2
votes
1answer
37 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 ...
0
votes
0answers
10 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
23 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
27 views

weakly etale maps

Let $k$ be an algebraically closed field. Consider the map $\phi:X:=\mathbb{A}^{1}\times\mathbb{G}^{m}\times\mathbb{A}^{\mathbb{N}}\rightarrow Y:=\mathbb{A}^{\mathbb{N}}$ given by $(\lambda, ...
-6
votes
1answer
35 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.
-4
votes
0answers
25 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?
4
votes
1answer
32 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 ...
-2
votes
0answers
125 views

How this conjecture is false and it was published in this journal as a true conjecture? [on hold]

This question is related to my recent question, i accrossed this conjecture in this pdf :http://www.anubih.ba/Journals/vol.8,no-2,y12/11Ladas-Lugo-Palladino.pdf, page (05).conjecture 08 as an ...
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 ...
7
votes
0answers
111 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
26 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
36 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 ...
-1
votes
0answers
49 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
58 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 ...
5
votes
1answer
90 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
26 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
26 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
44 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
27 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
48 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
77 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
31 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
16 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, ...
1
vote
2answers
115 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
66 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
13 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
58 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
82 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
20 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
63 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
42 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 $\mathcal{A}$ ...
-2
votes
0answers
22 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 ...
-3
votes
0answers
49 views

How to evalute: $\int_0^1 \frac{e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}((1-x)\cos (\pi x) + \frac{3}{\pi} \sin(\pi x)) dx$ and $a, b >0$ [on hold]

How to evalute: $$\int_0^1 \left[ \frac{e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}\left((1-x)\cos (\pi x) + \frac{3}{\pi} \sin(\pi x)\right) \right] dx$$ and $a, b >0$
-1
votes
0answers
34 views

how to solve a linear equation (Ax-b)T- lamda(c)? [on hold]

I'm trying to solve an linear optimization problem, it's first order Lagrange condition leas to this equation $$ (Ax-b)^T A- \lambda C = 0. $$ Here $A$ is an $m\times n$ matrix, $m>n$, $C$ ...
6
votes
1answer
121 views

Question about zeta function of function field in 1 variable over $\mathbb{F}_q$

From my previous question, I know that$$\zeta_X(s) = {{P(u)}\over{(1-u)(1-qu)}}$$for some polynomial $P(u)$ of degree $2g$, where $X$ is the set of all places of $F$, a function field in one ...
3
votes
3answers
245 views

How many primes have the form $(2^p+1)/3$?

Assuming that $p$ is an odd prime. How many primes have the form $(2^p+1)/3$? Is the number finite? Mathematica calculation shows that there are 23 such primes when $p$ ranges over the first 500 ...
-3
votes
0answers
72 views

Quotient group of an amalgam [on hold]

If a quotient of a group G is an amalgam then the group G is an amalgam. Is this true or false? How can we describe a quotient of an amalgam?
2
votes
0answers
47 views

Rational curves through a fixed number of points

Let us fix two positive integers $d$, and $N$. Can we determine a third integer $n$ such that given $n$ general points $p_1,...,p_n\in\mathbb{P}^N$ there exists a unique rational curve of degree $d$ ...
9
votes
0answers
137 views

On sentences true in all finite groups (revisited)

Let $w$ be a group word with variables $\bar x, \bar y$, where $\bar x=(x_1,\dots ,x_m)$ and $\bar y=(y_1,\dots ,y_n).$ I am interested in the following questions. (1) Is the sentence $(\forall\bar ...
1
vote
0answers
16 views

Difference of affine monoids with trivial intersection

Let $A,B \subset \mathbb N_0^d$ be affine monoids which satisfy $A \cap B = \{0\}$. If some $x \in \mathbb Z^d$ satisfies $x = a_1-b_1=a_2-b_2$ for some $a_1,a_2 \in A$ and $b_1,b_2 \in B$, is it true ...
7
votes
1answer
208 views

Reference request, zeta function is rational function via Riemann-Roch?

I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch ...
-1
votes
0answers
17 views

Definition of Category of Hypergraphs [migrated]

I have a question concerning the definition of Hypergraphs in category theory, which I adopted from "A category-theoretical approach to hypergraphs" by W.Dörfler and D.A.Waller: ...
5
votes
1answer
114 views

Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound. Question: Suppose $\mathbb{P}$ is a separative partial order which is ...
3
votes
1answer
119 views

Quotients of finitely generated nilpotent groups

I am currently writing my thesis and looking for a reference (or a short proof) to the following fact: Let $N$ be a finitely generated nilpotent group, and denote its lower central ...
1
vote
0answers
40 views

Is the set of singularities of a meromorphic function on a domain in $\mathbb{C}^n$ an analytic variety?

Let $f$ be meromorphic on a domain $D\subset \mathbb{C}^n$ ($n>1$), and let $S\subset D$ be the smallest set such that $f$ is holomorphic on $D\setminus S$. Is the set $S$ an analytic variety?
0
votes
1answer
79 views

Moment maps and flat degenerations of toric varieties

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$. How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...
1
vote
0answers
33 views

Multivariable polynomial interpolation via evaluations from entrywise powers of a point

I am interested in multivariate polynomial interpolation. Within computational complexity theory, I use it to create efficient reductions between counting problems. In the univariate case, there is ...
1
vote
0answers
56 views

Find a TSP tour passing through at least one node in each set of nodes [on hold]

Given a graph $G$ and a number of node sets, each consisting a number of nodes in $G$. The question is to find the shortest path passing through at least one node in each node set. If each node set ...
-1
votes
0answers
15 views

graph edge partitioning for isomorphism testing

by a theorem of P. Rowlinson a graph of diameter D is D-walk-regular if and only if it is distance-regular. See e.g. C. Dalfo, E.R. van Dam, M.A. Fiol, E. Garriga, and B.L. Gorissen, On almost ...

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