0
votes
0answers
11 views

Continuity of bounded sublinear or quasilinear operators

Let $\mathcal{T}$ be an operator defined on a linear space of complex-valued measurable functions on a measure space $(X,\mu)$ and taking values in the set of all complex-valued finite almost ...
0
votes
0answers
24 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
0answers
53 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, ...
1
vote
0answers
19 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
0answers
39 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
2answers
35 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
0answers
51 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
2answers
54 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
1answer
89 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 ...
2
votes
0answers
17 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
0answers
30 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
0answers
30 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
0answers
97 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
0answers
87 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
0answers
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
0answers
70 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
1answer
67 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 ...
11
votes
0answers
111 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
1answer
143 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
0answers
23 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
1answer
60 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
0answers
75 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
1answer
56 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
0answers
38 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
1answer
77 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
0answers
6 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
0answers
161 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
1answer
48 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
0answers
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
0answers
56 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
0answers
74 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
1answer
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
0answers
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
0answers
54 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
0answers
56 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
0answers
36 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
1answer
63 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
0answers
95 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
0answers
44 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
0answers
25 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
2answers
128 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 ...
11
votes
0answers
82 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
0answers
17 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
1answer
86 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
1answer
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
0answers
24 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
1answer
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
0answers
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
0answers
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 $\mathcal{A}$ ...
-1
votes
0answers
27 views

simplifying an equation that has infinitesimals [on hold]

I'm trying to understand an equation with infinitesimal changes: 8*X*dX = d(4*X^2) I think this can be written $8X\Delta X = \Delta (4(X^{2}))$ I'm guessing going from 8 outside the differential ...

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