3
votes
1answer
271 views

Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...
0
votes
0answers
27 views

Stochastic Pontryagin Principle with a final state constraint

I am searching for information about the Stochastic Pontryagin Principle with a final state constraint. Someone knows a paper or a book where this case is treated in depth?
-4
votes
0answers
42 views

Global dimension of matrix algebra [on hold]

Let $k$ be a field, and $A=T_{n}(k)$. $gldim(A) = 1$, and if $B = A/rad(A)^{2}$, then $gldim(B) = n-1$. Some indication!! How can I prove that $gldim(A) = 1$, and $gldim(B) = n-1$ ? Thank you!
5
votes
1answer
113 views

Example to $2^\kappa\nrightarrow (3)^2_\kappa$, plus closed walks of odd length?

Let $\kappa$ be an infinite cardinal. Consider the following example to $2^\kappa\nrightarrow (3)^2_\kappa$. $V$ is a set of vertices, each of which is an element of $2^\kappa$. Color the edge ...
1
vote
0answers
16 views

Upper bound on the number of convex connected components of the complement of the zero set of a polynomial

The classic Milnor-Thom upper bound on the sum of the Betti numbers of real algebraic sets (for a nice exposition and references, see e.g. N.R. Wallach, On a Theorem of Milnor and Thom, in S. Gindikin ...
1
vote
0answers
24 views

Multivariate analogue of Jackson's inequality's modulus of continuity form

According to Jackson's inequality, there is $c > 0$ s.t. for any continuous function $f: S^1 \rightarrow \mathbb{R}$ (where $S^1 := \mathbb{R} / \mathbb{Z}$ is the circle) and integer $n$ there is ...
13
votes
1answer
139 views

Why should intersection cohomology and quantum cohomology be related for a symplectic resolution?

In http://arxiv.org/pdf/1410.6240.pdf M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the ...
0
votes
0answers
94 views

Additional condition to the Bollobas theorem (Sperner's therorem) in extremal set theory

The Bollobas'1965 theorem is the following: If $A_1,...,A_n$ and $B_1,...,B_n$ are two sequences of subsets of $X=\{1,...,r\}$ such that $A_i\cap B_j = \emptyset$ if and only if $i=j$, then ...
-1
votes
0answers
69 views

Twisted Hodge numbers in a family

It is well known (e.g. Voisin's book) that for a smooth family $\pi: \mathcal{X} \to B$ of smooth projective varieties (and projectively normal) over $\mathbb{C}$, the Hodge numbers $h^{p,q}(X_b)$ are ...
1
vote
0answers
143 views

Calabi-Yau theorem on Arithmetic Variety

Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kaehler current of $\mathcal X(\mathbb C)$. ...
0
votes
0answers
9 views

Understanding the steps taken in a calculation of the maximum profile likelihood of a simple ODE, given some data [on hold]

I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of ...
2
votes
0answers
35 views

Uniform convergence of long geodesic to Liouville measure

Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the ...
2
votes
1answer
70 views

Minimal Support Solutions of a Linear System (Dissertation)

For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times ...
3
votes
0answers
108 views

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the ...
2
votes
0answers
102 views

Local product structure of determinantal variety

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...
1
vote
0answers
17 views

What is the topology of teichmuller space of a given hyperhahler manifold

From verbitsky's paper, I read that the teichmuller space of a given hyperkahler manifold is not hausdorff but it's smooth. I am a bit confused about how to put a natural topology on the teichmuller ...
1
vote
0answers
49 views

Boundary of pseudospectra

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
7
votes
1answer
114 views

Schur multiplier of $Sp(2g, \mathbb{Z}_2)$ for $g \geq 3$

This question is about the computation of $H_2(Sp(2g, \mathbb{Z}_2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}_2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}_2$. With respect to ...
5
votes
0answers
59 views

General properties of the Ruelle operator

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics. I have been interested in some technical properties of the ...
3
votes
1answer
100 views

Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...
7
votes
0answers
45 views

Holomorphic vector fields on compact complex manifolds with trivial canonical bundle

Let $M$ be a compact complex manifold whose canonical bundle $K_M$ is holomorphically trivial. Is it possible for $M$ to admit a non-zero holomorphic vector field with zeroes? Equivalently, using a ...
2
votes
1answer
95 views

Duality argument to get $L^\infty-L^2$ inequality

In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that $$\lVert e^{-Ht}f \rVert_2 \leq c_1t^{-\mu/ 4}\lVert f \rVert_1$$ where the norms are $L^p$ norms. He states ...
4
votes
2answers
183 views

Lefschetz on étale fundamental group for quasi-projective varieties

If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$ $$\pi^1(H)\to\pi^1(X)$$ is an isomorphism, ...
0
votes
0answers
29 views

to find topological properties under a metric on a set [on hold]

we define a metric d on a set of composition operators on L2. I would like to find connected component and path connected component and other topological properties by d . Is there any book or paper ...
0
votes
0answers
11 views

On finite Uniform (Goldie) dimensions

1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions? 2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to ...
6
votes
1answer
184 views

$K$ theory and singular cohomology

For cell complexes${}^1$ $X$ we have an isomorphism $$ K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}), $$ which is induced by the Chern character. What is the analogous statement for $KO(X)$? ...
1
vote
2answers
64 views

Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, ...
0
votes
0answers
50 views

Exercise 5, chapter 4 in Do Carmo's “Riemannian Geometry” [on hold]

This is probably a silly question and maybe in the end the answer is trivial but I can't see it. The problem is the following. Let $M$ be a Riemannian manifold, $\gamma \colon[0,l]\to M$ be a ...
0
votes
0answers
18 views

Properties of a map regarding the space of invariant probability measures for controlled Markov process

Let us consider a controlled Markov process with the transition kernel $p(dy|x,\theta)$ ($\theta$ being the control parameter. Now, consider the map $\theta \to I(\theta)$ where $I(\theta)$ is the ...
2
votes
0answers
89 views

The homology of $\varinjlim SO(p,q)$

Is there a way to explicitly compute the homology of the space $$ \varinjlim_{(p,q)} SO(p,q)^+, $$ where each $SO(p,q)$ is the indefinite special orthogonal group, and $SO(p,q)^+$ its identity ...
0
votes
1answer
54 views

About $c(A)$ in $c(A)|A|\leq |A^{-1}A|$

Let $G$ be a finite group, $\emptyset\neq A\subseteq G$, $A^{-1}:=\{ a^{-1}:a\in A\}$, and put $$c(A):=\max\{t\in \mathbb{Z}: t|A|\leq |A^{-1}A|\}$$ It is clear that $1\leq c(A)\leq ...
-1
votes
0answers
32 views

weakly p- summable sequence

Let $ (x_{n}) $ be a weakly $ p$-summable sequence in $ X $ and $ ( x^{\ast}_{n})$ a weakly null sequence in $ X^{\ast} $. Let $ i_{n} : Y_{n}\rightarrow X$ be the natural injection and $ p_{n} : ...
11
votes
1answer
1k views

Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?

Wikipedia claims (permanent link) without reference: Testing whether one planar graph is dual to another is NP-complete. Another claim with reference: For any plane graph G, the medial graph ...
1
vote
1answer
40 views

Pragmatic Test for Total Unimodularity

I want perform a simple check for total unimodularity. Question: what, if anything, can be concluded from the fact, that $$det(A)=1,\ a_{ij}\in\{-1,0,+1\}\ \wedge\ a_{ij}^{-1}\in\{-1,0,+1\}$$ ...
1
vote
0answers
39 views

How to solve the following bivariate recurrence?

$$F(n,r) = (1-w(r))F(n-1, r) + w(r-1)F(n-1, r-1)$$ where $w(r)$ is monotonically non-increasing in $r$ and $0 \leq w(r) \leq 1$ with $0 \leq r$ Initial condition: \begin{eqnarray} F(0, r) & = ...
3
votes
0answers
77 views

Numerical and topological density

Let $\mathbb{N}$ denote the set of positive integers, and let's say that $A\subseteq \mathbb{N}$ is numerically dense if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 1.$$ Is there a ...
2
votes
0answers
105 views

Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation $$ A_1 +...+A_n ...
2
votes
0answers
59 views

Darboux-like coordinates on a Kähler manifold

If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...
3
votes
0answers
95 views

Degree formalism for line bundles on Deligne-Mumford stacks

Let $k$ be an algebraically closed field and let $\mathcal{C}$ be proper, Cohen-Macaulay, purely $1$-dimensional Deligne-Mumford stack over $k$. From looking at section 4.3 on page 135 of the paper ...
2
votes
0answers
72 views

functor from complex algebraic variety to constructible function

I asked the same question on mathstackexchange but didn't get any response so I guess I should ask here. If you think it is a duplicate and it is not appropriate to post it here, I will delete this ...
2
votes
0answers
40 views

Is it possible to write down the explicit expressions of some extensions of conformal vector fields on spheres?

Let X be a conformal vector field on the standard sphere $S^n$ with standard metric $g_{S^n}$, then there exists a unique conformal vector fields in the unit ball $B_1(0)\subset \mathbb{R}^{n+1}$, ...
0
votes
0answers
58 views

A comninatorical sum involving ratios of binomials [on hold]

Can anyone suggest how to prove the following (for $k \le n$): $\sum \limits_{s=0}^N \frac{\binom{n}{k} \binom{N-n}{s-k} }{\binom{N}{s}} = \frac{N+1}{n+1}$ I am assuming it to be true and possibly ...
0
votes
0answers
23 views

Mapping a grayscale image into weighted undirected graph

I am looking for a method in order to convert an image into a network. I have found the study Z. Wu, X. Lu, Y. Deng, Image edge detection based on local dimension: A complex networks approach. Physica ...
4
votes
1answer
170 views

Uniformizer for splitting field of p^{1/p^n} over p-adics

Does anyone know an explicit uniformizer for $\mathbf{Q}_p(\zeta_{p^n}, p^{\frac{1}{p^n}}) / \mathbf{Q}_p$? I was reading the question "adding an n-th root to Q_p" where dke mentions this question but ...
2
votes
0answers
139 views

About the limitation by size

This could be a big post, so I'll try to summarize my thoughts and divide them into several questions. When working in category theory, I used to choose the following definition. A category $C$ is ...
7
votes
1answer
104 views

Admissibility in Heegaard Floer, especially with torsion Spin^c structures

I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsváth-Szàbo's original definitions: ...
0
votes
0answers
46 views

How to find singularities from data and find monodromy group from singularities and differential system? [on hold]

update1 i use interpolation for time series find 2 singular points, one is infinity and negative infinity, and find a differential equation which stated a, b, etc, are singular point, if i let a = ...
2
votes
0answers
67 views

Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...
2
votes
1answer
82 views

Lifting a differential operator

Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over compact manifold $M$. If $M$ is not simply connected one can construct the universal ...
6
votes
1answer
117 views

Simply connected noncompact surfaces

Is there a theorem saying that any noncompact, simply connected topological surface is homeomorphic to the plane ? There seems to be many well-known results about the classification of compact ...

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