1
vote
1answer
55 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 ...
0
votes
1answer
7 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, ...
1
vote
0answers
49 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 ...
1
vote
0answers
8 views

$K$ theory and singular homology

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
votes
0answers
29 views

Negative Arithmetic Line bundle [on hold]

Let $X$ be an arithmetic variety and $L$ be an Arithmetic Line bundle, then how can we define a negative Arithmetic line bundle ?
0
votes
0answers
4 views

to find topological properties under a metric on a set

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 ...
-7
votes
0answers
41 views

Erdös is solved and Collatz is solved too? [on hold]

I mean. Cut and Fold the "Quadrant I positiv Numbers" to a Cone with 0 in the Center. A Erdös solved Number X are a Orbit arround the 0. The Orbit are not harmoniously. The Orbit are very Fractal ...
-5
votes
0answers
46 views

Erdös is solved and Collatz is solved too [on hold]

I mean. Cut and Fold the "Quadrant I positiv Numbers" to a Cone with 0 in the Center. A Collatz solved Number X are a Orbit arround the 0. The Orbit are not harmoniously. The Orbit are very Fractal ...
1
vote
0answers
19 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) & = ...
0
votes
0answers
13 views

Exercise 5, chapter 4 in Do Carmo's “Riemannian Geometry”

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
1answer
23 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 ...
0
votes
0answers
56 views

What is the upper-bound for this?

I am looking at a paper and am trying to understand how this bound was driven. The first part is clear, but not sure how you can extend it to the second part. So here is the first part: Assume ...
0
votes
0answers
7 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
65 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
26 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 ...
1
vote
0answers
4 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 $$det(A)=1,\ a_{ij}\in\{-1,0,+1\}\ \wedge\ a_{ij}^{-1}\in\{-1,0,+1\}$$ ...
2
votes
0answers
30 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 ...
2
votes
0answers
60 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
52 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 ...
6
votes
1answer
48 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 Ozsvàth-Szábo's original definitions: ...
8
votes
2answers
317 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 ...
9
votes
3answers
174 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 ...
18
votes
0answers
152 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 ...
3
votes
0answers
62 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 ...
2
votes
0answers
26 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 $B_1(0)\subset \mathbb{R}^{n+1}$, ...
-1
votes
0answers
47 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 ...
5
votes
1answer
83 views

The unique positive real root of summation function

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 is very complicated, involving Bounded variation and ...
2
votes
1answer
59 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 ...
0
votes
0answers
40 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 = ...
8
votes
3answers
437 views

Collection of dense subsets as a “fingerprint” for Hausdorff topologies?

Let $(X,\tau)$ be a Hausdorff space and let ${\cal D}$ denote the collection of dense subsets of $(X,\tau)$. Is it possible that there is another Hausdorff topology $\tau_1 \neq \tau$ on $X$ such that ...
3
votes
2answers
64 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
76 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 ...
2
votes
0answers
65 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 ...
5
votes
1answer
98 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 ...
0
votes
0answers
44 views

Cone of curves in blow-ups

Let $X_s = Bl_{L_1,...,L_s}\mathbb{P}^3$ be the blow-up of $\mathbb{P}^3$ in $s$ general lines $L_1,...,L_s$. The exceptional divisor $E_i$ over $L_i$ is isomorphic to ...
1
vote
0answers
31 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 $$ ...
4
votes
0answers
86 views

A solution for this equation with a certain condition

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,\\ ...
2
votes
0answers
83 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 ...
-3
votes
0answers
68 views

Differential geometry [on hold]

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 ...
9
votes
2answers
234 views

Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
6
votes
4answers
379 views

Direct axiomatization of ordinal and cardinal numbers

Again, this question is related (**) to a previous one: in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
6
votes
3answers
186 views

Numerical invariants for a graph or its complement that are bounded by some constant

I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but ...
7
votes
0answers
85 views

A seemingly simple combinatorial object that must have an easy generating function

One more question related to my earlier "Special" meanders. I am trying to isolate simplest problems related to it. Here is one. For a composition (i. e. a tuple of natural numbers) ...
1
vote
1answer
242 views

research articles in topology/geometry [on hold]

There is a saying "Do you read the masters?" I want to read some basic papers in Topology/geometry... I can not clearly state what is basic as of now... My back ground includes course in ...
4
votes
0answers
51 views

Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows: Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...
33
votes
5answers
2k views

The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic

Let $(a_{n})_{n \ge 1}$ be a sequence of integers such that for all $n \ge 2$: $0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1$. Prove that the sequence $(a_{n})$ is periodic. ...
-2
votes
0answers
44 views
+50

How one can use a real math function on transaction in Hybrid Petri Net fundamental equation?

Say we have a simple HPN with 2 continuous places $A$ and $B$ and one transition. We want a transition not only add and substract $N$ marks from $A$ and add $M$ to $B$ but use mathematical function ...
7
votes
0answers
59 views

Factoring a multiset into a product of two multisets

Given a multiset $S$ of $mn$ numbers, how hard is to find multisets $A$ and $B$, of $m$ and $n$ numbers respectively, such that $$ S = \{ xy \mid x\in A, y\in B \}~~~~\text{(multiset sense)}, $$ or ...
0
votes
3answers
161 views

Divergence of general random series and a special case

Is there any sufficient condition in terms of moments under which $$ \sum_{n=1}^{\infty} X_n$$ diverges a.s.?Here $X_n$ are not independent I am given that $\sum_n E[X_n]$ diverges. Actually, I am ...
9
votes
0answers
222 views

How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $ A_i= \biggl(\begin{matrix} C_i+E_i & B_i \\ B_i^T & D_i-F_i \end{matrix} \biggr) $, where $B_i$ is an arbitrary $n\times n$ ...

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