2
votes
0answers
16 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 ...
1
vote
1answer
33 views

A generalization of Erdős–Mordell inequality

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 on the plane. Let $d_i$ be the distances from $P$ to $...
0
votes
0answers
10 views

Hyperbola application [on hold]

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
9 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 ...
0
votes
0answers
35 views

Is (+, *) the only semiring over the positive reals? [on hold]

I know that there are others over the reals but I can't find any others over the positive reals.
0
votes
0answers
23 views

How many number of finite points exists inside the circle?

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
2answers
26 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 for that graph using its ...
0
votes
0answers
8 views

Proving that an integral of several cdf and pdf functions is increasing in a certain parameter

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the usual general CDF and PDF (no specified distribution here)....
1
vote
0answers
31 views

Gauge Fixing Problem on Cylindrical

For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold. Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian connection $A_0$, we know that all other Hermtian ...
3
votes
0answers
25 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 \...
1
vote
0answers
56 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
17 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
31 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 ...
14
votes
0answers
138 views

Applications of arithmetic topology to number theory

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

Sums of unit vectors has a binary span after constrained permutations

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, f_{2(m_1+m_2)}\}...
1
vote
0answers
66 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
61 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
50 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 ...
3
votes
1answer
52 views

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

Assume that $M$ is a manifold.Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?
0
votes
2answers
95 views

number of partitions from 0 to n^2

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:...
0
votes
0answers
25 views

Non symmetric, zero column sum, positive diagonal and negative off diagonal entries matrix bound?

Under which conditions can the A-inner product of a non symmetric, zero column sum, positive diagonal and negative off diagonal entries matrix be bounded by the L2-inner product? $A \in \mathbb{R}^{n ...
-2
votes
1answer
123 views

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

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
46 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}{\...
2
votes
1answer
90 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
34 views

Count Functional digraph

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
30 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 ...
5
votes
1answer
117 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 $...
1
vote
0answers
42 views

Gauss Bonnett on a flat surface with border

I'm reading the article "Euler Characteristics of Teichmuller Curves in genus two" by Matt Bainbridge and there's a point I don't understand in the proof of theorem 5.5. Maybe you can help me clarify ...
2
votes
0answers
53 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
1answer
55 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
31 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
41 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
21 views
4
votes
1answer
94 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? [on hold]

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
248 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 ...
1
vote
1answer
103 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 $\...
0
votes
0answers
20 views

A specific spanning property of a family of vectors

Let $v_2, v_2, v_3, \dots$ 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
61 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
24 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
35 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(\...
3
votes
0answers
73 views

Curves embedding in plane

Given two closed simple(no self-intersection point) curves $C_1,C_2$ in the plane $\mathbb R^2$, is there a good way to judge whether one curve could be embedded inside into the other one, here ...
1
vote
1answer
88 views

a space isomorphic to $S^{p+q}$

I've asked this question few days ago in MathStackExchange, but there was no result. So, I decided to ask it there. In one of the paper I have met that $$\mathbb{S}^{p+q} \cong \mathbb{S}^p \times \...
4
votes
1answer
181 views

Is the assignment of a root system to a complex semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
0
votes
0answers
35 views

Is it possible for a quantum group algebra $U_{q}\left(\mathfrak{g}\right)$ to have a diagonal universal $R$-matrix?

I am writing a research paper and have shown that in the special case when a quantum group algebra $U_{q}\left(\mathfrak{g}\right)$ with the quantum group parameter $q$ not a root of unity has a ...
1
vote
0answers
32 views

Concentration of functional of Gaussian random variable

Suppose I have two Gaussian distributions $p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...
6
votes
1answer
83 views

Chevalley restriction theorem for non-split Cartan

Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of ...
2
votes
0answers
42 views

Average range of Motzkin path

Motzkin path are paths from (0,0) to (n,0) in $\mathbb{Z}^2$ such that we are allowed to move SE, E and NE. More on this is here https://en.wikipedia.org/wiki/Motzkin_number I would like to know if ...
0
votes
0answers
24 views

Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories. More concretely, I want to obtain a SDE of type as ...
8
votes
1answer
182 views

Groups whose word problem can be solved in constant time

Given a finitely generated group $G$, define an encoding of $G$ to be a one-to-one function $\Phi:G\to \bigcup_n \{0,1\}^n$ that sends each group element to a unique finite word. For $a,b\in G$, ...

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