# All Questions

**3**

votes

**0**answers

220 views

### Is it possible to find explicit formula for the product $\prod_{d|n,\ d>1} (1-\mu(d)/\varphi(d))^{\varphi(d)}$?

I am trying to calculate the following product
$$
\prod_{d|n}_{d>1} \left( 1-\frac{\mu(d)}{\varphi(d)} \right)^{\varphi(d)}
$$
where the functions $\varphi$ and $\mu$ are Euler's totient and ...

**3**

votes

**2**answers

217 views

### Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$

I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, i.e. at the coordinate axes. The ...

**0**

votes

**0**answers

22 views

### Apply 'splitting' in regenerating codes in locally repairable codes

In network coding for distributed storage system, regenerating codes are known to exist only in theory due to its complex implementation in storage system. In the codes, 'splitting' is applied in the ...

**19**

votes

**1**answer

743 views

### A conjectured formula for Apéry numbers

A conjecture by the late Romanian mathematician Alexandru Lupas.
Posted in sci.math in 2005, but no proof was found.
Physicist Alan Sokal just reminded me of it, saying it was related to something he ...

**0**

votes

**0**answers

70 views

### Action of the (special) orthogonal group on differential forms

I was told that the following facts are true. I am looking for a reference to them.
1) The action of $O(n,\mathbb{C})$ on $\wedge^l\mathbb{C}^n$ is irreducible for any $l$.
2) The action of ...

**1**

vote

**1**answer

44 views

### invertible unbounded linear maps defined on a Hilbert space

It is well-known that, assuming the axxiom of choice, there are unbounded linear maps defined not only on a dense subset but on all of Hilbert space. Is it possible that such a map is invertible?

**2**

votes

**0**answers

149 views

### Bott's Formula for Grassmannians

Bott's Formula gives the dimension of the cohomology $H^{q}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{p}(k))$ of the $k$-twisted sheaf of $p$-differential forms on the projective space ...

**0**

votes

**2**answers

109 views

### Terminology for beads/necklace/bracelet problem [on hold]

I'm new to mathoverflow but hopefully anyone here can point me in the right direction.
The problem is as follows, imagine you have 4 beads, lets give them numbers 1,2,3,4. Now I want the unique ...

**0**

votes

**0**answers

79 views

### How many possibilities would you have in an android lock pattern, always using all 9 moves? [migrated]

We are doing some research and wanted to know how many possibilities you would have if you would use all 9 dots/options in an (android) swipe lock pattern.
What would the formula be to get to this ...

**5**

votes

**3**answers

343 views

### In the category of sets epimorphisms are surjective - Constructive Proof?

The statement that surjective maps are epimorphisms in the category of sets can be shown in a constructive way.
What about the inverse?
Is it possible to show that every epimorphism in the category ...

**-6**

votes

**0**answers

57 views

### Let (A,≼A) and (B,≼B) be partially ordered sets. Define C = A×B and define the relation ≼ on C by (a,b)≼(a′,b′) ⇐⇒ (a≼A a′)∧(b≼B b′) [on hold]

Let (A,≼A) and (B,≼B) be partially ordered sets.
Define C = A×B and define the relation ≼' on C by (a,b)≼'(a′,b′) ⇐⇒ (a≼A a′)∧(b≼B b′).
(a) Prove that ≼' is a partial order on C.
(b) Prove that if ...

**4**

votes

**1**answer

147 views

### Endomorphisms of a maximal ideal of a local ring

Let $R$ be a commutative local ring with maximal ideal $\mathfrak{m}$. Is it true in general that $\text{Hom}_R(\mathfrak{m},\mathfrak{m})\cong \text{Hom}_R(\mathfrak{m}, R)$? What if the Krull ...

**-2**

votes

**0**answers

52 views

### Convergence of empirical random variable

Let $X$ be a RV on the real line, of probability measure $P_X$, and let $X_n$ for $n=1,...,N$ be an iid sample from $P_X$.
The Glivenko-Cantelli theorem says that the empirical measure, $P_N$, ...

**6**

votes

**1**answer

301 views

### Integer valued polynomial through some points with rational coordinates

I asked this question on MSE about 5 months ago, but, even after offering a bounty, I didn't receive any answer, I hope this question isn't too easy for MO.
If we have a set of points $(x_i,y_i)$ ...

**3**

votes

**1**answer

105 views

### A query about Hatcher flow on arc complex

In the paper "Triangulations of Surfaces" Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes ...

**5**

votes

**0**answers

111 views

### Applications of anabelian geometry to Galois representations?

One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi_1(X_\overline K)$, where $X$ is a (anabelian) variety and ...

**2**

votes

**1**answer

74 views

### What is the minimal girth of a cayley graph for Alt(n) in which the girth relator is not a proper power?

First, a definition: a girth relator for a Cayley graph is a word that you get by reading the edge labels along a shortest loop. The girth relator is not usually unique, though it is often unique up ...

**0**

votes

**1**answer

90 views

### Dimension of binary motives of a quadric

Let $Q$ be a anisotropic quadric of dimension $d$ over $k$.
We work in the category of effective Chow-Motives over $k$.
Let $T$ be the Tate-Motive.
For a motive $M$ we write $M(l)$ for its $l$-th ...

**3**

votes

**3**answers

231 views

### Generic absence of non-trivial first integrals of geodesic flows

Is it true that given a smooth manifold M (with or without boundary), a "generic" metric g on M does not possess any non-trivial (non-constant) first integral for the geodesic flow induced by g on the ...

**8**

votes

**1**answer

319 views

### Notion of infinity in categories

Please excuse me if the question is too vague or uninteresting.
Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. Motivated by the equivalence of Dedekind-finiteness and finiteness ...

**1**

vote

**1**answer

204 views

### pontryagin dual and maps between spectra

Given two spectra $A$ and $B$, the set $[A,B]$ of homotopy classes of maps from $A$ to $B$ forms an abelian group. Can the dual abelian group $\text{Hom}([A,B],\mathbb{Q}/\mathbb{Z})$ be expressed as ...

**3**

votes

**0**answers

51 views

### Prove that the maximizing point configuration on the unit circle for a Vandermonde like functional is a picket fence

This is probably too hard for math.stackexchange, so I migrated it here.
For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | ...

**4**

votes

**2**answers

139 views

### Build a topological polytope with a specified CW-structure

I am a topologist and not quite familiar with the tools for building a polytope. I would like to build some topological polytope which is an somewhere in between permutohedron and associahedron which ...

**-3**

votes

**0**answers

39 views

### Recurrence relation of degree 4 [on hold]

how to get the general solution of a recurrence relation degree 4 having repeated root r=2; for a characteristic equation of r^4-8r^3+24r^2-32r+16=0 where a(0)=1, q(1)=4, a(2)=44, a(3)=272.2?

**2**

votes

**0**answers

101 views

### Can the commuting condition in Jordan-Chevalley decomposition be replaced with this global criterion?

Let $G$ be a reductive linear algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra.
The Jordan-Chevalley ...

**6**

votes

**0**answers

327 views

### A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{trivial}, \mathcal T_{discrete}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$\mathcal ...

**4**

votes

**4**answers

847 views

### How short can we state the Axiom of Choice?

How short can we state a principle which is equivalent with the Axiom of Choice under $ZF$? The principle should be a sentence in the language of set theory with only $\in$ and$=$ as extralogical ...

**-5**

votes

**0**answers

64 views

### Squares in a square grid [on hold]

How many $n$ square $a \times a$ permutation are in a grid $b \times b$? (They must never overap themselves)
The question seems simple, but it isn't at all.

**1**

vote

**0**answers

54 views

### Pseudomodules, “general coherence theorem”

A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...

**-3**

votes

**1**answer

69 views

### How to find a modular multiplicative inverse when GCD is not 1 [on hold]

I am working on a problem that requires finding a multiplicative inverse of two numbers, but my algorithm is failing for a very simple reason: the GCD of the two numbers isn't 1. I figured I must've ...

**0**

votes

**2**answers

76 views

### a dcpo seen as a category: when does a dcpo map induce a functor with an adjoint?

Take two posets $A, B$ (partially ordered sets). Now consider these posets to be categories $Cat(A), Cat(B)$ respectively. Consider a map from $A$ to $B$, $f: A \rightarrow B$. This can be seen as ...

**1**

vote

**1**answer

50 views

### W^{2,∞} regularity of solutions of Poisson's equation if the right hand side is in L^{∞}

Let $u$ be solution of $-\Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial n} = 0$ on $\partial \Omega$.
Is it true that if $f \in L^{\infty}(\Omega)$ then $u \in W^{2,\infty}(\Omega)$?
...

**2**

votes

**2**answers

186 views

### Are there fragments of set theory which are axiomatized with only bounded (restricted) quantifiers used in axioms?

Bounded quantifiers in set theory are represented as $(\forall x \in S)$ or $(\exists x \in S)$. But since the modifier "bounded" brings up an association with "bounded variable", I prefer the term ...

**6**

votes

**1**answer

223 views

### Cohen-Lenstra heuristics for totally complex fields

If a number field $K$ is a Galois extension of $\mathbb{Q}$, and $G = \operatorname{Gal}(K/\mathbb{Q})$, then the class group of $K$ is a $\mathbb{Z}[G]$-module, and since $N = \sum_{g \in G} g$ acts ...

**1**

vote

**0**answers

70 views

### Symmetric spaces which are compact modulo the unipotent radical are compact

Is the following true?
Let $X = G/H$ be a symmetric space of a reductive group over a p-adic field $F$.
Let $X^0$ be an open orbit w.r.t. the action of the minimal parabolic $B$ of $G$.
Let $U$ be ...

**3**

votes

**0**answers

76 views

### Full-rank factorization of the graph Laplacian

Is there a combinatorially meaningful full-rank factorization of the Laplacian matrix of a graph?
The usual factorization $L=BB^{T}$, where $B$ is an oriented incidence matrix, is full-rank if and ...

**1**

vote

**1**answer

121 views

### Directed subposet of a poset containing the minimal elements

The following appears naturally in a certain context:
Let $P$ be a partially ordered set which is bounded below in the sense that for each $x\in P$ there is a minimal element $m$ with $m\leq x$. Let ...

**5**

votes

**1**answer

208 views

### Kernel of the character of congruence groups

Let $\Gamma$ be a congruence subgroup of $SL_2(\mathbb Z)$ and $\chi:\Gamma \to \mathbb{C}^*$ a character of $\Gamma$ with finite image. Is then $\ker(\chi)$ also a congruence group? If not, can ...

**2**

votes

**1**answer

79 views

### Does the right adjoint of the category of simplices functor is “homotopicaly inverse” to the category of simplices functor?

Short Version (the question)
Let $\text{Cat}$ be the category of (small) categories and $\text{sSet}$ the category of simplicial sets. There is a functor $\Gamma:\text{sSet}\to \text{Cat}$ that takes ...

**1**

vote

**0**answers

101 views

### A differential equation on toric Kahler manifolds

Let $M$ be a toric Kahler manifold with $\text{dim}_{\mathbb{R}} = 4$. Let $V$ be a Killing vector field associated to the action of the two-torus $\mathbb{T}^2$ on $M$. We also assume the existence ...

**0**

votes

**0**answers

57 views

### Algebraic subgroup lattices [closed]

For which groups is the subgroup lattice algebraic?
Jiří Tůma has proved every algebraic lattice is an interval in a subgroup lattice. It seems there is close relation between algebraic lattices and ...

**2**

votes

**0**answers

58 views

### Highest weight spaces in arbitrary representations?

An isotypic (maybe reducible) representation V of GL(V) may be represented by its highest weight subspace HW(V). We have dim HW(V) equal to the multiplicity of the irreducible representation inside V ...

**9**

votes

**2**answers

231 views

### Faster formula to compute sum over partitions

Let $f$ be a function from the positive integers to the real numbers (or some ring...). Let
$$(\star) \quad F(n) = \sum_{n_1 \leq \cdots \leq n_j\atop n_1 + \cdots + n_j = n} f(n_1) \cdots f(n_j), $$
...

**5**

votes

**1**answer

171 views

### How to solve such an optimization problem

I encounter the following optimization problem, but I can't solve it.
Given $N$ variables satisfying $0 \leq x_1 < x_2 < x_3 < ... < x_N \leq 1$ and an integer $K$ no large than $N$, find ...

**0**

votes

**0**answers

15 views

### Can Cavity method to analyze graph with loops that are short?

In statistical physics,Cavity method can be regarded as a generalization of the Bethe Peierls iterative method in tree-like graphs to the case of graph with loops that are not too short. And I want ...

**0**

votes

**0**answers

48 views

### Unique Limits in T1 Spaces [closed]

It's intuitive to me that limits in T2 (Hausdorff) spaces are unique, but I'm not sure about what properties of T1 spaces allows limits to be non-unique. Could someone explain this to me and perhaps ...

**-2**

votes

**0**answers

21 views

### Can a very bad Coefficient of determination (R squared value) not be indicative of model performance? [migrated]

Thanks in advance for the advice.
I am trying to build a generalized linear model that has many predictors. The R squared value of the model is quite low (.21), but when I use the model to predict ...

**11**

votes

**1**answer

296 views

### Is there an arXiv for Beamer presentations of scientific work? [closed]

When I give a "Beamer talk" I put in a lot of effort making the slides, and trying to give an efficient presentation of my work. The end product is often around 20 pages of figures, definitions, ...

**3**

votes

**0**answers

88 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**3**

votes

**1**answer

143 views

### Linear dependency of real numbers with integer coefficients adding up to zero [on hold]

Let $x = (x_1, \dots, x_n)$ be a vector of real number. I was asking myself if there was an efficient way of telling whether there exists a non-zero vector of integers $z \in \mathbb Z$ such that both
...