Selections and hyperspaces of finite sets

It is demonstrated that the hyperspace of at most (n+1)-point sets has a Vietoris continuous selection if both the hyperspace of at most n-point sets and that of exactly (n+1)-point sets have Vietoris continuous selections. This result is applied to demonstrate that the hyperspace of at most (2n+2)-point sets has a Vietoris continuous selection provided that one of at most (2n+1)-point sets has such a selection. This settles some open questions.     

On Minimal Defining Sets of Full Designs and Self-Complementary Designs, and a New Algorithm for Finding Defining Sets of t-Designs

A defining set of a t-(v, k, λ) design is a partial design which is contained in a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper partial designs is a defining set. This paper proposes a new and more efficient algorithm that finds all non-isomorphic minimal defining sets of a given t-design. The complete list of minimal defining sets of 2-(6, 3, 6) designs, 2-(7, 3, 4) designs, the full 2-(7, 3, 5) design, a 2-(10, 4, 4) design, 2-(10, 5, 4) designs, 2-(13, 3, 1) designs, 2-(15, 3, 1) designs, the 2-(25, 5, 1) design, 3-(8, 4, 2) designs, the 3-(12, 6, 2) design, and 3-(16, 8, 3) designs are given to illustrate the efficiency of the algorithm. Also, corrections to the literature are made for the minimal defining sets of four 2-(7, 3, 3) designs, two 2-(6, 3, 4) designs and the 2-(21, 5, 1) design. Moreover, an infinite class of minimal defining sets for 2-((v) || 3){v\choose3} designs, where v ≥ 5, has been constructed which helped to show that the difference between the sizes of the largest and the smallest minimal defining sets of 2-((v) || 3){v\choose3} designs gets arbitrarily large as v → ∞. Some results in the literature for the smallest defining sets of t-designs have been generalized to all minimal defining sets of these designs. We have also shown that all minimal defining sets of t-(2n, n, λ) designs can be constructed from the minimal defining sets of their restrictions when t is odd and all t-(2n, n, λ) designs are self-complementary. This theorem can be applied to 3-(8, 4, 3) designs, 3-(8, 4, 4) designs and the full 3-(8 || 4)3-{8 \choose 4} design using the previous results on minimal defining sets of their restrictions. Furthermore we proved that when n is even all (n − 1)-(2n, n, λ) designs are self-complementary.     

Unification of generalized open sets on topological spaces

A new kind of sets called generalized μ-closed (briefly g μ-closed) sets are introduced and studied in a topological space by using the concept of generalized open sets introduced by á. Császár. The class of all g μ-closed sets is strictly larger than the class of all μ-closed sets. Furthermore, g-closed sets (in the sense of N. Levine [17]) is a special type of g μ-closed sets in a topological space. Some of their properties are investigated. Finally, some characterizations of μ _g -regular and μ _g -normal spaces have been given.     

The existence of large set of symmetric partitioned incomplete latin squares

In this paper, we investigate the existence of large sets of symmetric partitioned incomplete latin squares of type g^u (LSSPILSs) which can be viewed as a generalization of the well‐known golf designs. Constructions for LSSPILSs are presented from some other large sets, such as golf designs, large sets of group divisible designs, and large sets of Room frames. We prove that there exists an LSSPILS(g^u) if and only if u ≥ 3, g(u ? 1) ≡ 0 (mod 2), and (g, u) ≠ (1, 5).     

A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three

We construct a new family of cyclic difference sets with parameters ((3 ^d – 1)/2, (3 ^{d – 1} – 1)/2, (3 ^{d – 2} – 1)/2) for each odd d. The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti's hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets.     

Between closed sets andg-closed sets

Quite recently, by using semi-open (resp.α-open, preopen,β-open) sets in a topological space, the notions ofsg*-closed (resp.αg*-closed,pg*-closedβg*-closed) sets are indroduced and investigated in [8]. These subsets place between closed sets andg-closed sets due to Levine [5]. In this paper, we introduce the notion ofmg*-closed sets and obtain the unified theory for collections of subsets between closed sets andg-closed sets.     

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.     

Frequencies of digits, divergence points, and Schmidt games

Sets of divergence points, i.e. numbers x (or tuples of numbers) for which the limiting frequency of a given string of N-adic digits of x fails to exist, have recently attracted huge interest in the literature. In this paper we consider sets of simultaneous divergence points, i.e. numbers x (or tuples of numbers) for which the limiting frequencies of all strings of N-adic digits of x fail to exist. We show that many natural sets of simultaneous divergence points are (α, β)-wining sets in the sense of the Schmidt game. As an application we obtain lower bounds for the Hausdorff dimension of these sets.     

Convex Sets in Lexicographic Products of Graphs

Geodesic convex sets, Steiner convex sets, and J-convex (alias induced path convex) sets of lexicographic products of graphs are characterized. The geodesic case in particular rectifies Theorem 3.1 in Canoy and Garces (Graphs Combin 18(4):787–793, 2002).     

Sets of type (m,n) in the affine and projective planes of order nine

In this paper we give the results of exhaustive computer searches for sets of points of type (m, n) in the projective and affine planes of order nine. In particular, as the list of planes of order nine is known to be complete, our results are also complete. We also examine all known constructions of sets of type (m, n) that apply to the planes of order nine, in an attempt to summarise and extend all existing knowledge about such sets. The contrast between known constructions and our computer results leads us to conclude that sets of type (m, n) are far more numerous than was previously thought.     

An enumeration of graphical designs

Let Ψ(t,k) denote the set of pairs (v,λ) for which there exists a graphical t‐(v,k,λ) design. Most results on graphical designs have gone to show the finiteness of Ψ(t,k) when t and k satisfy certain conditions. The exact determination of Ψ(t,k) for specified t and k is a hard problem and only Ψ(2,3), Ψ(2,4), Ψ(3,4), Ψ(4,5), and Ψ(5,6) have been determined. In this article, we determine completely the sets Ψ(2,5) and Ψ(3,5). As a result, we find more than 270,000 inequivalent graphical designs, and more than 8,000 new parameter sets for which there exists a graphical design. Prior to this, graphical designs are known for only 574 parameter sets. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 70–85, 2008     

An analysis of some dual problems in multiobjective optimization (I)

In this work we study the duality for a general multiobjective optimization problem. Considering, first, a scalar problem, different duals using the conjugacy approach are presented. Starting from these scalar duals, we introduce six different multiobjective dual problems to the primal one, one depending on certain vector parameters. The existence of weak and, under certain conditions, strong duality between the primal and the dual problems is shown. Afterwards, some inclusion results for the image sets of the multiobjective dual problems (D ₁), (D _α) and (D_FL ) are derived. Moreover, we verify that the efficiency sets within the image sets of these problems coincide, but the image sets themselves do not.     

Maximal sets of non-polar points of quadrics and symplectic polarities over GF(2)

A cap on a non-singular quadric over GF(2) is a set of points that are pairwise non-polar; equivalently the join of any two of the points is a chord. A non-secant set of the quadric is a set of points off the quadric that are pairwise non-polar; equivalently the join of any two of the points is skew to the quadric. We determine all the maximal caps and all the maximal non-secant sets of all non-singular quadrics over GF(2); and also all the maximal sets of non-polar points for symplectic polarities over GF(2). The classification is in terms of caps of greatest size on elliptic quadrics Q _– ^8k+3 (2), hyperbolic quadrics Q ₊ ^8k+7 (2) and on quadrics Q _4k+2(2), and of non-secant sets of greatest size of Q _– ^8k+1 (2), Q ₊ ^8k+5 (2) and Q _4k (2), for all quadrics of these types that occur as sections of the parent quadric or belong to the symplectic polarity. The sets of greatest size for these types of quadrics are larger than for other types. The results have implications about the non-existence of ovoids and the exterior sets of Thas. Only one part of the simple geometric inductive argument extends to larger ground fields.     

Normed algebras of differentiable functions on compact plane sets

We investigate the completeness and completions of the normed algebras (D ⁽¹⁾(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D ⁽¹⁾(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D ⁽¹⁾(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ.     

Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets

This paper is a continuation of the author's first paper (Set-Valued Anal. 9 (2001), pp. 217–245), where the normed and partially ordered vector space of directed sets is constructed and the cone of all nonempty convex compact sets in R ⁿ is embedded. A visualization of directed sets and of differences of convex compact sets is presented and its geometrical components and properties are studied. The three components of the visualization are compared with other known differences of convex compact sets.     

Topological classification of (n−1)-convex sets

We investigate the properties of (n−1)-convex sets associated with the properties of conjugate sets. We give a complete topological classification of (n−1)-convex sets. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1236–1243, September, 1998.     

Special sets of the Hermitian surface and indicator sets

An interesting connection between special sets of the Hermitian surface of PG(3,q²), q odd, (after Shult 13 ) and indicator sets of line‐spreads of the three‐dimensional projective space is provided. Also, the CP‐type special sets are characterized. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 18–24, 2008     

On Value Sets of Polynomials over a Finite Field

We study value sets of polynomials over a finite field, and value sets associated to pairs of such polynomials. For example, we show that the value sets (counting multiplicities) of two polynomials of degree at mostdare identical or have at mostq−(q−1)/dvalues in common whereqis the number of elements in the finite field. This generalizes a theorem of D. Wan concerning the size of a single value set. We generalize our result to pairs of value sets obtained by restricting the domain to certain subsets of the field. These results are preceded by results concerning symmetric expressions (of low degree) of the value set of a polynomial. K. S. Williams, D. Wan, and others have considered such expressions in the context of symmetric polynomials, but we consider (multivariable) polynomials invariant under certain important subgroups of the full symmetry group.     

Finite groups acting 2-transitively on their sylow subgroups

In this paper we have completely determined: (1) all almost simple groups which act 2-transitively on one of their sets of Sylow p-subgroups. (2) all non-abelian simple groups T whose automorphism group acts 2-transitively on one of the sets of Sylow p-subgroups of T. (3) all finite groups which are 2-transitive on all their sets of Sylow subgroups. The first author acknowledges the support of OPR Scholarship of Australia The second author is supported by the National Natural Science Foundation of China. Thanks are also due to the Department of Mathematics, the University of Western Australia, where he did his part of this work for its hospitality     

The sigma-core of a cooperative game

This paper is concerned with the existence of (σ-additive) measures in the core of a cooperative game. The main theorem shows, for a capacityu on the Borel sets of a metric space, that to each additive set function, majorized byu and agreeing withu on a system of closed sets, there exists a measure having these same properties. This theorem is applied, in combination with known core theorems, to the case of a cooperative game defined on the Borel sets of a metric space and whose conjugate is a capacity.