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15 August 2018, Volume 35 Issue 4 Previous Issue    Next Issue

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A Global Optimization Algorithm for Indefinite Quadratically Constrained Quadratic Programs ZHAO Ying-feng, LIU San-yang, GE Li 2018, 35 (4):  367-374.  doi: 10.3969/j.issn.1005-3085.2018.04.001 Abstract ( 195 )   PDF (177KB) ( 304 )   Save Indefinite quadratically constrained quadratic programs are widely used in fields of chip design, wireless communication network, finance and many practical engineering problems. Up to now, there is still no general global convergence criteria for solving indefinite quadratically constrained quadratic programs, and it brings great challenges. In this paper, the matrix elementary transformation technique is used for transforming the original problem into an equivalent bilinear programming problem. The linear relaxation programming problem is constructed based on the characteristics of the equivalent problem, and by means of the sequential solutions of a series of linear programming problems, the global optimal solution is obtained. The global convergence is proved and some numerical comparison and random experiments are performed, numerical results show that the algorithm is efficient and feasible. Related Articles | Metrics

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Barrier Options' Pricing and Its Error Analysis Based on Perturbation Method DONG Yan 2018, 35 (4):  375-384.  doi: 10.3969/j.issn.1005-3085.2018.04.002 Abstract ( 196 )   PDF (175KB) ( 490 )   Save The Barrier option pricing problem is one of hot topics in modern finance, and also one of important fields in Mathematical finance. In this paper, the pricing problems of barrier options are discussed under the nonlinear Black-Scholes model. Firstly, the author uses the perturbation method of single-parameter to obtain asymptomatic formulae of barrier options pricing problems. Secondly, error estimates of these asymptotic solutions are illustrated by using the Feymann-Kac formula under the given condition. Finally, numerical experiments confirm the correctness of the proposed theoretical results as well as error estimation. Related Articles | Metrics

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Empirical Likelihood Confidence Intervals for Quantiles in the Presence of Auxiliary Information under Strong Mixing Samples LI Ling, LI Hua-ying, LUO Min, QIN Yong-song 2018, 35 (4):  385-407.  doi: 10.3969/j.issn.1005-3085.2018.04.003 Abstract ( 193 )   PDF (217KB) ( 301 )   Save Strong mixing random variable sequences are used widely in practice. For example, linear processes are strongly mixing under certain conditions. In addition, some continuous time diffusion models and stochastic volatility models are strongly mixing as well. In financial risk management, population quantiles are also called VaR (Value-at-Risk) which specifies the level of excessive losses at a given confidence level. In this paper, in the presence of auxiliary information and under  strong mixing samples, the log-empirical likelihood ratio statistics for quantiles are proposed and it is shown that these statistics asymptotically have the distribution of $\chi^2$. Based on this result, the empirical likelihood based confidence intervals for quantiles are constructed. A class of testing problems are also investigated. It is shown that the asymptotic power of the testing rule in the presence of auxiliary information is higher than that without auxiliary information, and the power is not decreased as more information is available. Related Articles | Metrics

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On Rational Interpolation to $|x|$ at the Dense Newman Nodes ZHANG Hui-ming, LI Jian-jun 2018, 35 (4):  408-414.  doi: 10.3969/j.issn.1005-3085.2018.04.004 Abstract ( 167 )   PDF (143KB) ( 259 )   Save Rational approximation is an important and very vital topic in the theory of function approximation. In this paper, we study the approximation of the nonsmooth function $|x|$ by the Newman rational operator, by increasing $n$ nodes near the zero of the Newman constructed nodes. First, we introduce some main achievements on the rational interpolation to $|x|$. Then, by improving the Newman inequality, it improves from the original $e^{-\sqrt{n}}$ to $8e^{-2\sqrt{n}}$. From this, the approximation order of Newman-type rational operator approximating $|x|$ is $O(e^{-2\sqrt{n}})$, the result is better than the classical results of Newman. Related Articles | Metrics

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Comparison of Staggered and Cell-centered Lagrangian Schemes for One-dimensional Compressible Flow XU Xiao, DAI Zi-huan, GAO Zhi-ming 2018, 35 (4):  415-426.  doi: 10.3969/j.issn.1005-3085.2018.04.005 Abstract ( 150 )   PDF (820KB) ( 367 )   Save According to different discrete locations of the kinematic variables on grids, the Lagrangian algorithm is divided into staggered schemes and cell-centered schemes. Both kinds of schemes achieve remarkable results in computational fluid dynamics, but few attentions are payed to their comparison. In this paper, various one dimensional numerical tests are conducted to study the characteristics of the staggered and cell-centered Lagrangian schemes, and the accuracy of this two type of schemes is compared in detail. The results show that both kinds of schemes can describe the flow field and capture the shock waves and contact discontinuities accurately. Due to the artificial viscosity, the accuracy at the discontinuities decreases in the staggered schemes, both the form and the parameters of the artificial viscosity term have much effect on the results. On the other hand, the cell-centered schemes can keep consistent accuracy at the discontinuities, but suitable reconstruction method and approximate Riemann solver need to be chosen for different problems. Related Articles | Metrics

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Stability and Hopf Bifurcation of an Eco-epidemiological Predator-prey Model with Stage-structure and Time Delay WANG Ling-shu, YAO Pei 2018, 35 (4):  427-444.  doi: 10.3969/j.issn.1005-3085.2018.04.006 Abstract ( 153 )   PDF (188KB) ( 262 )   Save In this paper, the stability and Hopf bifurcation of an eco-epidemiological model with a time delay and a stage structure is investigated. By analyzing the characteristic equations and applying Hurwitz criterion, the local stability of the boundary equilibria and the positive equilibrium are discussed, respectively. Moreover, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium. By using Lyapunov functions and LaSalle's invariance principle, the global stability of the boundary equilibria and the positive equilibrium is addressed, respectively. Therefore, the sufficient conditions are given for the permanence and extinction of the model. Related Articles | Metrics

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The $b$-chromatic Number of Some Corona Graphs LV Chuang, WANG Ke-lun 2018, 35 (4):  445-456.  doi: 10.3969/j.issn.1005-3085.2018.04.007 Abstract ( 189 )   PDF (146KB) ( 474 )   Save Let $\{V_{1}, V_{2},\cdots, V_{k}\}$ be a proper vertex coloring of a graph $G=(V,E)$, which is called a $b$-coloring of $G$, if for all $i, j: 1\leq i\neq j\leq k$, exists $u\in V_{i}, v\in V_{j}$, satisfying $uv\in E$. The maximum positive integer $k$ for a $b$-coloring $\{V_{1}, V_{2},\cdots, V_{k}\}$ on a graph $G$ is called the $b$-chromatic number, denoted by $b(G)$. A graph G is called $b$-continuity if for all $k:\chi(G) \leq k \leq b(G)$, there exists a $(k)b$-coloring on graph $G$. According to the structural characteristics of the Corona graphs, the  cyclic coloring schemes are constructed. Through the cyclic coloring on two kinds of vertices of Corona graphs, the $b$-chromatic number of several Corona graphs equalling to its $m$-degree is obtained, and all these Corona graphs are $b$-continuous. Related Articles | Metrics

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Growth of Solutions of Certain Linear Differential Equations with Entire Coefficients TU Hong-qiang, LIU Hui-fang, ZHANG Shui-ying 2018, 35 (4):  457-467.  doi: 10.3969/j.issn.1005-3085.2018.04.008 Abstract ( 243 )   PDF (163KB) ( 452 )   Save This paper is devoting to study the growth of solutions of some types of higher-order linear differential equations with entire coefficients. One of its coefficients is an entire function extremal for Denjoy's conjecture. By using the value distribution theory of meromorphic functions and the asymptotic value theory of entire functions, and comparing the size of the module of each item appeared in such equations, the estimation on the growth order of its solutions are obtained. It is proved that any nontrivial solution of the equation with one dominant coefficient is of infinite, when there exists one coefficient satisfying the second-order differential equation. The same result also holds for the equation with coefficients having the same growth order and the exponential expressions. The obtained results are the generalization and supplement of some previous results in linear differential equations. Related Articles | Metrics

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Graphs Determined by Second Smallest Distance Eigenvalues in Given Intervals ZHAI Dan-dan 2018, 35 (4):  468-478.  doi: 10.3969/j.issn.1005-3085.2018.04.009 Abstract ( 179 )   PDF (167KB) ( 270 )   Save Let $G=(V,E)$ be a simplified connected graph with $n$ vertices. For a simplified connected graph with $n$ vertices, it has $n$ distance eigenvalues. The second small distance eigenvalue is studied mainly in this paper. And method of disabling the subgraph is used to characterize trees with $\lambda_{n-1}(D(G))\in [-2.4295,0]$, unicyclic and bicyclic graphs with $\lambda_{n-1}(D(G))\in [-2,0]$. Related Articles | Metrics

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A New Hybrid Projection Method and Its Numerical Realization for Two Quasi-nonexpansive Mappings GAO Xing-hui, MA Le-rong 2018, 35 (4):  479-488.  doi: 10.3969/j.issn.1005-3085.2018.04.010 Abstract ( 117 )   PDF (147KB) ( 425 )   Save In Hilbert spaces, a new hybrid projection method is proposed to approximate common fixed points of two quasi-nonexpansive mappings. A strong convergence theorem of the common fixed points is proved by using the quasi-nonexpansive mapping, the projection operator and other analysis techniques. Some numerical experiments are also included to explain the effectiveness of the proposed methods. Related Articles | Metrics