# Volume 02 (1998) No. 1

## Volume 02 (1998) No. 1

### Configuration spaces

Samuel Gitler

**Abstract:**

This article is intended as a brief introduction to the theory of configuration spaces as well as to some of the basic techniques used in Algebraic Topology. Some recent results about loop spaces of configuration spaces are also presented at the end.

### A conjecture on cycle-pancyclism in tournaments

Hortensia Galeana-Sánchez and Sergio Rajsbaum

**Abstract:**

Let $T$ be a Hamiltonian tournament with $n$ vertices and $\gamma$ a Hamiltonian cycle of $T$. In previous works we introduced and studied the concept of cycle--pancyclism to capture the following question: What is the maximum intersection with $\gamma$ of a cycle of length $k$? More precisely, for a cycle $C_k$ of length $k$ in $T$ we denote ${\cal I}_{\gamma} (C_k)=|A(\gamma )\cap A(C_k) |$, the number of arcs that $\gamma$ and $C_k$ have in common. Let $f(k,T,\gamma )=\max\{ {\cal I}_{\gamma}(C_k)|C_k\subset T\}$ and $f(n,k)=\min\{ f(k,T,\gamma )|T$ is a Hamiltonian tournament with $n$ vertices, and $\gamma$ a Hamiltonian cycle of $T\}$. In previous papers we gave a characterization of $f(n,k)$. In particular,

the characterization implies that $f(n,k)\geq k-4$. The purpose of this paper is to conjecture that for any vertex $v$ there exists a cycle of length $k$ containing $v$ with $f(n,k)$ arcs in common with $\gamma$. We present various particular cases in which this equality holds.

### Unique factorization in cartesian products

Eduardo Santillán

**Abstract:**

In this paper we partially answer the following question. "If the topological space $H$ can be decomposed as the Cartesian product $H= A \times X$, when is this factorization unique?"

### Ergodic decomposition of Markov operators on signed measures

César E. Villarreal

**Abstract:**

Let $X$ be a Polish space, and let $M_{\Sigma}$ be the Banach space of finite signed measures on the Borel ${\sigma}$-algebra $\Sigma$ of $X$. Given a quasi-constrictive Markov operator $P^*: M_{\Sigma} \rightarrow M_{\Sigma}$, we use the spectral decomposition of $P^*$ to determine the set of $P^*$-invariant distributions in $M_{\Sigma}$ and the set of $P^*$-ergodic distributions.

### Movimiento Browniano fraccionario

José Villa Morales

**Abstract:**

Se demostrará que solamente tiene sentido definir un Movimiento Browniano Fraccionario con espacio paramétrico $T$ si y sólo si $T$ es un espacio con producto interno. Además, se dará una representación integral de este proceso.

### Intercalate matrices and algebraic varieties

Francisco Javier Zaragoza Martínez

**Abstract:**

We give a characterization of intercalate matrices as an algebraic variety over a finite field. We also prove Yuzvinsky's conjecture on the minimum number of colors in an intercalate matrix for matrices with 5 or less rows. Finally, we obtain a set of 37 cases which, if established, would verify Yuzvinsky's conjecture as true for matrices up to order 32 $\times$ 32.