3
votes
1answer
93 views

Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$

I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...
3
votes
0answers
123 views

Cancellation and splitting theorems for vector bundles etc over schemes

It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This ...
0
votes
0answers
32 views

A hyperbolic partial differential equation

How solve this equation (numeral or analytical)? $u(t,x)=\int_{t-x}^{t}{a \cdot e^{b \cdot s} \cdot \int_{0}^{s-(t-x)}{u(s,z)dz}ds}+\int_{t-x}^{t}{c \cdot e^{d \cdot s} \cdot ...
0
votes
0answers
30 views

Does log(1+|x|) be a sub-analytic function? [on hold]

Does log(1+|x|) be a sub-analytic function? Then I wonder more examples of sub-analytic functions. And how to easily check a function is or not. Thanks!
3
votes
1answer
85 views

Why is SL(n,Z)[p] modulo the group normally generated by elementary matrices abelian?

I'm trying to understand part of Bass-Milnor-Serre's paper on the congruence subgroup problem. I'm pretty sure that the following statement is proved in there, but I'm having trouble find (and/or ...
1
vote
0answers
103 views

Strong Dependence

I don't know if this definition has been already given. Suppose $X$ and $Y$ are two random variables over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. We say $Y$ is strongly dependent on $X$ if ...
3
votes
1answer
80 views

Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...
6
votes
1answer
153 views

Do the terms of an iid sequence whose law has infinite expected value necessarily exceed the partial sums of the sequence infinitely often?

Let $\mu$ be a probability measure on $(0,\infty)$, and let $(\mathbf X_n)_1^\infty$ be a sequence of independent $\mu$-distributed random variables. Fix $\kappa > 0$, and consider A) $\int x \; ...
-1
votes
0answers
117 views

Functoriality of the right Kan extension [on hold]

I'm trying to understand how the right Kan extension functors can satisfy the equation $Ran_{LK}T = Ran_L Ran_KT$ (assuming existence) and that their universal arrows are equal, for functors $T : M ...
0
votes
0answers
99 views

A question on a toric singularity

Recently I was told that the threefold singularity $$ x^2+y^2+z^2+w^{2n}=0 $$ in $\mathbb{C}^4$ is a toric variety. How can I see this? What are the generators of the toric fan? Moreover, is it true ...
1
vote
0answers
60 views

Real algebraic groups and pseudo-finiteness

What is the relationship between pseudo-finite groups and real algebraic groups? Could you provide an example of a pseudo-finite real algebraic group and of a non pseudo-finite one, if any? Thank ...
2
votes
0answers
70 views

$k$-Disk algebras versus $E_k$ algebras

Background: The little $k$-cubes operad is the $(\infty,1)$-operad defined by embedding disjoint unions of $k$-dimensional open cubes rectilinearly into one another, that is using maps ...
3
votes
1answer
153 views

Preservation Results for Iterations of Non-Proper Forcing

Suppose $\mathbb{P}$ is a forcing with the following properties: Let $G \subseteq \mathbb{P}$ be filter generic over $V$, then there exists $A \in V[G]$ such that $V[G]$ thinks $A$ is countable and $A ...
2
votes
0answers
70 views

Stricter Notion of Crossing in a Partition

Let $k$ be an integer. Traditionally a partition $\pi=V_1\cup \dots \cup V_n$ of the set $[k]:=\{1,\dots, k\}$ is called crossing when there exist $a,c\in V_i$ and $b,d\in V_j\not= V_i$ such that ...
0
votes
0answers
24 views

contraction of tubular neighborhoods in Berkovich spaces

a question about Berkovich spaces: There is "Tubular Theorem, 1" on page 27 of the slide http://math.univ-lyon1.fr/~thuillier/articles/Toric_Leuven.pdf which says: If D be a simple normal crossing ...
2
votes
0answers
161 views

Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
4
votes
0answers
115 views

Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of ...
-2
votes
0answers
57 views

Switching Algebra Simplification - Beginner - how to continue with this problem? [on hold]

I'm currently trying to simplify the following expression: f = w'xz + wy + wy'z'+ w'x'z' + xyz There are five terms, nine literals. Assume this notation: xyz = x*y*z, and z' = the complement of z. ...
1
vote
0answers
54 views

Can any reductive $k$-group be written as a semidirect product of $k$-linear groups?

This might be an easy question for the experts, so I apologise in advance. By a reductive group over a field $k$, I mean a linear algebraic group (not necessarily connected) such that the unipotent ...
3
votes
0answers
79 views

Density of function spaces

Let $\Omega$ be a subset of either $\mathbb{R}^n$ ($n\geq 3$, if it matters) or of a compact manifold. In either case, we'll call the manifold $M$. Let $V_i\subset V_{i+1} \subset \Omega$ be an ...
4
votes
2answers
85 views

Number of edges in linklessly embeddable graphs

Consider graphs over $n$ nodes. What is the maximum number of edges of a linklessly embeddable graph? A more general question is the following. Given $\mu$ what is the maximum number of edges of ...
8
votes
0answers
116 views

Reference for maps whose pushouts are also homotopy pushouts

Consider a category C with weak equivalences, e.g., a model category. For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) ...
0
votes
0answers
34 views

Derivative of softmax loss function [on hold]

I am trying to wrap my head around backpropagation in a neural network with a softmax classifier, which uses the softmax function: \begin{equation} p_j = \frac{e^o_j}{\sum_k e^{o_k}} \end{equation} ...
2
votes
1answer
36 views

Linear least squares with unordered response variable

In the classical linear regression model one considers the equation $$ y = X \beta + \epsilon.$$ I was wondering whether there are also results when the ordering of the response variable $y$ is not ...
1
vote
1answer
141 views

pencil of quadrics consisting of singular quadrics

A pencil $l$ of quadrics in $\mathbb{P}^4_{<x_0,\cdots,x_4>}$ consists of singular quadrics only if: (1) quadrics in $l$ have a common singular point; or (2) quadrics in $l$ contain a common ...
4
votes
1answer
63 views

metrizable neighborhoods of compact subsets

This is a question about general topology: Assume we are given a first countable Hausdorff space and a compact subset K. Is it possible to find a countable basis of open neighborhoods of K ? ...
0
votes
0answers
18 views

Coupled recurrence relations for generating functions, involving squared arguments

I'm trying to find a solution for the following system of three equations in terms of three bivariate generating functions. $G(x,y)=b(x,y) \cdot \left( G(x,y^2) + I(x,y^2) \right) +y ;$ $H(x,y)=x ...
3
votes
2answers
106 views

Finite Dimensional Simple nonunital associative Algebras

I have the following problem: Let K be any field. An finite dimensional associative non-unital algebra A is a vector space A, togeter with a K-biliniear associative operation such that there is NO ...
0
votes
0answers
22 views

Optimization problem involving an entrywise function

Let $X$ a $n\times p$ real-valued matrix and $Y$ a $p\times q$ real-valued matrix. Let $\phi:\mathbb{R} \to \mathbb{R}$ a function. What is the appropriate way to deal with the following optimization ...
0
votes
0answers
156 views

Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves

Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...
3
votes
1answer
68 views

Existence of ind-right adjoint functor for semi-simple category?

I'm just reading a lemma in Yves ANDRE's seminar on finite dimensional motives. Soit $Σ:Rep_F G→T$ un ⊗-foncteur vers une categorie F-tensorielle T …… where $G$ is a pro-reductive group scheme, ...
1
vote
2answers
202 views

An interpretation of not-Con(PA)

Edit After Andreas Blass answer below and comments below the original post I have changed it to accommodate posters' remarks. I hope it is clear and makes more sense now. Let $\mathrm{PA}$ be the ...
-5
votes
0answers
137 views

Why so much graph theory? [closed]

I am a Computer Science student. In my college, we study many aspects of Discrete Mathematics mainly Graph Theory. I understand that, for a Computer Scientist, it is important to know about graphs, ...
2
votes
1answer
73 views

Is barycentric subdivision of a collapsible, regular CW complex collapsible (non-evasive)?

Let $X$ be a finite, regular CW complex, and let $X'$ be its barycentric subdivision (i.e. the order complex of the face poset of $X$). Assume $X$ is collapsible. Is $X'$ collapsible? Is $X'$ ...
6
votes
3answers
406 views

sum of binary and ternary digits

A problem in group theory (indices of imprimitive groups) gives rise to the following conjectures in number theory. Suppose a positive integer $n$ has binary and ternary expansions ...
9
votes
0answers
127 views

Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

On the last page of Schmid's article "Discrete Series", he says "In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure ...
6
votes
1answer
363 views

Is there a generalized Birkhoff ergodic theorem?

Is there a Birkhoff ergodic theorem for two measure preserving transformations $T$ and $S$ where $S\circ T= T \circ S$ so that $\frac{1}{n+1}\frac{1}{m+1}\sum_{i=0}^{n}\sum_{j=0}^{m}f \circ T^{i}\circ ...
0
votes
0answers
24 views

a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...
0
votes
0answers
49 views

Bases of surface groups with length restrictions

This question asks for a generalization of Bases of surface groups following the notation and definitions given therein. Let $\Gamma_g$ be a surface group of genus $g \geq 2$, $B$ a surface basis of ...
3
votes
1answer
103 views

Conditional Form of Rosenthal's Inequality

Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following: If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and ...
1
vote
0answers
35 views

Handelman's positivstellensatz for symmetric matrix-valued polynomials

For certain classes of sets $S \subseteq \mathbb{R}^n$, there exist algebraic characterizations of real valued polynomials $p: \mathbb{R}^n \rightarrow \mathbb{R}$ that are positive on $S$. Several ...
-2
votes
0answers
35 views

Does the algorithm to construct the edge-colored graphs with this special property have any importance? [closed]

I found a semi-general solution to the following open-ended question and obtained the explicit algorithms to construct the edge-colored graphs with the following special property. But does my solution ...
0
votes
0answers
72 views

How do we represent one-way “equality” in mathematical notation? [closed]

Here's an example: You have 20 dollars A full coke = 2$ A full coke yields an empty bottle and a bottle cap 4 empty bottles yields a full coke 2 bottle caps yields a full coke Let full coke = F, ...
-7
votes
0answers
67 views

Two graph theory questions, really hurry. Thanks a lot [closed]

Q2: Draw the `breadth-first searches' for the Tower of Hanoi for the cases with 1, 2 and 3 discs. Conjecture a strategy and show using a depth-first search, that this strategy works for 4 discs. Q3: ...
3
votes
0answers
114 views

What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...
1
vote
0answers
76 views

Dislocations,Disclinations Latices, Displacement fields and scaling

I am looking up Frank, and Burger vectors and associated material on dislocation/disclination. It seems straightforward describing a lattice and what dislocation means. It is even possible to restrict ...
-1
votes
0answers
70 views

On the lattice of submodules of a module over a simple ring [closed]

We suppose that all rings are left Artinian simple rings and all modules over a ring are of finite length. Let $M$ be a left module over a ring $R$. By Wedderburn theorem, $R$ is a matrix ring over a ...
10
votes
1answer
527 views

Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?

I am wondering if there is a multi-dimensional analog of the Birch/Swinnerton-Dyer (BSD) conjecture. The recent famous result inching toward resolution of that conjecture is: Bhargava, Manjul, and ...
0
votes
0answers
95 views

Irreducible action of an algebraic group

Is the following claim true?: Let $G$ be an algebraic group such that $G^\circ$ is reductive. Suppose $G$ acts irreducibly on $V$. Is it true that $V$ is decomposed (written as direct sum) into ...
1
vote
0answers
69 views

Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers. Can one classify all reductive group schemes over $C$? Certainly, you have the trivial ones (coming from pulling-back ...

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