A Helly-Type Theorem for Intersections of Compact Connected Sets in the Plane

For a family C of nonempty compact sets in the plane, the following conditions are equivalent:(1) Every two (not necessarily distinct) members of C have a connected union and every three (not necessarily distinct) members of C have a simply connected union.(2) C is a family of simply connected sets such that every two (not necessarily distinct) members of C have a connected intersection and every three (not necessarily distinct) members of C have a nonempty intersection.If either set of conditions is satisfied, then { C : C in C } is nonempty, simply connected, and connected. Furthermore, if the collection C is finite, then it is also true that { C : C in C } is simply connected.     

自相似集的质量分布原理与Hausdorff测度及其应用

本文提出了满足开集条件的自相似集的质量分布原理.作为应用,得到了计算一类满足开集条件的自相似集的Hausdorff测度的准确值的方法,并举例说明了此方法对于计算一类满足开集条件的自相似集的Hausdorff测度的准确值是行之有效的.     

On the relationship between fixed points and iteration in admissible set theory without foundation

In this article we show how to use the result in Jäger and Probst [7] to adapt the technique of pseudo-hierarchies and its use in Avigad [1] to subsystems of set theory without foundation. We prove that the theory KPi⁰ of admissible sets without foundation, extended by the principle (Σ-FP), asserting the existence of fixed points of monotone Σ operators, has the same proof-theoretic ordinal as KPi⁰ extended by the principle (Σ-TR), that allows to iterate Σ operations along ordinals. By Jäger and Probst [6] we conclude that the metapredicative Mahlo ordinal φω00 is also the ordinal of KPi⁰+(Σ-FP). Hence the relationship between fixed points and iteration persists in the framework of set theory without foundation.The author is supported by the Swiss National Science Foundation.     

A relational perspective of attribute reduction in rough set-based data analysis

Attribute reduction is very important in rough set-based data analysis (RSDA) because it can be used to simplify the induced decision rules without reducing the classification accuracy. The notion of reduct plays a key role in rough set-based attribute reduction. In rough set theory, a reduct is generally defined as a minimal subset of attributes that can classify the same domain of objects as unambiguously as the original set of attributes. Nevertheless, from a relational perspective, RSDA relies on a kind of dependency principle. That is, the relationship between the class labels of a pair of objects depends on component-wise comparison of their condition attributes. The larger the number of condition attributes compared, the greater the probability that the dependency will hold. Thus, elimination of condition attributes may cause more object pairs to violate the dependency principle. Based on this observation, a reduct can be defined alternatively as a minimal subset of attributes that does not increase the number of objects violating the dependency principle. While the alternative definition coincides with the original one in ordinary RSDA, it is more easily generalized to cases of fuzzy RSDA and relational data analysis.     

基于改进集的集值Ekeland变分原理

Ekeland变分原理在最优化理论及应用研究中具有十分重要的作用.利用非线性标量化函数及相应的非凸分离定理建立了基于改进集的集值Ekeland变分原理.新的Ekeland变分原理包含了一些经典的Ekeland变分原理作为其特例.     

Learning from imprecise and fuzzy observations: Data disambiguation through generalized loss minimization

Methods for analyzing or learning from “fuzzy data” have attracted increasing attention in recent years. In many cases, however, existing methods (for precise, non-fuzzy data) are extended to the fuzzy case in an ad-hoc manner, and without carefully considering the interpretation of a fuzzy set when being used for modeling data. Distinguishing between an ontic and an epistemic interpretation of fuzzy set-valued data, and focusing on the latter, we argue that a “fuzzification” of learning algorithms based on an application of the generic extension principle is not appropriate. In fact, the extension principle fails to properly exploit the inductive bias underlying statistical and machine learning methods, although this bias, at least in principle, offers a means for “disambiguating” the fuzzy data. Alternatively, we therefore propose a method which is based on the generalization of loss functions in empirical risk minimization, and which performs model identification and data disambiguation simultaneously. Elaborating on the fuzzification of specific types of losses, we establish connections to well-known loss functions in regression and classification. We compare our approach with related methods and illustrate its use in logistic regression for binary classification.     

On the points realizing the distance to a definable set

We prove a definable/subanalytic version of a useful lemma, presumably due to John Nash, concerning the points realizing the Euclidean distance to an analytic submanifold of Rn. We present a parameter version of the main result and we discuss the properties of the multifunction obtained.     

Extremality,Controllability, and Abundant Subsets of Generalized Control Systems

The generalized control system that we consider in this paper is a collection of vector fields, which are measurable in the time variable and Lipschitzian in the state variable. For such system, we define the concept of an abundant subset. Our definition follows the definition of an abundant set of control functions introduced by Warga. We prove a controllability–extremality theorem for generalized control systems, which says, in essence, that either a given trajectory satisfies a type of maximum principle or a neighborhood of the endpoint of the trajectory can be covered by trajectories of an abundant subset. We apply the theorem to a control system in the classical formulation and obtain a controllability–extremality result, which is stronger, in some respects, than all previous results of this type. Finally, we apply the theorem to differential inclusions and obtain, as an easy corollary, a Pontryagin-type maximum principle for nonconvex inclusions.     

Iterative Schemes for Nonconvex Variational Inequalities

In this paper, we suggest and analyze some iterative methods for solving nonconvex variational inequalities using the auxiliary principle technique, the convergence of which requires either only pseudomonotonicity or partially relaxed strong monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving general variational inequalities involving convex sets.     

Extensions of barrier sets to nonzero roots of the matching polynomial

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to their connection to maximum matchings in a graph. For a root θ of the matching polynomial, we define θ-barrier and θ-extreme sets. We prove a generalized Berge-Tutte formula and give a characterization for the set of all θ-special vertices in a graph.     

Polynomial configurations in difference sets

We prove a quantitative result on the existence of linearly independent polynomial configurations in the difference set of sparse subsets of the integers. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sárközy and then applying a simple lifting argument.     

Large Deviations of the Finite Cluster Shape for Two-Dimensional Percolation in the Hausdorff and L1 Metric

We consider supercritical two-dimensional Bernoulli percolation. Conditionally on the event that the open cluster C containing the origin is finite, we prove that: the laws of C/N satisfy a large deviations principle with respect to the Hausdorff metric; let f(N) be a function from to such that f(N)/ln N+ and f(N)/N0 as N goes to the laws of {x ² : d(x, C)f(N)}/N satisfy a large deviations principle with respect to the L ¹ metric associated to the planer Lebesgue measure. We link the second large deviations principle with the Wulff construction.     

Reasoning with doubly uncertain soft constraints

We describe the basic ideas of the theory of approximate reasoning and indicate how it provides a framework for representing human sourced soft information. We discuss how to translate linguistic knowledge into formal representations using generalized constraints. We consider the inference process within the theory of approximate reasoning and introduce the entailment principle and describe its centrality to this inference process. Next we introduce the idea of doubly uncertain statements such as John’s friend is young. In these statements there exists uncertainty both with respect to value of the age, young, and the object associated with the age, John’s friend. We suggest a method for representing these complex statements and investigate the problem of making inferences about specific objects.     

Nonconvex functions and variational inequalities

In this paper, we study some properties of a class of nonconvex functions, called semipreinvex functions, which includes the classes of preinvex functions and arc-connected convex functions. It is shown that the minimum of an arcwise directionally differentiable semi-invex functions on a semi-invex set can be characterized by a class of variational inequalities, known as variational-like inequalities. We use the auxiliary principle technique to prove the existence of a solution of a variational-like inequality and suggest a novel iterative algorithm.     

Generalized subdifferential of the distance function

We derive the proximal normal formula for almost proximinal sets in a smooth and locally uniformly convex Banach space. Our technique leads us to show the generic Fréchet smoothness of the distance function in the case the norm is Fréchet smooth, and we derive a necessary and sufficient condition for the convexity of a Chebyshev set in a Banach space with norms on and locally uniformly convex.

     

Internal sets and internal functions in Colombeau theory

This paper is devoted to the study of generalized functions as pointwise functions (so-called internal functions) on certain sets of generalized points (so-called internal sets). We treat the case of the Colombeau algebras of generalized functions, for which these notions have turned out to constitute a fundamental technical tool. We provide general foundations for the notion of internal functions and internal sets and prove a saturation principle. Various applications to Colombeau algebras are given.     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

Exponential number of inequivalent difference sets in ℤ

Kantor [ 5 ] proved an exponential lower bound on the number of pairwise inequivalent difference sets in the elementary abelian group of order 2^2s+2. Dillon [ 3 ] generalized a technique of McFarland [ 6 ] to provide a framework for determining the number of inequivalent difference sets in 2‐groups with a large elementary abelian direct factor. In this paper, we consider the opposite end of the spectrum, the rank 2 group ? , and compute an exponential lower bound on the number of pairwise inequivalent difference sets in this group. In the process, we demonstrate that Dillon difference sets in groups ? can be constructed via the recursive construction from [ 2 ] and we show that there are exponentially many pairwise inequivalent difference sets that are inequivalent to any Dillon difference set. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 249–259, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10046     

General decay to a von Kármán system with memory

In this paper we study the von Kármán plate model with long-range memory and we show the general decay of the solution as time goes to infinity. This result generalizes and improves on earlier ones in the literature because it allows certain relaxation functions which are not necessarily of exponential or polynomial decay.     

An adjustable approach to intuitionistic fuzzy soft sets based decision making

Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. There has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. In this paper we generalize the adjustable approach to fuzzy soft sets based decision making. Concretely, we present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and give some illustrative examples. The properties of level soft sets are presented and discussed. Moreover, we also introduce the weighted intuitionistic fuzzy soft sets and investigate its application to decision making.