# Volume 08 (2004) No. 2

## Volume 08 (2004) No. 2

### Homotopy triangulations of a manifold triple

Rolando Jimenez and Yuri V. Muranov

**Abstract:**

The set of homotopy triangulations of a given manifold fits into a surgery exact sequence which is the main tool for the classification of manifolds. In the present paper we describe relations between homotopy triangulations of different manifolds for a given manifold triple and its connection to surgery theory. We introduce a group of obstructions to split a homotopy equivalence along a pair of submanifolds and study its properties. The main results are given by commutative diagrams of exact sequences.

### On information measures and prior distributions: a synthesis

Francisco Venegas-MartÃnez

**Abstract:**

This paper suggests a new approach to reconciling, in a systematic way, all inferential methods that maximize a specific criterion functional to produce *non-informative* and *informative* priors. In particular, Good's (1968) Minimax Evidence Priors (MEP), Zellner's (1971) Maximal Data Information Priors (MDIP) and Bernardo's (1979) Reference Priors (RP) are seen as special cases of maximizing a more general criterion functional. In a unifying approach Good-Bernardo-Zellner's priors are introduced and applied to a number of Bayesian inference problems, including the Kalman filter and Normal linear model. Moreover, the paper focuses, under plausible conditions, on the existence and uniqueness of the solutions of the derived optimization problems.

### On a problem of Steinhaus concerning binary sequences

Shalom Eliahou and Delphine Hachez

**Abstract:**

A finite $\pm 1$ sequence $X$ yields a binary triangle $\Delta X$ whose first row is $X$, and whose $(k+1)$st row is the sequence of pairwise products of consecutive entries of its $k$th row, for all $k \ge 1$. We say that $X$ is *balanced* if its derived triangle $\Delta X$ contains as many $+1$'s as $-1$'s. In 1963, Steinhaus asked whether there exist balanced binary sequences of every length $n \equiv 0$ or 3 mod 4. While this problem has been solved in the affirmative by Harborth in 1972, we present here a different solution. We do so by constructing *strongly balanced* binary sequences, *i.e.* binary sequences of length $n$ all of whose initial segments of length $n- 4t$ are balanced, for $0 \le t \le n/4$. Our strongly balanced sequences do occur in every length $n \equiv 0$ or 3 mod 4. Moreover, we provide a complete classification of sufficiently long strongly balanced binary sequences.