Soft almost disjoint families

An almost disjoint family is said to be soft if there is an infinite set that meets each in a nonempty but finite set. We consider the associated cardinal invariant defined to be the minimal cardinality of an almost disjoint family that is not soft. We show that this cardinal coincides with J. Brendle's cardinal .

     

Homogeneity and the disjoint arcs property

Some previous results of the author towards a classification of homogeneous metric continua are improved. The disjoint arcs property is fully revealed in this context. In particular, closed -manifolds, , are characterized as those homogeneous continua which do not have the disjoint arcs property.

     

Disjoint Paths in Graphs III,Characterization

Let G be a graph, $ \{a, b, c\}\subseteq V(G) $, and $ \{a', b', c'\}\subseteq V(G) $ such that $ \{a, b, c\}\neq \{a', b', c'\} $. We say that $ (G, \{a, c\}, \{a', c'\}, (b, b')) $ is an obstruction if, for any three vertex disjoint paths from {a, b, c} to {a', b', c'} in G, one path is from b to b'. In this paper we characterize obstructions. As a consequence, we show that no obstruction can be 8-connected, unless b = b' or {a, c} = {a', c'}.AMS Subject Classification: 05C38.     

Localization-delocalization transition in chains with long-range correlated disorder

We investigated numerically localization properties of electron eigenstates in a chain with long-range correlated diagonal disorder. A tight-binding one-dimensional model with on-site energies exhibiting long-range correlated disorder (LCD) was used with various disorder strength W. LCD was defined so that it gave a power-law spectral density of the form S(k)αk^-p, where p determines the roughness of the potential landscape. Numerical results on the correlation length ξ of eigenstates shows the existence of the localization-delocalization transition at p=2. It is found that the critical values for disorder strength W_c and also the critical exponent ν for localization length change with the values of p.     

2‐factors with prescribed and proscribed edges

Let G be a finite k‐edge‐connected simple graph. We consider when a set of independent edges can be extended to a 2‐factor such that this 2‐factor avoids a fixed set of independent edges. A complete characterization is provided in those cases, where this is feasible. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 48–58, 2005     

Semiring unions of a ring and a half-body

We describe the structure of semirings which are unions of a ring and a half-body.     

Mobility edges and reentrant localization in one-dimensional dimerized non-Hermitian quasiperiodic lattice

The mobility edges and reentrant localization transitions are studied in one-dimensional dimerized lattice with nonHermitian either uniform or staggered quasiperiodic potentials.We find that the non-Hermitian uniform quasiperiodic disorder can induce an intermediate phase where the extended states coexist with the localized ones,which implies that the system has mobility edges.The localization transition is accompanied by the PT symmetry breaking transition.While if the non-Hermitian quasiperiodic disorder is staggered,we demonstrate the existence of multiple intermediate phases and multiple reentrant localization transitions based on the finite size scaling analysis.Interestingly,some already localized states will become extended states and can also be localized again for certain non-Hermitian parameters.The reentrant localization transitions are associated with the intermediate phases hosting mobility edges.Besides,we also find that the non-Hermiticity can break the reentrant localization transition where only one intermediate phase survives.More detailed information about the mobility edges and reentrant localization transitions are presented by analyzing the eigenenergy spectrum,inverse participation ratio,and normalized participation ratio.     

Constructions of optimal difference systems of sets

Difference systems of sets (DSSs) are combinatorial configurations which were introduced in 1971 by Levenstein for the construction of codes for synchronization. In this paper, we present two kinds of constructions of difference systems of sets by using disjoint difference families and a special type of difference sets, respectively. As a consequence, new infinite classes of optimal DSSs are obtained.     

Random graphs containing few disjoint excluded minors

The Erd?s‐Pósa theorem (1965) states that in each graph G which contains at most k disjoint cycles, there is a ‘blocking’ set B of at most f(k) vertices such that the graph G – B is acyclic. Robertson and Seymour (1986) give an extension concerning any minor‐closed class of graphs, as long as does not contain all planar graphs: in each graph G which contains at most k disjoint excluded minors for , there is a set B of at most g(k) vertices such that G – B is in . In an earlier paper (Kurauskas and McDiarmid, Combin, Probab Comput 20 (2011) 763–775), we showed that, amongst all graphs on vertex set which contain at most k disjoint cycles, all but an exponentially small proportion contain a blocking set of just k vertices. In the present paper we build on the previous work, and give an extension concerning any minor‐closed graph class with 2‐connected excluded minors, as long as does not contain all fans (here a ‘fan’ is a graph consisting of a path together with a vertex joined to each vertex on the path). We show that amongst all graphs G on which contain at most k disjoint excluded minors for , all but an exponentially small proportion contain a set B of k vertices such that G – B is in . (This is not the case when contains all fans.) For a random graph R_n sampled uniformly from the graphs on with at most k disjoint excluded minors for , we consider also vertex degrees and the uniqueness of small blockers, the clique number and chromatic number, and the probability of being connected. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 240‐268, 2014     

涉及四个单形的一类不等式

本文研究了涉及四个单形的一类不等式问题.利用距离几何的理论和方法获得了两个单形的棱长与另两个单形的内点、中线、高、外接超球半径、内切超球半径、旁切超球半径以及n-1维侧面的体积、外接超球半径、内切超球半径的一类新的几何不等式.推广了文献([5])中的全部结果.