0
votes
0answers
5 views

Estimating the shift in the spectral radius of a matrix under a diagonal perturbation

Given a matrix $A$ and a diagonal matrix $D$, what ways do we have to estimate, $\lambda_{max}(A+D) - \lambda_{max}(A)$? (Feel free to make other assumptions about the matrices that they are all ...
0
votes
0answers
4 views

largest subgroup of $Out(\hat{F_2})$ which preserves the Nielsen invariant

Let $x,y$ be generators for the free group $F_2$. It's known that $Aut(F_2)$, and hence $Out(F_2)$ preserves the conjugacy class of the subgroup $\langle[x,y]\rangle$ generated by $[x,y]$ (This ...
4
votes
1answer
65 views

Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help. In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...
0
votes
0answers
16 views

Intercahging the tensor product with infinite product

Let $R$ be a $k$-algebra (non necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested on the class of objects $V$ of $\mathbf{D}(R^{op})$ having the ...
12
votes
1answer
141 views

Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb{R}^3$ simply connected? Reading around I found plenty of articles discussing the path connectedness $\mathbb{R}^2 \setminus \mathbb{Q}^2$ ...
1
vote
0answers
49 views

Statements which are logically equivalent with P vs. NP

$P$ vs. $NP$ is the open problem which asks, roughly, whether every statement whose solution can be quickly verified by a turing machine with efficient bounds can also be quickly solved by a turing ...
0
votes
0answers
24 views

What is number of faces in a k-ary n-dim cube? [on hold]

What is the number of $(n-r)$ dim faces for a $k$-ary $n$-dim cube ? Definition of k-ary cube: In a $k$- ary $n$- cube , each node is identified by an $n$-bit base-$k$ address $b_{n − ...
-2
votes
0answers
31 views

Prime ideals decomposition [on hold]

How to prove the decomposition law for prime ideals in finite separable extensions of number fields? How we use the "conductor condition"?
1
vote
0answers
26 views

Doubling theorem for Alexandrov spaces

Is there a user friendly exposition of the notion of boundary of an Alexandrov space with curvature bounded from below and of the Doubling theorem? The only reference I am aware of is the original ...
2
votes
0answers
34 views

Roller's problem on median groups

At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...
4
votes
1answer
65 views

Time for brownian motion to cross a coordinate plane

Can I get a reference or some insight into the following? Suppose a particle moves by Brownian motion, starting from a point $P$ in $\mathbf{R}^{n}$. What can we say about the distribution of the ...
1
vote
0answers
30 views

Approximating a divergent integral with modified Bessel functions of the first and second kinds

I am a physicist who needs to evaluate the following (divergent at the origin) integral involving the modified Bessel functions of the first and second kinds $I = \int_0^{\infty} \frac{\cos(ax)}{x} ...
6
votes
1answer
123 views

Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?

I'm asking a question about Lie group representation. Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...
-1
votes
0answers
18 views

Any polynomial-time algorithm for hypergraph bisection? [on hold]

I work with hypergraph partitioning. I want to divide a complete weighted hypergraph into 2 parts using cut-net metric, a sum of all edges cut, and connectivity metric. Is there a polynomial-time ...
5
votes
0answers
103 views

Pointfree probability theory

I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like pointfree topology, where one basically replaces topological spaces by their locales ...
0
votes
0answers
93 views

A finite group with O_{p}(G)=1

Let $G$ be a finite group of order $p(p^2-1)/2$, where $p$ is prime number. If $O_{p}(G)=1$, then what is the number of Sylow $p$-subgroups G?
1
vote
1answer
100 views

Notation: $Sigma$ and $Pi$ of intersections

In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly. For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...
0
votes
0answers
18 views

Algorithm for 2-coloring classes of 3-uniform hypergraphs

Hi all and thanks in advance for your efforts. I'm interested in 2-coloring 3-uniform hypergraphs. I know that in general, the problem of deciding if a 3-uniform hypergraph is 2-colorable is ...
1
vote
0answers
40 views

What characteristic of a graph depend on the vertex labeling?

Different labeling on a graph produces class of isomorphic graphs. Two isomorphic graphs possess similar characteristic such as connectivity, degree distribution of vertices, equality of spectrum and ...
1
vote
1answer
55 views

Estimate $\sup_n\left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|$

Let $\mu$ be a singular Borel probability measure on $[0, 1)$, and $f\in L^2(\mu)$. Estimate $$\sup_n\left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|$$ where $\alpha_n\in\mathbb R$ and ...
-1
votes
0answers
36 views

About perturbation of spectral radius of a matrix because of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (you can assume the diagonal matrices to be such that ...
4
votes
2answers
170 views

cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...
-5
votes
0answers
36 views

Function that outputs only 1 or 0 depending on sign of variable? [on hold]

Is there a single variable (preferably simple) function which equals 0 for any positive input and 1 for any negative input, or vice versa?
5
votes
1answer
267 views

A question on Ramanujan's $1/\pi$ formulas

It is known that Ramanujan discovered a number of formulas fo $1/\pi$. All of these formulas are of the form $$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(1/2)_n(s)_n(1-s)_n}{(1)_n^3}(a+bn)z^n,$$where ...
5
votes
0answers
117 views

definition of “immersion” of schemes (without open or closed)

On Prop. 1.7 (a) on page 5 of Milne's Etale Cohomology book states: Any immersion is quasi-finite. A google search turned up definitions for "open immersion" and "closed immersion", never just ...
-4
votes
0answers
83 views

Prove $R$ is a finite ring [on hold]

Suppose that $R$ is a ring and $D(R)$ is the set of all elements $x\in R$ such that there exist $b,c\neq 0$ with $xb=cx=0$. Prove that if $1<|D(R)|<\infty$ then $|R|<\infty$. ($|S|<\infty$ ...
3
votes
1answer
143 views

The structure map of topological K-theory

This may be a silly question but I don't know the answer. I know the construction of (equivariant) K-spectrum $KU_G$ and the periodicity of (equivariant) K-theory. But I don't know its structure maps ...
0
votes
0answers
35 views

Choosing the weights of a Voronoi diagram — is this function always the gradient of another function?

This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = ...
-1
votes
0answers
26 views

Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane

We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$. What is ...
8
votes
3answers
576 views

May integration spoil real-analyticity?

Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function $$ g:(a,b)\to\mathbb{R},\qquad ...
3
votes
0answers
98 views

Commuting diagram, algebraic cycles and K-theory

What is the easiest way to see the veracity of the following commutative diagram?$$\require{AMScd} \begin{CD} K(X) \otimes K(Y) @>\text{ch}(-) \otimes \text{ch}(-)>> A(X, \mathbb{Q}) \otimes ...
4
votes
1answer
147 views

Improper integral $\int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx$ with $-a$ and $b$ positive

Is the following function real analytic in $t>0$: $$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$ where $-a$ and $b$ are positive, and $c\not=a$? ...
9
votes
2answers
358 views

Learning the exponents in a sum of two modular roots of unity

$\newcommand{\Z}{\mathbb{Z}}$ Suppose that $n$ is a large and known integer (say, with 100 digits) and that you are given access to a function $$f(x) = x^a + x^b$$ with unknown exponents $a,b \in ...
1
vote
0answers
70 views

Rank of the Jacobian of a family of hyperelliptic curves of genus 2

Assume tha $C$ be the hyperelliptic curve $y^2 = (x-a_1)\cdots (x-a_5)$ of genus $g=2$ and $a_i \in \mathbb{Z}$ and we know that the integers $a_i$ has the form $a_i= d_1^2 - d_i^2$ for some positive ...
0
votes
1answer
196 views

Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?

$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times. I am studying some function arising from symplectic geometry which happens in my case to be ...
2
votes
2answers
58 views

Basic question about discrete minimal surfaces

Let $P$ be a convex polygon with $n > 3$ vertices $v_1, \ldots, v_n \in \mathbb{R}^2$, let $x$ be a point in the interior of $P$, and let $u$ be a function with prescribed values at the vertices of ...
7
votes
0answers
134 views

Tangent space of Hilbert scheme

We have the following theorem: Let $X$ be a projective scheme over an algebraically closed field $k$, and $Y \subset X$ a closed subscheme with Hilbert polynomial $P$. Then$$T_{[Y]}\text{Hilb}_P (X) ...
1
vote
0answers
36 views

Avoiding the range of a bivariate integer function or Diophantine function

I have a bivariate integer function $f(x,y)=5+23x+7y+30xy$ where $x,y \geq 0$ and are integers. The lattice points of this function, or its range contain a large number of values. I'm trying to see if ...
1
vote
2answers
113 views

Automorphism group of the affine groups AGL(n,q), ASL(n,q)

I have a question. The automorphism group of the linear groups $GL(n,q)$, the group of linear transformations of $V = \mathbb{F}_q^n$, and $SL(n,q)$, the subgroup of $GL(n,q)$ consisting of elements ...
0
votes
0answers
144 views

How do mathematicians find the underlying idea? [on hold]

While reading through the lecture notes here (http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week2.pdf , page 22, last paragraph), I came across the following " Thus there must be some ...
2
votes
2answers
242 views

Trivial zeroes of the Riemann Zeta function are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...
3
votes
1answer
54 views

Number of linearly bisected subsets in finite vector space $F_2^n$

We consider the $n$-dimenstional finite vector space $\mathbb{F}_2^{n}$ over the finite field of two elements. For a subset $A\subseteq \mathbb{F}_2^{n}$ of even size $|A|=2m$ and a linear form ...
0
votes
0answers
27 views

Bounding Expected Value of a piecewise function [on hold]

Let X and Y to be two independent random variables with known pdfs. Get a bound for the expected value of the following expression in terms of $E[X]$, $E[Y]$, VAR[X] and VAR[Y]: \begin{equation} ...
2
votes
0answers
27 views

Changing the sign in the definition of the cocommutator of a coboundary Lie bialgebra

A Lie bialgebra is a Lie algebra additionally equipped with a 1-cocycle $\delta: {\mathfrak g}\to \Lambda^2 {\mathfrak g}$ that satisfies the co-Jacobi identity. Non-trivial Lie bialgebras can be ...
2
votes
0answers
80 views

What are the restrictions in the ramification behavior of a Galois extension of number fields imposed by the Galois group of the extension?

Studying class field theory, I have come across the following Proposition: Proposition. Let $K/E$ be an extension of number fields so that there is no nontrivial unramified subextension $F/E$ with ...
1
vote
0answers
48 views

A compact Alexandrov space with curvature bounded below has curvature bouneded above?

For a compact Riemannian manifold, Since the curvature tensor is continuous, we know that the sectional curvature is bounded, i.e. bounded above and below. Now let $M$ be a compact Alexandrov space ...
3
votes
0answers
80 views

Understanding Segal's definition of conformal field theory

I have a fundamental problem in understanding Segal's definition of a conformal field theory: On the one hand his monoidal CFT-functor is a formalization of the fact that, physically, the integrand ...
4
votes
1answer
183 views

$Ext$-algebra generated by $Hom$ and $Ext^1$ as $A_\infty$-algebra?

In [Keller: A-infinity algebras in representation theory, Proposition 1(b)], Keller states that for an associative algebra the $Ext$-algebra of the simples is generated by $Ext^1(S,S)$ as an ...
0
votes
0answers
13 views

MLE of Gamma when only given observations [on hold]

i've been given 10 observations of X, where X is a random variable. the observations are 141 16 46 40 351 259 317 1511 107 567 and now given they are gamma distributed, how do you find the MLE using ...
-2
votes
0answers
101 views

Elementary question of Group cohomology [on hold]

Let $G$ be a finite group. Assume $G$ acts on finite abelian module $M$ such that $(|G|,|M|)=1$. Question: Why $H^i(G,M) = 0$ for $i > 0$? Pierre MATSUMI

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