# All Questions

**-1**

votes

**0**answers

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

**1**answer

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of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**4**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**4**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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 ...