Some remarks on the genericity of unique range sets for meromorphic functions

In this paper we relate the study of unique range sets for meromorphic functions (URSM) with the hyperbolic hypersurfaces and give some remarks on the genericity of unique range sets for meromorphic functions.

     

On sensitivity analysis of parametric set-valued equilibrium problems under the weak efficiency

In the paper, we first extend calculus rules of variational sets, known as a kind of generalized derivatives for set-valued maps, from the first order to the second order. Then, we study sensitivity analysis of parametric set-valued equilibrium problems under the weak efficiency in terms of these sets.

     

Dual Representations of Convex Sets and Gâteaux Differentiability Spaces

The duality of two kinds of representations of convex sets is studied: the tangential representation of a convex body and the representations of its polar or negative polar by means of their weak^* exposed points. The equivalence of the representations is proved and a condition for their validity is obtained for individual sets (the case of arbitrary locally convex space) and for classes of sets (the case of Gâteaux differentiability locally convex space). Properties of Gâteaux differentiability locally convex spaces are studied and some examples of such spaces are given.

     

NOTE ON BRIDGING THEOREMS FOR SEMISIMPLICIAL SETS

Abstract

In this note we present a variant of bridging theorems in abstract homotopy theory which applies to the category of semisimplicial sets yielding bridging theorems for semisimplicial sets which might prove useful in semi-algebraic topology.     

Nonsequential approximate solutions in abstract problems of attainability

The problem of constructing attraction sets in a topological space is considered in the case when the choice of the asymptotic version of the solution is subject to constraints in the form of a nonempty family of sets. Each of these sets must contain an “almost entire” solution (for example, all elements of the sequence, starting from some number, when solution-sequences are used). In the paper, problems of the structure of the attraction set are investigated. The dependence of attraction sets on the topology and the family determining “asymptotic” constraints is considered. Some issues concerned with the application of Stone-Čech compactification and the Wallman extension are investigated.

     

Similar Transposed Sets of Polynomials

This paper studies the effectiveness of another kind of transposed sets of polynomials of one complex variable in closed regions, open discs, at the origin and for all entire functions. In addition, an upper bound for the order of these sets in this case is obtained.     

Irrational rotations motivate measurable sets

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Irrational rotations motivate measurable sets

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Zero sets of polynomials: one versus two variables

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Zero sets of polynomials: one versus two variables

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Zur Existenz freier Algebren einer r-dimensionalen Theorie

In [2] Linton introduced the notion of theory in such a way that he was able to indicate explicitly the leftadjoint of the underlying functor from the category of algebras over the theory in the sets. His method defines, in the case of the restriction on a r-ary theory [2] or a algebraische Theorie der Dimension r [3] (r an infinite cardinal number), the leftadjoint only for sets with cardinality less than r. In this paper it will be purely categorically shown what from the point of view of universal algebra is wellknown, that the remaining free algebras over a given r-ary theory can be obtained as injective limits of free algebras over sets with cardinality less than r. The proof stems from the following two observations: first that all sets are injective limits of the r-directed system of their subsets with cardinality less than r and secondly that for all sets m with cardinality less than r the Hom-functor preserves injective limits of r-directed systems of sets. In an appendix the endofunctors of the category of sets which preserve injective limits of r-directed systems of sets are characterized by the same property that the functor-part of a triple in the sets has by definition, if r is the rank of the triple [4].

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Supported by the National Research Council of Canada     

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.     

Some Geometric Properties of Julia Sets and filled-in Julia Sets of Polynomials

In this paper, some geometric properties of connected Julia sets and filled-in Julia sets of polynomials are given.     

Qualitative Properties of the Solution Set for Time-Delayed Discontinuous Dynamics

This article presents a survey of several properties of the set of solutions for a differential inclusion involving a time-delayed component and with right-hand side parametrized by either an upper semicontinuous or lower semicontinuous multifunction. Our results include: existence of solutions, compactness and contractibility of the solution and reachable sets in the upper semicontinuous case, precompactness and connectedness of the solution and reachable sets in the lower semicontinuous case, regularity of the solution and reachable mappings with respect to parameters, and existence of solutions for a dynamic optimization problem.

     

Projection theorems in hyperbolic space

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of the hyperbolic n-space by a geometric argument. Moreover, we obtain a Besicovitch–Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.

     

NON-ARCHIMEDEAN KÖTHE SPACES

Abstract

Köthe type non-Archimedean spaces are introduced in this paper. Bounded sets, compactoid sets, dual spaces, nuclearity, reflexivity and other properties of such spaces are investigated.     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

Boundedness of precompact sets of metric measure spaces

We give a detailed proof to Gromov’s statement that precompact sets of metric measure spaces are bounded with respect to the box distance and the Lipschitz order.

     

Convex-cone-based comparisons of and difference evaluations for fuzzy sets

Abstract

This paper focuses on how to compare two fuzzy sets and, from the viewpoint of set optimization, proposes eight types of fuzzy-set relations based on a convex cone as new comparison criteria of fuzzy sets. Then, difference evaluation functions for fuzzy sets are introduced. Under suitable assumptions of certain compactness and stability of fuzzy sets, we show that these functions correspond well to the fuzzy-set relations. In addition, through transforming these functions stepwise, we deal with numerical calculation methods of them in particular cases. Consequently, we can judge whether each fuzzy-set relation holds or not for given two fuzzy sets with the aid of computers.     

Analyticity and Singularity Sets

The problem of approximating singularity sets by analytic varieties has been considered by several authors. The present work deals with singularity sets projecting over a finitely connected domain. The results obtained in these cases allow us to prove the existence of a Riemann surface in a neighborhood of a two-sheeted singularity set.     

EQUIMEASURABLE SETS IN RIESZ SPACES

Abstract

We discuss the notion of equimeasurability in the general setting of Riesz spaces and obtain a characterization for (Carleman) abstract kernel operators in terms of equimea=surable sets.     

Error bounds for inequality systems defining convex sets

The main goal in this paper is to devise an approach to explicitly calculate the constant in the Hoffman’s error bound for (not necessarily convex) inequality systems defining convex sets. We give a constructive proof of the Hoffman’s error bound and show that we can use our method to calculate the constant at least in simple cases.