Invariant group orderings and Galois conjugates

This paper investigates conditions under which a given automorphism of a residually torsion-free nilpotent group respects some ordering of the group. For free groups and surface groups, this has relevance to ordering the fundamental groups of three-dimensional manifolds which fiber over the circle.     

Fuzzy preference orderings in group decision making

In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.     

On group stable commutative separable semisimple subalgebras

In a G-algebra A of a p-group G over a perfect ground field there is a commutative separable semisimple G-subalgebra such that any traces on relative projectivity in A can be controlled in the subalgebra; and in a residually central sense, the maximal such subalgebras form exactly one -conjugacy class. Applying to modules, the induced module of an indecomposable module of a subnormal subgroup of p- primary index is characterized. Received: 13 April 2002 / Published online: 2 December 2002     

Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data

In this paper, we establish some results for the increasing convex comparisons of generalized order statistics. First, we prove that if the minimum of two sets of generalized order statistics are ordered in the increasing convex order, then the remaining generalized order statistics are also ordered in the increasing convex order. This result is extended to the increasing directionally convex comparisons of random vectors of generalized order statistics. For establishing this general result, we first prove a new result in that two random vectors with a common conditionally increasing copula are ordered in the increasing directionally convex order if the marginals are ordered in the increasing convex order. This latter result is, of course, of interest in its own right.     

Weakening partial orderings to lattice and semilattice orderings

A partial ordering on a set P can be weakened to an upper or lower semilattice ordering, respectively a lattice ordering, if and only if P is filtered in the appropriate direction(s).This work was done while the author was partly supported by NSF contract DMS 85-02330.     

Product decompositions of the symmetric group induced by separable permutations

In this paper we consider the rank generating function of a separable permutation π in the weak Bruhat order on the two intervals [id,π] and [π,w₀], where w₀=n,n−1,…,1. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on [id,π] and [π,w₀], leading to the rank-symmetry and unimodality of the two graded posets.     

Real-local fields and the canonical operation of the automorphism group on the space of orderings

The object of our investigation is the canonical operation of the automorphism group of a formally real field F on X_F , the space of orderings of F. For a naturally distinguished class of formally real fields, the so-called real-local fields, the Baer-Krull-bijection induces on X_F the structure of a module over the endomorphism ring of the group of archimedean classes of F. We show that Aut F acts on X_F by affinities with respect to that module structure. Subsequently, this “arithmetization” of the operation is exemplarily applied to the question of transitivity (“When can any two orderings of F be transformed into each other by some automorphism of F?"), and to the investigation of the subgroup of Aut F generated by all order automorphism groups of F.     

Functionally separable solutions to nonlinear wave equations by group foliation method

We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: utt=A(x)(Dx(u)ux)+B(x)Q(u), Ax≠0. A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.     

On the left regular representation of a separable locally compact group

For an arbitrary separable locally compact group G we exhibit a canonical Borel subset $\text{G}\text{?}{}_{\mathrm{\Delta }}$ of the quasi-dual $\text{G}\text{?}\text{of}\text{G}$ (with the Mackey Borel structure), such that $\text{G}\text{?}{}_{\mathrm{\Delta }}$ is a standard Borel space in the induced Borel structure, and such that the canonical measure for the left regular representation λ_Gof G is concentrated on $\text{G}\text{?}{}_{\mathrm{\Delta }}$. On the basis of this we discuss the (non-unimodular) “Plancherel theorem.”     

Conditional orderings and positive dependence

Every univariate random variable is smaller, with respect to the ordinary stochastic order and with respect to the hazard rate order, than a right censored version of it. In this paper we attempt to generalize these facts to the multivariate setting. It turns out that in general such comparisons do not hold in the multivariate case, but they do under some assumptions of positive dependence. First we obtain results that compare the underlying random vectors with respect to the usual multivariate stochastic order. A larger slew of results, that yield comparisons of the underlying random vectors with respect to various multivariate hazard rate orders, is given next. Some comparisons with respect to the orthant orders are also discussed.     

A separable normal topological group which is not Lindelöf

We construct in ZFC the example mentioned in the title.