# All Questions

**0**

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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

**0**answers

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**

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**0**answers

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 ...

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votes

**0**answers

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**

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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 $a_{ijk}\neq0$ ...

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**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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 ...

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**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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 ...

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**0**answers

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

**0**answers

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

**0**answers

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

**4**answers

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

**1**answer

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**

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**0**answers

24 views

### Generating sequence [on hold]

I was trying to generate a sequence of numbers.
Let it be (i j k),
...

**-1**

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**0**answers

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

**1**answer

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**

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**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

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**0**answers

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

**1**answer

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, ...

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**0**answers

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**

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**0**answers

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 ...

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votes

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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 ...

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**0**answers

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

**0**answers

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

**1**answer

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

**6**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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 ...