers $\mathbb{Q}$. Let $E/\mathbb{Q}$ be a field ...
6
votes
0answers
80 views

Fibrations of the injective model structure on G-simplicial sets

Let $G$ be a discrete group. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which ...
3
votes
0answers
90 views

On a problem of Berkovich

What is the real history of the following problem proposed by Berkovich [Y. Berkovich, Z.Janko, Groups of prime power order. Volume 2, Expositions in Mathematics, 56, Walter de Gruyter, New York, ...
0
votes
0answers
36 views

Combining binary numbers [on hold]

Assume you have n n-digit binary numbers, for example: ...
4
votes
1answer
136 views

Shift-invariant symmetric functions in representation theory?

The connection between symmetric functions and representation theory is well-known. Now consider the subspace of symmetric functions that are shift-invariant, that is, functions satisfying ...
4
votes
1answer
87 views

Euler characteristic of open varieties as degree of Chern class of logarithmic differentials

Let $U$ be a smooth variety over a subfield $k$ of $\mathbb{C}$. Let $X$ be a smooth projective variety containing $U$ as the complement of a normal crossings divisor $D$. Denote by $\chi(U)$ the ...
1
vote
0answers
43 views

The normalization axiom of a quantization

Guillemin, Ginzburg and Karshon explain a quantization in their book [Chap 6,MR1929136] as follows. The quantization is a process which associates to a symplectic manifold $M$ a Hilbert space ...

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