Score sets in oriented 3-partite graphs

Let D(U, V, W) be an oriented 3-partite graph with |U|=p, |V|=q and |W|= r. For any vertex x in D(U, V, W), let d x and d-x be the outdegree and indegree of x respectively. Define aui (or simply ai) = q r d ui - d-ui, bvj(or simply bj) = p r d vj - d-vj and Cwk (or simply ck) = p q d wk - d-wk as the scores of ui in U, vj in V and wk in Wrespectively. The set A of distinct scores of the vertices of D(U, V, W) is called its score set. In this paper, we prove that if a1 is a non-negative integer, ai(2≤i≤n - 1) are even positive integers and an is any positive integer, then for n≥3, there exists an oriented 3-partite graph with the score set A = {a1,2∑i=1 ai,…,n∑i=1 ai}, except when A = {0,2,3}. Some more results for score sets in oriented 3-partite graphs are obtained.     

Approximation of holomorphic maps with a lower bound on the rank

Let be a closed polydisc or ball in , and let be a quasi-projective algebraic manifold which is Zariski locally equivalent to , or a complement of an algebraic subvariety of codimension in such a manifold. If is an integer satisfying , then every holomorphic map from a neighborhood of to with rank at every point of can be approximated uniformly on by entire maps with rank at every point of .

     

Some new results in multiplicative and additive Ramsey theory

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including and . It was recently shown that sets in which are multiplicatively large must contain arbitrarily large geoarithmetic progressions, that is, sets of the form , as well as sets of the form . Consequently, given a finite partition of , one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups and , derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of there must be, for each , sets of the form together with , the arithmetic progression , and the geometric progression in one cell of the partition. More generally, we show that, if is a commutative semigroup and a partition regular family of finite subsets of , then for any finite partition of and any , there exist and such that is contained in a cell of the partition. Also, we show that for certain partition regular families and of subsets of , given any finite partition of some cell contains structures of the form for some .

     

A fuzzy goal programming method with imprecise goal hierarchy

Two most widely used approaches to treating goals of different importance in goal programming (GP) are: (1) weighted GP, where importance of goals is modelled using weights, and (2) preemptive priority GP, where a goal hierarchy is specified implying infinite trade-offs among goals placed in different levels of importance. These approaches may be too restrictive in modelling of real life decision making problems. In this paper, a novel fuzzy goal programming method is proposed, where the hierarchical levels of the goals are imprecisely defined. The imprecise importance relations among the goals are modelled using fuzzy relations. An additive achievement function is defined, which takes into consideration both achievement degrees of the goals and degrees of satisfaction of the fuzzy importance relations. Examples are given to illustrate the proposed method.     

Rough set based rule induction from two decision tables

We study rule induction from two decision tables as a basis of rough set analysis of more than one decision tables. We regard the rule induction process as enumerating minimal conditions satisfied with positive examples but unsatisfied with negative examples and/or with negative decision rules. From this point of view, we show that seven kinds of rule induction are conceivable for a single decision table. We point out that the set of all decision rules from two decision tables can be split in two levels: a first level decision rule is positively supported by a decision table and does not have any conflict with the other decision table and a second level decision rule is positively supported by both decision tables. To each level, we propose rule induction methods based on decision matrices. Through the discussions, we demonstrate that many kinds of rule induction are conceivable.     

Moments of an exponential functional of random walks and permutations with given descent sets

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ₁ + ξ₁ξ₂ + ξ₁ξ₂ξ₃ + ⋯ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μ _k = E(ξ ^k ) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle. This revised version was published online in August 2006 with corrections to the Cover Date.     

无限差集类及其应用

若S为Z+的一无限子集,称S-S={n-m|n,m∈S,n>m}为一无限差集.本文研究Z+的子集族F生成的无限差集类(?)F-△={S-S|S∈F)及其对偶族K(?)F-△的性质,并讨论它们在动力系统研究中的应用.     

Separating the solution sets of analytical and polynomial systems

A linear inequality system with infinitely many constraints is polynomial (analytical) if its index set is a compact interval of the real line and all its coefficients are polynomial (analytical, respectively) functions of the index on this interval. This paper provides an example of analytical system whose solution set cannot be the solution set of any polynomial system. Research supported by DGES of Spain and FEDER of UE, Grant BFM2002-04114-C02-01. Research supported by CONACyT of Mexico, Grant 130036. Research partially supported by CONACyT of Mexico, Grant 44003.     

有关拓扑学与覆盖粗糙集的一些结论

引入了拓扑覆盖的概念,并结合最小描述元对有限论域上的拓扑覆盖加于研究,得出了拓扑覆盖的最简覆盖和基与最小描述元之间的关系.介绍了在基于有限论域U上的覆盖,构造U上的一个拓扑的方法.并且在最小描述元的基础上将划分下的粗糙隶属函数推广至一般覆盖下的粗糙隶属函数,而后介绍了其相关运用.     

LK property for σ-ideals

An ideal J of subsets of a Polish space X has (LK) property whenever for every sequence (An) of analytic sets in X, if lim supn∈HAn∉J for each infinite H then _?n∈G∉J for some infinite G. In this note we present a new class of σ-ideals with (LK) property.