Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF _σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.     

Diameter of the set of midpoints of chords of a bounded closed set

The estimate is obtained for the diameter d(S_n(a)) of the set S_n(a) of midpoints of chords of length ≥a(0n, namely $$d(S_n (a)) \leqslant \left\{ \begin{gathered} 1 - a^2 /2, n = 2, \hfill \\ \sqrt {1 - a^2 /2,} n \geqslant 3, \hfill \\ \end{gathered} \right.$$ and it is shown that the inequality cannot be improved.     

Approximating the Pareto optimal set using a reduced set of objective functions

Real-world applications of multi-objective optimization often involve numerous objective functions. But while such problems are in general computationally intractable, it is seldom necessary to determine the Pareto optimal set exactly. A significantly smaller computational burden thus motivates the loss of precision if the size of the loss can be estimated. We describe a method for finding an optimal reduction of the set of objectives yielding a smaller problem whose Pareto optimal set w.r.t. a discrete subset of the decision space is as close as possible to that of the original set of objectives. Utilizing a new characterization of Pareto optimality and presuming a finite decision space, we derive a program whose solution represents an optimal reduction. We also propose an approximate, computationally less demanding formulation which utilizes correlations between the objectives and separates into two parts. Numerical results from an industrial instance concerning the configuration of heavy-duty trucks are also reported, demonstrating the usefulness of the method developed. The results show that multi-objective optimization problems can be significantly simplified with an induced error which can be measured.     

Density of the set of positive proper minimal points in the set of minimal points

One of the important problems of vector optimization concerns the density of the set of positive proper minimal points in the set of minimal points. We use the concepts of dentable point and approximating cones to derive sufficient conditions guaranteeing that the set of minimal points is contained in the closure of the set of positive proper minimal points. The result can be applied to obtain a density result for the unit ball in 1 ^p , 1<p<+, which does not follow from any other well-known density theorem.The author would like to thank Professor W. T. Fu for helpful comments. Moreover, the author is grateful to Professor H. P. Benson and the referees for valuable remarks and suggestions concerning a previous draft of this paper.     

拟变分不等式解集的极小本质集及应用

引入了拟变分不等式解集的极小本质集的概念，并证明了每个拟变分不等式(满足一定条件)的解集至少存在一个极小本质集．作为应用，还证明了大多数(在Baire分类意义下)拟-似变分不等式问题的解集是稳定的；每个拟-似变分不等式(满足一定条件)的解集至少存在一个本质连通区．     

On the packing dimension of the Julia set and the escaping set of an entire function

Let f be a transcendental entire function. We give conditions which imply that the Julia set and the escaping set of f have packing dimension 2. For example, this holds if there exists a positive constant c less than 1 such that the minimum modulus L(r, f) and the maximum modulus M(r, f) satisfy log L(r, f) ≤ c logM(r, f) for large r. The conditions are also satisfied if logM(2r, f) ≥ d logM(r, f) for some constant d greater than 1 and all large r.     

Distributionally scrambled set and minimal set

We investigate the relation between distributional chaos and minimal sets, and discuss how to obtain various distributionally scrambled sets by using least and simplest minimal sets. We show: i) an uncountable extremal distributionally scrambled set can appear in a system with just one simple minimal set: a periodic orbit with period 2; ii) an uncountable dense invariant distributionally scrambled set can occur in a system with just two minimal sets: a fixed point and an infinite minimal set; iii) infinitely many minimal sets are necessary to generate a uniform invariant distributionally scrambled set, and an uncountable dense extremal invariant distributionally scrambled set can be constructed by using just countably infinitely many periodic orbits.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...     

The union of a Riesz set and a Lust-Piquard set is a Riesz set

We prove that the union of a Riesz set and a Lust-Piquard set is a Riesz set. This gives as corollaries known results of Y. Katznelson, R.E. Dressler-L. Pigno, and D. Li. Moreover, we give an example of a Rosenthal set which is dense in Z for the Bohr topology.     

Partitions of a set satisfying certain set of conditions

In this paper we present a new combinatorial class enumerated by Catalan numbers. More precisely, we establish a bijection between the set of partitions π₁π₂?π_n of [n] such that π_i+1−π_i≤1 for all i=,1,2…,n−1, and the set of Dyck paths of semilength n. Moreover, we find an explicit formula for the generating function for the number of partitions π₁π₂?π_n of [n] such that either π_i+?−π_i≤1 for all i=1,2,…,n−?, or π_i+1−π_i≤m for all i=1,2,…,n−1.     

A unified treatment of fuzzy set theory and Boolean-valued set theory—Fuzzy set structures and normal fuzzy set structures

This is Part I of a two-part paper; the purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of infinite-dimensional programming, and (b) to obtain fruitful applications in continuous time programming. Part I deals with the development of continuous time analogues to those concepts which are the cornerstones of finite-dimensional programming theory. Specifically, a constraint qualification analogous to that found in finite-dimensional programming and a continuous time version of Farkas' theorem are developed. The latter result, stated in terms of convex and polar cones, is then employed in conjunction with the newly-developed constraint qualification to establish necessary conditions and a duality theory for a class of nonlinear continuous time programming problems. This approach to duality permits the imposition of assumptions that are less stringent than those needed for duality in previous formulations of the nonlinear problem.     

Conformal dimension of the antenna set

We show that the self-similar set known as the ``antenna set' has the property that (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space.

     

On the complement of the Mandelbrot set

We study the uniformization function of the Mandelbrot set via the behavior of multipliers of periodic orbits     

On the diameter of the Banach-Mazur set

On every subspace of l _∞(ℕ) which contains an uncountable ω-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of l _∞(ℕ) is infinite. This provides a partial answer to a question asked by Johnson and Odell.     

The area of the Mandelbrot set

Summary We obtain upper bounds for the area of the Mandelbrot set. An effective procedure is given for computing the coefficients of the conformal mapping from the exterior of the unit circle onto the exterior of the Mandelbrot set. The upper bound is obtained by computing finitely many of these coefficient and applying Green's Theorem. The error in such calculations is estimated by deriving explicit formulas for infinitely many of the coefficients and comparing.Partially supported by a grant from the National Science Foundation     

Interpretations of the alternative set theory

Summary We show an axiom A such that there is no nontrivial interpretation of the alternative set theory (AST) inAST+A keeping , sets and the class of all standard natural numbers. Furthermore, there is no interpretation ofAST inAST without the prolongation axiom, but there is an interpretation ofAST in the theory having the prolongation axiom and the basic set-theoretical axioms only.     

Facets of the independent set polytope

Theoretical results pertaining to the independent set polytope P^ISP=conv{x{0,1}ⁿ:Axb} are presented. A conflict hypergraph is constructed based on the set of dependent sets which facilitates the examination of the facial structure of P^ISP. Necessary and sufficient conditions are provided for every nontrivial 0-1 facet-defining inequalities of P^ISP in terms of hypercliques. The relationship of hypercliques and some classes of knapsack facet-defining inequalities are briefly discussed. The notion of lifting is extended to the conflict hypergraph setting to obtain strong valid inequalities, and back-lifting is introduced to strengthen cut coefficients. Preliminary computational results are presented to illustrate the usefulness of the theoretical findings.Mathematics Subject Classification (2000): 90C11, 90C57, 90C35