**7**

votes

**1**answer

143 views

### Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G ...

**5**

votes

**0**answers

79 views

### What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory

I would like a reference/argument for the truth/falsity of the following statement:
The etale sheafification of the unramified Milnor-Witt K-theory (Nisnevich) sheaves are the (etale sheafification ...

**2**

votes

**0**answers

17 views

### Joint cumulants of $Z_2^n$ characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus ...

**33**

votes

**12**answers

3k views

### Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career,
collected their thoughts on mathematics (its aesthetic, purposes,
methods, etc.) and on the work of a mathematician in written ...

**0**

votes

**0**answers

44 views

### Factorial Sums over Compositions or ``Unlabeled Permutations"

Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$
In a divergent sum, the sequence
$$
a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i!
$$
frequently shows up and one ...

**0**

votes

**0**answers

30 views

### Systems of linear modular equations with unknowns in the moduli

I am interested in systems of linear modular equations, where the unknowns also appear in the moduli. The general form would be:
$A \vec{x}= \vec{b} \;\textrm{mod} \; (C \vec{x}+\vec{d})$
where A ...

**0**

votes

**0**answers

20 views

### References on Lorentzian geometry with non-vanishing torsion [on hold]

For my thesis I have to study Lorentzian geometry with non-vanishing torsion. Do you know any references on this? 'Riemannian geometry' with non-vanishing torsion will also be usefull.

**-3**

votes

**0**answers

44 views

### What are eigenbasis of binary brownian motion? [on hold]

I am seeking for a closed-form expression of eigenbasis for binary brownian motion sign(cumsum(randn(n,1))). Eventually, I need an associate fast transform for this ...

**4**

votes

**2**answers

132 views

### Differential structures and K-homology groups

What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...

**-6**

votes

**0**answers

138 views

### Formal generic fibre and Fermat's Last Theorem

Set $A_{n} \colon= {\Bbb F}_p[[S_1,...,S_n]]$ and
$A_{n,d} \colon= {\Bbb F}_p[[S_1,...,S_n]][[X_1,...,X_d]]$ be a $d$-variables formal power series ring over $A_n$. We denote by $K$ the fractional ...

**2**

votes

**0**answers

38 views

### Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...

**8**

votes

**1**answer

288 views

### Distribution of the number of prime factors

Count the number of prime factors of a number $n$
to include multiplicity,
so that
$$n=24=2^3 \cdot 3 = 2 \cdot 2 \cdot 2 \cdot 3$$
has $4$ prime factors, and
$$n =
6500 =
2^2 \cdot 5^3 \cdot 13 =
2 ...

**10**

votes

**1**answer

184 views

### What can we say about tropical maps $\mathbb{P}^1 \to A$ for an Abelian variety $A$?

It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus).
Is there any similar statement in the tropical case? Naively, the ...

**-2**

votes

**0**answers

43 views

### Is D646 a Boolean Algebra? [on hold]

I read here: How to recognize if a lattice is distributive? that N5 and M3 lattices are not distributive. So I concluded that these lattices are not Boolean Lattices (Correct me If I am wrong!)
My ...

**32**

votes

**4**answers

4k views

### Are dagger categories truly evil?

Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the ...

**2**

votes

**0**answers

80 views

### Which sheaves on a projective bundle are flat over the base scheme?

Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.
Are there any ...

**8**

votes

**1**answer

112 views

### Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...

**3**

votes

**1**answer

104 views

### In what sense is $\Omega_p$ universal?

In Chapter 3 of the book ''A Course in $p$-adic Analysis'' A.M Robert defines the field $\Omega_p$. He calls the field ''universal'' but doesn't show a universal property. I would have guessed that ...

**4**

votes

**1**answer

55 views

### Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
...

**2**

votes

**0**answers

58 views

### Inverse limits of schemes and open subsets

Let $R$ be a discrete valuation ring, $\{A_i\}_{i \in I}$ be a direct system of $R$-algebras and $A$ the limit of the system. Let $X$ be a noetherian projective scheme over $\mathrm{Spec}(R)$. ...

**0**

votes

**1**answer

66 views

### Boolean algebras and free filters generated by chains

Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the ...

**2**

votes

**0**answers

38 views

### Gonality and Clifford dimension of curves on a K3 surface

Let $X$ be a K3 surface. Let $L$ be an ample line bundle on $X$. When/how can we say that any smooth curve $C\in |L|$ has maximal gonality $k=[\frac{g+3}{2}]$ and Clifford dimension 1. Is there some ...

**4**

votes

**0**answers

136 views

### Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = ...

**11**

votes

**1**answer

229 views

### Generating function of the Thue-Morse sequence

Let $T$ be the generating function of the Thue-Morse sequence; thus,
$T(x)=x+x^2+x^4+x^7+\dotsb$. It is known that $T$ satisfies the nice
congruence
$$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 ...

**4**

votes

**1**answer

97 views

### Extensions of $\Bbb Z_3$ by $PGL(2,q)$ where $q$ is odd

Let $q$ be odd. If $G$ is a finite group such that $G$ has a normal subgroup $H$ of order $3$ such that $G/H\cong {\rm PGL}(2,q)$, what can we say about $G$. Is it true in general that $G\cong {\Bbb ...

**3**

votes

**0**answers

53 views

### Semi-continuity of the dimension of the null space

Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...

**0**

votes

**0**answers

46 views

### Nonnegative matrix times positive diagonal matrix's spectral bound

Assume A is a matrix. Denote $s(A)$ the spectral bound of A, which means the maximum real part of the eigenvalues and donate $r(A)$ the spectral radius. Now suppose $M$ is nonnegative $n\times n\ $ ...

**6**

votes

**0**answers

217 views

### Is Frac $\mathbb{Z}((x))$ Hilbertian?

Note that Frac $\mathbb{Z}((x)) \ne\mathbb{Q}((x))$.
As a result of Some questions about the ring Z((x)), we know that it is a Dedekind domain with uncountably many primes, each of which is of the ...

**2**

votes

**1**answer

201 views

### $dd^\mathbb{C}$-lemma on pair $(X,D)$

Let $X$ be a Kähler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kähler forms in the same Kähler class then have we $dd^\mathbb{C}$-lemma ...

**3**

votes

**1**answer

224 views

### Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?

http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least ...

**0**

votes

**0**answers

65 views

### Showing a permutation module is reducible [on hold]

CROSS POST
For a permutation module $V$, which is a permutation module if it has the basis $B = \{v_1,...,v_n\}$ such that the matrix of every $g \in G$ with respect to this basis is a permutation ...

**0**

votes

**1**answer

145 views

### A question on the integrability of eigenfunctions of the Laplacian

Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction,
$$\Delta u=-\lambda u.$$
I was wondering under what condition (for example, spaces ...

**-5**

votes

**0**answers

27 views

### Problem on Shell Method [on hold]

I've been on this problem for awhile and have no idea how to figure it out. Some help would be greatly appreciated.
Thank you! http://i.stack.imgur.com/I4dP7.png

**2**

votes

**1**answer

66 views

### Veronese embeddings and locally free resolutions

Let $i : \mathbf P^1 \to \mathbf P^2$ be the second Veronese embedding. Clearly, $i_\star \mathcal O_{\mathbf P^1}$ has a locally free resolution of the form
$$
0 \to \mathcal O_{\mathbf P^2} (-2) ...

**3**

votes

**0**answers

70 views

### Plurisubharmonic functions on Kähler manifolds, intuition?

As the question suggests, what is the intuition for working with plurisubharmonic functions on Kähler manifolds?

**2**

votes

**2**answers

164 views

### Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.
Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...

**2**

votes

**0**answers

64 views

### Which self-reference restrictions can be weakened in probabilstic logic?

This work suggests that there is some generalization of Truth in terms of probability, which can be definable within the logic itself.
Is where any other thorems on self-reference restrictions, which ...

**3**

votes

**1**answer

412 views

### What is the Euler characteristic of a mapping space?

Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for ...

**8**

votes

**1**answer

306 views

### Do the complex zeros of the sum/difference of these series all reside on the line $\Re(s)=\frac12$?

The following series seems convergent for all $s\in \mathbb{C}$:
$$\displaystyle f(s):=\sum_{n=1}^\infty \frac{(-1)^n}{(n+s)^{n+s}}$$
The function itself does not appear to have any real or complex ...

**-3**

votes

**0**answers

138 views

### Has this special functor a left and a right adjoint? [on hold]

I would like to know if there exists the left and the right adjoint functors of the functor : $ X \to \displaystyle \bigoplus_{ n \geq 0 } H^n ( X , \mathbb{Q} ) = \displaystyle \bigoplus_{ n \geq 0 } ...

**2**

votes

**1**answer

115 views

### A question about simple closed curves in 3-dimensional Euclidean space

Let E(3) be 3-dimensional Euclidean space. I have submitted the following question to Mathstackexchange and other mathematical websites, but have never received any responses-not even rejections on ...

**9**

votes

**2**answers

231 views

### Traces of operators in nuclear spaces

I am currently reading up on nuclear spaces in Yarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states:
Let $F$ be a nuclear ...

**-4**

votes

**0**answers

62 views

### Proof equation is of O(log(n)) [on hold]

I am following a course of CS and we are getting Big Oh Notation ( discrete math)
We have to proof certain equations are of O(n^2) etc
I can solve easy equations like 3N + 4 and (n +1)^2 = o(n^2).
...

**21**

votes

**2**answers

424 views

### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**-1**

votes

**0**answers

27 views

### Bayesian inference on gamma distribution

The likelihood of an observation $x$ under a gamma distribution is
$$L(x | \alpha, \beta) \propto \beta^\alpha x^{\alpha-1} \frac {\exp(-x\beta)} {\Gamma(\alpha)}$$
Suppose I have some observations ...

**5**

votes

**1**answer

353 views

### Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...

**7**

votes

**0**answers

133 views

### A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...

**-3**

votes

**0**answers

38 views

### Similarity estimation [on hold]

http://www.diku.dk/summer-school-2014/course-material/mikkel-thorup/bottomk-exercise.pdf Can somebody help with exercise 4 in chapter 2.2? Any hints would be highly appreciated.

**1**

vote

**0**answers

73 views

### When does $R [x]/I $ have infinitely many idempotents in special case?

At < When does $R [x]/I $ have infinitely many idempotents? >, Er_Ro asked the following question.
Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking ...

**1**

vote

**0**answers

150 views

### the first chern class of complex vector bundles [on hold]

Let $\xi^\mathbb{C}$ be a complex vector bundle over a manifold $M$ (or $CW$-complex $B$).
Case~1: $\xi^\mathbb{C}$ is a complex line bundle. Then the first Chern class
$c_1(\xi^\mathbb{C})$ is zero ...