0
votes
1answer
56 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 ...
-2
votes
0answers
39 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} : ...
12
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
43 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
42 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
82 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
111 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
68 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
112 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
85 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
68 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 $B_1(0)\subset \mathbb{R}^{n+1}$, ...
2
votes
0answers
69 views

A combinatorial 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 ...
0
votes
0answers
26 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
203 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
153 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
108 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
48 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
77 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
86 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 ...
7
votes
1answer
126 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 ...
4
votes
2answers
140 views

Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
0
votes
0answers
24 views

Critical probability, bond percolation on triangular lattice

Let G a graph and $p_c(G)$ the critical probability of bond percolation. Let G be a triangular lattice then $p_c(G) = 2\sin(\frac{\pi}{18})$ (Grimmett, Percolation p.65). Ramanujan find this formula : ...
10
votes
2answers
415 views

Blinking graphs

For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$. Call this a bit assignment for $G$. Now, generate a new bit assignment as follows: Each node $x$'s bit is replaced by $1$ if the ...
24
votes
0answers
255 views

What is the “real” meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...
6
votes
1answer
144 views

The unique positive real root of summation function

update: add one condition according to answer below. I post this question in MSE a week ago. I thought this should be an easy freshman exercise, but it turns out not easy... The original question ...
2
votes
0answers
54 views

Christoffel symbols of a moduli of smooth curves

The Setting: Let $H$ be the Hilbert space of all class $C^k$-curves into $\mathbb{R}$ with inner product: \begin{equation} <f,g>:=\int_{\mathbb{R}} f'(x)g'(x) e^{-x}dx \end{equation} ...
0
votes
0answers
42 views

Example of a compact geodesic space, which is not doubling [on hold]

Are all compact geodesic spaces doubling? If not, could you give an example?
-2
votes
0answers
32 views

How to draw hypocycloids using a program? [closed]

How can I program Mathematica (or any other program able to serve this function) to draw hypotrochoid curves in which I am able to control the radii of both the inner and outer circle, as well as "d", ...
2
votes
1answer
62 views

Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1 $$ ...
1
vote
0answers
116 views

universality for large deviations?

This is a question about universality in probability theory, with combinatorics in mind. Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...
3
votes
0answers
91 views

Intersection patterns of loops on surfaces

Let $a,b$ be to simple closed loops on a surface $S$ with homologically trivial intersection (more generally I'm also interested in the case when $b$ is 1-codimensional). Denote their intersection on ...
1
vote
1answer
34 views

Relative local compactness for locales?

I am looking for informations on the relative version of local compactness for locales: If $f:X \rightarrow Y$ is a morphism of locales I want to say that $f$ is relatively locally compact if ...
2
votes
0answers
145 views

Groups with isomorphic quotients [on hold]

Assume we have a finitely presented group $G$ and a non-trivial normal subgroup N. How can one decide that $G/N$ is isomorphic to $G$ or not? $G$ is given as a presentation and $N$ as a set of words.
6
votes
1answer
270 views

Solutions of equations characterizing a complex structure

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ ...
0
votes
0answers
54 views

“Dimension” of ideals in $F_q[x]/\langle x^n-1\rangle$? [closed]

I'm very much confused by algebra. Hoping to get a bit more comfortable I tried to compute different things and see what happens... Let $F_q$ be the finite field with $q$ elements and ...
0
votes
0answers
68 views

Circle actions on graph C*-algebras [on hold]

Do all graph C*-algebras admit actions of the circle? Suppose we have a graph C*-algebra which we know is the quotient of a graph algebra by a circle action. Is it possible to read off the original ...
2
votes
2answers
112 views

A reference about Grassmannian over finite fields

Suppose $Gr_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F_{q}$, we can ...
0
votes
0answers
15 views

Approximation of a volume preserving Hölder homeo by diffeomorphisms?

Is it known whether a volume preserving Hölder homeomorphism of an arbitrary manifold can be approximated by a volume preserving diffeomorphism? The answer is clearly no if the volume preserving ...
10
votes
3answers
232 views

Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture: A non-trivial connected sum $M_1\# M_2$ admits a ...
3
votes
1answer
235 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...
0
votes
0answers
39 views

Open Hamiltonian Gromov-Witten Invariants

Both open Gromov-Witten invariants and Hamiltonian Gromov-Witten invariants have been studied. I am interested in knowing whether anyone has considered open Hamiltonian Gromov-Witten invariants ...
0
votes
0answers
147 views

research statement for assistant professorship [closed]

What should a research statement for an application for an assistant professorship contain (pure mathematics in Germany)? How long should it be?
0
votes
1answer
40 views

A question on decreasing function [closed]

Let $t\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} + f(t) = 0$ $f(t) $ is continuous decreasing function of $t$. $a_i\ge0$ for all $i$. $y(t)$ is positive real zero of the first equition. Can ...
-3
votes
0answers
92 views

Differential geometry [closed]

If we have integrable distribution D of rank k on a manifold, and we have k functions which are zero homogeneous and constant on the leaves (basic functions). Can we glue together these functions to ...
1
vote
1answer
95 views

$L^1$ convergence to equilibrium of solutions of heat equation

Let $u$ and $v$ be the weak solutions of $$u_t - \Delta u = f$$ $$u(0)=u_0$$ and $$-\Delta v = f$$ $$|\Omega|^{-1}\int_\Omega v =0$$ on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous ...
0
votes
0answers
114 views

Advice on dealing with the gap [closed]

A young mathematician AA is writing a paper proving a property X for a certain model. There have been quite a few articles proving the property X for various models. One of the first ones, let's call ...
0
votes
0answers
51 views

Separating the points of projective spaces with real-analytic functions

Is there an easy way to separate the points of $\Bbb C \Bbb P^n$ or $\Bbb R \Bbb P^n$ (viewed as real-analytic manifolds) with real-analytic functions? If two points lie in a coordinate patch where a ...
2
votes
1answer
105 views

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
-4
votes
0answers
46 views

Probability Inequality when X > Y > 0 [closed]

I want to know whether the following statement is true or not, and the proof. Let X, Y be random variable, satisfying X > Y > 0, and have finite variance, $Var(X) < \infty$ and $Var(Y) < ...
3
votes
0answers
141 views

splitting property of etale covering

Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely ...

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