0
votes
1answer
428 views

Maximal score for the 2048 game [duplicate]

t's been weeks (months?) since the 2048 game--by Gabriele Cirulli--took Internet by storm. I have an explicit integer $X$ which is greater or equal than any score of this game. Possibly my $X$ is the ...
1
vote
0answers
84 views

Families of curves with “almost-general” moduli

The Brill-Noether theorem says that, if $\rho(d, g, r) := (r + 1)d - rg - r(r + 1) \geq 0$, then there exists a unique component of the Hilbert scheme of curves of degree $d$ and genus $g$ in ...
0
votes
0answers
7 views

Recursively calculate Tikhonov regularizer in b-spline objective function

I'm trying to write a program to calculate cubic b-spline based on set of inputs. But I can't figure out how to calculate value of Tikhonov regularizer. My b-spline function is this: I have ...
3
votes
0answers
41 views

Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
4
votes
1answer
222 views

Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...
1
vote
0answers
50 views

coxeter element of a reflection group (reference request)

I am reading reflection groups and coxeter groups book by Humphreys. now I want to learn more about "coxeter element" of a reflection group. Can any body suggests me some good books to read more about ...
1
vote
0answers
124 views

Continuous dependence of the roots of a polynomial on its coefficients

In their article "The roots of a polynomial vary continuously as a function of the coefficients" Gary Harris and Clyde Martin give a topological proof of the well-known theorem that the roots of a ...
3
votes
0answers
46 views

Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets?

Is the following statement true, and if it is, does someone have a reference? Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, ...
2
votes
1answer
80 views

Fractional Brownian motion via Hilbert space

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms: Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...
0
votes
1answer
97 views

Open subset of the moduli space of stable sheaves on a noetherian scheme

This is my question: Given a projective noetherian scheme $X$, the structural sheaf $\mathcal{O}_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable ...
3
votes
1answer
323 views

How to prove a Proposition of Rouquier?

Proposition 7.15 in Rouquier's paper (see Publication paper) "Dimension of triangulated categories J. K-theory, 1 (2008) 193-256" as follows, for details please see Rouquier's paper (or arXiv): ``Let ...
3
votes
0answers
215 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
2answers
216 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
0answers
20 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
1answer
719 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
0answers
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
1answer
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
0answers
144 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
2answers
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
0answers
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
3answers
334 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
0answers
56 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
1answer
145 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
0answers
50 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
1answer
256 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
1answer
104 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
0answers
109 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
1answer
73 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
1answer
89 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 ...
2
votes
3answers
215 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
1answer
316 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
1answer
201 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
0answers
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
2answers
137 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
0answers
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
0answers
93 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
0answers
319 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
4answers
828 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
0answers
62 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
0answers
52 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
1answer
66 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
2answers
57 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
1answer
48 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
2answers
181 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
1answer
221 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
0answers
66 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
0answers
74 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
1answer
110 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
1answer
205 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
1answer
78 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 ...

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