Paths in graphs and minimal π-sequences in semigroups

By taking a graph theoretic point of view, the author's theory of minimal sequences in semi groups is generalized to minimal π-sequences, where π is any positive quasi-order on the semi group. This generalization is especially complete when π is a linear quasi-order. Finally some possible future applications to the social sciences are considered.     

Axioms for function semigroups with agreement quasi-order

The agreement quasi-order on pairs of (partial) transformations on a set X is defined as follows: ${(f, g) \preceq (h, k)}$ if whenever f, g are defined and agree, so do h, k. We axiomatize function semigroups and monoids equipped with this quasi-order, thereby providing a generalisation of first projection quasi-ordered ${\cap}$ -semigroups of functions. As an application, axiomatizations are obtained for groups and inverse semigroups of injective functions equipped with the quasi-order of fix-set inclusion. All axiomatizations are finite.     

拟序下非连续增算子新的广义不动点定理

运用序方法研究了拟序空间X中具A=CB型非连续增算子的广义不动点的存在性问题,证明了在拟序空间X中具A=CB型非连续增算子存在最小广义不动点和最大广义不动点.所得结果推广了相关结论.     

Join-Irreducible Boolean Functions

This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting poset [(W)\tilde]\tilde{\Omega}. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield join-irreducible members of [(W)\tilde]\tilde{\Omega} are the − 2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of [(W)\tilde]\tilde{\Omega}.     

A nonlinear inequality of Moser-Trudinger type

In this note, we will prove an inequality for almost plurisubharmonic functions on any K?hler-Einstein manifolds with positive scalar curvature. This inequality generalizes the stronger version of the so called Moser-Trudinger-Onofri inequality on , which was proved in [Au], and also refines a weaker inequality found by the first author in [T2]. Received: May 27, 1997 / Accepted: June 11, 1999     

Rings with kernel inclusion quasiorder

Building on previous work for semigroups of functions and binary relations, we axiomatize structures consisting of endomorphisms of abelian groups equipped with composition, the usual pointwise operations, and the quasi-order of kernel inclusion. The resulting structures are associative rings enriched by a quasiorder satisfying a finite set of laws. More generally, we axiomatize the kernel inclusion quasi-order on the ring R induced by a right R-module, and we call the resulting abstract structures rings with ker-order. A characterisation of the possible ker-orders on a fixed ring is given in terms of certain families of its right ideals. The lattice of all ker-orders on the ring of rational integers is described.     

An internal characterisation of structural matrix rings

It is known that structural matrix rings pro-vide a natural passage from complete matrix rings to upper and lower triangular matrix rings, and they often explain the peculiarities regarding certain properties of complete matrix rings on the one hand and of triangular matrix rings on the other hand. In this paper the concept of a set of matrix units in a ring associated with a quasi-order relation is introduced and used to provide an internal char-acterisation of structural matrix rings.     

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.     

Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator

We establish, by simple semigroup arguments, a Lévy-Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian, isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.Oblatum 19-VI-1995     

Exponential inequalities for dynamical measures of expanding maps of the interval

 We prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval. As an immediate corollary we obtain a concentration inequality. We apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure. Received: 14 August 2001 / Revised version: 13 February 2002 / Published online: 1 July 2002     

Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator

 We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature. Oblatum 19-VI-1995     

An extension of Harnack type determinantal inequality

We revisit and comment on the Harnack type determinantal inequality for contractive matrices obtained by Tung in the nineteen sixties and give an extension of the inequality involving multiple positive semidefinite matrices .     

Minimal Antichains in Well-founded Quasi-orders with an Application to Tournaments

We investigate the minimal antichains (in what is essentially Nash-Williams' sense) in a well-founded quasi-order. We prove the following finiteness theorem: If Q is a well-founded quasi-order and k a fixed natural number, then there is a finite set Λ_k of minimal antichains of Q with the property that for any ideal I of Q obtained by excluding at most k elements of Q, I is well-quasi-ordered if and only if its intersection with each antichain in Λ_k is finite. When applied in a suitably sharpened form to an algorithmic problem arising in model theory, this yields a strengthening of the main result of [18].     

The Hijazi inequality on conformally parabolic manifolds

We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequality on conformally parabolic manifolds if the spin analog to the Yamabe invariant is positive.     

The Priestley Separation Axiom for Scattered Spaces

Let R be a quasi-order on a compact Hausdorff topological space X. We prove that if X is scattered, then R satisfies the Priestley separation axiom if and only if R is closed in the product space X×X. Furthermore, if X is not scattered, then we show that there is a quasi-order on X that is closed in X×X but does not satisfy the Priestley separation axiom. As a result, we obtain a new characterization of scattered compact Hausdorff spaces.     

Inequalities Involving Superquadratic Functions and Operators

By the use of some integral inequalities containing superquadratic functions, we obtain an inequality which generalizes some previous results. We also present an inequality for positive linear mappings of operators on Hilbert spaces. Some applications and examples are given as well.     

三条路并的极值能量及一类图的能量排序

拟序是图能量排序中一种有效方法,基于该方法,已经得到了大量图类的极值能量排序的结果.Gutman给出了点数和为n的两条路的并的能量排序,而三条路的并的能量排序没有一个理想的结论.本文利用拟序法给出点数和为n的三条路的并的极值能量及一类图能量的排序.     

Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains

We consider the oblique derivative problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz cylinders. We derive an optimal elliptic-type Harnack inequality for positive solutions of this problem and use it to show that each positive solution exponentially dominates any solution which changes sign for all times. We show several nontrivial applications of both the exponential estimate and the derived Harnack inequality.     

Scattered subsets of\mathbb{Q}

A linear order isscattered if it contains no copy of the rationals. The scattered subsets of form a better quasi-order under embeddability via order automorphisms.Research supported by NSERC grant #9848.     

Realizing Quasiordered Sets by Subspaces of ‘Continuum-Like’ Spaces

Given an ordered set E and a topological space X, we say that E can be realized within X if there is an injection j from E into the class of (homeomorphism classes of) subspaces of X such that, for x, y in E, x y if and only if j(x) is homeomorphically embeddable into j(y). It is known, for instance, that transfinite induction demonstrates that every partially-ordered set of cardinality c (and some larger ones) can be realized within the real line. We explore aspects of the realizability problem, indicating, in particular, how to weaken the hypothesis on E from partial- to quasi-order, and seeking to isolate the characteristics of the real line that are relevant here.