3
votes
0answers
12 views

Decomposition of an induced representation of $GL(2, q)$

Denote $G = GL(2, q) = GL_2(\mathbb{F}_q)$, $B$ its Borel subgroup of upper triangular matrices, $T$ its splitting torus of diagonal matrices. The object I am interested in is $Ind_B^G\rho$, where $\...
1
vote
0answers
15 views

A (second-order) axiomatic characterization of the integers which rules out surreal/hyperreal versions

I've seen it stated, for example here, that the integers are the unique commutative ordered ring with identity whose positive elements are well-ordered. I understand why the integers are the ...
2
votes
0answers
17 views

Reference for Using Group Cohology to calculate Etale Cohomology

I'm looking for a reference for the following statement: Let $X$ be a variety (over an algebraically closed field $k$), and let $F$ be a locally constant etale sheaf. Let $x \in X(k)$. Then $ \mathrm{...
-2
votes
0answers
21 views

I need to know the most active research topic which depend on Real analysis and functional analysis? [on hold]

I need to know the most current topic in pure math with depend mainly on real analysis and functional analysis and not need a good knowledge in algebra and geomtry ? is delay differntial equations is ...
1
vote
0answers
14 views

Unique Stationary Distribution of A Markov Chain

I have a Markov Chain like $Y_i=\sum_n\pi_{n,i}(Y)Y_n$, i=1,2,3...N. So the Markov chain has N states and the transition matrix depends on the vector $\textbf{Y}$. I am wondering what conditions $\pi$ ...
2
votes
0answers
25 views

Randomly put $k$ balls in $2n$ circular boxes, pick $n$ consecutive boxes such that the number of balls is minimum!

You are given $2n$ boxes that are arranged circular (you can imagine all boxes are on the edge of a circular table). Then randomly, you put $k$ balls in the boxes such that each box is containing ...
0
votes
1answer
31 views

Random variable of random variable [on hold]

This is confusing and difficut, but I hope it makes a sence. I am interested in kind of like Random variable of Random variable. This issue might've been mentioned below before. The Probability ...
0
votes
0answers
9 views

Generalizing approximate $\mathbb{Z}$-equivariance of a simple function

Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. http://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...
2
votes
0answers
27 views

Importance of $E_n$-algebras over ring structures on $\pi_*(E)$

Hopefully this question is not too vague to be closed. I am looking for examples of when a construction/theorem that involves $E$-(co)homology or even simply the ring $E_*$ requires an understanding ...
1
vote
0answers
19 views

An asymptotic formula involving the $2$-torsion subgroup of the class group of real quadratic fields

Let $R$ be an order in some number field $K$ (not necessarily maximal). Then the class number $\text{Cl}(R)$ is equal to the cardinality of the Picard group of $R$, which is the group of equivalence ...
-1
votes
0answers
42 views

An Example to Marsden-Weinstein Theorem

Suppose that the action of a compact Lie group $G$ on the closed symplectic manifold $(M,\omega)$ is Hamiltonian, with moment map $\mu : M\to \mathfrak{g}^*$. From the Hamiltonian condition it ...
2
votes
1answer
46 views

Bounding a distance function on polynomials with real roots

Consider univariate polynomials $f(x)=\sum_{i=0}^{r_f}f_ix^i$ and $g(x)=\sum_{i=0}^{r_g}g_ix^i$ in $\Bbb Z[x]$ with only distinct real roots $\alpha_1,\dots,\alpha_{r_f}\in\Bbb R$ and $\beta_1,\dots,\...
-3
votes
0answers
36 views

MDAS solution glitch [on hold]

I always get confused with this kind of math problem, How do we solve this problem? Example: 1 * 5 - 10 + 2 = ? Solution 1 [My solution]: Of couse I use MDAS ...
1
vote
1answer
26 views

Sobolev regularity for systems of elliptic boundary value problems

My question is about Sobolev estimates near the boundary for elliptic systems (equivalently, elliptic boundary-value problems for vector-valued functions). Note, results for the scalar case are ...
0
votes
0answers
41 views

Riemannian manifolds conformally diffemorphic to sphere [on hold]

let a Riemannian manifold be conformally diffeomorphic to a sphere. Is its scalar curvature constant?
1
vote
1answer
71 views

Counting tournaments with ties

An improper tournament, or tournament with ties, is a graph in which every pair of nodes is connected by a single uniquely directed edge or by a single undirected edge. There are 1, 2, and 7 improper ...
3
votes
1answer
42 views

Uncountable divisible groups and the existence of order-preserving ismorphisms of their subsetss

Let $(G,+,0,<)$ be an ordered divisible group of uncountable dimension. Consider the subset $G^{<0}$ of $G$. Question: Are $G$ and $G^{<0}$ isomorphic as ordered sets? Does there exists an ...
3
votes
0answers
65 views

Whether a given algebra is the algebra of endomorphisms for a vector space

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
0
votes
0answers
26 views

Sub-matrices with a real spectrum

This question arises from the study of PT-symmetric quantum mechanics. Let $A$ be an $n\times n$ complex-valued matrix with a real spectrum. If $A$ is Hermitian, then any sub-matrix corresponding to ...
2
votes
1answer
46 views

Elementary question: Curvature change under Complexified Gauge Transformation

Forgive me for this elementary question. Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ ...
4
votes
1answer
79 views

Section of ellipsoids

I am reading some survey about euclidean sections of convex bodies(http://arxiv.org/abs/1110.6401). It is written that any $k$-dimensional ellipsoid easily seen to have a $k/2$-dimensional section ...
1
vote
2answers
100 views

About consecutive integers covered by arithmetic progressions

Help me please to solve the following problem. There are $n$ arithmetic progressions of the form: $$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$ Initial integer terms $x_i \geq 0$ are varying. ...
1
vote
2answers
80 views

On the automorphism group of binary quadratic forms

This question is a continuation of the following two questions: Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion. On certain solutions of a quadratic form ...
1
vote
0answers
25 views

Functorial Path objects for bisimplicial sets

Let $s^2Set$ denote the category of bisimplicial sets with the Bousfield-Kan model structure. Recall that a path object is a factorization of the diagonal $X\rightarrow X\times X$ into $X\xrightarrow{...
-1
votes
1answer
94 views

Clarification of the proof of the main theorem of the paper of Hulse et al

I am trying to understand some open steps in the following article The Sign of Fourier coefficients of Half-integral Weight Cusp Form by Hulse, Kiral, Kuan, and Lim, I find the following : Let $f\...
0
votes
0answers
20 views

Comparison inequalities for Hamiltonian mechanics with convex potential - analogue to Rauch's theorem?

I'm asking about an area (Hamiltonian mechanics) that I don't know at all well; thus, I keep the question somewhat vague. In differential geometry, there are a number of results saying that geodesics ...
1
vote
0answers
39 views

The composition of a dissipative operator and a positive operator is dissipative?

Consider the following bilinear system on a open and bounded domain $\Omega$ \begin{equation} \left\{\begin{array}{r c l} \displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\ y(0) &...
1
vote
1answer
31 views

Non-lattice Veech groups

I was thinking of Veech surfaces, which are translation surfaces whose stabilizer under the $\mathrm{Sl}_2(\mathbb{R})$ action is a lattice in $\mathrm{Sl}_2(\mathbb{R})$. They seem to have been ...
4
votes
1answer
160 views

Hodge standard conjecture for étale cohomology

It is known that Hodge standard conjecture is true for étale cohomology for field $k$ in characteristic zero. It means that the following pairing $$(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\rangle$$ ...
1
vote
0answers
27 views

$L^\infty(\Omega)$-regularity for strongly damped wave equation

I am interested in the following IBVP for the strongly damped wave equation: \begin{equation} u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\ u=0 \quad \text{on} \ ...
2
votes
0answers
21 views

Matrix minimization

Is there an explicit solution to the problem of minimizing $||X-X_0||_F^2+||X^{-1}-Y_0||^2_F$, with respect to matrix X, where $X_0$ and $Y_0$ are given, and all matrices are real $n\times n$ and ...
1
vote
0answers
36 views

Parabolic characters of subgroups $\Gamma \subset \operatorname{SL}_2(\textbf{Z})$ generated by parabolic and elliptic elements [on hold]

In the paper Generalized Modular Forms from Knopp and Mason, one can read in page $6$: Remark. It is not too hard to prove that a subgroup $\Gamma$ of finite index in $\operatorname{SL}_2(\textbf{Z})$...
0
votes
0answers
20 views

Is the inverse of the criterion of fully-faithfulness of derived tensor product also true?

Let $A$ and $B$ be differential graded algebras over a field $k$ and $D(A)$ and $D(B)$ be the derived categories of differential graded modules. Let $N$ be an $A$-$B$ bimodule. Then $(-)\otimes^{\...
2
votes
1answer
75 views

Projectively equivalent toric varieties

Say I have two normal lattice polytopes $P$ and $Q$ in $\mathbb{R}^n$ (lattice $\mathbb{Z}^n$) with the same number of lattice points $N+1$. Then they define two toric varieties $X_P$ and $X_Q$ which ...
1
vote
1answer
35 views

Limit of stochastic subsequence of stationary ergodic sequence

Let $\{X_k\}_{k\in\mathbb{N}}$ be a stationary ergodic sequence on a probability space $(\Omega,\mathcal{F},P)$ with shift $T$. Also, let $\{v_k\}_{k\in\mathbb{N}}$ be a sequence of random variables ...
0
votes
0answers
15 views

Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$

Let $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ $(\Omega,\mathcal A,\...
7
votes
1answer
298 views

Algebraic structure on homotopy groups of spheres

It is about a "conjecture" I heard (when I was student). There would exist an algebraic structure on the homotopy groups of spheres such that this algebraic structure would be the free algebraic ...
0
votes
1answer
177 views

Number of fixed points in Zagier's involution (Fermat's Theorem) [on hold]

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...
1
vote
1answer
92 views

Why vanish the integer m of an ample line bundle in the Kodaira embedding theorem?

I try to understand the following version of the Kodaira embedding theorem: Let $X$ be a compact Kähler manifold. A line bundle $L$ is positiv if and only if it is ample. I have a problem with the '...
3
votes
1answer
96 views

Finitely generated subrings of $\mathbb{R}$ are finitely approximable

In Ivanov's Finite Approximability of Modular Teichmüller Groups, for the proof of Lemma 2, the following is stated: Let $G$ be a finitely generated group and $\tau: G \to \operatorname{PSL}(2,\...
-3
votes
0answers
58 views

Advice on Family Index theorem [on hold]

I am reading Bismut's paper Family Index and Heat equation, but I have knowledge on the probability or stochastic. Could anyone give some advice or introduce some ref. on probability to understand ...
4
votes
1answer
165 views

Fermionic Wick Theorem

Assume we are given are fermionic many-particle system with creation operators $c_1^\dagger,c_2^\dagger,...$ acting on some Hilbert space $\mathcal{H}$. That is, the $c_i^\dagger$ and their ...
0
votes
1answer
89 views

An inequality in product space $V$ [on hold]

I found an inequality as following: Let $x, y, z$ be three complex numbers then: \begin{equation*} \frac{1}{2}(|y+z-x|+|x+z-y| + |y+x-z|) \le |x| + |y|+|z|+\frac{1}{2}|x+y+z| \end{equation*} (1) The ...
1
vote
1answer
49 views

Fourier transform of complex functions [on hold]

I want to know what is the meaning of Fourier transform for a complex function, say function $F:\mathbb C \to \mathbb C$? Up to my knowledge, Fourier transform is defined for real-valued or complex-...
-5
votes
0answers
51 views

On the symmetricity of the Riemann zeta function [on hold]

Yesterday, i stumbled upon something quite interesting about the Riemann zeta function, and i posted it on MSE. However, i could not get a conclusive response, hence i decided to post it here, though ...
0
votes
0answers
28 views

Reference Request: “Resolutions” of $K_n$ for $n$ odd

A resolution (in the combinatorial design sense) of $K_{n}$ is a collection of sets of edges of $K_{n}$ so that within each set of edges, each vertex appears once, and over the entire collection, each ...
0
votes
0answers
26 views

Counting different residue classes of certain form modulo $4abc-1$?

I like to calculate number of different residue classes of the form $b^2c$ modulo $4abc-1$, where $a,b,c$ are positive integers and $\gcd(a,b)=1$. I could prove that if $abc$ is prime there are only 3 ...
6
votes
1answer
142 views

A $n$-gon is isospectral to a regular $n$-gon (Isospectral $\implies$ isometry ?)

If an $n$-gon $P$ is isospectral to a regular $n$-gon $Q$, what could we say about the shape of the $P$. Otherwise, what could we say about $Q$? In fact, some hints or simply some ideas should be ...
0
votes
0answers
73 views

construction of the group $O(2,1)$ [on hold]

I am reading a paper "Proper actions and Pseudo-Riemannian space forms" by R. Kulkarni, and I am interested are there any references/papers which decribe the construction of the group $O(2,1)$ ...
1
vote
1answer
84 views

Polish Group Topologies on PSL(2,C)

Does anyone know how many Polish group topologies (or where to begin to look for this information) can be put on $\text{PSL}_2(\mathbb C)$?

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