3
votes
1answer
136 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
0
votes
1answer
70 views

Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $0 \le a_{j,j} \le 1$ and $-1 \le ...
0
votes
0answers
32 views

Moduli space of Parabolic Vector Bundles with arbitrary parabolic weights

I have just posted another question related to Moduli Space of Parabolic Vector Bundles on Curve. The questions came up when I was trying to read the paper (Desingularisation of the Moduli Varieties ...
1
vote
1answer
71 views

Isofibrations and Diagonal Functors

Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id_c,id_c)$ be the diagonal functor. Recal that an isofibration is a functor p: E→B such that for any object $e\in E $ and any ...
-3
votes
0answers
53 views

Fredholm operators: how to calculate Coker and Ker [closed]

Exercise: Let $1\leq p \leq \infty$. For each $n\in\mathbb{Z}$ construct a Fredholm operator $F:l^p\to l^p$ whose index is $n$. Solution (given in the lecture classe): $F_n(x_i):=(0,\ldots, 0,x_1,...
-1
votes
0answers
40 views

Direct product between joins of subgroups [on hold]

Suppose that $G$ is a finite group with $A, B, H, K \leq G$. Suppose that $H\times A \leq G$ and $K\times B \leq G$. I want to show that $\langle H,K \rangle \times \langle A, B \rangle = \langle H \...
3
votes
1answer
95 views

Linear independency and compactness of the set of pure states of a $C^*$-algebra

Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states. Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\...
5
votes
0answers
140 views

An inequality in cyclic polygon and tangential polygon

I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following: Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let P be ...
3
votes
0answers
64 views

Extending homomorphisms between ordered abelian groups

Let $\Omega$ be a linearly (i.e. fully) ordered set, and let $\Lambda_{\Omega}$ be the ordered abelian group consisting of those $(\lambda_\omega)_{\omega\in\Omega}\in\mathbb{R}^{\Omega}$ with well-...
11
votes
1answer
461 views

Is a one-dimensional compact complex analytic space necessarily projective?

Let $X$ be a compact complex analytic space with singular locus $X^{\mathrm{sing}}$. Suppose that $X\setminus X^{\mathrm{sing}}$ is a Riemann surface. If $X^{\mathrm{sing}} = \emptyset$, then $X$ is ...
6
votes
1answer
150 views

Group bundles for topological spaces without universal cover

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. Let me first describe the situation: There are two approaches in defining Homology with local coefficients of a ...
0
votes
1answer
117 views

Hodge decomposition on open manifold

For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold. Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.
6
votes
0answers
230 views

Cohen's model yet again

It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g. A question about ordinal definable real numbers . A negative answer was obtained in Archive for ...
4
votes
1answer
233 views

A generalization of Erdős–Mordell inequality [on hold]

I proposed my conjecture generalization of Erdős–Mordell inequality as following: Let $A_1A_2....A_n$ be a polygon in a plane, $P$ be the point in $A_1A_2....A_n$. Let $d_i$ be the distances from $P$ ...
-1
votes
0answers
24 views

Hyperbola application [closed]

A curved mirror is placed in a store for a wide angle view of the room. the right hand branch of x squared over one minus y squared over three equals one models the curvature of the mirror. a small ...
0
votes
0answers
20 views

Algorithm for Longest Common Subtour

For a new kind of heuristic for the TSP I need to calculate the longest subtour, that is common to a set $T_1,\ ...,\ T_m$ of tours, that are "good" approximations of the optimal tour $T_{opt}$. By a ...
-1
votes
0answers
54 views

How many number of finite points exists inside the circle? [closed]

I am doing project on Image processing dealing with circular images. So I need an approximate number of pixels present inside circle image of radius R and Circle center of (x,y). Please give me the ...
0
votes
1answer
63 views

Computing canonical forms from orbit partitions

Suppose we know the orbit partition of the vertices of a graph (due to the action of its automorphism group). Is it easy (as in "polynomial time") to generate a canonical form (aka "canonical labeling"...
1
vote
1answer
84 views

Proving that an integral related to order statistics is increasing in a certain parameter

Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$. Does it always follow that $$...
0
votes
0answers
93 views

Gauge Fixing Problem on Cylindrical

For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold. If it is necessary, we could consider the $b_1(Y)=0$ case. Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
3
votes
0answers
62 views

Intersection multiplicity of limit linear spaces

Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$. Now let $\Lambda_{\xi, q}$, with $\xi \...
6
votes
1answer
247 views

The current status of the conjecture on algebraic matroids

Can anyone point out some articles for the conjecture: the dual of an algebraic matroid is algebraic? Thank you!
0
votes
0answers
26 views

On balanced bipartite graphs

Of the $2^{n^2}$ balanced bipartite graphs on $2n$ vertices how many of them have $i$ perfect matchings where $i\in\Bbb N\cup\{0\}$ and $0\leq i\leq n!$ holds?
1
vote
0answers
44 views

A priori estimates for elliptic operators

Suppose $L : L^{m,p}(M)\rightarrow L^p(M)$ is some elliptic operator of order $m$, and $(M,g)$ is a compact Riemannian manifold. Then it is known that there exists a constant $C$ such that we have the ...
21
votes
0answers
646 views

Applications of arithmetic topology to number theory

There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this ...
0
votes
0answers
94 views

Sum of unit vectors always has a binary span after constrained permutations

Conjecture: Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$. An enumeration $ E \cup -E = \{f_1, \ldots, ...
10
votes
0answers
202 views

A Combinatorial Game: the Snake and the Hunter

The Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \times 6$. In both ...
0
votes
0answers
99 views

Selmer and free rank of Elliptic Curves

If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal. This is one of the many equivalent formulations of the Birch and ...
3
votes
1answer
87 views

Isoperimetric inequality via Crofton's formula

I have seen various assertions that one can derive the isoperimetric inequality in the plane from Crofton's formula in geometric probability. Unfortunately, I have not managed to figure out such a ...
6
votes
1answer
209 views

The minimum codimension of Lie subalgebra of $\chi^{\infty}(M)$

Assume that $M$ is an arbitrary manifold. Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one? If not, what is a counter ...
-1
votes
1answer
135 views

number of partitions from 0 to n^2 [closed]

You are given the numbers 0 to n^2. You must use n numbers with no number greater than n to form all the partitions of the numbers 0 to n^2. For example with n=4 you want to find the partitions of 7:...
-3
votes
1answer
153 views

Does one need an external, peer-reviewed grant to become tenured faculty in this field? [closed]

As a secondary question, how important is it to be awarded grants to remain employed as a PhD mathematician at an academic research institution? Is it common or uncommon for PhD mathematicians to have ...
2
votes
0answers
79 views

Kahler Einstein metric with minimal singularities

Let $X$ be a Kahler variety with snc divisor $D$ such that $K_X+D$ is ample. then there is a Kahler metric $\omega_E$ such that $Ric(\omega_E)=-\omega_E$ on $E=X\setminus D$, then $h=\frac{1}{\...
3
votes
1answer
122 views

Elements in the group von Neumann algebra which are not summable

Let $G$ be a discrete group. I am wondering if there is a recipe which can be applied to find elements in the group von Neumann algebra which are not absolutely summable, i.e. $T \in VN(G)$ while $T \...
0
votes
1answer
50 views

Count Functional digraph [closed]

Given a set of nodes, how can I count the number of different functional digraph containing a specific number of connected components? With a restriction that no node can have an edge point to itself. ...
1
vote
0answers
46 views

Finding the analytic Zariski decomposition singular hermitian metric on a relative line bundle

Let $f:X\to S$ be a proper surjective projective morphism between complex manifolds with connected fibers and let $D$ be an effective $\mathbb Q$-divisor on $X$ such that $$S^°=\{s\in S| f\text{ is ...
8
votes
2answers
261 views

How to compute $[CP^2, G/PL]$?

Let $E^4$ be the two stage Postnikov space appearing in the homotopy type of the classifying space $G/PL$. One of its properties is that it only has two nontrivial homotopy groups $\pi_2(E)=Z/2Z$ and $...
2
votes
0answers
79 views

Some queries on Moduli Space of Parabolic Vector Bundles on curve

In "Moduli of Vector Bundles on curves with Parabolic Structures"-Bulletin of the American Mathematical Society Volume 83, Number 1, January 1977 the author announces the following result on moduli ...
2
votes
0answers
103 views

Are all zeros of $\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$ either real or on the line with $\Re(s)=\frac12$?

In this post it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$. I have been searching for similar reflexive $...
1
vote
0answers
67 views

Singular canonical hermitian metric

Let $M$ be a complex manifold , take $$K_{M,m}:=\sup \{|\sigma|^{\frac{2}{m}}; \sigma\in H^0(M,\mathcal O_M(mK_M)), |\int_M(\sigma\wedge\bar\sigma)^{\frac{1}{m}}|\leq 1\}$$ Let $$K_{M,\infty}:=\lim\...
0
votes
0answers
52 views

What is combinatorially possible, for a singular minimal surface in $\mathbf{R}^3$?

A cubic planar graph gives a cell decomposition of a two-sphere, whose dual complex is a triangulation. If I understand Plateau's laws here, a soap film gives a cell decomposition whose dual complex ...
-2
votes
0answers
23 views
6
votes
1answer
242 views

Stopping times for Brownian motion

Let $B_t, t\geq 0$ be standard Brownian motion. Let $\big(\mathcal{G}_t, t\geq 0\big)$ be the natural filtration, defined by $\mathcal{G}_t=\sigma(B_s, 0\leq s\leq t)$. Define also a filtration $\...
0
votes
0answers
37 views

How to build C-Corn for Coq from source? [closed]

Trying to install C-Corn via opam install. The problem is that since two weeks I do not see any progress, I see only Processing: make . I have i7 , 8 GB RAM. Is that normal or has something gone ...
6
votes
2answers
300 views

Simple proof for $\sum_{i=1}^{n} a^{gcd(i,n)} $ is divisible by n

Burnside's Lemma Deduce That: $$\sum_{i=1}^{n} a^{gcd(i,n)} $$ is divisible by n it's a beautiful result. but i want to prove it without any abstract algebraic tools such as Burnside's Lemma... is ...
2
votes
1answer
129 views

Affine open subsets for algebraic group actions

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero. Assume that $G$ acts on an affine variety $X$. Assume that $X$ contains an open orbit $U$ (so $\...
3
votes
1answer
120 views
+100

A specific spanning property of a family of vectors

Let $v_1, v_2, v_3, \dots$ be a family of vectors in $\mathbb R^n$ that span $\mathbb R^n$. Given this family, define the family of vectors \begin{align*} \begin{pmatrix} v_1 \\ \alpha_1 v_1 \end{...
0
votes
1answer
69 views

Heat kernel upper bounds on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, and $p(t, x, y)$ denotes its heat kernel. I am trying to find sufficient conditions for when the following holds: $$ p(t, x, y) \leq Ct^{-n/2}, \forall x, y, ...
0
votes
0answers
36 views

Evaluation modules of $U_q(L(sl_2))$

Let $a \in \mathbb{C}^{\times}$, $r \in N$. Let $W = V_q(r)$ be the $r$-dimensional irreducible type 1 representation of $U_q(gl_2(\mathbb{C}))$. In the usual basis $\{v_0, \ldots, v_r\}$, the action ...
0
votes
0answers
50 views

$L^\infty$-contractive semigroups

Let $L^\infty(\mathbb T)$ be the space of $2\pi$-periodic and bounded measurable functions and $\mathcal P$ be a pseudo-differential operator defined on $\mathcal D(\mathcal P)\subset L^\infty(\...

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