数学季刊

刘长河, 尚有林, 李振国

2015, 30(4): 475-494. doi:10.13371/j.cnki.chin.q.j.m.2015.04.001

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We establish polynomial complexity bounds of the Mehrotra-type predictorcorrector algorithms for linear programming over symmetric cones. We first slightly modify the maximum step size in the predictor step of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al[18]. Then, using the machinery of Euclidean Jordan algebras, we extend the modified algorithm to symmetric cones. Based on the Nesterov-Todd direction, we obtain O（r log ε-1） iteration complexity bound of this algorithm, where r is the rank of the Jordan algebras and ε is the required precision. We also present a new variant of Mehrotra-type algorithm using a new adaptive updating scheme of centering parameter and show that this algorithm enjoys the same order of complexity bound as the safeguard algorithm. We illustrate the numerical behaviour of the methods on some small examples.

热方程紧差分格式重叠型区域分解算法

张红梅

2015, 30(4): 495-502. doi:10.13371/j.cnki.chin.q.j.m.2015.04.002

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In this paper, a modified additive Schwarz finite difference algorithm is applied in the heat conduction equation of the compact difference scheme. The algorithm is on the basis of domain decomposition and the subspace correction. The basic train of thought is the introduction of the units function decomposition and reasonable distribution of the overlap of correction. The residual correction is conducted on each subspace while the computation is completely parallel. The theoretical analysis shows that this method is completely characterized by parallel.

与一族（1+1）维孤子方程相联系的有限维可积系

魏含玉, 郭汉东, 夏铁成

2015, 30(4): 503-514. doi:10.13371/j.cnki.chin.q.j.m.2015.04.003

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In this paper, a new spectral problem is proposed and the corresponding soliton equations hierarchy are also obtained. Under a constraint between the potentials and the eigenfunctions, the eigenvalue problem is nonlinearized so as to be a new finitedimensional Hamiltonian system. By resotring to the generating function approach, we obtain conserved integrals and the involutivity of the conserved integrals. The finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Finally, we show the decomposition of the soliton equations.

全平面上Dirichlet级数的级与余项估计

黄慧珺, 宁菊红

2015, 30(4): 515-523. doi:10.13371/j.cnki.chin.q.j.m.2015.04.004

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In this paper, firstly, the ρ order and ρ_βorder of Dirichlet series which converges in the whole plane are studied. Secondly, the equivalence relation between remainder logarithm ln E_n-1（f, α）, ln R_n（f, α） and coefficients logarithm ln |a_n| is discussed respectively. Finally,the theory of applying remainder to estimate ρ order and ρ_βorder can be obtained by using the equivalence relation.

可转包的双目标单机排序问题

陈荣军, 秦立珍, 唐国春

2015, 30(4): 524-531. doi:10.13371/j.cnki.chin.q.j.m.2015.04.005

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Scheduling with outsourcing is studied in this paper. It is assumed that both manufacturer and subcontractor have a single machine to process n jobs. The manufacturer needs to determine simultaneously a set of outsourced jobs and the schedule of the jobs in-house such that two criterias, i.e., outsourcing cost and production cost, are minimized.The production cost is measured by the number of tardy jobs or the total tardiness of jobs in-house, and the outsourcing cost is proportional to the total processing time of jobs outsourced. Two kinds of problems with different criterias are considered. We analyze the computational complexity and provide pseudo-polynomial time optimization algorithms for the NP-hard version of the problems.

由广义Dziok-Srivastava算子及其积分算子定义的亚纯多叶倒星象函数的某些新子类

马丽娜 , 李书海

2015, 30(4): 532-544. doi:10.13371/j.cnki.chin.q.j.m.2015.04.006

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In this paper, we introduce some new subclasses of meromorphically uniformly reciprocal starlike functions associated with the generalized Dziok-Srivastava operator and its corresponding integral operator defined by subordination. We obtain the inclusion relation,sufficient conditions and majorization property of the class. Moreover, we point out some new and interesting corollaries of our main result. These results generalize some known results.

混合幂丢番图不等式

牟全武, 吕晓东

2015, 30(4): 545-554. doi:10.13371/j.cnki.chin.q.j.m.2015.04.007

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It is proved that if λ₁, λ₂, ···, λ₇are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 < σ <1/16, the inequality |λ₁x₁²+ λ₂x₂²+∑7 i=3λ_ix_i⁴+ η| <（ max1≤i≤7|x_i|）^-σhas infinitely many solutions in positive integers λ₁, λ₂, ···, λ₇ Similar result is proved for |λ₁x₁²+ λ₂x₂²+ λ₃x₃²+ λ₄x₄⁴+ λ₅x₅⁴+ λ₆x₆⁴+ η| <（ max1≤i≤6|x_i|）^-σ.These results constitute an improvement upon those of Shi and Li.

鞍点问题PHSS方法预条件的新取法

孙佳, 王世恒, 王珂

2015, 30(4): 555-561. doi:10.13371/j.cnki.chin.q.j.m.2015.04.008

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Bai, Golub and Pan presented a preconditioned Hermitian and skew-Hermitian splitting（PHSS） method [Numerische Mathematik, 2004, 32: 1-32] for non-Hermitian positive semidefinite linear systems. We improve the method to solve saddle point systems whose（1,1） block is a symmetric positive definite M-matrix with a new choice of the preconditioner and compare it with other preconditioners. The results show that the new preconditioner outperforms the previous ones.

带跳一扩散风险过程保险人的最优控制

吴锟, 肖建武, 罗荣华

2015, 30(4): 562-569. doi:10.13371/j.cnki.chin.q.j.m.2015.04.009

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In this paper, the optimal XL-reinsurance of an insurer with jump-diffusion risk process is studied. With the assumptions that the risk process is a compound Possion process perturbed by a standard Brownian motion and the reinsurance premium is calculated according to the variance principle, the implicit expression of the priority and corresponding value function when the utility function is exponential are obtained. At last, the value function is argued, the properties of the priority about parameters are discussed and numerical results of the priority for various claim-size distributions are shown.

一类多线性Hardy型算子在变指数Herz-Morrey空间的有界性

程星星, 瞿萌, 束立生

2015, 30(4): 570-578. doi:10.13371/j.cnki.chin.q.j.m.2015.04.010

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In this paper, we will obtain that the boundedness of multilinear n-dimensional fractional Hardy operators of variable order β（x） on variable exponent Herz-Morrey spaces.

2 -（v,23,1）设计的传递自同构群

李上钊

2015, 30(4): 579-586. doi:10.13371/j.cnki.chin.q.j.m.2015.04.011

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This paper is a contribution to the study of the automorphism groups of 2-（v, k, 1） designs. Let D be a 2-（v, 23, 1） design and G a block-transitive and point-primitive group of automorphism of D. Then the socle of G is not Sz（q） and²G₂（q）.Key words: block-transitive; point-primitive; design; socle

一类q-Sz\acute{a}sz-Mirakyan-Baskakov算子的逼近性质

莫庆峰, 胡晓敏, 吴鹏

2015, 30(4): 587-595. doi:10.13371/j.cnki.chin.q.j.m.2015.04.012

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In this paper we propose the q analogues of modified Baskakov-Sz′asz operators.we estimate the moments and establish the direct results in term of modulus of continuity.An estimate for the rate of convergence and weighted approximation properties of the q operators are also obtained.

关于L_p-极投影体Shephard问题的一个类似

马统一

2015, 30(4): 596-609. doi:10.13371/j.cnki.chin.q.j.m.2015.04.013

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For p > 0, Lutwak, Yang and Zhang introduced the concept of L_p-polar projection body Γ_-pK of a convex body K in Rⁿ. Let p ≥ 1 and K, L \subset Rⁿbe two origin-symmetric convex bodies, we consider the question of whether Γ_-p K \subset Γ_-p L implies \Omega_p（L） ≤ \Omega_p（K）,where \Omega_p（K） denotes the L_p-affine surface area of K and K = Voln（K）^-1/pK. We prove a necessary and sufficient condition of an analog of the Shephard problem for the L_p-polar projection bodies.

一类不连续系统Φ-有界变差解对参数的连续依赖性

马学敏, 李宝麟, 林长伟

2015, 30(4): 610-619. doi:10.13371/j.cnki.chin.q.j.m.2015.04.014

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The functions of bounded Φ-variation are development and generalization of bounded variation functions in the usual sense.Henstock-Kurzweil integral is a very useful tool for some discontinuous systems. In this paper, by using Henstock-Kurzweil integral, we establish theorems of continuous dependence of bounded Φ-variation solutions on parameter for a class of discontinuous systems on the base of Φ-function. These results are essential generalizations of continuous dependence of bounded variation solutions on parameter for the systems.

关于图的强乘积的可迁性的研究

董丽欣, 李峰, 赵海兴

2015, 30(4): 620-623. doi:10.13371/j.cnki.chin.q.j.m.2015.04.015

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Since many large graphs are composed from some existing smaller graphs by using graph operations, say, the Cartesian product, the Lexicographic product and the Strong product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the Strong product of vertex-transitive graphs. In particular, we show that the Strong product of Cayley graphs is still a Cayley graph.

算子数值域的内点

窦艳妮 , 李陈心 , 李荣 , 杜鸿科

2015, 30(4): 624-632. doi:10.13371/j.cnki.chin.q.j.m.2015.04.016

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In this note, some properties of the interior of numerical ranges of operators are established, and an alternative proof of Embry’s theorem associated with the interior of a numerical ranges of an operator is given(see [3]).

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