0
votes
0answers
18 views

Standard conjectures on positive characteristic

In this MO answer of M. Bondarko, he says: "the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..." and in Remarks on Grothendieck's ...
1
vote
0answers
9 views

target category of extended field theory

An A-S TFT is a functor from $\text{Bord}_{<n−1,n>}(\mathcal{F})$ to $\text{Vect}$ where $\mathcal{F}$ denotes a set of background fields, eg a spin structure. An extended theory is a functor ...
0
votes
0answers
18 views

Game on the tree

There's a problem from programming competition which already finished: http://codeforces.com/contest/458/problem/F Two weeks already passed but still nobody solved it yet - in fact you can see here ...
1
vote
1answer
14 views

When does a hypergraph represent maximal independent sets?

Let $G = (V,E)$ be a simple graph. Then, we can view the set of maximal independent sets (or the set of maximal cliques) as a hypergraph $H = (V, E')$. This is quite a useful device when connecting ...
0
votes
1answer
53 views

Who defined and who coined “module”?

The title of my Q. says it all: QUESTION:   Who defined and who coined: module? Would it be Emmy Noether?
2
votes
0answers
41 views

How do I find coefficients of a product expansion

Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways: $$1 + \sum_{i=1}^\infty f_i t^i = \prod_{i=1}^\infty (1-t^i)^{-n_i}$$ Here, the $f_i$ and $n_i$ ...
3
votes
0answers
50 views

Who originated the standard symbols for Lie groups GL, SL, SU, etc.?

Who was first to use symbols GL, SL, O, SO, U, SU, Sp and their projective versions, and how did this notation become standard? The notation appears in fairly modern form in Weyl's "The Classical ...
2
votes
1answer
12 views

A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...
2
votes
0answers
45 views

Higher Degree Data in a Cosimplicial Quasicategory and Delooping

If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference. My question is regarding accessing data ...
0
votes
0answers
64 views

A System of Diophantine Equations [on hold]

$p^2+1=2y^2$ $p+1=2x^2$ $p$ is prime and $x,y$ are integers. I conjecture that this only has solution for $p=7$
2
votes
0answers
25 views

Extension of the product formula for valuations to a simultaneous completion

It is well known that $\mathbb{C}$ and $\mathbb{C}_p$ are "algebraically" isomorphic (that is, ignoring the topology), but an isomorphism depends on the axiom of choice and there is no canonical way ...
1
vote
0answers
24 views

Toda Flow Embeddings

What are strategies for generating the following types of picture: Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are: $$\frac{d}{dt}a_k=2(b_k^2-b_{k-1}^2),$$ ...
1
vote
2answers
102 views

For a defined set $M$ (see problem) do there exist $a,b$ natural numbers so that $a,ab+1 \in M$

Let $\rho \in \mathbb{R}\setminus \mathbb{Q}$ be a irrational nuber, and let $\varepsilon>0$ be arbitrarily small. Define $M=\{m \in \mathbb{N}: \exists k \in \mathbb{N}\hbox{ so that} |\rho m -k ...
6
votes
0answers
92 views

Continuous relations? [on hold]

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful? I have a vague idea of the possibility of ...
2
votes
0answers
32 views

Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$. On the other ...
2
votes
4answers
73 views

Approaching convex and discrete geometry from other disciplines

I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning ...
1
vote
1answer
69 views

Nearby cycles and specialisation - properties

I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
1
vote
2answers
84 views

Plucker embedding and tautological/universal quotient bundle

Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the ...
1
vote
0answers
97 views

A lifting problem

Let $E\overset{\pi'}{\longrightarrow} B'$ and $E\overset{\pi}{\longrightarrow} B$ be vector bundles. For $i=0,1$, let $f_i$ be a fiber-preserving open embeddings of $\pi'$ into $\pi$, with $g_i$ the ...
8
votes
1answer
54 views

Torsion-free group that is not of type F but is virtually of type F

Recall that a group $G$ is of type F if there exists a compact $K(G,1)$. There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...
3
votes
1answer
45 views

Reference for Elliptic PDE on $\mathbb{R}^d$

Could anyone suggest a textbook, article, or lecture notes that covers elliptic PDE theory (existence, uniqueness, regularity) on all of $\mathbb{R}^d$, as opposed to the Dirichlet or Neumann problem ...
1
vote
1answer
23 views

Eigendecomposition of analytic Hermitian matrix-valued functions of several variables

If $A(t)$ is an analytic, Hermitian matrix-valued function of a real variable $t$, then it is known that there are analytic functions $\lambda_i(t)$ and $x_i(t)$ corresponding to the eigenvalues and ...
-4
votes
0answers
48 views

Quantifier problem of equations in physics [on hold]

Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down. To assert that static energy equals mass ...
4
votes
0answers
58 views

Is the sequence of Apéry numbers a Stieltjes moment sequence?

Consider the sequence of Apéry numbers $$ A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3 = \sum_{k=0}^n \binom{k}{n}^2\binom{n+k}{k}^2 . $$ In an email, physicist Alan Sokal ...
1
vote
0answers
66 views

Group action induced on homology under change of coefficients

Let $M$ be a closed manifold equipped with a (continuous or smooth) $\mathbb{Z}_2$-action such that - for simplicity - both $H_{k-1}(M;\mathbb{Z})$ and $H_{k+1}(M;\mathbb{Z})$ are zero for some ...
0
votes
0answers
25 views

Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization

I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform Probability ...
1
vote
0answers
87 views

Operator norm vs spectral radius for positive matrices

I believe the following statement should be true but somehow I don't see an argument: For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...
0
votes
0answers
50 views

Semidirect products with braid groups and type $F_\infty$

Let $F$ be a group which is strongly type $F_\infty$ in the sense that every subgroup is of type $F_\infty$. Here, type $F_\infty$ means that the group admits a classifying space with compact skeleta. ...
4
votes
0answers
86 views

Without Skolem–Mahler–Lech Theorem? [on hold]

Using Skolem–Mahler–Lech theorem one can easily prove the $\displaystyle \lim_{n\to +\infty}\left|\Re\left(\frac{1+i\sqrt{7}}{2} \right)^n\right| =+\infty$. Is there a "simple way" to prove this ...
0
votes
0answers
15 views

Mixed Integer Quadratic Programming using Opti Toolbox in MATLAB [on hold]

I wish to solve a mixed integer quadratic program with linear constraints using OPTI toolbox in MATLAB. I want some of my decision variables to be continuous and some decision variables to be binary. ...
1
vote
1answer
29 views

Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton

I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations ...
2
votes
1answer
43 views

Find base of kernel with as many 0 as possible

I have a 400x132 rectangular matrix with only 0 and 1. I am looking for the linear combinations of the columns of the matrix that sum to 0. For example C1 + C2 - C3 = 0. I want to find the linear ...
2
votes
2answers
46 views

Proof the solution of von Neumann equation will fluctuate forever if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will fluctuate ...
2
votes
0answers
37 views

Linear transformation of a polyhedral cone

Let $C$ be a polyhedral cone in $\mathbb{R}^m$ with H- and V-representations $C = \{x : A x \le 0 \} = \{R y : y \ge 0\}.$ The pair $(A,R)$ is referred to as a double description (DD) pair of the ...
0
votes
0answers
25 views

Inequality related to regularized incomplete beta function

I am trying to analyze the following inequality related to regularized incomplete beta function $I$: $$p^{c}/2\geq I_{\frac{p}{1+p}}(\alpha,\alpha)= ...
0
votes
0answers
122 views

concepts in fields other than physics and computer science which touch on concepts that are fundamental in pure mathematics [on hold]

[ Contextual Information: Is mathematics held back by it's relative lack of ties to disciplines other than physics and computer science? Are there areas of mathematics which have gone underexplored ...
3
votes
0answers
85 views

How does the Atiyah-Singer index theorem in a relative setting related to “ringed spaces and pseudocoherent complexes of finite tor-dimension”?

I come across the following paragraph from the article Reminiscences of Grothendieck and His School, here is from the part of the interview by Luc Illusie,: " I was indeed looking for an ...
0
votes
0answers
63 views

What is free product of two k-algebra (k is a field) [on hold]

Let A, B be k-algebra (k is a field). What is free product of k-algebra A and k-algebra B? ($A \ast_{k}B$)
0
votes
0answers
124 views

natural map from $H^3(BG,\mathbb{Z})$ to $H^3(\underline E G,\mathbb{Z})$?

is there any natural map from natural map from $H^3(BG,\mathbb{Z})$ to $H^3(\underline E G,\mathbb{Z})$? where $G$ is a discrete group, $BG$ is the classifying space, and $\underline E G$ is the ...
2
votes
1answer
71 views

Smith Normal Form for block matrix

are there any known results on the smith normal form for block matrices over the integers? In particular I am interested in matrices of size (kr)x(ks) made of square blocks of size k such that each ...
5
votes
1answer
82 views

Aperiodic set of corner Wang Tile [on hold]

There is quite some reference on aperiodicity of the edge-type of Wang Tile. But I could not yet find aperiodic corner type of Wang Tiles... Could someone provide me some instances (better with ...
0
votes
0answers
96 views

“Open Points” in the 1983 proof of Szemerédi-Trotter theorem

I was reading through the 1983 paper "Extremal Problems in Discrete Geometry" and I was confused about the definition of "open point" appearing in this paper. By this point in the paper, the authors ...
0
votes
0answers
45 views

Radial Symmetry [on hold]

Let $B\subset\mathbf{R}^n$ ($n\geq2)$ the open ball of radius $R$ centred at the origin and $u\in C^2(\overline{B})$. Suppose that $$ v_{ij}=x_i\frac{\partial u}{\partial x_j}-x_j\frac{\partial ...
0
votes
0answers
67 views

Superelliptic Curves [duplicate]

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...
2
votes
1answer
93 views

Pseudo-automorphisms on Fano varieties

Is every pseudo-automorphism (self-birational map which does not contract any hypersurface) of a smooth Fano variety of Picard rank $1$ equal to a biregular automorphism? Remark: For $\mathbb{P}^n$, ...
6
votes
0answers
119 views

A question about cardinal numbers when the Axiom of Choice is absent

The Axiom of Choice constrains every set of cardinal numbers which is linearly ordered by size to be well-ordered. By contrast, does ZF-without the Axiom of Choice (but with the Axiom of ...
8
votes
0answers
124 views

Is there a “hereditary” construction for $L$?

Recall that $L$, Godel's constructible universe is constructed by defining the following hierarchy: $L_0=\varnothing$, for a limit ordinal $\delta$, $L_\delta=\bigcup_{\alpha<\delta}L_\alpha$, and ...
-3
votes
1answer
136 views

Continuity of Continuity [on hold]

I' read somewhere the following: Let $I=[0,1]$ and equip $\mathcal{C}(I,\mathbb{R})$ with the uniform convergence topology. If $f:I^2\rightarrow \mathcal{C}(I,\mathbb{R})$ is a continuous map, then ...
0
votes
0answers
108 views

Simultaneous root of polynomials — must it exist by continuity? [on hold]

Suppose we have $n$ polynomials in $n$ variables $p_1, \dots, p_n$ and $n$ scalars $y_1, \dots, y_n$ which are in the range $[0,1]$. These polynomials have all positive coefficients. We want to find a ...
2
votes
0answers
42 views

Why do rotationally ordered configurations have well defined distributon function?

Let $u=(u_{j})_{j \in \mathbb{Z}}$ where $u_{j}\in \mathbb{R}$ for all $j \in \mathbb{Z}$ be a rotationally ordered configuration i.e. $S_{n,m}u>u$ or $S_{n,m}u<u$ or $S_{n,m}u=u$ where ...

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