2017年, 第33卷, 第4期　刊出日期：2017-11-15

  

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  Analysis of Queues in a Random Environment with Impatient Customers

  Sen-lin YU, Zai-ming LIU

  应用数学学报(英文版). 2017, 33(4): 837-850. https://doi.org/10.1007/s10255-017-0701-2

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  We study M/M/c queues (c=1, 1 < c < ∞ and c=∞) in a Markovian environment with impatient customers. The arrivals and service rates are modulated by the underlying continuous-time Markov chain. When the external environment operates in phase 2, customers become impatient. We focus our attention on the explicit expressions of the performance measures. For each case of c, the corresponding probability generating function and mean queue size are obtained. Several special cases are studied and numerical experiments are presented.
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  Score Tests for Zero-inflated Double Poisson Regression Models

  Feng-chang XIE, Jin-guan LIN, Bo-cheng WEI

  应用数学学报(英文版). 2017, 33(4): 851-864. https://doi.org/10.1007/s10255-017-0702-1

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  Count data with excess zeros encountered in many applications often exhibit extra variation. Therefore, zero-inflated Poisson (ZIP) model may fail to fit such data. In this paper, a zero-inflated double Poisson model (ZIDP), which is generalization of the ZIP model, is studied and the score tests for the significance of dispersion and zero-inflation in ZIDP model are developed. Meanwhile, this work also develops homogeneous tests for dispersion and/or zero-inflation parameter, and corresponding score test statistics are obtained. One numerical example is given to illustrate our methodology and the properties of score test statistics are investigated through Monte Carlo simulations.
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  A New Class for Large Sets of Almost Hamilton Cycle Decompositions

  Hong-tao ZHAO

  应用数学学报(英文版). 2017, 33(4): 865-870. https://doi.org/10.1007/s10255-017-0703-0

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  A k-cycle system of order v with index λ, denoted by CS(v, k, λ), is a collection A of k-cycles (blocks) of Kv such that each edge in Kv appears in exactly λ blocks of A. A large set of CS(v, k, λ)s is a partition of the set of all k-cycles of Kv into CS(v, k, λ)s, and is denoted by LCS(v, k, λ). A (v -1)-cycle in Kv is called almost Hamilton. The completion of the existence problem for LCS(v, v-1, λ) depends only on one case:all v ≥ 4 for λ=2. In this paper, it is shown that there exists an LCS(v, v -1, 2) for all v ≡ 2 (mod 4), v ≥ 6.
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  Valuation and Hedging Strategy of Currency Options under Regime-Switching Jump-Diffusion Model

  Shou-ting CHEN, Xun-di DIAO, Ai-lin ZHU

  应用数学学报(英文版). 2017, 33(4): 871-892. https://doi.org/10.1007/s10255-017-0704-z

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  The main purpose of this thesis is in analyzing and empirically simulating risk minimizing European foreign exchange option pricing and hedging strategy when the spot foreign exchange rate is governed by a Markov-modulated jump-diffusion model. The domestic and foreign money market interest rates, the drift and the volatility of the exchange rate dynamics all depend on a continuous-time hidden Markov chain which can be interpreted as the states of a macro-economy. In this paper, we will provide a practical lognormal diffusion dynamic of the spot foreign exchange rate for market practitioners. We employing the minimal martingale measure to demonstrate a system of coupled partial-differential-integral equations satisfied by the currency option price and attain the corresponding hedging schemes and the residual risk. Numerical simulations of the double exponential jump diffusion regime-switching model are used to illustrate the different effects of the various parameters on currency option prices.
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  Analyzing Right-Censored Length-Biased Data with Additive Hazards Model

  Mu ZHAO, Cun-jie LIN, Yong ZHOU

  应用数学学报(英文版). 2017, 33(4): 893-908. https://doi.org/10.1007/s10255-017-0705-y

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  Length-biased data are often encountered in observational studies, when the survival times are left-truncated and right-censored and the truncation times follow a uniform distribution. In this article, we propose to analyze such data with the additive hazards model, which specifies that the hazard function is the sum of an arbitrary baseline hazard function and a regression function of covariates. We develop estimating equation approaches to estimate the regression parameters. The resultant estimators are shown to be consistent and asymptotically normal. Some simulation studies and a real data example are used to evaluate the finite sample properties of the proposed estimators.
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  Counting Rooted 4-regular Unicursal Planar Maps

  Shu-de LONG, Jun-liang CAI

  应用数学学报(英文版). 2017, 33(4): 909-918. https://doi.org/10.1007/s10255-017-0706-x

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  A map is 4-regular unicursal if all its vertices are 4-valent except two odd-valent vertices. This paper investigates the number of rooted 4-regular unicursal planar maps and presents some formulae for such maps with four parameters:the number of edges, the number of inner faces and the valencies of the two odd vertices.
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  Lattices Generated by Joins of the Flats in Orbits under Finite Affine-singular Symplectic Group and its Characteristic Polynomials

  You GAO, Yan-yan XUE, Yu-ting XIAO, Xue-mei LIU

  应用数学学报(英文版). 2017, 33(4): 919-932. https://doi.org/10.1007/s10255-017-0707-9

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  Let ASG(2ν + l, ν;F_q) be the (2ν + l)-dimensional affine-singular symplectic space over the finite field F q and ASp_2ν+l,ν(F_q) be the affine-singular symplectic group of degree 2ν + l over F_q. Let O be any orbit of flats under ASp_2ν+l,ν(F_q). Denote by L^J the set of all flats which are joins of flats in O such that O ⊆ L^J and assume the join of the empty set of flats in ASG(2ν + l, ν;F_q) is Ø. Ordering L^J by ordinary or reverse inclusion, then two lattices are obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice L^J, when the lattices form geometric lattice, lastly gives the characteristic polynomial of L^J.
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  Conditional Connectivity of Bubble Sort Graphs

  Ling-sheng SHI, Peng WU

  应用数学学报(英文版). 2017, 33(4): 933-944. https://doi.org/10.1007/s10255-017-0708-8

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  A subset F ⊂ V (G) is called an R^k-vertex-cut of a graph G if G -F is disconnected and each vertex of G -F has at least k neighbors in G -F. The R^k-vertex-connectivity of G, denoted by κ^k(G), is the cardinality of a minimum R^k-vertex-cut of G. Let B_n be the bubble sort graph of dimension n. It is known that κ_k(B_n)=2^k(n -k -1) for n ≥ 2k and k=1, 2. In this paper, we prove it for k=3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.

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  Optimal Investment and Premium Control in a Nonlinear Diffusion Model

  Ming ZHOU, Kam Chuen YUEN, Chuan-cun YIN

  应用数学学报(英文版). 2017, 33(4): 945-958. https://doi.org/10.1007/s10255-017-0709-7

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  This paper considers the optimal investment and premium control problem in a diffusion approximation to a non-homogeneous compound Poisson process. In the nonlinear diffusion model, it is assumed that there is an unspecified monotone function describing the relationship between the safety loading of premium and the time-varying claim arrival rate. Hence, in addition to the investment control, the premium rate can be served as a control variable in the optimization problem. Specifically, the problem is investigated in two cases:(i) maximizing the expected utility of terminal wealth, and (ii) minimizing the probability of ruin respectively. In both cases, some properties of the value functions are derived, and closed-form expressions for the optimal policies and the value functions are obtained. The results show that the optimal investment policy and the optimal premium control policy are dependent on each other. Most interestingly, as an example, we show that the nonlinear diffusion model reduces to a diffusion model with a quadratic drift coefficient when the function associated with the premium rate and the claim arrival rate takes a special form. This example shows that the model of study represents a class of nonlinear stochastic control risk model.
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  Chung-type Law of the Iterated Logarithm for Continuous Time Random Walk

  Wen-sheng WANG

  应用数学学报(英文版). 2017, 33(4): 959-966. https://doi.org/10.1007/s10255-017-0711-0

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  A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish a Chung-type law of the iterated logarithm for continuous time random walk with jumps and waiting times in the domains of attraction of stable laws.
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  Bifurcations and Chaos in Mira 2 Map

  Tao JIANG, Zhi-yan YANG

  应用数学学报(英文版). 2017, 33(4): 967-978. https://doi.org/10.1007/s10255-017-0710-1

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  In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And the conditions of the existence for chaos in the sense of Marroto are obtained. Numerical simulation results not only show the consistence with the theoretical analysis but also display complex dynamical behaviors, including period-n orbits, crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.
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  A Class of Spline Curves with Four Local Shape Parameters

  Yuan-peng ZHU, Xu-li HAN

  应用数学学报(英文版). 2017, 33(4): 979-988. https://doi.org/10.1007/s10255-017-0712-z

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  A class of spline curves with four local shape parameters, which includes the quartic spline curves with three local shape parameters given in Han [Xuli Han. A class of general quartic spline curves with shape parameters. Comput. Aided Geom. Design, 28:151-163 (2011)], is proposed. Without solving a linear system, the spline curves can be used to interpolate sets of points with C² continuity partly or entirely. The shape parameters have a predictable adjusting role on the spline curves.
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  Extending Slow Manifold Near Generic Transcritical Canard Point

  Hai-bo LU, Ming-kang NI, Li-meng WU

  应用数学学报(英文版). 2017, 33(4): 989-1000. https://doi.org/10.1007/s10255-017-0713-y

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  We consider the dynamics of planar fast-slow systems near generic transcritical type canard point. By using geometric singular perturbation theory combined with the recently developed blow-up technique, the existence of canard cycles, relaxation oscillations and solutions near the attracting branch of the critical manifold is established. The asymptotic expansion of the parameter for which canard exists is obtained by a version of the Melnikov method.
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  2-Connected Factor-critical Graphs G with Exactly|E(G)|+1 Maximum Matchings

  Ming-hua LI, Yan LIU

  应用数学学报(英文版). 2017, 33(4): 1001-1014. https://doi.org/10.1007/s10255-017-0715-9

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  A connected graph G is said to be a factor-critical graph if G-v has a perfect matching for every vertex v of G. In this paper, the 2-connected factor-critical graph G which has exactly |E(G)|+1 maximum matchings is characterized.
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  An Improved Approximation Algorithm for the k-Level Facility Location Problem with Soft Capacities

  Chen-chen WU, Da-chuan XU

  应用数学学报(英文版). 2017, 33(4): 1015-1024. https://doi.org/10.1007/s10255-017-0714-x

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  We consider the k-level facility location problem with soft capacities (k-LFLPSC). In the kLFLPSC, each facility i has a soft capacity ui along with an initial opening cost f_i ≥ 0, i.e., the capacity of facility i is an integer multiple of ui incurring a cost equals to the corresponding multiple of f_i. We firstly propose a new bifactor (ln(1/β)/(1 -β), 1 + 2/(1 -β))-approximation algorithm for the k-level facility location problem (k-LFLP), where β ∈ (0, 1) is a fixed constant. Then, we give a reduction from the k-LFLPSC to the k-LFLP. The reduction together with the above bifactor approximation algorithm for the k-LFLP imply a 5.5053-approximation algorithm for the k-LFLPSC which improves the previous 6-approximation.
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  Asymptotic Behaviors of a Size-structured Population Model

  Xian-long FU, Qiong WU

  应用数学学报(英文版). 2017, 33(4): 1025-1042. https://doi.org/10.1007/s10255-017-0717-7

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  In this paper we devote ourselves to the study of the asymptotic behavior of a size-structured population dynamics with random diffusion and delayed birth process. Within a semigroup framework, we discuss the local stability and asynchrony respectively for the considered population system under some conditions. We use for our discussion the techniques of operator matrices, Hille-Yosida operators, positivity, spectral analysis as well as Perron-Frobenius theory.
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  Existence of Nontrivial Solutions for Superlinear Three-Point Boundary Value Problems

  Hong-yu LI

  应用数学学报(英文版). 2017, 33(4): 1043-1052. https://doi.org/10.1007/s10255-017-0718-6

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  In this paper, we investigate the existence of nontrivial solutions for some superlinear second order three-point boundary value problems by applying new fixed point theorems in ordered Banach spaces with the lattice structure derived by Sun and Liu.
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  A Mixed-finite Volume Element Coupled with the Method of Characteristic Fractional Step Difference for Simulating Transient Behavior of Semiconductor Device of Heat Conductor And Its Numerical Analysis

  Yi-rang YUAN, Qing YANG, Chang-feng LI, Tong-jun SUN

  应用数学学报(英文版). 2017, 33(4): 1053-1072. https://doi.org/10.1007/s10255-017-0721-y

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  The mathematical system is formulated by four partial differential equations combined with initialboundary value conditions to describe transient behavior of three-dimensional semiconductor device with heat conduction. The first equation of an elliptic type is defined with respect to the electric potential, the successive two equations of convection dominated diffusion type are given to define the electron concentration and the hole concentration, and the fourth equation of heat conductor is for the temperature. The electric potential appears in the equations of electron concentration, hole concentration and the temperature in the formation of the intensity. A mass conservative numerical approximation of the electric potential is presented by using the mixed finite volume element, and the accuracy of computation of the electric intensity is improved one order. The method of characteristic fractional step difference is applied to discretize the other three equations, where the hyperbolic terms are approximated by a difference quotient in the characteristics and the diffusion terms are discretized by the method of fractional step difference. The computation of three-dimensional problem works efficiently by dividing it into three one-dimensional subproblems and every subproblem is solved by the method of speedup in parallel. Using a pair of different grids (coarse partition and refined partition), piecewise threefold quadratic interpolation, variation theory, multiplicative commutation rule of differential operators, mathematical induction and priori estimates theory and special technique of differential equations, we derive an optimal second order estimate in L²-norm. This numerical method is valuable in the simulation of semiconductor device theoretically and actually, and gives a powerful tool to solve the international problem presented by J. Douglas, Jr.
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  Admissibility of Linear Estimators with Respect to Inequality Constraints Under Some Loss Functions

  Shang-li ZHANG, Hong QIN, Chang-chun WU

  应用数学学报(英文版). 2017, 33(4): 1073-1082. https://doi.org/10.1007/s10255-017-0719-5

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  In this paper, we study the issue of admissibility of linear estimated functions of parameters in the multivariate linear model with respect to inequality constraints under a matrix loss and a matrix balanced loss. Under the matrix loss, when the model is not constrained, the results in the class of non-homogeneous linear estimators [Xie, 1989, Chinese Sci. Bull., 1148-1149; Xie, 1993, J. Multivariate Anal., 1071-1074] showed that the admissibility under the matrix loss and the trace loss is equivalent. However, when the model is constrained by the inequality constraints, we find this equivalency is not tenable, our result shows that the admissibility of linear estimator does not depend on the constraints again under this matrix loss, but it is contrary under the trace loss [Wu, 2008, Linear Algebra Appl., 2040-2048], and it is also relative to the constraints under another matrix loss [He, 2009, Linear Algebra Appl., 241-250]. Under the matrix balanced loss, the necessary and sufficient conditions that the linear estimators are admissible in the class of homogeneous and non-homogeneous linear estimators are obtained, respectively. These results will support the theory of admissibility on the linear model with inequality constraints.
Original Articles
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  Generalized Latin Matrix and Construction of Orthogonal Arrays

  Shan-qi PANG, Li-yan CHEN

  应用数学学报(英文版). 2017, 33(4): 1083-1092. https://doi.org/10.1007/s10255-017-0720-z

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  In this paper, generalized Latin matrix and orthogonal generalized Latin matrices are proposed. By using the property of orthogonal array, some methods for checking orthogonal generalized Latin matrices are presented. We study the relation between orthogonal array and orthogonal generalized Latin matrices and obtain some useful theorems for their construction. An example is given to illustrate applications of main theorems and a new class of mixed orthogonal arrays are obtained.

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  Optimal Eighth Order Convergent Iteration Scheme Based on Lagrange Interpolation

  Rajni SHARMA, Ashu BAHL

  应用数学学报(英文版). 2017, 33(4): 1093-1102. https://doi.org/10.1007/s10255-017-0722-x

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  In this paper, based on fourth order Ostrowski method, we derive an optimal eighth order iteration scheme for obtaining simple roots of nonlinear equations using Lagrange interpolation and suitable weight functions. The scheme requires three evaluations of the function and one evaluation of the first derivative per iteration. Numerical examples are included to confirm the theoretical results and to show the competitive performance of the proposed iteration scheme.