Structure of stability sets and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: I

We consider families of linear differential systems continuously depending on a real parameter. The stability (respectively, asymptotic stability) set of such a family is defined as the set of all values of the parameter for which the corresponding systems in the family are stable (respectively, asymptotically stable). We show that a set on the real axis is the stability (respectively, asymptotic stability) set of some family of this kind if and only if it is an F _σ -set (respectively, an F _σδ -set). For families in which the parameter occurs only as a factor multiplying the matrix of the system, their stability sets are exactly F _σ -sets containing zero on the real line. The asymptotic stability sets of such families will be described in the second part of the present paper.     

Stability Constants in Linear Spaces

We investigate the stability constants of convex sets in linear spaces. We prove that the stability constants of affinity and of the Jensen equation are of the same order of magnitude for every convex set in arbitrary linear spaces, even for functions mapping into an arbitrary Banach space. We also show that the second Whitney constant corresponding to the bounded functions equals half of the stability constant of the Jensen equation whenever the latter is finite. We show that if a convex set contains arbitrarily long segments in every direction, then its Jensen and Whitney constants are uniformly bounded. We prove a result that reduces the investigation of the stability constants to the case when the underlying set is the unit ball of a Banach space. As an application we prove that if D is convex and every δ-Jensen function on D differs from a Jensen function by a bounded function, then the stability constants of D are finite.     

The problem of applying representative sets in problems of constructing domains of stability and quality of dynamic control systems

We introduce the concept of a representative set as the set of values of the stability and quality characteristics possessing the properties of informativeness and nonredundancy. We obtain relations that determine representative sets for problems of constructing domains of stability and quality for linear stationary systems and retarded systems. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 16–20.     

Multi-fuzzy sets: An extension of fuzzy sets

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multifuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory.     

Ned sets on a hyperplane

Under study are the sets in ℝ ⁿ (NED sets) each of which does not affect the conformal capacity of any condenser with connected plates disjoint from this set. These sets are removable singularities of quasiconformal mappings, which explains our interest in them. For compact sets on a hyperplane we obtain a geometric criterion of the NED property; we point out a simple sufficient condition for an NED set in terms of the connected attainability of its points from its complement in the hyperplane. For compact sets on a hypersphere we obtain a criterion for an NED set in terms of the reduced module at a pair of points in its complement. We establish that a compact set on a hypersphere S, removable for the capacity in at least one spherical ring concentric with S and containing S, is an NED set.     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves

Minkowski geometric algebra is concerned with sets in the complex plane that are generated by algebraic combinations of complex values varying independently over given sets in ℂ. This algebra provides an extension of real interval arithmetic to sets of complex numbers, and has applications in computer graphics and image analysis, geometrical optics, and dynamical stability analysis. Algorithms to compute the boundaries of Minkowski sets usually invoke redundant segmentations of the operand-set boundaries, guided by a “matching” criterion. This generates a superset of the true Minkowski set boundary, which must be extracted by the laborious process of identifying and culling interior edges, and properly organizing the remaining edges. We propose a new approach, whereby the matching condition is regarded as an implicit curve in the space ℝⁿ whose coordinates are boundary parameters for the n given sets. Analysis of the topological configuration of this curve facilitates the identification of sets of segments on the operand boundaries that generate boundary segments of the Minkowski set, and rejection of certain sets that satisfy the matching criterion but yield only interior edges. Geometrical relations between the operand set boundaries and the implicit curve in ℝⁿ are derived, and the use of the method in the context of Minkowski sums, products, planar swept volumes, and Horner terms is described.     

On the range sets of variational inequalities

This paper investigates the closedness and convexity of the range sets of the variational inequality (VI) problem defined by an affine mappingM and a nonempty closed convex setK. It is proved that the range set is closed ifK is the union of a polyhedron and a compact convex set. Counterexamples are given such that the range set is not closed even ifK is a simple geometrical figure such as a circular cone or a circular cylinder in a three-dimensional space. Several sufficient conditions for closedness and convexity of the range set are presented. Characterization for the convex hull of the range set is established in the case whereK is a cone, while characterization for the closure of the convex hull of the range set is established in general. Finally, some applications to stability of VI problems are derived.This work was supported by the Australian Research Council.We are grateful to Professors M. Seetharama Gowda, Olvi Mangasarian, Jong-Shi Pang, and Steve Robinson for references. We are thankful to Professor Jim Burke for discussions on Theorem 2.1 and Counterexample 3.5.     

A new class of generalized f-projected dynamical system in Banach spaces

In this article, a new class of generalized f-projected dynamical systems is introduced and studied in Banach spaces. A global existence and uniqueness result of generalized f-projected dynamical system is proved, which generalizes the existence result of Xia and Vincent [Y.S. Xia and T.L. Vincent, On the stability of global projected dynamical systems, J. Optim. Theory Appl. 106 (2009), pp. 129–150]. The global convergence stability of the generalized f-projected dynamical system and the sensitivity result of solutions set with perturbations of the constraint sets are also obtained under some suitable conditions.     

Locating and paired-dominating sets in graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998), 199–206]. A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. We consider paired-dominating sets which are also locating sets, that is distinct vertices of G are dominated by distinct subsets of the paired-dominating set. We consider three variations of sets which are paired-dominating and locating sets and investigate their properties.     

On some subclasses of well-covered graphs

A set of points in a graph is independent if no two points in the set are adjacent. A graph is well covered if every maximal independent set is a maximum independent set or, equivalently, if every independent set is contained in a maximum independent set. The well-covered graphs are classified by the W_n property: For a positive integer n, a graph G belongs to class W_n if ≥ n and any n disjoint independent sets are contained in n disjoint maximum independent sets. Constructions are presented that show how to build infinite families of W_n graphs containing arbitrarily large independent sets. A characterization of W_n graphs in terms of well-covered subgraphs is given, as well as bounds for the size of a maximum independent set and the minimum and maximum degrees of points in W_n graphs.     

The stability of the solution sets for set optimization problems via improvement sets

ABSTRACT

The aim of this paper is to investigate the stability of the solution sets for set optimization problems via improvement sets. Firstly, we consider the relations among the solution sets for optimization problem with set optimization criterion. Then, the closeness and the convexity of solution sets are discussed. Furthermore, the upper semi-continuity, Hausdorff upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets are established under some suitable conditions. These results extend and develop some recent works in this field.     

Badly approximable vectors on fractals

For a large class of closed subsetsC of ℝ ⁿ , we show that the intersection ofC with the set of badly approximable vectors has the same Hausdorff dimension asC. The sets are described in terms of measures they support. Examples include (but are not limited to) self-similar sets such as Cantor’s ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson’s open set condition.     

Spaces in Which Countable Closed Sets Have Countable Character

It is proved that in a T ₃ space countable closed sets have countable character if and only if the set of limit point of the space is a countable compact set and every compact set is of countable character. Also, it is shown that spaces where countable sets have countable character are WN-spaces and are very close to M-spaces. Finally, some questions of Dai and Lia are discussed and some questions are proposed.     

Effectively and Noneffectively Nowhere Simple Sets

R. Shore proved that every recursively enumerable (r. e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice ? of all r. e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r. e. sets which can be split into two (non) effectively nowhere simple sets, and r. e. sets which can be split into two r. e. non-nowhere simple sets. We show that every r. e. set is either the disjoint union of two effectively nowhere simple sets or two noneffectively nowhere simple sets. We characterize r. e. sets whose every nontrivial splitting is into nowhere simple sets, and r. e. sets whose every nontrivial splitting is into effectively nowhere simple sets. R. Shore proved that for every effectively nowhere simple set A, the lattice L* (A) is effectively isomorphic to ?*, and that there is a nowhere simple set A such that L*(A) is not effectively isomorphic to ?*. We prove that every nonzero r. e. Turing degree contains a noneffectively nowhere simple set A with the lattice L*(A) effectively isomorphic to ?*. Mathematics Subject Classification: 03D25, 03D10.     

Prime Orders All of Whose Prime Suborders Are Selfdual

Let P be an order on a set V. A subset A of V is autonomous in P if every element of V not in A is either less than or greater than or incomparable to all elements of A. The empty set, the singletons from V and the set V are autonomous sets and are called trivial. Call an order prime if all its autonomous sets are trivial. We give the complete list of all finite prime orders all of whose prime suborders are selfdual.     

Bounded Immunity and Btt-Reductions

We define and study a new notion called k-immunity that lies between immunity and hyperimmunity in strength. Our interest in k-immunity is justified by the result that θ does not k-tt reduce to a k-immune set, which improves a previous result by Kobzev [7]. We apply the result to show that Φ′ does not btt-reduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new characterization of effectively simple sets and show that simple sets are not btt-cuppable.     

Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in ⁿ which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism . A Delone set of finite type is a Delone set X such that X-X is a discrete closed set. A Meyer set is a Delone set X such that X-X is a Delone set. Delone sets of finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on ^s and ⁿ , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map and a constant C such that ||ϕ (x) - (x)|| for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | η; and if X is a Meyer set, then all algebraic conjugates | η ' | 1. Received May 9, 1997, and in revised form March 5, 1998.     

Two-dimensional orthogonal polynomials,their associated sets and the co-recursive sets

This paper deals with the general theory of sets of polynomials verifying a (d+1)-order recurrence. The cased=2 is specially carried out. First, we introduce the notion of associated set of a set of monic polynomials. A general formula for successive associated polynomials is given. The co-recursive sets of a two-dimensional orthogonal set are introduced. We calculate the corresponding formal Stieltjes functions.Finally, we determine the self-associated two-dimensional orthogonal sets and we show they are classical two-dimensional orthogonal sets, that is to say, their set of derivatives is also a two-dimensional orthogonal set.