On <Emphasis Type="Italic">k</Emphasis>-Spaces and <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-Spaces

In this note we study the relation between k _R -spaces and k-spaces and prove that a k _R -space with a σ-hereditarily closure-preserving k-network consisting of compact subsets is a k-space, and that a k _R -space with a point-countable k-network consisting of compact subsets need not be a k-space. This work was supported by the NSF of China (10271056).     

强∑~*空间是D空间

研究了强Σ~*空间和D空间的关系,证明了强Σ~*空间是D空间,改进了R.Z.Buzyakova等的结果.     

Spaces With Compact-Countable k-Systems

In this paper the relations among k-covers, cs ^*-covers and k-systems are discussed. The following question is partially answered: Does every separable k'-space with a point-countable k-system have a countable k-system?     

On k-primitivity of two classes of digraphs

In this paper, we study two classes of primitive digraphs, and show that there are k-colorings that are k-primitive.     

Dense trees: a new look at degenerate graphs

Dense trees are undirected graphs defined as natural extensions of trees. They are already known in the realm of graph coloring under the name of k-degenerate graphs. For a given integer k1, a k-dense cycle is a connected graph, where the degree of each vertex is greater than k. A k-dense forest F=(V,E) is a graph without k-dense cycles as subgraphs. If F is connected, then is a k-dense tree. 1-dense trees are standard trees. We have |E|k|V|−k(k+1)/2. If equality holds F is connected and is called a maximal k-dense tree. k-trees (a subfamily of triangulated graphs) are special cases of maximal k-dense trees.We review the basic theory of dense trees in the family of graphs and show their relation with k-trees. Vertex and edge connectivity is thoroughly investigated, and the role of maximal k-dense trees as “reinforced” spanning trees of arbitrary graphs is presented. Then it is shown how a k-dense forest or tree can be decomposed into a set of standard spanning trees connected through a common “root” of k vertices. All sections include efficient construction algorithms. Applications of k-dense trees in the fields of distributed systems and data structures are finally indicated.     

The Width-Volume Inequality

We prove that a bounded open set U in has k-width less than C(n) Volume(U) ^k/n . Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in . In particular, we estimate the smallest (n – 1)-dilation of any degree 1 map between two n-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant C(n). We give examples in which the (n – 1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor. Received: January 2006, Revision: May 2006, Accepted: June 2006     

Orderings on Graphs and Game Coloring Number

Many graph theoretic algorithms rely on an initial ordering of the vertices of the graph which has some special properties. We discuss new ways to measure the quality of such orders, give methods for constructing high quality orders, and provide applications for these orders. While our main motivation is the study of game chromatic number, there have been other applications of these ideas and we expect there will be more. This revised version was published online in June 2006 with corrections to the Cover Date.     

The <Emphasis Type="Italic">k</Emphasis>-domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-dominating set of the graph G is a subset D of V (G) such that every vertex of V (G)-D is adjacent to at least k vertices in D. A k-domatic partition of G is a partition of V (G) into k-dominating sets. The maximum number of dominating sets in a k-domatic partition of G is called the k-domatic number d _k (G). In this paper, we present upper and lower bounds for the k-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number d(G) = d ₁(G).      

On a conjecture concerningk-Hamilton-nice sequences

In this paper, we prove that a non-negative rational number sequence (a ₁,a ₂, ...,a _k+1) isk-Hamilton-nice, if (1)a _k+12, and (2) _j ^=1/h (i _j –1)k–1 implies for arbitraryi ₁,i ₂,...i _h {1,2,... ,k}. This result was conjectured by Guantao Chen and R.H. Schelp, and it generalizes several well-known sufficient conditions for graphs to be Hamiltonian.This project is supported by the National Natural Science Foundation of China.     

On Pseudocompact, Countably Compact, Locally Bicompact Mappings, and k-Mappings

We consider various definitions of a pseudocompact mapping and the basic properties of pseudocompact mappings. Moreover, we consider the definition of countable compactness of a continuous mapping and study the properties of a countably compact mapping similar to the corresponding properties for countably compact spaces and also the interrelation between countable compactness and pseudocompactness of mappings. We also extend the notions of local bicompactness and k-space to continuous mappings.     

The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics

In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.     

Map Preserving Local Properties of a Digital Image

In order to study digital topological properties of a k-surface in Z ⁿ , we generalize the topological number in Bertrand (Pattern Recogn. Lett. 15:1003–1011, 1994). Furthermore, we show that a local (k ₀,k ₁)-isomorphism preserves some digital-topological properties, such as a generalized topological number and a simple k ₀-point, and prove that a local (k ₀,k ₁)-isomorphism takes a simple k ₀-surface in into a simple k ₁-surface in .      

New algorithms for lineark-matroid intersection and matroidk-parity problems

We present algorithms for thek-Matroid Intersection Problem and for the Matroidk-Parity Problem when the matroids are represented over the field of rational numbers andk > 2. The computational complexity of the algorithms is linear in the cardinality and singly exponential in the rank of the matroids. As an application, we describe new polynomially solvable cases of thek-Dimensional Assignment Problem and of thek-Dimensional Matching Problem. The algorithms use some new identities in multilinear algebra including the generalized Binet—Cauchy formula and its analogue for the Pfaffian. These techniques extend known methods developed earlier fork = 2.A preliminary version of this paper appeared in the Proceedings of the Second IPCO Conference [2].Supported by the Mittag-Leffler Institute and KTH, Stockholm.     

<Emphasis Type="Italic">k</Emphasis>-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ⩾ 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ⩾ m + 1 (resp., k ⩾ m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ⩾ 3. Supported by NSF (grant Nos. DMS 0455881 and DMS-0700811). (A. Yu. Olshanskii, M. V. Sapir) Supported by RFBR project No. 08-01-00573. (A. Yu. Olshanskii) Supported by BSF grant (USA–Israel). (M. V. Sapir) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 245–257, March–April, 2009.     

Norm of pre-Schwarzian derivative for the class of k-uniformly convex and k-starlike functions

A quasiconformal extension for the class of k-uniformly convex functions, denoted , and for the class of k-starlike functions, denoted is provided. Also, estimation of the norm of pre-Schwarzian derivative in is given.     

Some comments on a paper onk-HNBUE life distributions

Summary In a recent paper by A. P. Basu and N. Ebrahimi (1984, Ann. Inst. Statist. Math., A, 36, 87–100) a new class of life distributions calledk-HNBUE (with dualk-HNWUE) is introduced. Closure properties and bounds on the moments and on the survival function to ak-HNBUE (k-HNWUE) life distribution are presented. However, some of the results presented are incorrect. This research was supported by Swedish Natural Science Research Council Post Doctoral Fellowship F-PD 1564-100.     

The construction of infinite families of any κ-tight optimal and singular κ-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.     

Transient and Stationary Distributions for the GI/G/k Queue with Lebesgue-Dominated Inter-Arrival Time Distribution

In this paper, the multi-server queue with general service time distribution and Lebesgue-dominated iid inter-arival times is analyzed. This is done by introducing auxiliary variables for the remaining service times and then examining the embedded Markov chain at arrival instants. The concept of piecewise-deterministic Markov processes is applied to model the inter-arrival behaviour. It turns out that the transition probability kernel of the embedded Markov chain at arrival instants has the form of a lower Hessenberg matrix and hence admits an operator–geometric stationary distribution. Thus it is shown that matrix–analytical methods can be extended to provide a modeling tool even for the general multi-server queue.     

k-Arcs,Hyperovals, Partial Flocks and Flocks

Some recent results on k-arcs and hyperovals of PG(2,q),on partial flocks and flocks of quadratic cones of PG(3,q),and on line spreads in PG(3,q) are surveyed. Also,there is an appendix on how to use Veronese varieties as toolsin proving theorems.     

Estimation of parameters in the discrete distributions of order k

This paper considers estimating parameters in the discrete distributions of order k such as the binomial, the geometric, the Poisson and the logarithmic series distributions of order k. It is discussed how to calculate maximum likelihood estimates of parameters of the distributions based on independent observations. Further, asymptotic properties of estimators by the method of moments are investigated. In some cases, it is found that the values of asymptotic efficiency of the moment estimators are surprisingly close to one.