On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

A rough set approach to intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. Since its appearance, there has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. The intuitionistic fuzzy soft set is a combination of an intuitionistic fuzzy set and a soft set. The rough set theory is a powerful tool for dealing with uncertainty, granuality and incompleteness of knowledge in information systems. Using rough set theory, this paper proposes a novel approach to intuitionistic fuzzy soft set based decision making problems. Firstly, by employing an intuitionistic fuzzy relation and a threshold value pair, we define a new rough set model and examine some fundamental properties of this rough set model. Then the concepts of approximate precision and rough degree are given and some basic properties are discussed. Furthermore, we investigate the relationship between intuitionistic fuzzy soft sets and intuitionistic fuzzy relations and present a rough set approach to intuitionistic fuzzy soft set based decision making. Finally, an illustrative example is employed to show the validity of this rough set approach in intuitionistic fuzzy soft set based decision making problems.     

Tight Gabor Sets on Discrete Periodic Sets

This paper addresses Gabor analysis on a discrete periodic set. Such a scenario can potentially find its applications in signal processing where signals may present on a union of disconnected discrete index sets. We focus on the Gabor systems generated by characteristic functions. A sufficient and necessary condition for a set to be a tight Gabor set in discrete periodic sets is obtained; discrete periodic sets admitting a tight Gabor set are also characterized; the perturbation of tight Gabor sets is investigated; an algorithm to determine whether a set is a tight Gabor set is presented. Furthermore, we prove that an arbitrary Gabor frame set can be represented as the union of a tight Gabor set and a Gabor Bessel set.     

Simplifying essential competencies for Taiwan civil servants using the rough set approach

Applying competency models to identify and develop capabilities of civil servants is now a leading strategy for every government. However, an ideal competency model usually contains too many intended competencies, impeding implementation. Recently, some scholars and experts argued that there is a maximum of eight competencies for effective assessment. Hence, how to simplify a set of competencies becomes an important issue. This study is presented as a test case to extend practical applications of rough set theory (RST) in the human resource field of Government. A well-known data mining technique, RST is a relatively new approach to this problem and is good at data reduction in qualitative analysis. Hence, the rough set approach is suitable for dealing with the qualitative problem in simplifying a set of competencies. This paper slimmed a set of competencies using RST, thus helping the Taiwan Government to better understand the perceived competency levels of its civil servants. Using the rough set analysis, this paper successfully reduced the numerous essential competencies into a more compact set, by omitting low-consensus competencies.     

Projective Minkowski set

The Minkowski set or the central symmetry set (CSS) of a smooth curve Γ on the affine plane is the envelope of chords connecting pairs of points such that the tangents to Γ at them are parallel. Singularities of CSS are of interest, in particular, for applications (for example, in computer graphics). A generalization of the Minkowski set is considered in the paper, namely, the projective Minkowski set with respect to a line on the plane; in the case of general position, we describe its singularities and the bifurcation set of lines corresponding to lines defining the projective Minkowski set having singularities being more degenerate than those of the Minkowski set for a generic line.     

Construction of the Solvability Set in Differential Games with Simple Motion and Nonconvex Terminal Set

We consider planar zero-sum differential games with simple motion, fixed terminal time, and polygonal terminal set. The geometric constraint on the control of each player is a convex polygonal set or a line segment. In the case of a convex terminal set, an explicit formula is known for the solvability set (a level set of the value function, maximal u-stable bridge, viability set). The algorithm corresponding to this formula is based on the set operations of algebraic sum and geometric difference (the Minkowski difference). We propose an algorithm for the exact construction of the solvability set in the case of a nonconvex polygonal terminal set. The algorithm does not involve the additional partition of the time interval and the recovery of intermediate solvability sets at additional instants. A list of half-spaces in the three-dimensional space of time and state coordinates is formed and processed by a finite recursion. The list is based on the polygonal terminal set with the use of normals to the polygonal constraints on the controls of the players.     

Uniqueness of non-Archimedean entire functions sharing sets of values counting multiplicity

A set is called a unique range set for a certain class of functions if each inverse image of that set uniquely determines a function from the given class. We show that a finite set is a unique range set, counting multiplicity, for non-Archimedean entire functions if and only if there is no non-trivial affine transformation preserving the set. Our proof uses a theorem of Berkovich to extend, to non-Archimedean entire functions, an argument used by Boutabaa, Escassut, and Haddad to prove this result for polynomials

     

The set of all nondominated solutions in linear cases and a multicriteria simplex method

In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.     

The construction of stable bridges in differential games with phase constraints

A differential approach-and-evasion game in a finite time interval is considered [1]. It is assumed that the positions of the game are constricted by certain constraints which represent a closed set in the space of the positions. In the case of the first player, it is necessary to ensure that the phase point falls into the terminal set at a finite instant of time and, in the case of the second player, that this terminal set is evaded at this instant [1]. A method is proposed for the approximate construction of the positional absorption set, that is, the set of all positions belonging to a constraint from which the problem of approach facing the first player is solvable. Relations are written out which determine the system of sets which approximates the positional absorption set. The main result is a proof of the convergence of the approximate system of sets to the positional absorption set and the construction of a computational procedure for constructing the approximate system of sets.     

The complexity of dissociation set problems in graphs

A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set problem is NP-hard for planar line graphs of planar bipartite graphs. In addition, we describe several polynomially solvable cases for the problem under consideration. One of them deals with the subclass of the so-called chair-free graphs. Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and bipartite graphs, are derived. Finally, we provide inapproximability results for the dissociation set problems mentioned above.     

Element number of the Platonic solids

Let Σ be a set of polyhedra. A set Ω of polyhedra is said to be an element set for Σ if each polyhedron in Σ is the union of a finite number of polyhedra in Ω. We call each polyhedron of the element set Ω an element for Σ. In this paper, we determine one element set for the set Π of the Platonic solids, and prove that this element set is, in fact, best possible; it achieves the minimum in terms of cardinality among all the element sets for Π. We also introduce the notion of indecomposability of a polyhedron and present a conjecture in Sect. 3.     

3-连通3-正则图中的可序子集

图G的k元点集X={x1,x2,…,xk}被称为G的k-可序子集,如果X的任意排列都按序排在G的某个圈上.称G是k-可序图,如果G的每一个k元子集都是G的k-可序子集.称G为k-可序Hamilton图,如果X的任意排列都位于G的Hamilton圈上.研究了3-连通3-正则图的可序子集的存在性问题.     

Topological equisingularity of hypersurfaces with 1-dimensional critical set

We focus on topological equisingularity of families of holomorphic function germs with 1-dimensional critical set. We introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular. We show that if a family of germs with 1-dimensional critical set has constant generic Lê numbers then it is equisingular at the critical set, and hence topologically equisingular (answering a question of D. Massey [13]). It is worth to remark that this does not happen for higher dimensional critical set [5]. We use these topological triviality results to modify the definition of singularity stem present in the literature, introducing and characterising topological stems (being this concept closely related with Arnold?s series of singularities). We provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly. Finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non-flat deformation with singular special fibre and smooth generic fibre.     

A note on correlated equilibrium

The set of correlated equilibria for a bimatrix game is a closed, bounded, convex set containing the set of Nash equilibria. We show that every extreme point of a maximal Nash set is an extreme point of the above convex set. We also give an example to show that this result is not true in the payoff space, i.e. there are games where no Nash equilibrium payoff is an extreme point of the set of correlated equilibrium payoffs.     

Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems

In this paper, a notion of Levitin–Polyak (LP in short) well-posedness is introduced for a vector optimization problem in terms of minimizing sequences and efficient solutions. Sufficient conditions for the LP well-posedness are studied under the assumptions of compactness of the feasible set, closedness of the set of minimal solutions and continuity of the objective function. The continuity assumption is then weakened to cone lower semicontinuity for vector-valued functions. A notion of LP minimizing sequence of sets is studied to establish another set of sufficient conditions for the LP well-posedness of the vector problem. For a quasiconvex vector optimization problem, sufficient conditions are obtained by weakening the compactness of the feasible set to a certain level-boundedness condition. This in turn leads to the equivalence of LP well-posedness and compactness of the set of efficient solutions. Some characterizations of LP well-posedness are given in terms of the upper Hausdorff convergence of the sequence of sets of approximate efficient solutions and the upper semicontinuity of an approximate efficient map by assuming the compactness of the set of efficient solutions, even when the objective function is not necessarily quasiconvex. Finally, a characterization of LP well-posedness in terms of the closedness of the approximate efficient map is provided by assuming the compactness of the feasible set.     

Dynamical level set method for parameter identification of nonlinear parabolic distributed parameter systems

This article considers a dynamical level set method for the identification problem of the nonlinear parabolic distributed parameter system, which is based on the solvability and stability of the direct PDE (partial differential equation) in Sobolev space. The dynamical level set algorithms have been developed for ill-posed problems in Hilbert space. This method can be regarded as a asymptotical regularization method as long as a certain stopping rule is satisfied. Hence, the convergence analysis of the method is established similar to the proof of convergence of asymptotical regularization. The level set converges to a solution as the artificial time evolves to infinity. Furthermore, the proposed level set method is proved to be stable by using Lyapunov stability theorem, which is constructed in my previous article.Numerical tests are discussed to demonstrate the efficacy of the dynamical level set method, which consequently confirm the level set method to be a powerful tool for the identification of the parameter.     

On the theory of Banach space valued multifunctions. 2. Set valued martingales and set valued measures

This second part of the work on Banach space valued multifunctions begins with a detailed study of set valued martingales, which have their values in a Banach space. Several new convergence theorems are established for different modes of convergence. The profile of a multifunction in connection with set valued martingales is also studied. The notion of weak convergence of multifunctions is introduced and used to obtain additional convergence theorems for set valued martingales. In the last two sections of the paper set valued measures dealt with and an integral with respect to a set valued measure is introduced.     

Hesitant fuzzy information aggregation in decision making

As a generalization of fuzzy set, hesitant fuzzy set is a very useful tool in situations where there are some difficulties in determining the membership of an element to a set caused by a doubt between a few different values. The aim of this paper is to develop a series of aggregation operators for hesitant fuzzy information. We first discuss the relationship between intutionistic fuzzy set and hesitant fuzzy set, based on which we develop some operations and aggregation operators for hesitant fuzzy elements. The correlations among the aggregation operators are further discussed. Finally, we give their application in solving decision making problems.     

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality is proposed in this paper. The addressed approach is based on the fact that zonotopes are more flexible for representing sets than boxes in interval analysis. Then the solver of set inversion via interval analysis is extended to set inversion via zonotope geometry by introducing the novel idea of bisecting zonotopes. The main feature of the extended solver of set inversion is the bisection and the evolution of a zonotope rather than a box. Thus the shape of admissible domains for set inversion can be broadened from boxes to zonotopes and the wrapping effect can be reduced as well by using the zonotope evolution instead of the interval evolution. Combined with global optimization via interval analysis, the extended solver of set inversion via zonotope geometry is further applied to design control invariant sets for constrained nonlinear discrete-time systems in a numerical way. Finally, the numerical design of a control invariant set and its application to the terminal control of the dual-mode model predictive control are fulfilled on a benchmark Continuous-Stirred Tank Reactor example.