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.     

RELIABILITY OF LINEAR （n,f, k） SYSTEMS WITH WEIGHTED COMPONENTS

The (n, f, k): F(G) system consists of n components and the system fails (works) if and only if there are at least f failed (working) components or at least k consecutive failed (working) components. These system models can be used in electronic equipment, automatic payment systems in banks, and furnace systems. In this paper we introduce and study the (n, f, k):F and (n, f, k): G systems consisting of weighted components. Recursive equations are presented for reliability evaluation of these new models. We also provide some conditions on the weights to represent weighted-(n, f, k) systems as usual (n, f, k) systems.     

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.     

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.     

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 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.     

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.     

An improved algorithm for solving all <Emphasis Type="Italic">d</Emphasis>-MPs in multi-state networks

Reliability is a desirable performance indicator of many real-world systems to measure the quality level. One general method for evaluating multi-state reliability is using d-minimal paths (d-MPs). However, being an NP-hard problem, searching for all d-MPs is a rather challenging task. This paper proposes an improved algorithm to solve the d-MP problem. To reduce the search space of d-MPs, a concept of lower capacity bound is introduced into the d-MP problem, and an effective technique is developed to find lower capacity bounds. Meanwhile, the fast enumeration method which is a recent improvement to the traditional enumeration method is employed to solve d-MPs. In addition, by introducing the operation of transforming undirected edges into directed edges, the proposed algorithm is applicable to solving both directed networks and undirected networks. Through numerical experiments, it is found that the proposed algorithm holds a distinct advantage over the existing methods in solving all d-MPs.     

Effective lower bounds of wrap-around <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>-discrepancy on three-level combined designs

How to obtain an effective design is a major concern of scientific research. This topic always involves high-dimensional inputs with limited resources. The foldover is a quick and useful technique in construction of fractional designs, which typically releases aliased factors or interactions. This paper takes the wrap-around L₂-discrepancy as the optimality measure to assess the optimal three-level combined designs. New and efficient analytical expressions and lower bounds of the wraparound L₂-discrepancy for three-level combined designs are obtained. The new lower bound is useful and sharper than the existing lower bound. Using the new analytical expression and lower bound as the benchmarks, the authors may implement an effective algorithm for constructing optimal three-level combined designs.     

The equivalent representation of the breadth-one <Emphasis Type="Italic">D</Emphasis>-invariant polynomial subspace and its discretization

This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the authors solve the discrete approximation problem in ideal interpolation for the breadth-one D-invariant subspace. Namely, the authors find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given D-invariant subspace, as the evaluation points all coalesce at one point.     

ELIMINATION IN WEYI ALGEBRA AND q-IDENTITIES

In this paper, the following are introduced briefly： the basic concept of q-proper-hypergeometric; an algorithmic proof theory for q-proper-hypergeometric identities; and elimination in the non- commutative Weyl algebra. We give an algorithm for proving the single-variable q-proper-hypergeometric identities that is based on Zeilberger＇s approach and the elimination in Weyl algebra. Finally, we test several examples that have been proven by D. Zeilberger and H. Will using the WZ-pair method and Gosper algorithm.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-estimates of solutions for damped wave equations with variable coefficients

The authors establish weighted L²-estimates of solutions for the damped wave equations with variable coefficients u _tt ? divA(x)?u+au _t = 0 in ? ⁿ under the assumption a(x) ≥ a₀[1+ρ(x)]^?l, where a₀ > 0, l < 1, ρ(x) is the distance function of the metric g = A^?1(x) on ? ⁿ . The authors show that these weighted L²-estimates are closely related to the geometrical properties of the metric g = A^?1(x).     

ON THE TRANSIENT DEPARTURE PROCESS OF M^x/G/1 QUEUEING SYSTEM WITH SINGLE SERVER VACATION

This paper studies the transient departure process of M^x/G/1 queueing system with single server vacation. We present a simple probability decomposition method to derive the expected number of departures occurring in finite time interval from any initial state and the asymptotic expansion of the expected number. Especially, we derive some more practical results for some special cases.     

ON APPROXIMATION BY REPRODUCING KERNEL SPACES IN WEIGHTED L^p SPACES

In this paper, we investigate the order of approximation by reproducing kernel spaces on （-1, 1） in weighted L^p spaces. We first restate the translation network from the view of reproducing kernel spaces and then construct a sequence of approximating operators with the help of Jacobi orthogonal polynomials, with which we establish a kind of Jackson inequality to describe the error estimate. Finally, The results are used to discuss an approximation problem arising from learning theory.     

Queue size distribution of Geo/G/1 queue under the Min(<Emphasis Type="Italic">N,D</Emphasis>)-policy

This paper considers a discrete-time Geo/G/1 queue under the Min(N,D)-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D, whichever occurs first (Min(N,D)-policy). By using renewal process theory and total probability decomposition technique, the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursive expression of the z-transformation of the transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n ⁺. Meanwhile, the authors obtain the explicit expressions of the additional queue length distribution. Furthermore, the important relations between the steady state queue length distributions at different time epochs n ^-, n and n ⁺ are also reported. Finally, the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution, and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.     

A perishable inventory system with service facilities, <Emphasis Type="Italic">MAP</Emphasis> arrivals and <Emphasis Type="Italic">PH</Emphasis> — Service times

We consider a continuous review perishable （s,S） inventory system with a service facility consisting of a waiting hall of finite capacity and a single server. We assume two types of customers, ordinary and negative, arrive according to a Markovian Arrival Process （MAP）. An ordinary customer joins the queue and a negative customer instead of joining the queue removes one ordinary customer from the queue. The removal rule adopted in this paper is RCE （removal of a customer from the end）. The individual customer＇s unit demand is satisfied after a random time of service which is assumed to have a phase-type distribution. The life time of each item and the lead time of the reorders have been assumed to be independent exponential distributions. The joint probability distribution of the number of customers in the system and the inventory level is obtained for the steady state case. Various stationary system performance measures are computed and the total expected cost rate is calculated. The results are illustrated numerically.     

The recursive solution of queue length for <Emphasis Type=Italic>Geo/G</Emphasis>/1 queue with <Emphasis Type=Italic>N</Emphasis>-policy

This paper considers a discrete-time queue with N-policy and LAS-DA(late arrival system with delayed access) discipline.By using renewal process theory and probability decomposition techniques,the authors derive the recursive expressions of the queue-length distributions at epochs n~-,n~+,and n.Furthermore,the authors obtain the stochastic decomposition of the queue length and the relations between the equilibrium distributions of the queue length at different epochs(n~-,n~+,n and departure epoch D_n).