Topological representation of posets

There is a canonical imbedding of a poset into a complete Boolean lattice and hence into a Boolean lattice. This gives it a representation as a collection of clopen sets of a Boolean space. There are reflective functions from a category of distributive posets to the subcategories of distributive and Boolean lattices and consequently a topological dual equivalence that extends the Stone duality of Boolean lattices.Presented by B. Jonsson.     

A Priori and A Posteriori: A Bootstrapping Relationship

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning, and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification.     

Remarks on pseudo-valuation rings

A prime ideal P of a ring A is said to be a strongly prime ideal if aP and bA are comparable for all a,b ε A. We shall say that a ring A is a pseudo-valuation ring (PVR) if each prime ideal of A is a strongly prime ideal. We show that if A is a PVR with maximal ideal M, then every overring of A is a PVR if and only if M is a maximal ideal of every overring of M that does not contain the reciprocal’of any element of M.We show that if R is an atomic domain and a PVD, then dim(R) ≤ 1. We show that if R is a PVD and a prime ideal of R is finitely generated, then every overring of R is a PVD. We give a characterization of an atomic PVD in terms of the concept of half-factorial domain.     

Kripke submodels and universal sentences

We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with (homo)morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms (including ? and ⊥) using ∧, ∨, ?, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model‐consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory Δ if and only if every rooted Kripke model of Δ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On dominance-based multiobjective local search: design, implementation and experimental analysis on scheduling and traveling salesman problems

This paper discusses simple local search approaches for approximating the efficient set of multiobjective combinatorial optimization problems. We focus on algorithms defined by a neighborhood structure and a dominance relation that iteratively improve an archive of nondominated solutions. Such methods are referred to as dominance-based multiobjective local search. We first provide a concise overview of existing algorithms, and we propose a model trying to unify them through a fine-grained decomposition. The main problem-independent search components of dominance relation, solution selection, neighborhood exploration and archiving are largely discussed. Then, a number of state-of-the-art and original strategies are experimented on solving a permutation flowshop scheduling problem and a traveling salesman problem, both on a two- and a three-objective formulation. Experimental results and a statistical comparison are reported in the paper, and some directions for future research are highlighted.     

Necessary and sufficient existence conditions for a boundary control of vibrations of a string and a spherical layer for small times

We find necessary and sufficient conditions for the existence of a boundary control of vibrations of a string or a spherical layer for critical and subcritical times. We completely analyze the existence of a boundary control of vibrations of a spherical layer by a force on two spheres. We find necessary and sufficient existence conditions for the control. Along with the control problem for vibrations of a spherical layer, we consider a similar control problem for string vibrations.     

A note on subsymmetric renormings of Banach spaces

We revisit the concept of a subsymmetric norm and construct a subsymmetric renorming of a Banach space with a subsymmetric basis. As a by-product of our work we introduce the concept of a lower symmetric basis and investigate its connection with subsymmetric bases and subsymmetric renormings.     

Subgradient Projectors: Extensions,Theory, and Characterizations

Subgradient projectors play an important role in optimization and for solving convex feasibility problems. For every locally Lipschitz function, we can define a subgradient projector via generalized subgradients even if the function is not convex. The paper consists of three parts. In the first part, we study basic properties of subgradient projectors and give characterizations when a subgradient projector is a cutter, a local cutter, or a quasi-nonexpansive mapping. We present global and local convergence analyses of subgradent projectors. Many examples are provided to illustrate the theory. In the second part, we investigate the relationship between the subgradient projector of a prox-regular function and the subgradient projector of its Moreau envelope. We also characterize when a mapping is the subgradient projector of a convex function. In the third part, we focus on linearity properties of subgradient projectors. We show that, under appropriate conditions, a linear operator is a subgradient projector of a convex function if and only if it is a convex combination of the identity operator and a projection operator onto a subspace. In general, neither a convex combination nor a composition of subgradient projectors of convex functions is a subgradient projector of a convex function.     

Rings with Elementary Reduction of Matrices

We establish necessary and sufficient conditions under which a quasi-Euclidean ring coincides with a ring with elementary reduction of matrices. We prove that a semilocal Bézout ring is a ring with elementary reduction of matrices and show that a 2-stage Euclidean domain is also a ring with elementary reduction of matrices. We formulate and prove a criterion for the existence of solutions of a matrix equation of a special type and write these solutions in an explicit form.     

Partnership formation based on multiple traits

A model of partnership formation based on two traits, called beauty and character, is presented. There are two classes of individual and partners must be of different classes. Individuals prefer prospective partners with a high beauty measure and of a similar character. This problem may be interpreted as e.g. a job search problem in which the classes are employer and employee, or a mate choice problem in which the classes are male and female. Beauty can be observed instantly. However, a costly date (or interview) is required to observe the character of a prospective partner. On observing the beauty of a prospective partner, an individual decides whether he/she wishes to date. During a date, the participants observe each other’s character and then decide whether to form a pair. Mutual acceptance is required both for a date to occur and pair formation. On finding a partner, an individual stops searching. Beauty has a continuous distribution on a finite interval, while character ‘forms a circle’ and has a uniform distribution. Criteria based on the concept of a subgame perfect Nash equilibrium are used to define a symmetric equilibrium of this game. It is argued that this equilibrium is unique. When dating costs are high, this equilibrium is a block separating equilibrium as in more classical formulations of two-sided job search problems. However, for sufficiently small dating costs the form of this equilibrium is essentially different.     

Partially isometric dilations of noncommuting -tuples of operators

Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.

     

Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.     

Banach空间中非膨胀映象和α-逆强增生算子的强收敛性

在2-一致光滑的Banach空间中,引入一种新的迭代算法研究非膨胀映象的不动点集与α-逆强增生算子的变分不等式解集的公共元素,并获得了迭代算法的强收敛性定理.而且应用这些结果考虑了非膨胀映象和严格伪压缩映象公共不动点的收敛性问题.     

On the intersections of circuits and cocircuits in matroids

Seymour has shown that a matroid has a triad, that is, a 3-element set which is the intersection of a circuit and a cocircuit, if and only if it is non-binary. In this paper we determine precisely when a matroidM has a quad, a 4-element set which is the intersection of a circuit and a cocircuit. We also show that this will occur ifM has a circuit and a cocircuit meeting in more than four elements. In addition, we prove that if a 3-connected matroid has a quad, then every pair of elements is in a quad. The corresponding result for triads was proved by Seymour.     

Skew category, Galois covering and smash product of a -category

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.

     

一致覆盖和度量空间的紧映象

本文利用一致覆盖的概念,讨论了度量空间的序列覆盖紧映象的结构.主要结果有: (1)空间X是局部可分度量空间的序列覆盖紧映象当且仅当X具有由cosmic子空间构成的一致sn网; (2)空间X是局部可分度量空间的序列覆盖,商紧映象当且仅当X是度量空间的序列覆盖,商紧映象且是局部cosmic空间.     

Conceptual issues in understanding the inner logic of statistical inference: Insights from two teaching experiments

We report on a sequence of two classroom teaching experiments that investigated high school students’ understandings as they explored connections among the ideas comprising the inner logic of statistical inference—ideas involving a core image of sampling as a repeatable process, and the organization of its outcomes into a distribution of sample statistics as a basis for making inferences. Students’ responses to post-instruction test questions indicate that despite understanding various individual components of inference—a sample, a population, and a distribution of a sample statistic—their abilities to coordinate and compose these into a coherent and well-connected scheme of ideas were usually tenuous. We argue that the coordination and composition required to assemble these component ideas into a coherent scheme is a major source of difficulty in developing a deep understanding of inference.     

A relaxed extragradient-like method for a generalized mixed equilibrium problem,a general system of generalized equilibria and a fixed point problem

Very recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025–1033] suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a Hilbert space. In this paper, based on Takahashi–Takahashi’s iterative method and well-known extragradient method we introduce a relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a Hilbert space and then obtain a strong convergence theorem. Utilizing this theorem, we establish some new strong convergence results in fixed point problems, variational inequalities, mixed equilibrium problems and systems of generalized equilibria.     

A Characterization of Semirings Which Are Subdirect Products of a Distributive Lattice and a Ring

E -inversive semiring and a Clifford semiring and show that a semiring S is a subdirect product of a distributive lattice and a ring if and only if S is an E-inversive strong distributive lattice of halfrings. Further a Clifford semiring which is, in fact, an inversive subdirect product of a distributive lattice and a ring, is characterized as a strong distributive lattice of rings. Finally, as a consequence of these results we extend a result of Galbiati and Veronesi [2] in the case of Boolean semirings.     

An algebraic characterization of planar graphs

A cycle in a graph is a set of edges that covers each vertex an even number of times. A cocycle is a collection of edges that intersects each cycle in an even number of edges. A bicycle is a collection of edges that is both a cycle and a cocycle. The cycles, cocycles, and bicycles each form a vector space over the integers modulo two when addition is defined as symmetric difference of sets. In this paper we examine the relationship between the left-right paths in a planar graph and the cycle space, cocylce space, and bicycle space. We show that planar graphs are characterized by the existence of a diagonal—a double cover by tours that interacts with the cycle space, cocycle space, and bicycle space in a special manner. This generalizes a result of Rosenstiehl and Read that characterized those planar graphs with no nonempty bicycles. © 1995 John Wiley & Sons, Inc.