强Raney偏序集

引入强Raney偏序集的概念,讨论了强Raney偏序集的一些性质,证明了强Raney偏序集为超代数偏序集,定向完备的偏序集为强Raney偏序集当且仅当它既是Raney偏序集也是A-偏序集.     

H-代数偏序集

本文引入了H-代数偏序集的概念,讨论了它的一些基本性质。得到如下主要结果:(1)举例说明了代数偏序集未必是H-代数偏序集;(2)偏序集是H-代数偏序集当且仅当强紧元是它的强基;(3)偏序集是H-代数偏序集当且仅当它的局部Scott拓扑是强代数格。     

FS-偏序集和连续L-偏序集

引入了FS-偏序集和连续L-偏序集概念,探讨了FS-偏序集和连续L-偏序集的性质.主要结果有(1)每一FS-偏序集都是有限上集生成的,因而是Scott紧的;(2)证明了FS-偏序集(连续L-偏序集)的定向完备化是FS-偏序集(连续L-偏序集);(3)一个偏序集是一个FS-Domain当且仅当它为Lawson紧的FS-偏序集;(4)FS-偏序集(连续L-偏序集)去掉部分极大元后还是FS-偏序集(连续L-偏序集).     

偏序集的φ-(交)连续性及其主理想刻画

证明φ-完备偏序集是（强）P连续的当且仅当该偏序集的任一主理想是（强）φ-连续的。在φ-完备偏序集中利用φ-S集族生成f-Scott拓扑，并由此引入φ-交连续偏序集概念。证明φ-完备偏序集是P交连续的当且仅当该偏序集的任一主理想是φ-交连续的。     

带Morita对偶的偏序集环

设Ｒ是有１的结合环，Ｉ是任意偏序集，ＲＩ是Ｒ上Ｉ的偏序集环．本文考虑了带对偶的偏序集环，得到：ＲＩ带Ｍｏｒｉｔａ对偶当且仅当Ｒ带Ｍｏｒｉｔａ对偶．推广了已有的在Ｒ是有限偏序集时的有关结果     

可数S_2-拟连续偏序集北大核心CSCD

作为广义可数逼近偏序集与S2-拟连续偏序集的共同推广,引入了可数S2-拟连续偏序集的概念并讨论了它的一些性质.本文的主要结果:(1)可数S2-拟连续偏序集上的可数way below关系满足插入性质;(2)可数S2-拟连续偏序集关于其上的弱σ-Scott拓扑为局部紧致的可数sober空间;(3)偏序集P为可数S2-连续偏序集当且仅当P为可数S2-交连续的可数S2-拟连续偏序集.     

可数S_2-拟连续偏序集

作为广义可数逼近偏序集与S2-拟连续偏序集的共同推广,引入了可数S2-拟连续偏序集的概念并讨论了它的一些性质.本文的主要结果:(1)可数S2-拟连续偏序集上的可数way below关系满足插入性质;(2)可数S2-拟连续偏序集关于其上的弱σ-Scott拓扑为局部紧致的可数sober空间;(3)偏序集P为可数S2-连续偏序集当且仅当P为可数S2-交连续的可数S2-拟连续偏序集.     

交C-连续偏序集

利用偏序集上的半拓扑结构,引入了交C-连续偏序集概念,探讨了交C-连续偏序集的性质、刻画及与C-连续偏序集、拟C-连续偏序集等之间的关系.主要结果有:(1)交C-连续的格一定是分配格;(2)有界完备偏序集(简记为bc-poset)L是交C-连续的当且仅当对任意x∈L及非空Scott闭集S,当∨S存在时有x∧∨S=∨{x∧s:s∈S};(3)完备格是完备Heyting代数当且仅当它是交连续且交C-连续的;(4)有界完备偏序集是C-连续的当且仅当它是交C-连续且拟C-连续的;(5)获得了反例说明分配的完备格可以不是交C-连续格,交C-连续格也可以不是交连续格.     

C_Z-偏序集

我们将强Z-连续偏序集推广到了CZ-偏序集,并讨论了CZ-偏序集和强Z-连续偏序集之间的关系。同时我们定义了CZ-偏序集上的CZ-连续映射,得到CZ-偏序集在该映射下的像集仍是CZ-偏序集。最后,我们讨论了CZ-偏序集上的基及其相关性质。     

偏序集的内蕴拓扑连通性北大核心

从序与拓扑的交叉考虑,进一步研究偏序集在多种内蕴拓扑下的连通性和局部连通性.主要结果有:(1)一个偏序集是序连通的当且仅当它赋予Alexandrov拓扑是连通的,也当且仅当它赋予Scott拓扑是连通的;(2)每一偏序集赋予Alexandrov拓扑是局部连通的,每一偏序集赋予Scott拓扑是局部连通的;(3)如果拓扑空间的特殊化偏序集序连通,则该拓扑空间是连通的;(4)构造反例说明了存在偏序集赋予下拓扑后是连通空间,但该偏序集本身不是序连通的.     

Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.     

Density and unique decomposition theorems for the lattice of cellular classes

A classC of pointed spaces is called a cellular class if it is closed under weak equivalences, arbitrary wedges and pointed homotopy pushouts. The smallest cellular class containingX is denoted byC(X), and a partial order relation ≪ is defined by:X ≪Y ifY εC(X). In this text we investigate the sub partial order sets generated respectively by simply connected finite CW-complexes and by rational spaces. For rational spaces we prove a unique decomposition theorem, a density theorem and the existence of infinitely many non-comparable elements. We then prove the density theorem for a generic class of finite CW-complexes.     

Discrete unbounded sets in a finite dimensional space and beyond

Discrete infinite sets in a finite dimensional space, i.e., infinite sets without finite limit points appear in various branches of analysis (zero and pole sets of meromorphic functions, various models in the mathematical theory of quasicrystals, and so on). Here we introduce some notions and present some new theorems connected with such sets.     

Connected Domatic Number in Planar Graphs

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.     

Finite-Dimensional Spaces where the Class of Chebyshev Sets Coincides with the Class of Closed and Monotone Path-Connected Sets

Mathematical Notes - In a two-dimensional Banach space $$X$$ , the class of Chebyshev sets coincides with the class of closed and monotone path- connected sets if and only if $$X$$ is strictly...     

Convex Sets in Acyclic Digraphs

A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by and its size by co(D) (cc(D)). Gutin et al. (2008) conjectured that the sum of the sizes of all convex sets (connected convex sets) in D equals Θ(n · co(D)) (Θ(n · cc(D))) where n is the order of D. In this paper we exhibit a family of connected acyclic digraphs with and . We also show that the number of connected convex sets of order k in any connected acyclic digraph of order n is at least n − k + 1. This is a strengthening of a theorem of Gutin and Yeo.     

Minimal balanced sets and finite geometries

A balanced set is a collection of subsets of a finite set that can be weighted so as to cover the whole set uniformly. Minimal balanced sets are of interest in the theory of n-person games, in particular for the existence of outcomes that cannot be improved upon by any coalition (core of the game).The object of this paper is to determine the finite geometries which are minimal balanced sets. We prove that the dual of any t-design with t ? 2 is a minimal balanced set. In particular symmetrical 2-designs (as projective spaces, biplanes, etc.) are always minimal balanced sets. For 1-designs the problem becomes much more difficult, but it is for instance easy to prove that any partial geometry which is not the dual of a 2-Steiner system is never a minimal balanced set; in particular generalized quadrangles are never minimal balanced sets. For linear graphs the problem is completely solved: the dual of a connected linear graph is a minimal balanced set if and only if this linear graph is not bichromatic.     

Permanence of dispersal population model with time delays

Permanence of a dispersal single-species population model is considered where environment is partitioned into several patches and the species requires some time to disperse between the patches. The model is described by delay differential equations. The existence of food-rich patches and small dispersions among the patches are proved to be sufficient to ensure partial permanence of the model. It is also shown that partial permanence ensures permanence if each food-poor patch is connected to at least one food-rich patch and if each pair in food-rich patches is connected. Furthermore, it is proved that partial persistence is ensured even under large dispersion among food-rich patches if the dispersion time is relatively small.     

Collet, Eckmann and Hölder

We prove that Collet-Eckmann condition for rational functions, which requires exponential expansion only along the critical orbits, yields the H?lder regularity of Fatou components. This implies geometric regularity of Julia sets with non-hyperbolic and critically-recurrent dynamics. In particular, polynomial Collet-Eckmann Julia sets are locally connected if connected, and their Hausdorff dimension is strictly less than 2. The same is true for rational Collet-Eckmann Julia sets with at least one non-empty fully invariant Fatou component. Oblatum 22-III-1996 & 15-VII-1997     

Poset extensions, convex sets, and semilattice presentations

The paper is devoted to an algebraic and geometric study of the feasible set of a poset, the set of finite probability distributions on the elements of the poset whose weights satisfy the order relationships specified by the poset. For a general poset, this feasible set is a barycentric algebra. The feasible sets of the order structures on a given finite set are precisely the convex unions of the primary simplices, the facets of the first barycentric subdivision of the simplex spanned by the elements of the set. As another fragment of a potential complete duality theory for barycentric algebras, a duality is established between order-preserving mappings and embeddings of feasible sets. In particular, the primary simplices constituting the feasible set of a given finite poset are the feasible sets of the linear extensions of the poset. A finite poset is connected if and only if its barycentre is an extreme point of its feasible set. The feasible set of a (general) disconnected poset is the join of the feasible sets of its components. The extreme points of the feasible set of a finite poset are specified in terms of the disjointly irreducible elements of the semilattice presented by the poset. Semilattices presented by posets are characterised in terms of various distributivity concepts.