4
votes
1answer
14 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 ...
2
votes
3answers
57 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 ...
-1
votes
0answers
25 views

Quotient group of an amalgam

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
18 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 $d$ ...
2
votes
0answers
38 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
6 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 ...
5
votes
1answer
58 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
16 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
59 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
0answers
35 views

Reference Request - 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
24 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
0answers
38 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
44 views

Pure spinors and $SU(n)$ structures

Proposition 9.16 of Lawson's Spin Geometry book reads as follows: Let $X$ be an $2m$-dimensional manifold. Proposition 9.16: Each globally defined pure spinor $\sigma$ on $X$ determines a unique ...
1
vote
0answers
24 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 ...
0
votes
0answers
19 views

Find a TSP tour passing at least one nodes 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 at least one node in each node set. If each node set consists ...
-1
votes
0answers
11 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 ...
-4
votes
0answers
19 views

Show that the mapping A linear. Lays down rules for adjoint transformation A * [on hold]

Let V n-expansive real vector space with scalar product, a and b given linearly independent vectors from the space V. mapping A: V -> V is given by Regulations Ax = (x, a), * b Assign eigen values ...
0
votes
0answers
47 views

The kernel of the natural map $\pi_k(BU(r)) \to \pi_k (BU)$

Is this group known outside of the stable range? If so, what is it? If not, what is known about it?
2
votes
0answers
44 views

Condition number after preconditioning

Suppose $A$ and $P$ are symmetric, positive definite matrices and that we factor $P^{-1}=EE^\top.$ Is it true that the condition number of $PA$ is upper-bounded by the condition number of ...
-4
votes
0answers
17 views

How many are there orthogonal transformations? [on hold]

which transform the line x = y / 2 = z line -x / 2 = y = z and the line x = -y = with the line x = y = z? Find the matrix of any of them in a standard basis I have tried to equal equations.
-2
votes
0answers
20 views

embedding dimension of normal surface singularity [on hold]

$0\in X$ : a normal surface singularity. $C_1, C_2$ : hypersurface sections of $0\in X$ and $C_1 \cdot C_2=2$. $\stackrel{(1)}{\Longrightarrow}$ $C_i$ is a reduced curve singularity of multiplicity ...
0
votes
0answers
15 views

Extract a “fraction” bipartite subgraph from a given graph

Question: Do there exist real constants $k,c$ such that: for every graph $G$ of large minimum degree, there exists a spanning subgraph $H$ of $G$ such that $H$ is bipartite and ...
1
vote
1answer
31 views

Problem about the existence of a continuous surjective map

Let $F$ be a closed set in $\Bbb R^2$, $F\neq \varnothing,\Bbb R^2$, and $F^\circ\neq \varnothing$, does there exist a continuous surjective map from $\Bbb R\times \Bbb Z$ to $F^{\circ-}-F^\circ$?
1
vote
0answers
31 views

How can I calculate the adjoint of the wave operator $\square_{g}$ in Sobolev Spaces?

I have a three part question, which I could only received an answer for the first part here. The Laplace-Beltrami operator is an operator which is the typical example of a self-adjoint operator in ...
2
votes
0answers
55 views

Rational curves in projective spaces

Let $X\subset(\mathbb{P}^{N})^n$ be the variety defined as follows: $(p_1,...,p_n)\in (\mathbb{P}^{N})^n$ such that there exists a rational curve $C$ of degree $d$ with $p_1,...,p_n\in C$. Is there a ...
0
votes
0answers
9 views

VC-dimension for conjunctions with negations [migrated]

I need any hint in the following problem. Let $\mathcal{F}_k$ be a set of all possible conjunctions of binary variables $x_1, \dots, x_k$ and their negations. How could I prove that ...
5
votes
2answers
143 views

Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold

It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the ...
0
votes
0answers
25 views

Probability distribution [on hold]

I have 2 groups of elements, say A={a} (only 1 element) and B = {b1, b2,..., bn}. Now the take the probability of picking "a" is 0.3 and the probability of picking an element in B is 0.7. An element ...
12
votes
0answers
196 views

The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture. (The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
6
votes
1answer
101 views

When does the free loop space fibration split?

This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that. Let $X$ be ...
1
vote
0answers
41 views

Concentration and Correlation for Magnitudes of Gaussian Vectors

Suppose we have a large collection of standard normal random variables $a_i\in\mathbb{R}^n$. We know by standard concentration results that if we take $m \geq C\left(t/\epsilon\right)^2n$ samples, ...
2
votes
0answers
46 views

Generalized Hurwitz Spaces

In this question all the varieties are over $\mathbb{C}$. Classic Hurwitz spaces $\mathcal{H}_{g,r}$ are moduli spaces of simple branched coverings $f \colon X \to \mathbb{P}^1$ of degree $d$, where ...
5
votes
1answer
89 views

Does the property of being a local homeomorphism descend through split surjections?

Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps (between topological spaces). Assume these hypotheses: $f : X \to Y$ is a split surjection, i.e. has a section. $g \circ f : X \to Z$ is a ...
-4
votes
0answers
23 views

system of linear equations [on hold]

These are the two known equations (I2+I3)-(I1+I4)/(I1+I2+I3+I4) = 2x/L (I2+I4)-(I1+I3)/(I1+I2+I3+I4) = 2y/L where I know x,y,L values. How can I find the values of I1,I2,I3,I4?
0
votes
1answer
37 views

Hadamard Product and Eigendecomposition

I just found this related question in here Q1. Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary ...
1
vote
0answers
14 views

Oscillating Markovprocess Transition Probabilities

Suppose we have an irreducible positive-recurrent Markov process $\{X(t), t\geq0\}$ with generator $G$. Let $P(t)$ be its transition probability matrix and $\pi$ its stationary distribution. Then we ...
0
votes
0answers
14 views

Pseudo-braided fusion categories

A few definitions first, please replace with the standard terminology (and correct me if I confuse all the by-names of fusion categories :-) I call a complex number $z$ pseudo-cyclotomic if $|z|=1$. I ...
7
votes
0answers
106 views

How many Fréchet manifolds are there?

Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small. ...
1
vote
0answers
70 views

Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video. From 33:00 to 37:00, it is said that after changing of variables, ...
2
votes
0answers
54 views

Analytic continuation of intertwining operator

I was trying to understand the paper "Form of GL(2) from analaytic point of view", by Gelbart and Jacquet. On Page 226 in Remark (4.13) they mention that the kernel of the local intertwining operator ...
0
votes
0answers
52 views

canonical divisors of a resolution of a normal surface singularity

Let $(0\in X)$ be the germ of a normal surface singularity and let $f: Y \to X$ be the minimal resolution. Questions> (1) How can I define a map $f_*\mathcal{O}_Y(K_Y)\hookrightarrow ...
10
votes
1answer
257 views

Are symplectic methods used in (classical) Economics?

The tl;dr question is this: are economists using coordinate-free formulations in studying theory? Borrowing from classical mechanics, the framework I have in mind for classical economics--involving ...
2
votes
2answers
89 views

If $X$ is compact, is $[X]^2$ compact, too?

Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in ...
4
votes
1answer
112 views

Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ ...
-4
votes
0answers
20 views

log in calculating depreciation rate [on hold]

I can't understand log160=log2000+10log(1-r); 2.2041=3.3010+10log(1-r); 10log(1-r)=2.2041-3.3010; 10log(1-r)=-1.10969; ...
3
votes
0answers
117 views

Where can I find a proof of this result on optimal tessellation of a unit square?

Here is an excerpt from the paper "The Hexagon Theorem" by Donald J.Newman Does anyone know where I can find a proof of the underlined statement? Newman states it without a proof, and I could get ...
5
votes
1answer
122 views

Map between homotopy groups of O, related to J-homomorphism and K-theory of Z

Let $s \geq 0$ be fixed. The $J$-homomorphism includes $\pi_{8s+1}(SO) = \mathbb Z/2$ in $\pi_{8s+1}^s$, the $(8s+1)$-th stable homotopy group of spheres. Now regard $\pi_{8s+1}^s = \pi_{8s+1} ...
1
vote
0answers
27 views

Examples of non-extremal subfactors

Every subfactor $(N \subset M)$ in this post are supposed to be finite index inclusion of ${\rm II}_1$ factors. Definition (here p64): Such a subfactor is called extremal if $tr_{N'} = tr_M$ on ...
1
vote
1answer
59 views

finite generation of a certain type of subring

Let $k$ be a field, and let $R$ be a finitely generated $k$-algebra. (If it helps, you may assume $R$ is an integral domain.) Let $I$ be an ideal of finite colength. Note that $A:=k+I$ is a subring ...
0
votes
0answers
21 views

Which properties determine the uniqueness of the local Artin map?

Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...

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