-1
votes
0answers
68 views

Are Modular Collatz Graphs strongly connected? [on hold]

A while ago, I stumbled on the idea of representing the Collatz function, modulo a prime $p$, as a directed graph. Define, as usual $$ T(x) = \begin{cases} (3x+1)/2 & \text{if $x$ is odd,} \\ x/2 ...
3
votes
1answer
143 views

Spectrum of the Laplacian on p-forms on the sphere

In this paper the authors give an explicit description of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...
4
votes
0answers
64 views

Geometric interpretation of the Desnanot-Jacobi Identity

Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the i-th row and j-th column of $M$. The Desnanot-Jacobi Identity states ...
2
votes
0answers
39 views

Is the Quot scheme of finite length quotients with prescribed composition factors projective?

Assume we have a scheme $X$ over a field, say $\mathbb{C}$, and a "nice" sheaf of ring $R$ on it. $E$ denotes a left $R$-module. We denote by $Q:=Quot_R(E,n)$ the scheme classifying quotients ...
-2
votes
0answers
42 views

What kind of matrix is similar to an irreducible matrix? [on hold]

For example, we know that the set of positive definite matrix is similar to an irreducible matrix. Therefore, the set of such matrices should include the positive definite matrix cone. What kind of ...
0
votes
1answer
46 views

How many edges guarantee an expander?

Complete graphs are good expanders. Deleting few edges from a complete graph leaves a good expander. What's the proportion of edges that one needs to remove to make the graph a bad expander? Or, is ...
0
votes
0answers
38 views

Densely-defined operator with closed range: conditions for operator closed

Suppose we have Banach spaces (or Hilbert spaces) $X$ and $Y$, and a densely-defined linear operator $A : \operatorname{dom}(A) \subset X \rightarrow Y$ that is densely-defined and with closed range. ...
-1
votes
0answers
39 views

Is there a stochastic process with zero mean but nonzero value in each time instant? [on hold]

Background: In a signal processing application, we want to randomized a signal by 'whiten' it, that is to say we want the 'whitened' signal with zero mean. However, there is a constraint on the signal ...
2
votes
1answer
134 views

Deformation of curves and closed immersions

Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow ...
2
votes
1answer
144 views

Old Peano theorem (demonstration is missing details) [on hold]

Let $A, B :[a,b]\subset\mathbb{R}\to\mathbb{R}^2$ be two functions of class $C^{1}([a,b])$ such that two segments (or intervals) $[A(t_1),B(t_1)]$ and $[A(t_2), B(t_2)]$ never intersect for $t_1, ...
2
votes
1answer
68 views

Generators of the colored braid group (two colors), reference

I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white. It is easy to find a set of generators for $B_{n,n}$: $$ \begin{cases} ...
2
votes
0answers
91 views

Reference to parabola lemma

I am looking for a previous reference to the following very simple geometric lemma, which I use in my paper [arXiv:1503.03462]: Let $P$ be the parabola $y=x^2$. Let $a,b,c,d$ be four points on $P$ ...
4
votes
1answer
140 views
+50

Radius of the largest enclosed ball in the convex hull of an algebraic variety

Let $\mathcal{V}\subset\mathbb{R}^n$ be a real compact algebraic variety. Let $\mathcal{V}^c$ be the convex hull of $\mathcal{V}$ and let us assume that $\mathcal{V}^c$ has nonzero n-dimensional ...
2
votes
1answer
139 views

Does hyperconnected imply path-connected

If a space is hyperconnected (that is, the every non-empty open sets intersect), is it also path-connected?
-1
votes
0answers
60 views

Non Satisfiability of disjuction [on hold]

Problem: If $S_1$ and $S_2$ are (possibly infinite) sets of propositional formulas where their union, $S_1\cup S_2$, is not satisfiable, prove that there exists an $\psi$ such that $S_1\models \psi$ ...
-5
votes
0answers
57 views

vector bundle and characteristic classes [on hold]

why every orient able line bundle is trivial? I give an orientation s(x) to each fib re such that s(x).s(x)=1 using the euclidean metric but I can not understand why this section becomes continuous
0
votes
0answers
36 views

projective representation of supergroup

In fact, I am not very clear about what I am asking, but I am looking for a concrete example of supergroup which has non-trivial projective representation(some supergroup similar to usual Lie group ...
1
vote
1answer
53 views

Alexandrov spaces of constant curvature

Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian ...
4
votes
0answers
43 views

Strongly minimal set with DMP

Recall that a strongly minimal theory $T$ has the Definable Multiplicity Property (DMP) if for all natural $k$, $m$ and $\varphi(\bar{x},\bar{b})$ of rank $k$, multiplicity $m$, there exists a formula ...
11
votes
1answer
192 views

Fair cake-cutting between groups

The cake-cutting game is usually played between individuals. What if we try to play it between groups? A certain land has to be divided between two states. ‎There are $n$ citizens in each state. ...
3
votes
0answers
123 views

Using $\mathcal{U(H)}$ as a model for $EG$ and working with the Fredholm Operators

Let $\mathcal{H}$ be a unitary universe for some group $G$. As $\mathcal{H}$ is a faithful representation the representation map is an injection $G \to \mathcal{U(H)}$, so there's a free $G$ action on ...
7
votes
2answers
144 views

Decomposition of a cross-polytope into simplices

Consider the standard embedded $n$-cross polytope $P_n$ with vertices $\pm e_i \in \mathbb R^n$. Let us consider decompositions of this polytope into $2^{n-1}$ simplices, such that these simplices ...
0
votes
0answers
23 views

What does it mean for a prime ideal to divide a natural number m? [migrated]

In Cassels and Frohlich (Algebraic Number Theory) Exercise 1, one is asked to derive some properties of the power residue symbol. It begins by stating the following: Let $m$ be a fixed natural ...
9
votes
4answers
419 views

Rate of convergence in the Law of Large Numbers

I'm working on a problem where I need information on the size of $E_n=|S_n-n\mu|$, where $S_n=X_1+\ldots+X_n$ is a sum of i.i.d. random variables and $\mu=\mathbb EX_1$. For this to make sense, the ...
3
votes
1answer
155 views

Moser estimates?

Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
0
votes
0answers
66 views

First order elliptic pseudodifferential operator and Sobolev space [on hold]

Let $P$ be an elliptic pseudodifferential operator of order 1 on a compact manifold $M$. Also let the spectrum of $P$ lie inside $(0, \infty)$. I am trying to see how one could prove that $P : H^1(M) ...
4
votes
1answer
138 views

Sets of natural numbers such that sums of a bounded number of its elements form a semigroup

This is a naive question and I'm afraid it might be better placed on math.se. I would like to leave it to your judgement. I would like to know what is known about sets $A$ of natural numbers such ...
5
votes
1answer
102 views

Coverings/Cech cohomology of totally disconnected spaces

For any topological space $X$ we have a natural functor $\text{Cov}_X \rightarrow \text{Fun}(\pi_1(X),\text{Set})$ from the category of coverings of $X$ to the category of functors $\pi_1(X) ...
40
votes
4answers
1k views

When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true. For example, it gives some evidence that there are finitely many ...
3
votes
1answer
196 views

$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...
2
votes
0answers
69 views

Motivation for the existence of periodic solutions [on hold]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form ...
2
votes
1answer
63 views

Proof of eigenvalue stability inequality via Courant-Fischer min-max theorem

Dr. Tao in his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality. Specifically, I am looking for proof of Eq. (13) where Dr. Tao ...
0
votes
0answers
142 views

Complex submanifold has minimal volume

I know that the following theorem is true: If $W$ is a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$, $sngW$ is the set of its singular points, $V \subset W$ is open, ...
2
votes
0answers
98 views

Simply connected Kahler manifold without any effective divisor

Does anyone know an example of a simply-connected compact Kahler manifold without an effective divisor? Does anyone know a reference on this topic? Thanks!
-2
votes
0answers
42 views

$u_{xx}+u_{yy}=0,1<r<3,0<\theta<\frac{\pi}{2}$ [on hold]

$u_{xx}+u_{yy}=0,1<r<3,0<\theta<\frac{\pi}{2}$ $u(1,\theta)=u(3,\theta)=0,0\leq\theta\leq\frac{\pi}{2}$ $u(r,0)=(r-1)(r-3),u(r,\frac{\pi}{2})=0,1\leq r\leq3$ I have no idea how to solve ...
0
votes
0answers
44 views

Dimension of a trajectory in $\mathfrak{su}(n)$

Consider $a,b \in \mathfrak{su}(n)$, a fixed positive real number $t$ and the equation: $\frac{d U(s,t)}{ds} = \left(a + \omega(s)b \right) U(s, t)$ (where $U(t,t)=I$) and the definition: $B(s) = ...
1
vote
2answers
113 views

Sum of two unbounded self-adjoint operators

Let $A$ and $B$ be two unbounded self-adjoint operators. From this mathoverflow post, for instance, we know that $A + B$ is self-adjoint on $\mathcal{D}(A) \cap \mathcal{D}(B)$ if $A$ and $B$ are ...
1
vote
0answers
113 views

Vector bundle is semistable if only if it's pull back is semistable?

If $X$ is a smooth projective variety and $D$ is a divisor on $X$, and let $i:D\longrightarrow X$ be the closed immersion. Let $E$ be a vector bundle on $X$. Are there any theorems which say that $E$ ...
4
votes
1answer
179 views

Young tableau with no i in row i, name that derangement

This question has its genesis in a group assignment: $k$ students are to be given oral exams. Each student will be asked one distinct question from $n$ questions given to them earlier, no two students ...
0
votes
0answers
35 views

Initial value problem with unique solution and rear wheel of a bike problem [on hold]

Consider $I\subseteq\mathbb{R}$ an arbitrary interval (it can be of the following types: $[a,b], [a,b), (a,b], [a,+\infty),(a,+\infty),(-\infty,b), (-\infty, b], \mathbb{R}$). Let ...
13
votes
1answer
398 views

When complex conjugation lies in the center of a Galois group

Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...
0
votes
0answers
28 views

Reparametrization with non-vanishing lateral derivatives [on hold]

Let $\mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2$ be a $C^{1}(I)$ function, non-constant on any subinterval, such that $\forall\ t_0\in I$, the following limits exist: $\lim\limits_{t\nearrow ...
-2
votes
0answers
35 views

Find a shortest way between nodes in graph [on hold]

I have a next structure : Each node in graph may have more than 2 links. I want to find a shortest way with node 1 and 13. ...
6
votes
2answers
85 views

Reference request: monochromatic paths in edge-colored complete graphs

Given $k,c \in \mathbb{N}$, let $P(k,c)$ be the minimum $n$ such that no matter how we color the edges of the complete graph $K_n$ with $c$ colors, there is always a monochromatic path of length $k$. ...
1
vote
1answer
85 views

Push-forward of sum of two maps

Let $X=R^n$ and $Y=R^m$ are two Euclidean spaces with $m<n$. Let $\varphi$ and $\phi$ are two (smooth) maps from $X$ to $Y$ and $\mu$ is a probability measure on $X$. Is there any relationship ...
1
vote
1answer
90 views

Are sections of $\tau M$ differential operators on the exterior algebra?

Let $M$ be a smooth manifold and $\tau M$ its second-order tangent bundle. A second-order vector field $A\in \Gamma(\tau M)$ can locally be expressed as a finite sum of operators ...
-6
votes
0answers
59 views

What math do I need to know in order to understand basic trigonometry? [on hold]

Im reaching a point in programing where I need to create basic shapes which I simply cant since my math skills are very bad. After finding out that the skills required are trigonometry I read a few ...
0
votes
0answers
52 views

Log-convexity preserved by sum? [migrated]

I'm currently reading Boyd's book on Convex Optimization (free book which can be found here : http://stanford.edu/~boyd/cvxbook/) , and there is one proof I do not understand : p. 105, the book says ...
2
votes
0answers
24 views

The Stratonovich formulation of the Double Mean Reverting Model

I am writing my Bachelor's Thesis on the fast Ninomiya-Victoir calibration of the Double Mean Reverting model and have a question to its Stratonovich formulation. I am new to mathoverflow and a novice ...
-2
votes
0answers
109 views

How to use the derived functor to write down a lot of natural morphisms or isomorphisms? [on hold]

In the Kashiwara’s book:Sheaves on manifolds, the derived functor is defined in the abstract form, you can see some discussing at here: Derived functor Kashiwara’s book gives a lot of natural ...

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