Nonseparable UHF algebras I: Dixmier's problem

There are three natural ways to define UHF (uniformly hyperfinite) C^∗-algebras, and all three definitions are equivalent for separable algebras. In 1967 Dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras. We give a complete answer to this question. More precisely, we show that in small cardinality two definitions remain equivalent, and give counterexamples in other cases. Our results do not use any additional set-theoretic axioms beyond the usual axioms, namely ZFC.     

Refinements of rationalizability for normal-form games

There exist three equivalent definitions of perfect Nash equilibria which differ in the way “best responses against small perturbations” are defined. It is shown that applying the spirit of these definitions to rationalizability leads to three different refinements of rationalizable strategies which are termed perfect (Bernheim, 1984), weakly perfect and trembling-hand perfect rationalizability, respectively. We prove that weakly perfect rationalizability is weaker than both perfect and proper (Schuhmacher, 1995) rationalizability and in two-player games it is weaker than trembling-hand perfect rationalizability. By means of examples, it is shown that no other relationships can be found. Received: January 1997/final version: August 1998     

Affine Yangians and deformed double current algebras in type A

We study the structure of Yangians of affine type and deformed double current algebras, which are deformations of the enveloping algebras of matrix W1+∞-algebras. We prove that they admit a PBW-type basis, establish a connection (limit construction) between these two types of algebras and toroidal quantum algebras, and we give three equivalent definitions of deformed double current algebras. We construct a Schur-Weyl functor between these algebras and rational Cherednik algebras.     

Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds

In this paper we consider three problems, which are related to the classical Monges optimal mass transport problem and which are known to be equivalent when the ambient space is an open, convex and bounded subset of ⁿ; to these problems there correspond different definitions of particular measures (often called transport densities), which are also known to be equivalent. Here we will generalize the setting of these problems and the resulting definitions of transport densities to the case of a Riemannian manifold endowed with a finslerian semidistance, and we will see that the equivalences still hold. Mathematics Subject Classification (2000) 46G10, 49Q20, 58C35     

Nonequivalence of Various Definitions of Differentiability Directions for Vector Measures

It is proved that the definitions of differentiability directions for vector measures in various topologies, namely, the topology of convergence on a system measurable sets, the topology of convergence with respect to semivariation, and the topology of convergence in variation, are generally pairwise nonequivalent. It is also proved that, for measures with values in a Banach space with the Radon--Nikodym property, these definitions are equivalent.     

函数两种凸性定义等价性的今惑前世之初探

通过研究中点凸函数和一般凸函数这两种凸性定义的早期发展历史和凸性性质来探索两种凸性定义的等价性.结果表明,两种凸性定义不等价;但是,当函数满足连续、可微、半连续和有界这四个条件中的任何一个条件时,两种凸性定义等价.     

Equivalent definitions of the Ganea fibrations¶and cofibrations

The Ganea fibrations have been used to define Lusternik–Schnirelmann category. It is proved that three definitions of the Ganea fibrations are equivalent. Consequences are drawn for the category of a map and the notion of cocategory. In addition, Ganea cofibrations are briefly considered. Received: 2 April 1999     

Omega-limit sets and invariant chaos in dimension one

Omega-limit sets play an important role in one-dimensional dynamics. During last fifty year at least three definitions of basic set has appeared. Authors often use results with different definition. Here we fill in the gap of missing proof of equivalency of these definitions. Using results on basic sets we generalize results in paper [P. Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), no. 1, 31–43.] to the case continuous maps of finite graphs. The Li-Yorke chaos is weaker than positive topological entropy. The equivalency arises when we add condition of invariance to Li-Yorke scrambled set. In this note we show that for a continuous graph map properties positive topological entropy; horseshoe; invariant Li-Yorke scrambled set; uniform invariant distributional chaotic scrambled set and distributionaly chaotic pair are mutually equivalent.     

Definitions of criticality with respect to edge-coloring

Here we examine six definitions of criticality concerning the chromatic index (edge chromatic number) of a simple graph. Five of these turn out to be almost always almost equivalent. Some problems arise and some conjectures are posed.     

Choice between opportunity sets: A characterization of welfarist behaviour

Welfarist behaviour in choosing between opportunity sets is defined in a number of alternative ways. Under the condition of consistent behaviour these definitions are shown to be equivalent. Two conditions involving revealed preference relations are proposed and shown to constitute a characterization of welfarist behaviour.     

Notions of automaticity in semigroups

We will investigate various possible notions of automaticity in semigroups. We point out that it makes a difference which side we choose for multiplication by generators and also which side we pad pairs of strings of unequal length (unlike the situation in groups, where the different choices give rise to equivalent definitions) and we investigate the properties of semigroups satisfying these definitions.     

On different notions of tameness in arithmetic geometry

The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these definitions are essentially equivalent. Furthermore, we prove finiteness theorems for the tame fundamental groups of arithmetic schemes.     

Note on the Norm Involved in the Definition of the ε-spectrum

There exist many definitions of the ε-spectrum. Unlike the spectrum, the ε-spectrum definitions depend on the choice of the norm. We propose to study the conditions on the norm that make all these definitions equivalent. The spectral radius of a related matrix is a useful tool in the eigenvalues perturbation analysis. The study of its connection with the ε-spectrum provides a new definition of the ε-spectrum. In most articles dealing with this subject, the chosen norm is the spectral one. In this paper, we intend to justify this choice.     

Defining continuity of real functions of real variables

Continuity of a real function of a real variable has been defined in various ways over almost 200 years. Contrary to popular belief, the definitions are not all equivalent, because their consequences for four somewhat pathological functions reveal five essentially different cases. The four defensible ones imply just two cases for continuity on an interval if that is defined by using pointwise continuity at each point. Some authors had trouble: two different textbooks each gave two arguably inconsistent definitions, three more changed their definitions in their second editions, two more claimed continuity at a point for functions not defined there, and one gave a definition implying it for a function with no limit there.     

Equivalents of the finitary non-deterministic inductive definitions

We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the notion of biclosed subset. Our result suggests that proving these statements in constructive topology requires genuine extensions of CZF with the elementary or finitary NID.     

Notions of Automaticity in Semigroups

Abstract. We will investigate various possible notions of automaticity in semigroups. We point out that it makes a difference which side we choose for multiplication by generators and also which side we pad pairs of strings of unequal length (unlike the situation in groups, where the different choices give rise to equivalent definitions) and we investigate the properties of semigroups satisfying these definitions.     

格序代数的商及其代数性质

研究了格序代数的商及其一些重要的代数性质.给出了格序代数的商的概念,定义了商f-代数、商几乎f-代数和商d-代数,并给出其等价刻画.讨论了这三类格序代数的商的半素性和交换性质,得到了这些性质的若干刻画.     

多目标半定规划的互补弱鞍点和G-鞍点最优性条件

对于含矩阵函数半定约束和多个目标函数的多目标半定规划问题，给出Lagrange函数在弱有效意义下的互补弱鞍点和Geofrrion恰当有效意义下的G-鞍点的定义及其等价定义．然后，在较弱的凸性条件下，利用含矩阵和向量约束的择一性定理，建立多目标半定规划的互补弱鞍点和G-鞍点充分必要条件．     

Convergence of continued fraction type algorithms and generators

The concept of convergence of continued fraction type algorithms has been defined a number of times in the literature. We investigate the relation between these definitions, and show that they do not always coincide. We relate the definitions to the question whether or not the natural partition of the underlying dynamical system is a generator. It turns out that the right definition of convergence is equivalent to this partition being a generator. The second definition of convergence is shown to be equivalent only under extra conditions on the transformation. These extra conditions are typically found to be satisfied when the second definition is used in the literature.     

直观模糊集的熵、距离测度、相似测度及它们的关系

熵、距离测度和相似测度是模糊集理论中的三个重要概念.首先系统地给出了直观模糊集的熵、距离测度和相似测度的公理化定义,并讨论了它们之间的一些基本关系.然后在距离测度公理化定义的基础上产生了一些新的直观模糊集的熵公式.