Testing <Emphasis Type="Italic">k</Emphasis>-edge-connectivity of digraphs

This paper presents an algorithm that tests whether a given degree-bounded digraph is k-edge-connected or ɛ-far from k-edge-connectivity. This is the first testing algorithm for k-edgeconnectivity of digraphs whose running time is independent of the number of vertices and edges. A digraph of n vertices with degree bound d is ɛ-far from k-edge-connectivity if at least ɛdn edges have to be added or deleted to make the digraph k-edge-connected, preserving the degree bound. Given a constant error parameter ɛ and a degree bound d, our algorithm always accepts all k-edge-connected digraphs and rejects all digraphs that is ɛ-far from k-edge-connectivity with probability at least 2/3. It runs in $ O\left( {d\left( {\frac{c} {{\varepsilon d}}} \right)^k log\frac{1} {{\varepsilon d}}O} \right) $ O\left( {d\left( {\frac{c} {{\varepsilon d}}} \right)^k log\frac{1} {{\varepsilon d}}O} \right) (c > 1 is a constant) time when input digraphs are restricted to be (k-1)-edge connected and runs in $ O\left( {d\left( {\frac{{ck}} {{\varepsilon d}}} \right)^k log\frac{k} {{\varepsilon d}}O} \right) $ O\left( {d\left( {\frac{{ck}} {{\varepsilon d}}} \right)^k log\frac{k} {{\varepsilon d}}O} \right) (c > 1 is a constant) time for general digraphs.     

A class of constacyclic codes over ring <Emphasis Type="Italic">R</Emphasis> + <Emphasis Type="Italic">vR</Emphasis>

By constructing a Gray map, a class of constacyclic codes over ring R = R+vR is studied. Using cyclic codes and negacyclic codes of length p ^s over ring R, the structure of (1?2v)-constacyclic codes and dual codes of length p ^s over ring R are given, the Gray images of (1 ? 2v)-constacyclic codes in a particular case are also studied. It is shown that linear codes of length p ^s over ring R are (1?2v)-constacyclic codes if and only if their Gray images are distance-invariant cyclic codes of length 2p ^s over ring R.     

Construction of one-Lee weight and two-Lee weight codes over F<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> + <Emphasis Type="Italic">v</Emphasis>F<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is devoted to the construction of one-Lee weight codes and two-Lee weight codes over F _p + vF _p (v ² = v) with type \({p^{2{k_1}}}{p^{{k_2}}}{p^{{k_3}}}\) based on two different distance-preserving Gray maps from ((F _p + vF _p ) ⁿ , Lee weight) to (F _p ²ⁿ , Hamming weight), where p is a prime. Moreover, the authors prove that the obtained two-Lee weight codes are projective only when p = 2.     

On Fuzzy <Emphasis Type="Italic">h</Emphasis>-Ideals of Hemirings

The concept of quasi-coincidence of a fuzzy interval value in an interval valued fuzzy set is considered. In fact, this concept is a generalized concept of the quasi-coincidence of a fuzzy point in a fuzzy set. By using this new concept, the authors define the notion of interval valued （∈, ∈ Vq）fuzzy h-ideals of hemirings and study their related properties. In addition, the authors also extend the concept of a fuzzy subgroup with thresholds to the concept of an interval valued fuzzy h-ideal with thresholds in hemirings.     

Strong approximation method and the (functional) law of iterated logarithm for <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 queue

In this paper, a unified method based on the strong approximation (SA) of renewal process (RP) is developed for the law of the iterated logarithm (LIL) and the functional LIL (FLIL), which quantify the magnitude of the asymptotic rate of the increasing variability around the mean value of the RP in numerical and functional forms respectively. For the GI/G/1 queue, the method provides a complete analysis for both the LIL and the FLIL limits for four performance functions: The queue length, workload, busy time and idle time processes, covering three regimes divided by the traffic intensity.     

The structure of departure process and optimal control strategy <Emphasis Type="Italic">N</Emphasis>* for <Emphasis Type="Italic">Geo/G</Emphasis>/1 discrete-time queue with multiple server vacations and Min(<Emphasis Type="Italic">N,V</Emphasis>)-Policy

This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n⁺ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0⁺, n⁺] from any initial state. Meanwhile, the relationship among departure process, server’s state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures, including the expected length of server busy period, server’s actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N* for minimizing the system cost under a given cost structure.     

The generalized Serre problem over <Emphasis Type="Italic">K</Emphasis>-Hermite rings

This paper investigates the completion and the zero prime factorization for matrices over a K-Hermite ring. The authors have generalized Kaplansky’s related results, solved similar Lin-Bose problem and generalized Serre problem over a K-Hermite ring.     

THE M/M/c QUEUE WITH （e,d） SETUP TIME

The authors present a new queueing model with （e, d） setup time. Using the quasi-birth-and-death process and matrix-geometric method, the authors obtain the stationary distribution of queue length and the LST of waiting time of a customer in the system. Furthermore, the conditional stochastic decomposition results of queue length and waiting time are given.     

Distributed Consensus-Based <Emphasis Type="Italic">K</Emphasis>-Means Algorithm in Switching Multi-Agent Networks

This paper discusses a distributed design for clustering based on the K-means algorithm in a switching multi-agent network, for the case when data are decentralized stored and unavailable to all agents. The authors propose a consensus-based algorithm in distributed case, that is, the double-clock consensus-based K-means algorithm (DCKA). With mild connectivity conditions, the authors show convergence of DCKA to guarantee a distributed solution to the clustering problem, even though the network topology is time-varying. Moreover, the authors provide experimental results on various clustering datasets to illustrate the effectiveness of the fully distributed algorithm DCKA, whose performance may be better than that of the centralized K-means algorithm.     

Efficient Asymmetrical Extended Designs Under Wrap-Around <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>-Discrepancy

The purpose of the present article is to introduce a class of mixed two- and three-level extended designs obtained by adding some new runs to an existing mixed two-and three-level design. A formulation of wrap-around L₂-discrepancy for the extended designs is developed. As a benchmark of obtaining (nearly) uniform asymmetrical extended designs, a lower bound to the wrap-around L₂-discrepancy for our proposed designs is established. Thorough numerical results are displayed, which provide further corroboration to the derived theoretical results.