# All Questions

**3**

votes

**0**answers

21 views

### Existence of finite set of points in the revolving circles

Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ ...

**1**

vote

**0**answers

24 views

### Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true?

I apologize if this is not a research level question (already tried asking http://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange with ...

**1**

vote

**0**answers

60 views

### On the use of the symmetric asymptotic upper density on $\mathbf Z$

The upper asymptotic density on $\mathbf Z$, viz. the function
$$
{\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n},
$$
has a ''symmetric ...

**-4**

votes

**0**answers

58 views

### How to prove equality fo inner product <Ax, x>= <Bx, x> then A = B? [on hold]

Let be a vector space with the inner product $<.,.>$ And A, B: V ---> The self adjoint linear mapping. Prove: if for every vector x is elem of V holds the equality $<Ax, x> = <Bx, ...

**2**

votes

**0**answers

27 views

### Norm of $n$-linear symmetric forms

Let $B$ be a symmetric bilinear form over a Euclidean space $E$. Say that $|B(v,v)|\le c\|v\|^2$ for every $v\in E$, for some $c\ge0$. Then
$$4B(v,w)=B(v+w)+B(v-w)$$
yields $2|B(v,w)|\le ...

**1**

vote

**0**answers

57 views

### Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$.
There is a candidate ...

**1**

vote

**2**answers

36 views

### u.p (unique product) group which is not Right ordered ($RO$)

I am looking for an example of a u.p (unique product) group which is not Right ordered. Almost any group I pick up (obviously torsion free, as u.p. group cannot have torsion element, so no use ...

**3**

votes

**0**answers

68 views

### Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...

**0**

votes

**2**answers

79 views

### Special connection of vector bundle over real manifold

Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...

**6**

votes

**1**answer

113 views

### Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...

**3**

votes

**0**answers

23 views

### Value of prolate speroidal wave function at 0

I have a very basic question about prolate speroidal wave functions that can be defined as eigenfunctions to the following integral equation:
$$
\lambda\cdot \psi(x) = \int_{-1}^{1}\frac{\sin ...

**1**

vote

**0**answers

31 views

### Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?

Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$.
But do we have any quantitative ...

**0**

votes

**0**answers

38 views

### There is no quasiregular diffeomorphism from punctured ball into ring (on the plane)

The idea is to use l2 cohomology as a quasiregular map invariant.
It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form
$f_1(x,y)dx ...

**0**

votes

**0**answers

106 views

### Construction of Highly Structured Quotient Objects in Quasicategories

Given a symmetric monoidal quasicategory $C$ and a morphism of $E_n$-algebras $f:A\to B$ in $C$ we can attempt to construct the quotient object $A/B$. If $A$ is augmented, and we're in a discrete ...

**3**

votes

**0**answers

97 views

### Contractibility of regular CW sphere minus open star

Let $S$ be any regular CW decomposition of (a space homeomorphic to) the $n$-sphere, and consider a cell $\sigma$ of dimension $d \in \{0,\ldots,n\}$. Let $S'$ be the regular CW complex which remains ...

**0**

votes

**0**answers

26 views

### The Free Loop Space of a Manifold $M$ when $M$ is not compact

In Klingenberg's Lectures on Closed Geodesics, before constructing the differentiable structure of the free loop space of a compact manifold $M$, he states that:
A large part of the construction ...

**-1**

votes

**0**answers

79 views

### Non-principal ultrafilters preserving infinite joins/meets

There is a theorem (of Tarski, and Rasiowa-Sikorski) that if we consider countably many infinite joins (or meets), call them Q, in a boolean algebra A, then there is an ultrafilter U which preserves ...

**3**

votes

**1**answer

68 views

### Topologies on spaces of distributions and test functions

Let $X$ be an open subset of $\mathbb{R}^n$. Following the notation of Schwartz, we denote $\mathcal{D}$ the space of compactly supported complex-valued smooth functions on $X$ equipped with the ...

**12**

votes

**0**answers

122 views

### Finding a path through real rooted polynomials

This is a lemma I wanted in order to solve Patrizio Neff's conjecture. It turned out to be the wrong way to think about it, but I am still curious if it is true.
Let $z^n+a_{n-1} z^{n-1} + \cdots + ...

**4**

votes

**1**answer

147 views

### Reference for an unbiased definition of a symmetric monoidal category

In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor
$$
\mathcal{C}\times ...

**1**

vote

**0**answers

26 views

### About properties of polynomials with common interlacing

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order)
Can ...

**3**

votes

**1**answer

65 views

### Segal maps for Segal precategories

A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal ...

**-1**

votes

**0**answers

81 views

### weakly etale maps [on hold]

Let $k$ be an algebraically closed field.
Consider the map $\phi:X:=\mathbb{A}^{1}\times\mathbb{A}^{\mathbb{N}}\rightarrow Y:=\mathbb{A}^{\mathbb{N}}$
given by $(\lambda,x(t))\mapsto (t-\lambda) ...

**-6**

votes

**1**answer

58 views

### Find the probability that the product of these numbers is a multiple of 3 [on hold]

From the sequence of natural numbers randomly select a pair of numbers. Find the probability that the product of these numbers is a multiple of 3.

**-3**

votes

**0**answers

39 views

### What is the idea behind a projection operator?what does it do? [on hold]

I need the idea behind this not the definitions of the examples can someone help?

**8**

votes

**1**answer

79 views

### Computing a transversal of a subgroup $H$ of $G$ in expected $O(|G : H|^2 \log |G : H| + |H|)$ time

I have the book "Handbook of Computational Group Theory", by Derek Holt, and in it is a section on finding the transversal of a subgroup. Recall a transversal of a subgroup $H$ of $G$ is a single ...

**0**

votes

**0**answers

7 views

### Laplacian Matrix for weighted Adjacency? [on hold]

I have seen definitions for Laplacian matrix in many resources as follows :
L = D − A
where D and A are the degree and ...

**9**

votes

**0**answers

165 views

### Excellent rings

If A is an excellent commutative ring and G is a finite group of automorphisms of A, is the invariant subring A^G still excellent ? I think this is false -- because if not it would probably be written ...

**5**

votes

**1**answer

49 views

### Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator
$$D:= \frac{d}{dt}+A(t)$$
and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...

**-4**

votes

**0**answers

47 views

### TOPOLOGY DATA ANALYSIS [on hold]

actually i am PHD student and my research in TOPOLOGY DATA ANALYSIS (TDA)
what I dream about to understand this artcile cran.r-project.org/web/packages/TDA/vignettes/article.pdf may you help me:
1- MY ...

**-2**

votes

**0**answers

59 views

### Relation between Kahler form and Kahler potential [on hold]

Let us consider an example. Take $\mathbb{C}^m$ which is identified with $\mathbb{R}^{2m}$. Now, the Kahler form is given by
$$\Omega = \frac{i}{2} \sum_{i=1}^m dz^i \wedge d\bar{z}^i$$
Now, how can ...

**2**

votes

**0**answers

77 views

### The existence of proper schemes under complection

Let $R$ be a regular local ring, $\hat{R}$ be its completion, $X$ be a proper scheme over $\text{Spec}(\hat{R})$. In what case there exist a proper scheme $Y$ over $\text{Spec}(R)$, such that $X$ is ...

**8**

votes

**1**answer

132 views

### Direct proof that $U$ is an $E_\infty$-space

An immediate consequence of Bott periodicity is that the infinite unitary space is an infinite loop space and so an $E_\infty$-space. I wonder if there is a direct proof (not using $U = \Omega^2 U$) ...

**1**

vote

**0**answers

32 views

### why group completion of configuration space is the iterated suspension space

In Lecture notes in mathematics Vol. 533, The homology of $C_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 226, Corollary 3.3:
$\alpha_{n+1}: C(\mathbb{R}^{n+1};X)\to \Omega^{n+1}\Sigma^{n+1}X$ is a ...

**4**

votes

**0**answers

59 views

### Endomorphism algebras of abelian surfaces with real multiplication

Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider ...

**-5**

votes

**0**answers

57 views

### Has solution of Brocard's Equation n!=m^2-1 [on hold]

Brocard conjecture in 1904 that the only solution of are n=4,5,7. There are no other solutions with .(Berndt and Galway n.d).Another of Brocard’s conjecture is that there are at least four primes ...

**4**

votes

**0**answers

37 views

### Sobolev-Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...

**6**

votes

**1**answer

64 views

### Infinite graphs with “similar” Hom-sets

Let $G,H$ be infinite simple undirected graphs with the property that for any graph $X$ we have $|\text{Hom}(X,G)| = |\text{Hom}(X,H)|$. Does this imply that $G$ is isomorphic to a subgraph of $H$, ...

**8**

votes

**0**answers

98 views

### The multiplicative group generated by shifted primes

I am asking for references about the following problem.
In particular, it is still open? If not, what is the state of the art result?
Problem 1. Let $\Gamma$ be the multiplicative subgroup of ...

**3**

votes

**0**answers

47 views

### Sparsifiers for 3-term arithmetic progressions

Let $G$ be a finite abelian group of odd order, let $D\subseteq G$, and $\epsilon \in (0,1)$.
For $S\subseteq G$ define
$$
\Lambda(S) = \frac{1}{|S||G|} \sum_{s\in S}\sum_{g\in G} ...

**0**

votes

**0**answers

26 views

### Stochastic gradient descent interleaved with deterministic optimization

I wish to solve
$\min_{x, y_k} \frac{1}{n} \sum_{k=1}^n f_k(x, y_k)$.
where $f_k$ are all smooth and convex.
Using standard stochastic gradient descent (SGD), each iteration I sample a k from $\{1, ...

**2**

votes

**2**answers

130 views

### Algebraic groups “generated” by a Lie algebra element

Here is a definition which I invented and which I would like to understand better.
Let $ A $ be a complex affine algebraic group. Let $ X \in \mathfrak g $ be an element in its Lie algebra. We say ...

**12**

votes

**0**answers

111 views

### Quickest and/or most elementary proof of “principal iff splits completely”?

Let $L$ be the Hilbert class field of a number field $K$, and let $\mathfrak{p}$ be a prime ideal of $K$. Then $\mathfrak{p}$ splits completely in $L$ if and only if $\mathfrak{p}$ is a principal ...

**1**

vote

**0**answers

18 views

### Find paths in a graph that any 2 vertices can be reached through N of them

Given a undirected weighted graph.
I would like to find a finite set of paths (consecutive vertices and edges)
each shorter than L
any two vertices can be reached through at most N(in my case N=4) ...

**5**

votes

**1**answer

88 views

### Iterated sumset inequalities in semigroups

This question is motivated by the following well-known theorems:
Thm (Plünnecke): If $A$ is a finite nonempty subset of an abelian group, then for every $n$ we have $|A^n| \le ...

**1**

vote

**1**answer

83 views

### iterated loop spaces and configuration spaces [on hold]

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map
$$
\phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y)
$$
is defined. And a map
$$
...

**1**

vote

**0**answers

26 views

### Symmetry conjecture for minimal dilatation pseudo Anosov mapping classes

The conjecture is something like the following:
The minimal dilatation among pseudo-Anosov mapping classes on a surface $S_{g,n}$ is realized by $\rho\circ\omega$ where $\omega$ is supported on a ...

**2**

votes

**1**answer

69 views

### References about spectral theory on Hyperbolic space

Can anyone suggest me some books or papers that include details about spectral theory on Hyperbolic spaces or related topics such as scattering theory on Hyperbolic spaces?
After some googling, I ...

**0**

votes

**0**answers

24 views

### Meaning of k-connected directed graphs [on hold]

Is there any existing definition for "k-connected directed graphs"?
Any reference paper?

**-3**

votes

**0**answers

43 views

### If there is in a category $\mathcal{A}$ finite products and equalizers then it has pullbacks [on hold]

My homework consist in showing that "If there is in a category $\mathcal{A}$ finite coproducts and coequalizers then it has pushouts" based on the proof that "If there is in a category $\mathcal{A}$ ...