-1
votes
0answers
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
2answers
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
0answers
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
0answers
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
2answers
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 ...
0
votes
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
5answers
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
3answers
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
1answer
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
0answers
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
1answer
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
2answers
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
4answers
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
3answers
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
1answer
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
1answer
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
0answers
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 ...

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