0
votes
0answers
9 views

Reference for analyticity of $K$-theory

This is a follow-up to my last question, Homotopy excision for structured ring spectra -- reference?. The immediate reason why I care about Blakers-Massey theorems for ring spectra is to prove that ...
0
votes
0answers
18 views

Most Regularity of a Polygon

Conseider $n$ electrons in an empty sphere. What structure do they make? This question have two cases: (i) if electrons should be sit on the boundry of sphere (one can suppose that the boundry of ...
0
votes
0answers
22 views

Carlsson's spectrum BG^-V

In the appendix to Carlsson's "Equivariant stable homotopy and Segal's Burnside ring conjecture," he introduces a spectrum BG^-V associated to a G-representation V. It is like a Thom spectrum of the ...
0
votes
0answers
32 views

Albanese map on a universal jacobian associated to a curve on an abelian surface

Given an abelian surface $A$ and a curve $C$ on it, consider the component $\{C\}$ of the Hilbert scheme of curves with the same cohomology class of $C$. I would say that it has a fibration structure ...
1
vote
0answers
63 views

Error in trace of operator

Imagine you have a Schroedinger operator $H:=-\frac{d^2}{dx^2}+V$ with $V \in C[a,b]$ on some compact interval $[a,b]$. The boundary conditions are supposed to be taken in such a way that this ...
-4
votes
0answers
34 views

What do rank-2 tensor entries represent? What's their geometrical meaning? [on hold]

given a vector (with, say, 3 dimensions), if I multiply that with a second-rank tensor, I can make it change in both direction and magnitude (for that 2nd rank tensors are involved in vector-vector ...
0
votes
0answers
77 views

Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf ...
1
vote
0answers
45 views

Singularities of secant varieties of rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper: ...
1
vote
1answer
78 views

Linear map with two “incompatible” representations

Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ ...
3
votes
1answer
60 views

What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
0
votes
1answer
50 views

Is there some lattice not rigid

I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...
0
votes
0answers
20 views

Representation of piecewise functions using variational inequalities

I need some help about how to represent piecewise smooth functions defined over N subintervals by using variational inequalities. From a paper I'm analyzing I get: "If we can find a function ...
5
votes
1answer
84 views

Homotopy excision for structured ring spectra — reference?

I'm looking for a reference for analogues of the Blakers-Massey triad connectivity theorem (and its higher-order generalization) for ring spectra. That is: Suppose that $A\to A_1$ is a ...
0
votes
0answers
27 views

quasiprimitive unsoluble groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
2
votes
0answers
41 views

Which valuations of a field yield codimension $1$ subschemes of their 'models'

For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It ...
0
votes
1answer
50 views

On two notions of 'generators' for a 'large' triangulated category

Let $C$ be a triangulated category that is closed with respect to arbitrary small coproducts; let $D$ be some class of objects of $C$. Then it would be natural to say that $D$ generates $C$ either if ...
0
votes
0answers
26 views

Interior gradient estimate for uniformly elliptic equations

I am struggling with a problem like this: In dimension $n\geq 3$, For the following uniformly elliptic equation, do we have interior gradient estimates? $$a^{ij}(x)u_{ij}(x)+u_{nn}=0.$$, where ...
5
votes
1answer
196 views

The (un)decidability of Robinson-Arithmetic-without-Multiplication?

I asked this over at math.stackexchange, and though a number of people were interested enough to vote up the question, I didn't get an answer -- which makes me wonder whether it isn't quite so ...
2
votes
0answers
38 views

Vanishing patterns of minors of matrix

Let $M$ be a $m\times n$ matrix with entries in, say $\mathbb{C}$; assume $n\leq m$. Denote by $I\subseteq\{1,2,\ldots, n\}$ a subset of the columns of M. I am interested in positive results to the ...
7
votes
1answer
107 views

A strengthening of base 2 Fermat pseudoprime

If $n$ is a prime then for all $k$ with $1 \le k \le [n/2]$, $k$ divides ${n-1 \choose 2k-1}$ because of the identity ${n-1 \choose 2k-1} \frac{n}{k}=2{n \choose 2k}$. My question is whether an ...
0
votes
1answer
24 views

Fitting a quadratic using regression when the y-intercept needs to be 0 [on hold]

I'm trying to fit a quadratic $a_0 + a_1x + a_2x^2$ by Polynomial Regression: $$ \begin{pmatrix} n & \Sigma x_i & \Sigma x_i\\ \Sigma x_i & \Sigma x_i^2 & \Sigma x_i^3\\ \Sigma ...
0
votes
0answers
64 views

A linear/Lie algebra problem

Let $\mathfrak{g}$ be a complex linear Lie algebra of dimension $n$. If there exists a basis $\{e_1,\dots,e_n\}$ of $\mathfrak{g}$ such that ...
11
votes
2answers
537 views

Why do Lie algebras pop up, from a categorical point of view?

Groups pop up as automorphism groups in any category. Rings pop up as endomorphism rings in any additive category. Is there a similar way to attach a Lie algebra to an object in a category of a ...
1
vote
0answers
43 views

If a compact real submanifold of $\mathbb{CP}^n$ is approximable by complex algebraic curves, is it algebraic?

To make this into a separate question: If the supports of a sequence of complex algebraic curves in $\mathbb{CP}^n$ (images of non-constant holomorphic maps from compact Riemann surfaces) converge to ...
0
votes
0answers
56 views

a question on sum of Gaussian binomial coefficients

I was trying to calculate something and at some point I get the following sum: \begin{equation} \sum_{t=0}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n \brack 2i}x^t ...
2
votes
2answers
182 views

$K$-homology of $BG$

Let $G$ be a finite group. Atiyah proved that the $K$-cohomology of $BG$ vanishes in odd degrees and in even degrees is the completion of the representation ring of $G$ at the augmentation ideal. ...
1
vote
0answers
16 views

Degenerations and spanning monomials

Let $R = \mathbb{C}[x_1,…,x_n]$, let $J\subset R$ be a graded ideal, and consider the initial monomial ideal $\operatorname{in}(J)$ with respect to some term order. Suppose that we are given a linear ...
-4
votes
0answers
56 views

Video Lecture for topolgy [on hold]

want to study general topology. i find video lecture at nptel, but instructor's pronunciation is hard to understand. so other course to study general topology? thankyou!
0
votes
0answers
18 views

Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Denote by $p(x)$ the input probability distribution. The mutual information between the input and the output of ...
1
vote
0answers
110 views

Subset of the integers with certain properties

How would one find the maximal $n$ such that there exists an $n$-subset $S$ of $\mathbb{Z}^+$ such that $\forall A\subseteq S, \sum_{a\in A}a$ is either a perfect square or a perfect cube, or can one ...
0
votes
0answers
45 views

How many edges are in the kth-iterated common neighborhood knight's graph?

For terminology see https://www.academia.edu/2180382/The_Common_Neighborhood_Graph_and_Its_Energy) It has been known for quite some time by players that a vertex whose corresponding square is in the ...
1
vote
0answers
87 views

Complex structures on Riemann surfaces

This is cross posted from math.SE: http://math.stackexchange.com/q/876432/9 Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a map ...
13
votes
0answers
278 views

The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with ...
1
vote
1answer
128 views

Milnor's exact sequence and a certain proof

Please forgive me if this is not the right forum for this question. Let $$ X = \cdots \rightarrow X_n \rightarrow X_{n-1} \rightarrow \cdots \rightarrow X_0 = \ast$$ be a tower of fibrations of ...
-2
votes
0answers
94 views

P != #P ? And its implications [on hold]

Is it known that p != #p ? And what would such a proof imply? I've been reading Valiant's paper on the permanent, and this isn't clear to me.
0
votes
0answers
35 views

Uniqueness of a quadratic time-dependent matrix equation

Let $v: [0,1] \to \mathbb R^n, t \mapsto v(t)$ continuously differentiable with the property that for any constant vector $h \in \mathbb R^n$ the fact that $v(t)^{\top} h = 0$ for all $t \in [0,1]$ ...
-1
votes
0answers
30 views

Two-Dimensional Recurrence Solving [on hold]

Can someone help me to solve this two dimensional recurrence: f(n,k)=f(n-k,1)+f(n-k,2)+f(n-k,3)+...+f(n-k,n-k); f(n,1)=1; f(n,n)=1; Thanks
1
vote
1answer
103 views

Brun's algorithm

Does anyone have an exact reference for the weak convergence (convergence in angle) of Brun's subtractive multi-dimensional continued fractions algorithm (in all dimensions)? I have been given ...
4
votes
1answer
84 views

non-negative random variable

Let $X$ be a real-valued random variable with $X \geq 0$ and $\mathbb E X >0$. I would like to bound $\mathbb P(X >0)$ from below using information about the first few moments of the variable. ...
2
votes
1answer
134 views

When are isotrivial families split by a finite base-change?

A well-known theorem of Grauert and Fischer states that a smooth proper family of complex manifolds is a locally trivial fibration as soon as all the fibers are isomorphic. It is also easy to obtain ...
0
votes
1answer
79 views

How do eigenvalues change if we duplicate a row and column of a symmetric matrix

Let ${\bf A}$ be a size $n \times n$ symmetric positive semidefinite matrix with the first column being ${\bf a}_1$. If we define a new matrix, \begin{align} {\bf B} = \left[\begin{array}{cc} a_{11} ...
4
votes
1answer
111 views

$RO(G)$-Graded Cohomology Theories

Let $G$ be a compact Lie group with real representation ring $RO(G)$. Recently, I have been learning about some aspects of $RO(G)$-graded cohomology theories (for a precise definition, see Chapter ...
1
vote
1answer
180 views

Example of a specific manifold

I want to find a example of a manifold that has positive scalar curvature but is not half conformally flat. Does there exists such manifolds? Thanks.
0
votes
0answers
18 views

Proximal operator of modified L1 matrix norm

In literature proximal operator $prox_{\lambda f} : R^n \rightarrow R^n$ of $f$ is defined as: $prox_{\lambda f}(V) = argmin(X) (f(X) + (1/2 \lambda)||X-V||^2_2)$ Consider now $g(X) = ...
6
votes
1answer
198 views

The definition of < in Robinson's Q

I recently had to explain how the basic axioms in Simpson's Subsystems of Second Order Arithmetic were interpretable in Robinson's Q. Most of the axioms are actually the same, except that Simpson ...
1
vote
0answers
66 views

A conjecture about the entropy of matrix vector products

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
1
vote
0answers
62 views

Question about the specialization map for Etale Fundamental Groups

Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...
3
votes
0answers
121 views

Both NP-hard but different [on hold]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?
10
votes
0answers
132 views

The definition of Reedy category

The common definition of Reedy category seems to be this one that a Reedy category is a small category $R$ with two wide subcategories $R_+$ and $R_-$ and an ordinal-valued degree function on its ...
8
votes
1answer
120 views

Max flow, min cut on manifolds

If a graph has some half edges marked "input" and some half edges marked "output", it is well known that the smallest number of edges which must be cut to disconnect input from output is equal to the ...

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