Preparation of papers

LetF be a family of real-valued maps onR ⁿ, and letY be a subset ofR ⁿ. Denote byS(Y|F) the set of ally* Y such that, for somef F,f(y)f(y*) for ally inY. Let us say thatF is a scalarization family if, for any subsetY,S(Y|F) is equal to the set of properly efficient points inY. General conditions forF to be a scalarization family were given in Ref. 1. However, scalarization families must contain nondifferentiable functions. In this note, it is shown that, if the condition of Ref. 1 which forces nondifferentiability is dropped, thenS(Y|F) is dense in the set of properly efficient points.     

An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables

Let B be the unit ball in C ⁿ and let U be the unit disc in C. The aim of this work is to construct a family of operators Ψ _n,α that provide a way to extend a locally univalent function ? ? H(U) to a locally univalent mapping F_α ? H(B), where α ? (0,1]. If ? is normalized univalent, then F_α can be imbedded in a Loewner chain. Also if ? ? S?, then F_α is starlike. We show that if ? belongs to a class of univalent functions which satisfy growth and distortion results, then the mapping F_α satisfies similar growth and distortion results. Also we study the concept of linear-invariant families as it relates to families generated by the operator Ψ _n,0, and we obtain in this way another example of a L.I.F. that has minimum order (n + 1)/2 and is not a subset of the normalized convex mappings in the unit ball of C ⁿ (for n ≤ 2.)     

Non commutative unique factorization rings

Let R and S be two rings. Each category equivalence between a torsion class of left (right) R-modules and a torsion-free class of left (right) S-modules is represented by a left (right) quasi-tilting triple. Suppose we have a pair of equivalences T ? Y and X F between the torsion class T of R-modules and the torsion-free class Y of S-modules and between the torsion class X of S-modules and the torsion-free class F of R-modules. Denote by (R, V, S) and (S, U, R) the quasi-tilting triples representing these equivalences. We say that (R, V, S) and (S, U, R) are complementary if T, F) and X, Y) are torsion theories in R-Mod and S-Mod, respectively. We find necessary and sufficient conditions on the bimodules _RV_S and _SU_R to have the complementarity of (R, V, S) and (S, U, R).     

On <Emphasis Type="Italic">n</Emphasis>-commuting and <Emphasis Type="Italic">n</Emphasis>-skew-commuting maps with generalized derivations in prime and semiprime rings

Let R be a ring with center Z(R), let n be a fixed positive integer, and let I be a nonzero ideal of R. A mapping h: R → R is called n-centralizing (n-commuting) on a subset S of R if [h(x),x ⁿ ] ∈ Z(R) ([h(x),x ⁿ ] = 0 respectively) for all x ∈ S. The following are proved:

Bernstein�CWalsh Type Theorems for Real Analytic Functions in Several Variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝ ^N . In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure F=g[`(h)]F=g\overline{h}, where g and h are holomorphic in a neighborhood of a compact set K⊂ℂ <sup>N</sup> . To this end the maximal convergence number ρ(S <sub>c</sub> ,f) for continuous functions f defined on a compact set S <sub>c</sub> ⊂ℂ <sup>N</sup> is connected to a maximal convergence number ρ(S <sub>r</sub> ,F) for continuous functions F defined on a compact set S <sub>r</sub> ⊂ℝ <sup>N</sup> . We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions F=g[`(h)]F=g\overline{h} if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.     

Higher-order symmetric duality in nondifferentiable multiobjective programming problems

In this paper, a pair of nondifferentiable multiobjective programming problems is first formulated, where each of the objective functions contains a support function of a compact convex set in Rⁿ. For a differentiable function h :Rⁿ×Rⁿ→R, we introduce the definitions of the higher-order F-convexity (F-pseudo-convexity, F-quasi-convexity) of function f :Rⁿ→R with respect to h. When F and h are taken certain appropriate transformations, all known other generalized invexity, such as η-invexity, type I invexity and higher-order type I invexity, can be put into the category of the higher-order F-invex functions. Under these the higher-order F-convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems related to a properly efficient solution.     

On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Let H¹(U) denote the space of all pointwise limits of bounded sequences from H(U), where H(U) consists of all continuous functions on the closure [`(U)]\overline{U} of a bounded open set U⊂ℝ^m that are harmonic on U. It is shown that the space H¹(U) is a lattice in the natural ordering if and only if the set ∂_regU of all regular points of U is an F_σ-set.     

Evasive random walks and the clairvoyant demon

A pair of random walks (R, S) on the vertices of a graph G is successful if two tokens moving one at a time can be scheduled (moving only one token at a time) to travel along R and S without colliding. We consider questions related to P. Winkler's clairvoyant demon problem, which asks whether for random walks R and S on G, Pr[(R, S)is successful] >0. We introduce the notion of an evasive walk on G: a walk S so that for a random walk R on G, Pr[(R, S)is successful] >0. We characterize graphs G having evasive walks, giving explicit constructions on such G. Also, we show that on a cycle, the tokens must collide quickly with high probability. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 220–229, 2002     

Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M and I^?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I^?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.     

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.     

Intersection formulae for the kernel of a cone

A subset S of Euclidean space is called a cone if it is the union of a set of halflines having the same endpoint called the apex of the cone, and the set of all such apices is denoted by ker _R S and called the R-kernel or, when it does not lead to any confusion with the kernel of a starshaped set, simply the kernel of S. ker _R S is shown to be the intersection of a family of flats passing through some selected boundary points of S. Three independent formulae of this type are established, respectively: for an arbitrary proper subset S, for S closed, and for S closed connected and nonconvex.The author is with the Department of Mathematics, University of Notre Dame, Indiana, on leave from the Mathematical Institute of the Polish Academy of Sciences.     

Algebraic aspects of generalized approximation spaces

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of R-open sets, R-closed sets and regular sets of a generalized approximation space (U,R) are introduced. Algebraic structures of various families of subsets of (U,R) under the set-inclusion order are investigated. Main results are: (1) The family of all R-open sets (respectively, R-closed sets, R-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation R is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation R is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive.     

Variable neighborhood search for extremal graphs. 22. Extending bounds for independence to upper irredundance

A set of vertices S in a graph G is independent if no neighbor of a vertex of S belongs to S. A set of vertices U in a graph G is irredundant if each vertex v of U has a private neighbor, which may be v itself, i.e., a neighbor of v which is not a neighbor of any other vertex of U. The independence number α (resp. upper irredundance number IR) is the maximum number of vertices of an independent (resp. irredundant) set of G. In previous work, a series of best possible lower and upper bounds on α and some other usual invariants of G were obtained by the system AGX 2, and proved either automatically or by hand. These results are strengthened in the present paper by systematically replacing α by IR. The resulting conjectures were tested by AGX which could find no counter-example to an upper bound nor any case where a lower bound could not be shown to remain tight. Some proofs for the bounds on α carry over. In all other cases, new proofs are provided.     

Construction method for some real division algebras with <Emphasis Type="Italic">su</Emphasis>(3) as derivation algebra

We obtain a family of eight-dimensional unital division algebras over a field F out of a separable quadratic field extension S of F, a three-dimensional anisotropic hermitian form h over S of determinant one and an element c ∈ S ^× not contained in F. These algebras are not third-power associative.     

Automorphism groups of certain simple 2-(q,3,λ) designs constructed from finite fields

Let F be a finite field of characteristic not 2, and SF a subset with three elements. Consider the collection

S={S·a+b | a,bF, a≠0}.

Then (F,S) is a simple 2-design and the parameter λ of (F,S) is 1, 2, 3 or 6. We find in this paper the full automorphism group of (F,S). Namely, if we put U={r | {0,1,r}S} and K the subfield of F generated by U, then the automorphisms of (F,S) are the maps of the form xg(α(x))+b, xF, where bF, α : F→F is a field automorphism fixing U, and g is a linear transformation of F considered as a vector space over K.     

Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ?-ideals contained in the maximal ?-ideal M _x  = {f ? C(X):f(x) = 0}. The space X has finite rank if there is an n ? N such that every point x ? X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ? X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.     

On co-semihereditary rings

A ringR is left co-semihereditary (strongly left co-semihereditary) if every finitely cogenerated factor of a finitely cogenerated (arbitrary) injective leftR-module is injective. A left co-semihereditary ring, which is not strongly left co-semihereditary, is given to answer a question of Miller and Tumidge in the negative. If _R U _S defines a Morita duality,R is proved to be left co-semihereditary (left semihereditmy) if and only ifS is right semihereditary (right co-semihereditary). Assuming thatS⩾R is an almost excellent extension,S is shown to be (strongly) right co-semihereditary if and only ifR is (strongly) right co-semihereditary. Project supported by the National Natural Science Foundation of China.     

Estimates for maximal functions associated with hypersurfaces in ℝ<Superscript>3</Superscript> and related problems of harmonic analysis

We study the boundedness problem for maximal operators M \mathcal{M} associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we prove that if no affine tangent plane to S passes through the origin and S is analytic, then the associated maximal operator is bounded on L^p( \mathbbR³ ) {L^p}\left( {{\mathbb{R}^3}} \right) if and only if p > h(S), where h(S) denotes the so-called height of the surface S (defined in terms of certain Newton diagrams). For non-analytic S we obtain the same statement with the exception of the exponent p = h(S). Our notion of height h(S) is closely related to A. N. Varchenko’s notion of height h(ϕ) for functions ϕ such that S can be locally represented as the graph of ϕ after a rotation of coordinates.     

Gabor frame sets for subspaces

This paper investigates Gabor frame sets in a periodic subset \mathbb S\mathbb S of \mathbb R\mathbb R. We characterize tight Gabor sets in \mathbb S\mathbb S, and obtain some necessary/sufficient conditions for a measurable subset of \mathbb S\mathbb S to be a Gabor frame set in \mathbb S\mathbb S. We also characterize those sets \mathbb S\mathbb S admitting tight Gabor sets, and obtain an explicit construction of a class of tight Gabor sets in such \mathbb S\mathbb S for the case that the product of time-frequency shift parameters is a rational number. Our results are new even if \mathbb S=\mathbb R\mathbb S=\mathbb R.