# All Questions

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
80 views

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 ...
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?
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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 ...
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 $u \in W^{2,\infty}(\Omega)$? ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...