1
vote
0answers
30 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 $U(\mathfrak{g})$ ...
2
votes
0answers
47 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
36 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?
0
votes
0answers
17 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 ...
0
votes
0answers
6 views

Any polynomial-time algorithm for hypergraph bisection?

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
2answers
405 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 ...
0
votes
0answers
12 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 ...
3
votes
1answer
119 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 $c\not=a$? ...
-1
votes
0answers
7 views

Convolution between two exp RV

I have two exponential random variables, x~ exp(mu) y~ exp(lambda) and I'm looking for the density function of, z=x-y I'm trying to it by the Convolution form but I'm not sure about the way. ...
1
vote
1answer
42 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 ...
2
votes
1answer
104 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 ...
4
votes
0answers
161 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 ...
9
votes
2answers
326 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 ...
-4
votes
0answers
79 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 $|R|<\infty$. ($|S|<\infty$ ...
-5
votes
0answers
30 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?
-1
votes
0answers
21 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 ...
3
votes
1answer
137 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 ...
5
votes
0answers
98 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 ...
21
votes
4answers
1k views

Why should we care about “higher infinities” outside of set theory?

Let's say you are a prospective mathematician with some addled ideas about cardinality. If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :) If you ...
1
vote
2answers
100 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
1answer
189 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
226 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 ...
51
votes
39answers
6k views

Important formulas in Combinatorics

Motivation: The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...
3
votes
1answer
51 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 ...
6
votes
2answers
216 views

For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature? An obvious case is when $Y$ ...
4
votes
0answers
136 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 ...
26
votes
3answers
529 views

The coupon collector's earworm

I thank Nicolas Dupont for the following question (and for permission to disseminate it further): I have a playlist with, say, $N$ pieces of music. While using the shuffle option (each such ...
1
vote
0answers
31 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 ...
4
votes
2answers
285 views

Bateman-Horn, continued even further

As before, consider the "singular series", which shows up in the Bateman-Horn conjecture: for an irreducible polynomial $f,$ this is equal to $$ s(f) = \prod_p \frac{1-\frac{n_f(p)}p}{1-\frac1p}, $$ ...
3
votes
1answer
118 views

On the search for an explicit form of a particular integral

Let $f$ be integrable over the interval $(0, 1)$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
9
votes
1answer
253 views

Is an irreducible ideal in $R$ also irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
4
votes
0answers
85 views

Vanishing natural transformation and strong generator

Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a given finite number ...
-4
votes
0answers
72 views

Non-standard numbers and exponential form of Zeta function

Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...
1
vote
0answers
62 views

Properties of Coefficients of Order Polynomials

I am working on a problem involving determining the order polynomial $\Omega_P(k)$ of a partial order $P$, which counts the number of order-preserving transformations/maps from $P$ to the $k$-chain ...
3
votes
1answer
132 views

Derivatives of theta functions at zero

Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions $$ \theta_s(z;\tau)_k=\sum_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, ...
11
votes
1answer
337 views

How big is the lattice of all functions?

Define the lattice $(\mathcal{L},\prec)$ as the set of all function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ ...
12
votes
1answer
187 views

Which sequential colimits commute with pullbacks in the category of topological spaces?

This question was asked on math.stackexchange.com without a reaction. Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ ...
4
votes
2answers
270 views

Riemannian metrics preserved by diffeomorphisms

Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$? Note that ...
36
votes
3answers
961 views

Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?

Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that $$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$ for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the ...
4
votes
0answers
134 views

object in D^bCoh(P^2) with prescribed RHom's

Let $\mathbb{C}P^2$ denote the projective plane. From reading the section of http://homepages.math.uic.edu/~coskun/gokova.pdf which surveys Gieseker stable sheaves, I have understood that there are ...
14
votes
2answers
474 views

Matrix equation $XAXBXC=I$

Let $A,B,C$ be unitary matrices. Does there always exist a unitary matrix $X$ such that $$(XA)(XB)(XC)=I,$$ where $I$ is the identity matrix? The quadratic equation $(XA)(XB)=I$ has the solution ...
1
vote
0answers
77 views

“GraphI Individualization” referece request [on hold]

Context: I am studying Weisfeiler Lehman method(WL method) and have clear idea about 1 and 2 dimensional WL method. I was wondering about the individualization process described below- Defined ...
9
votes
2answers
193 views

A generalization of Chebyshev polynomials

What is the monic polynomial $p(x)$ of degree $n$ which minimizes $\max_{x \in [-1,1]} |p(x)|$? The answer is the Chebyshev polynomial, and its largest value on $[-1,1]$ is $1/2^{n-1}$. Now suppose ...
2
votes
1answer
273 views

Quadratic Diophantine equation in $\mathbb Z[T]$

I am trying to solve the following quadratic diophantine equation in $\mathbb Z[T]$: $$((T+1)X+TY-1-Z)((T+1)X+TY-1+Z)=24XY$$ One has the following trivial solutions: $(X,Y,Z)=(0,Y,\pm(1-TY))$, ...
-1
votes
0answers
56 views

A finite field $0$-free restrictive sumset problem

Fix $m\in\Bbb N$. Let $\Bbb F_2^{\big\lceil\log^d m\big\rceil}$ be $\Big\lceil\log^dm\Big\rceil$-dimensional vector space over $\Bbb F_2$. What is minimum $d$ (if it exists - need sufficient ...
6
votes
1answer
240 views

Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$?

Let $k$ be an algebraically closed field of characteristic $p>0$. All the examples of non-smooth algebraic group schemes over $k$ that I have seen (apart from "artificial" examples; see below) have ...
488
votes
199answers
124k views

Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
3
votes
1answer
185 views

coproduct of the homology of iterated loop space on spheres

Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...
3
votes
1answer
125 views

Paper of Boutot-Carayol in `Courbes modulaires et courbes de Shimura'

I am trying to obtain a copy of the following J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfel'd , Astérisque No. 196-197 ...
8
votes
1answer
161 views

Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...

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