0
votes
0answers
7 views

Two questions on the Schur multiplier of groups of order $p^4$

I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$. The case in which $p=2$ is well known, see e.g. Table 1 at ...
0
votes
0answers
7 views

pushing out families of curves

Let $f:X\rightarrow Y$ be a morphism of schemes with smooth curves as fibers. Let $g:X\rightarrow Z$ be a family of smooth or nodal curves with $Z$ a regular scheme. Does the push-out $Z\coprod_X Y$ ...
0
votes
0answers
10 views

Distinct determinants of circulants

If $M$ is a circulant integer matrix of size $n\times n$ whose entries are randomly chosen from $\{0,1\}$ value, how many different determinants does $M$ possibly take value in? If $M$ is symmetric ...
0
votes
0answers
12 views

Nontrivial norm values in rings of integers

Let $L=Q(\sqrt{d})$ for some SQF integer $d\equiv_4 3$ (the same can be asked for $d\equiv_4 1$). In this case the ring of integers of $L$ is $O_L=\{x+\sqrt{d}y \mid x,y\in Z\}$ so the norms of $O_L$ ...
2
votes
0answers
62 views

Distinct Numbers

Let $x_{ij}\in\{0,1\}$ for ${i=1}$ to ${m}$ and for ${j=1}$ to $n$. How many different values does $$\prod_{i=1}^m\sum_{j=1}^nx_{ij}$$ cover? Is there an $a_{ijk}\in\Bbb R$ (there is a $a_{ijk}\neq0$ ...
0
votes
0answers
32 views

Generalized Dedekind Sum Reciprocity Law

Is there a reciprocity law for generalized Dedekind sums of the form: $$S(a,b;x,y;c)=\sum_{k \mod c}\tilde{B}_1\left(\frac{ak+x}{c}\right)\tilde{B}_1\left(\frac{bk+y}{c}\right)$$ such that the other ...
1
vote
0answers
16 views

Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution ...
2
votes
2answers
35 views

Measurable representations of semi simple Lie groups

Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple Lie groups, ...
3
votes
0answers
40 views

Which known theorems of Lie algebras are still valid for Leibniz algebras?

Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. Thus, it is common to see a lot of papers which topic is about a generalization of a classic theorem of Lie algebras ...
1
vote
0answers
38 views

Limiting Entropy of deterministic sequences - 1

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution ...
4
votes
0answers
39 views

Retractions of Yoneda are retractors, i.e., left adjoints?

Background It is fairly well known that if a full subcategory $i: C \hookrightarrow D$ has a left adjoint $F: D \to C$, then the canonical counit $F i(c) \to c$ is an isomorphism. (A classical ...
-1
votes
0answers
80 views

Fundamental group of connected sum for non-orientable manifolds

For orientable manifolds, $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$, fundamental group of connected sum is free product of fundamental groups. As far as I understand, for non-orientable manifolds connected sum ...
2
votes
1answer
71 views

Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface)

What is the co-rank of a group $$G=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation? By co-rank, I mean the ...
2
votes
1answer
104 views

Stable Household Formation

I want to model the problem of household formation by a finite number of individuals, each of whom has preferences over sets of housemates. A collection of households is unstable if there is a set ...
3
votes
0answers
43 views

Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...
6
votes
2answers
157 views

Cubic-exponential enumerative combinatorics

There are many quantities in enumerative combinatorics that grow roughly exponentially, like the Fibonacci numbers, the Catalan numbers, and the factorials; indeed, most of the functions that arise in ...
1
vote
0answers
41 views

Bounded functions dense in Sobolev Spaces

Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...
0
votes
0answers
13 views

Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem: Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation: $Z_{k+1}-Z_k=P_k(1-2Z_k)$ where $P_k=0$ with probability ...
-1
votes
0answers
14 views

Finite time optimization problem of a linear time varying discrete(LTV) multi input multi output(mimo) system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot ...
2
votes
0answers
89 views

Is a quasi-coherent sheaf of ideals with free stalks of rank 1 a Cartier divisor?

Let $X$ be a scheme and let $\mathscr{I} \subset \mathscr{O}_X$ be a quasi-coherent sheaf of ideals. Suppose that for each $x \in X$, the stalk $\mathscr{I}_x$ is generated by an element $f_x \in ...
2
votes
4answers
322 views

A generalized diagonal?

A simple question. Let $ f:X\to Y $ be a function and let $ E_f:=\{(x, y): f (x)=f (y)\}\subset X\times X $. What is the name of the set $ E(f) $? It would be nice to have some reference also. It ...
2
votes
1answer
105 views

Trying to prove a congruence for Stirling numbers of the second kind

This a repost of a question I asked at Stack Exchange, but I got no answer so far, so I am trying here, even though it may not suit the "research level" requirement. Proposition: When $n$ and $m$ are ...
-3
votes
0answers
24 views

Generating sequence [on hold]

I was trying to generate a sequence of numbers. Let it be (i j k), ...
-1
votes
0answers
31 views

How to derive a sympletic form of a Hamiltonian in terms of wedge products

I know a Hamiltonian in $\mathbb{R}^{2N}$ can be represented as a sympletic form: $$\omega(X_h, v)= \langle DH,v \rangle$$ Could anyone tell me how to derive the following formula of $\omega$: ...
2
votes
1answer
64 views

Classical deformation of algebras

Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$. ...
2
votes
0answers
62 views

Preservation of universes in presheaves

In Lifting Grothendieck universes Hofmann and Streicher construct a universe in the category of presheaves over a small category given a Grothendieck universe in $\mathbf{Set}$. Suppose now I have ...
4
votes
1answer
98 views

The generic fiber pullback for $p$-divisible groups in characteristic $p$

Let $R$ be a discrete valuation ring with the field of fractions $K$ and the residue characteristic $p$. If $K$ is of characteristic $0$, then a celebrated theorem of Tate says that the pullback ...
-1
votes
1answer
73 views

Ext functor for more than two modules? [on hold]

The question is natural. Let's just work in the category of modules over a ring. Pick three modules $M_1, M_2, M_3$. Consider consecutive extensions of these modules, i.e., consider M, such that we ...
2
votes
1answer
79 views

Characterizing residually amenable groups

Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as ...
-1
votes
0answers
57 views

The functional $L(\varphi)=\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$

Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$. Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$. Let $G\triangleq\{\varphi\in ...
12
votes
1answer
277 views

Unifying Geometry for Characteristic Classes

When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches: Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...
0
votes
0answers
79 views

Banach space of discontinuous functions on a product space

Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define a semi ...
2
votes
0answers
147 views

Is the “algebraic closure” of the quaternions, finite dimensional?

This post is a sequel of: What's the algebraic closure of the quaternions? $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$, but it is not for the polynomials ...
-8
votes
0answers
32 views

How plot in logiciel R [on hold]

I have an estimator T(n,k) i.e. dependent in n and k where n is sample size and k=k(n) is a function n. How to plot (code in logiciel R) T(n,k) as function k for N=100 samples of size n=1000. ...
1
vote
0answers
51 views

A bilinear estimate in Lp space

Let $\varphi(D)$ be a Fourier multiplier with symbol $\varphi(\xi) = \xi/(1+|\xi|^2)$. It's easy to prove that \begin{equation} \|\varphi(D)u^2\|_{H^s(R)}\lesssim \|u\|^2_{H^s(R)} \quad (*) ...
-3
votes
0answers
37 views

Proof the monotonic of a iterated function equation [on hold]

Assume $f(x)\in C[0,1]$,and $f(0)=0,f(1)=1,f(f(x))=x$,Proof $f(x)$is monotonic in $[0,1]$. Above is a question in my homework and it is easy. My question is : (1) what if $f(f(x))= Arbitrary ...
6
votes
1answer
115 views

A version of Wald identity

Let $W$ be a standard one-dimensional Brownian motion. Let $T$ be a stopping time with $\mathbb{E}\sqrt{T}<+\infty$. Then $$\mathbb{E}W_T=0\quad \mathbb{E}W^2_T=\mathbb{E}T$$ I can prove these ...
0
votes
0answers
97 views

A question on Ito integral

Let $W$ be a standard one-dimensional Brownian motion and $0<T<+\infty$. Then $$\lim_{\beta\to+\infty}\sup_{0\le t\le T}|e^{-\beta t}\int_0^te^{\beta s}\mathrm{d}W_s|=0\quad \text{a.s.}$$ Could ...
-1
votes
0answers
97 views

Secret Santa Assignments! [on hold]

I'm trying to figure out how many ways there are to assign Secret Santas to a set of N (distinguishable) people. For those who don't know what this is, it's a gift exchange where each person is ...
0
votes
1answer
112 views

Sending one curve on a surface to the other by a homeomorphism [on hold]

Consider two arbitrary simple closed curves on a closed orientable surface. Does there always exist a homeomorphism of a surface, sending one curve to the other?
23
votes
6answers
2k views

Companion to theoretical physics for working mathematicians

In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...
0
votes
0answers
45 views

Asymptotic pseudo orbit of an action

Let $G$ be finitely generated group (i.e $G= <S>$ $S=\{ s_1, ..., s_n\}$) and $\varphi:G\times M\longrightarrow M$ is an action then $f:G\longrightarrow M$ is called $\delta$- pseudo orbit if ...
0
votes
0answers
24 views

Multidimensional Filters

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then ...
3
votes
1answer
211 views

On the elliptic curve $x(x+a^2)(x+b^2) = y^2$

Ajai Choudhry showed that special cases of the elliptic curve, $$x(x+a^2)(x+b^2)=y^2\tag1$$ can be used to prove that, $$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$ has an infinite number of primitive ...
0
votes
0answers
72 views

Codim r strata in a scheme

My question is what is the definition of codimension r strata for a scheme $X$, with NCD $D$. I heard the term in a lecture but I could not find it nowhere.
2
votes
1answer
140 views

Splitting varieties of two Galois cohomology symbols

One characteristic property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$, meaning: For some $\alpha \in H^n(k,\mu_p)$ ...
-1
votes
0answers
73 views

Riemann Mapping Theorem [on hold]

It is well known that the Riemann mapping theorem asserts that for any open simply connected $G\subset \mathbb{C}$ and $z_{0}\in G$, there exists a unique bijective analytic function $f:G\to ...
11
votes
0answers
215 views

Evaluating products of cyclotomic polynomials at roots of unity

Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are ...
0
votes
0answers
39 views

What we can say about the behavior of the solution of an ODE? [on hold]

Let $\mu_1>\rho>0>\mu_2$, $\lambda_i>0$ and $\sigma>0$. Consider the ODE: $\begin{equation} \frac{1}{2}\left(\frac{\mu_1-\mu_2}{\sigma}\right)^2p^2(1-p)^2 \frac{d^2u}{dp^2}(p)+ ...
6
votes
0answers
103 views

Complex structures on $\Bbb R^4$

Calabi & Eckmann proved that $S^{2p+1} \times S^{2q+1}$ admits an integrable complex structure fibred by holomorphic tori, and this implies that $\Bbb R^{2p+2q+2}$, obtained by removing a point in ...

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