2018年, 第34卷, 第1期　刊出日期：2018-02-15

  

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  Adjoining Batch Markov Arrival Processes of a Markov Chain

  Xiao-yun MO, Xu-yan XIANG, Xiang-qun YANG

  应用数学学报(英文版). 2018, 34(1): 1-10. https://doi.org/10.1007/s10255-018-0724-3

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  A batch Markov arrival process (BMAP) X^*=(N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly:given a Markov chain J, can we deploy a process N such that the 2-dimensional process X^*=(N, J) is a BMAP? The process X^*=(N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D_k, k=0, 1, 2…) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.
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  Optimality Conditions of the Set-valued Optimization Problem with Generalized Cone Convex Set-valued Maps Characterized by Contingent Epiderivative

  Zhi-ang ZHOU, Xin-min YANG, Qiu-sheng QIU

  应用数学学报(英文版). 2018, 34(1): 11-18. https://doi.org/10.1007/s10255-018-0727-0

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  In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.
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  Disjoint Cliques in Claw-free Graphs

  Su-yun JIANG, Jin YAN

  应用数学学报(英文版). 2018, 34(1): 19-34. https://doi.org/10.1007/s10255-018-0737-y

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  A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_1,3. Let s and k be two integers with 0 ≤ s ≤ k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n ≥ 3s+4(k-s) and d(x)+d(y) ≥ n-2s+2k+1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K₃s and k-s disjoint K₄s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.
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  An Efficient Risk Estimator with External Information Under Additive Hazards Model

  Xin WANG, Xiao-ming XUE, Jie ZHOU, Liu-quan SUN

  应用数学学报(英文版). 2018, 34(1): 35-50. https://doi.org/10.1007/s10255-018-0726-1

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  Rare event data is encountered when the events of interest occur with low frequency, and the estimators based on the cohort data only may be inefficient. However, when external information is available for the estimation, the estimators utilizing external information can be more efficient. In this paper, we propose a method to incorporate external information into the estimation of the baseline hazard function and improve efficiency for estimating the absolute risk under the additive hazards model. The resulting estimators are shown to be uniformly consistent and converge weakly to Gaussian processes. Simulation studies demonstrate that the proposed method is much more efficient. An application to a bone marrow transplant data set is provided.
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  Bayesian Planning of Optimal Step-stress Accelerated Life Test for Log-location-scale Distributions

  Qiang GUAN, Yin-cai TANG

  应用数学学报(英文版). 2018, 34(1): 51-64. https://doi.org/10.1007/s10255-018-0725-2

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  This paper introduces some Bayesian optimal design methods for step-stress accelerated life test planning with one accelerating variable, when the acceleration model is linear in the accelerated variable or its function, based on censored data from a log-location-scale distributions. In order to find the optimal plan, we propose different Monte Carlo simulation algorithms for different Bayesian optimal criteria. We present an example using the lognormal life distribution with Type-I censoring to illustrate the different Bayesian methods and to examine the effects of the prior distribution and sample size. By comparing the different Bayesian methods we suggest that when the data have large(small) sample size B₁(τ) (B₂(τ)) method is adopted. Finally, the Bayesian optimal plans are compared with the plan obtained by maximum likelihood method.
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  A Geometric Flow Approach for Region-based Image Segmentation-theoretical Analysis

  Zhu-cui JING, Juntao YE, Guo-liang XU

  应用数学学报(英文版). 2018, 34(1): 65-76. https://doi.org/10.1007/s10255-018-0723-4

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  In this paper, we analyze the well-posedness of an image segmentation model. The main idea of that segmentation model is to minimize one energy functional by evolving a given piecewise constant image towards the image to be segmented. The evolution is controlled by a serial of mappings, which can be represented by B-spline basis functions. The evolution terminates when the energy is below a given threshold. We prove that the correspondence between two images in the segmentation model is an injective and surjective mapping under appropriate conditions. We further prove that the solution of the segmentation model exists using the direct method in the calculus of variations. These results provide the theoretical support for that segmentation model.
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  Asymptotic Properties of Wavelet Estimators in Partially Linear Errors-in-variables Models with Long-memory Errors

  Hong-chang HU, Heng-jian CUI, Kai-can LI

  应用数学学报(英文版). 2018, 34(1): 77-96. https://doi.org/10.1007/s10255-018-0730-5

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  While the random errors are a function of Gaussian random variables that are stationary and long dependent, we investigate a partially linear errors-in-variables (EV) model by the wavelet method. Under general conditions, we obtain asymptotic representation of the parametric estimator, and asymptotic distributions and weak convergence rates of the parametric and nonparametric estimators. At last, the validity of the wavelet method is illuminated by a simulation example and a real example.
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  Ergodicity of the 2D Navier-Stokes Equations with Degenerate Multiplicative Noise

  Zhao DONG, Xu-hui PENG

  应用数学学报(英文版). 2018, 34(1): 97-118. https://doi.org/10.1007/s10255-018-0728-z

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  Consider the two-dimensional, incompressible Navier-Stokes equations on torus T²=[-π, π]² driven by a degenerate multiplicative noise in the vorticity formulation (abbreviated as SNS):dw_t=ν△w_tdt + B(Kw_t, w_t)dt + Q(w_t)dW_t. We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup {P_t}_t>0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility. Under almost the same Hypotheses as that given by Odasso, Prob. Theory Related Fields, 140:41-82 (2005) with a different method, we get an exponential ergodicity under a stronger norm.
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  A Primal-dual Large-update Interior-point Algorithm for P_* (κ)-LCP Based on a New Class of Kernel Functions

  Ping JI, Ming-wang ZHANG, Xin LI

  应用数学学报(英文版). 2018, 34(1): 119-134. https://doi.org/10.1007/s10255-018-0729-y

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  In this paper, we propose a large-update primal-dual interior point algorithm for P_*(κ)-linear complementarity problem. The method is based on a new class of kernel functions which is neither classical logarithmic function nor self-regular functions. It is determines both search directions and the proximity measure between the iterate and the center path. We show that if a strictly feasible starting point is available, then the new algorithm has O (1+2κ)p√n(1/p log n+1)² log n/ε iteration complexity which becomes O((1+2κ)√n log n log n/ε) with special choice of the parameter p. It is matches the currently best known iteration bound for P_*(κ)-linear complementarity problem. Some computational results have been provided.
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  Neighbor Sum Distinguishing Chromatic Index of Sparse Graphs via the Combinatorial Nullstellensatz

  Xiao-wei YU, Yu-ping GAO, Lai-hao DING

  应用数学学报(英文版). 2018, 34(1): 135-144. https://doi.org/10.1007/s10255-018-0731-4

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  Let φ:E(G) → {1, 2, …, k} be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if ∑_e∋u φ(e)≠∑_e∋v φ(e) for each edge uv ∈ E(G). The smallest value k for which G has such a coloring is denoted by χ'_∑(G), which makes sense for graphs containing no isolated edge (we call such graphs normal). It was conjectured by Flandrin et al. that χ'_∑(G) ≤ △(G) + 2 for all normal graphs, except for C₅. Let mad(G)=max { (2|E(H)|)/(|V (H)|)|H ⊆ G} be the maximum average degree of G. In this paper, we prove that if G is a normal graph with △(G) ≥ 5 and mad(G) < 3 -2/△(G), then χ'_∑(G) ≤ △(G) + 1. This improves the previous results and the bound △(G) + 1 is sharp.
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  Mean-square Stability of Stochastic Age-dependent Delay Population Systems with Jumps

  Qiang LI, Qi-min ZHANG, Bo-qiang CAO

  应用数学学报(英文版). 2018, 34(1): 145-154. https://doi.org/10.1007/s10255-018-0732-3

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  In this paper, we present the compensated stochastic θ method for stochastic age-dependent delay population systems (SADDPSs) with Poisson jumps. The definition of mean-square stability of the numerical solution is given and a sufficient condition for mean-square stability of the numerical solution is derived. It is shown that the compensated stochastic θ method inherits stability property of the numerical solutions. Finally, the theoretical results are also confirmed by a numerical experiment.
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  Em Algorithm of the Truncated Multinormal Distribution with Linear Restriction on the Variables

  Bai-suo JIN, Jing-jing HAN, Shu DING, Bai-qi MIAO

  应用数学学报(英文版). 2018, 34(1): 155-162. https://doi.org/10.1007/s10255-018-0733-2

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  A new expectation-maximization (EM) algorithm is proposed to estimate the parameters of the truncated multinormal distribution with linear restriction on the variables. Compared with the generalized method of moments (GMM) estimation and the maximum likelihood estimation (MLE) for the truncated multivariate normal distribution, the EM algorithm features in fast calculation and high accuracy which are shown in the simulation results. For the real data of the national college entrance exams (NCEE), we estimate the distribution of the NCEE examinees' scores in Anhui, 2003, who were admitted to the university of science and technology of China (USTC). Based on our analysis, we have also given the ratio truncated by the NCEE admission line of USTC in Anhui, 2003.
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  Hosoya and Merrifield-Simmons Indices in Random Polyphenyl Chains

  Wei-ling YANG

  应用数学学报(英文版). 2018, 34(1): 163-172. https://doi.org/10.1007/s10255-018-0734-1

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  The Hosoya index of a graph is the total number of matchings in it. And the Merrifield-Simmons index is the total number of independent sets in it. They are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In this paper, we obtain explicit analytical expressions for the expectations of the Hosoya index and the Merrifield-Simmons index of a random polyphenyl chain.
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  On Local Spectral Properties of Hamilton Type Operators

  Jun-li SHEN, Alatancang

  应用数学学报(英文版). 2018, 34(1): 173-182. https://doi.org/10.1007/s10255-018-0736-z

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  In this paper, we introduce the class of Hamilton type operators and study various properties of this class. We show that every Hamilton type operator with property (β) or (δ) is decomposable. In addition, we prove that a Hamilton type operator T satisfies property (β), Dunford's property (C) and Weyl's theorem if and only if its adjoint does.
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  Second-order Optimality Conditions for Cone-subarcwise Connected Set-valued Optimization Problems

  Zhen-hua PENG, Yi-hong XU

  应用数学学报(英文版). 2018, 34(1): 183-196. https://doi.org/10.1007/s10255-018-0738-x

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  The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.
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  A Distance Function for Computing on Finite Subsets of Euclidean Spaces

  Hajar Ghahremani-Gol, Farzad Didehvar, Asadollah Razavi

  应用数学学报(英文版). 2018, 34(1): 197-208. https://doi.org/10.1007/s10255-018-0735-0

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  In practical purposes for some geometrical problems, specially the fields in common with computer science, we deal with information of some finite number of points. The problem often arises here is:"How are we able to define a plausible distance function on a finite three dimensional space?" In this paper, we define such a distance function in order to apply it to further purposes, e.g. in the field settings of transportation theory and geometry. More precisely, we present a new model for traveling salesman problem and vehicle routing problem for two dimensional manifolds in three dimensional Euclidean space, the second problem on which we focus on this line is, three dimensional triangulation.