# All Questions

**-1**

votes

**0**answers

3 views

### Number of ways 3 people can take up to 3 stones from a bag containing 5 stones

Number of ways 5 people can take stones from a bag containing 5 stones where the first persons may take up to 3 stones (0 is allowed), second can take up to as many as the first person took, third may ...

**0**

votes

**2**answers

27 views

### Functions with scalar times orthogonal Jacobian

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix.
I've seen a similar ...

**1**

vote

**0**answers

275 views

### Elementary analytic number theory problem

$\forall k>10^2$ is there $m_k\in\Bbb N$ such that at infinitely many pairwise coprime $a,b,q$ with $q>m_k$ there is a $c\in\Bbb N$ such that
$$(1)\quad ...

**1**

vote

**0**answers

5 views

### On the numerical range of non-self adjoint Gaussian matrix

For a complex $n \times n$ matrix $A$, its numerical range is the set
$$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$
We can further define the ...

**10**

votes

**2**answers

836 views

### Translation length functions of non-simplicial trees

Let $G$ be a finitely generated group. By a theorem of Culler and Morgan, the set of non-abelian (not necessarily simplicial) minimal $\mathbb{R}$-trees with isometric $G$-action injects into the ...

**-5**

votes

**0**answers

12 views

### Complete the table by determining the value of each letter Explain what rule is and how you found it [on hold]

The Problem
Thank you in advanced :)

**0**

votes

**0**answers

9 views

### What is an upper bound for $\|E(X|\mathcal{A})-E(X)\|$?

Let $X$ be a random element in a Banach space with norm $\|\cdot\|$ less than 1, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $\|E(X|\mathcal{A})-E(X)\|$?
Existing results:
It ...

**1**

vote

**0**answers

75 views

### globally well-defined holomorphic vector field on a curve $y^N = x^2 - z^2$

Let us start with a multiple cover C of the x-plane branched at $z$ and $-z$, and so described by an equation $y^N = x^2 - z^2$.
For N=2, it is known that there are globally-defined holomorphic ...

**4**

votes

**1**answer

50 views

### I have a very large sparse matrix, 'A', in Ax = b. What work in advance of getting 'b' can be done to reduce solving time?

This question borders between a programming and math question (more math). I have a little matrix knowledge but this is past my ability, so any help is very much appreciated.
Question
I have a very ...

**1**

vote

**0**answers

102 views

### First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...

**1**

vote

**1**answer

210 views

### Natural number properties as uninterpreted functions in first order logic

Can we express the following property of natural numbers as FOL. The property given below is only indicative, I am more interested in knowing how the concepts such as "infinitely many X exists for so ...

**11**

votes

**5**answers

1k views

### Texts on the General History of Contemporary Combinatorics

I am looking for some core texts (books, book chapters, papers) about the general history of contemporary combinatorics, starting, say, from the interwar period up to today.
Texts about the history ...

**0**

votes

**0**answers

17 views

### (Topological) K-theory for commutative $C^*$-algebras: operator and standard approaches

Let $A$ be a commutative unital $C^*$ algebra. Then $A=C(X)$ for some compact Hausdorff space $X$. Topological $K$-theory group (namely $K_0$) is defined in terms of vector bundles as a Grothendieck ...

**7**

votes

**0**answers

54 views

### Which paths in a graph are orthogonal to all cycles?

Start with some standard stuff. Suppose we have a directed graph $\Gamma$. I'll write $e : v \to w \,$ when $e$ is an edge going from the vertex $v$ to the vertex $w$. We get a vector space of ...

**15**

votes

**2**answers

671 views

### A set of integers whose factorial can be written as a product of two factorials

I am trying to collect informations concerning the set
$$\mathcal{A}=\left\{n\in\mathbb{N} \mid (\exists k,l\in\{2,3,\dots,n-2\})(n!=k!l!)\right\}.$$
It seems not much is known about the set ...

**2**

votes

**1**answer

41 views

### relate shellability of a simplicial complex to the links of its faces

Reisner's criterion give a complete characterization of Cohen–Macaulay simplicial complexes, based on $link$s of faces of the simplicial complex. Is there a known fact that relate shellability of a ...

**1**

vote

**0**answers

26 views

### Determine Toeplitz matrix

For an arbitrary $NXN$ Hermitian matrix $A$. I want to derive a Toeplitz matrix from $A$ such that eigenvectors of both matrix has minimal change. Specifically I want find the Toeplitz matrix such ...

**2**

votes

**0**answers

21 views

### Groebner bases for differential operators with field coefficients (reference request)

Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements ...

**3**

votes

**0**answers

115 views

+100

### Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...

**2**

votes

**0**answers

57 views

### Is it true that irreducible smooth representations of $G_2(F)$ are self-dual?

Let $G_2$ be the split exceptional group of type $G_2$ and $F$ be a p-adic field. Is it true that every irreducible smooth representation of $G_2(F)$ is self-contragradient? If the answer is Yes, can ...

**4**

votes

**0**answers

50 views

### Computations in Weyl algebra with rational function coefficients

I am looking for a software to perform calculations with modules over the algebra $R_n=\mathbb{C}(x_1\ldots x_n)\langle \partial_1\ldots\partial_n\rangle$ of differential operators with rational ...

**0**

votes

**0**answers

43 views

### Dimension of a sheaf cohomology group on a genus 1 curve

Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...

**0**

votes

**0**answers

21 views

### Linearization of product of two variables

In the objective function of a mathematical programming model,we have an expression like this:
$$
\biggl(\biggl|X\biggl| \biggl) . Q
$$
in which both X and Q are continuous variables, and $||$ ...

**3**

votes

**1**answer

75 views

### Vanishing of power of nilpotent operator $\mathrm{ad} \, \;e$ in different characteritics

This question needs some background:
(1) In his influential 1959 paper here, Kostant studied the adjoint representation of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (which can be ...

**14**

votes

**1**answer

286 views

### Swan K-theory of Z/4

Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with ...

**3**

votes

**0**answers

30 views

### Solving algebraic recurrence relations on a cyclic graph

I have a set of $n$ variables $p_1, \ldots p_n$ with $0 \leq p_i \leq 1$ and a defining equation for each of one of the forms:
$p_i = 0$.
$p_i = 1$
$p_i = p_j p_k$ for some $j, k$ with $i, j, k$ all ...

**0**

votes

**1**answer

69 views

### Invariance of spin coefficients

I have a question about how spin coefficients (Newman Penrose formalism) transform.
I know that if we perform a tetrad rotation, say of Class III:
$(l,n,m,\overline{m})\mapsto \left(\frac{1}{A}l, ...

**1**

vote

**1**answer

142 views

### Algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $SL(2,C)$-character variety

Does the following statement:
"Let $G$ be a finitely generated
group and let $X(G)$ be the
$SL(2,\mathbb{C})$-character variety
of $G$. Suppose $X(G)$ contains an
irreducible component ...

**-6**

votes

**0**answers

29 views

### Problem in calculating integral [on hold]

[I tried a lot to calculate integral of this question and I also tried it with substitution method but I failed to calculate.]
[1][Question]: http://i.stack.imgur.com/ZLBZ7.png

**5**

votes

**0**answers

93 views

### Semi-continuity of intersection numbers

I always trusted the following quite vague statement:
If you have a family of divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say varying in ...

**1**

vote

**1**answer

105 views

### Spacing of the largest singular values of Wishart matrix

Let $X \in \mathbb{R}^{n \times p}$ consist of iid $\mathcal{N}(0,1)$. Assume that $n/p$ converges to a positive constant. Denote by $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_{\min(n,p)} \ge 0$ the ...

**3**

votes

**0**answers

25 views

### Is the restriction of a graded automorphism linearizable in characteristic zero?

This question follows up a previous one which was answered by Todd Leason. I want to impose two new requirements on the setup.
Let $k$ be a characteristic zero field. Let $A=k[x_1,\dots,x_n]$ be the ...

**1**

vote

**0**answers

12 views

### random consecutive decreasing subset/chain in point process

In my study of percolation system, I encounter a very interesting problem. I tried to map it into well-studied permutation problem but not very successful... I debrief it as follows:
imagine you have ...

**0**

votes

**1**answer

159 views

### Approximation of real numbers

Is there any function $f(x)$, such that for all real $\alpha$ and rational $p/q$
$$\left|\alpha-\frac{p}{q}\right|>\frac{1}{f(q)}.$$
or at least
...

**1**

vote

**1**answer

100 views

### A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ...

**3**

votes

**3**answers

309 views

### What is the group of automorphisms of $l^{\infty}$?

What is the group of automorphisms of $l^{\infty}$?
I think it would be the permutations of the integers. Is this right?

**1**

vote

**1**answer

88 views

### The reproducing kernel for harmonics on compact manifolds

Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...

**1**

vote

**0**answers

60 views

### extension for a complex operator

Let be $\lambda>0$. Put
$$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial ...

**5**

votes

**1**answer

165 views

### Examples to keep in mind while reading the book 'The Admissible Dual…' by Bushnell and Kutzko and the importance of Interwining of representations

I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko.
Let me first describe the book a ...

**7**

votes

**2**answers

227 views

### “Identity tensor transpose” as a map $M_n \hat{\otimes} M_n \to M_n \overline{\otimes} M_n$

Equipping $M_n$ with its usual operator space structure,
$\newcommand{\ptp}{\widehat{\otimes}}$
we can form the projective tensor product of operator spaces $M_n\ptp M_n$. In particular this puts a ...

**7**

votes

**4**answers

388 views

### Picard groups of quartic K3 surfaces

Does anyone know where I can find examples of quartic K3 surfaces for which the Picard group is known? I'm really interested in examples where there are explicit constructions of the divisors ...

**2**

votes

**2**answers

289 views

### Irreducible algebraic sets via irreducible polynomials

There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...

**5**

votes

**1**answer

247 views

### Milnor descent for ring spectra

Suppose given a homotopy cartesian square of (commutative) ring spectra (or (c)dgas)
$\begin{matrix}A & \to & A_1 \\
\downarrow & & \downarrow \\
A_2 & \to &A'\end{matrix}.$
...

**14**

votes

**1**answer

706 views

### Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in ...

**9**

votes

**1**answer

120 views

### Geometric quantization: why are the prequantum operators self-adjoint?

I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical ...

**5**

votes

**0**answers

59 views

### Integral representation of adjoint L-factor for GL(2)

My question is about a local computation in the paper of Gelbart and Jacquet, "A relation between automorphic representations of GL2 and GL3", from 1978.
Let $\sigma$ be an irreducible smooth complex ...

**41**

votes

**3**answers

4k views

### Is it possible to have a research career while checking the proof of every theorem that you cite?

A colleague raised the above question with me; more precisely he said:
Suppose that a mathematician were resolved not to publish any theorems
unless they had checked the proof of every theorem ...

**2**

votes

**1**answer

40 views

### How nontrivial can “central extensions of ribbon fusion categories” be?

In a sense, this is a follow up on this question, but one PhD programme later.
Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of ...

**1**

vote

**1**answer

175 views

### Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...

**2**

votes

**0**answers

148 views

### Number of critical points of a smooth function

Are there any examples of closed manifolds with the property that the minimal number of critical points (possibly degenerate) of a smooth function on this manifold is strictly bigger than the stable ...