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15 December 2017, Volume 34 Issue 6 Previous Issue    Next Issue

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Principle Component Analysis for Tensors and Compression Theory for High-dimensional Information XIA Zhi-ming, ZHAO Wen-zhi, XU Zong-ben 2017, 34 (6):  571-590.  doi: 10.3969/j.issn.1005-3085.2017.06.001 Abstract ( 157 )   PDF (242KB) ( 631 )   Save In this paper, we summarize the past, present of principle component analysis for tensors in the context of information compression, and show some untouched research fields. Firstly, we review the conception of tensors and tensor decomposition which can be expressed by an unified statistical model. Secondly, by the order of the classical principal component analysis, robust principal component analysis and sparse principal component analysis, we  summarize the development of relative statistical theories and algorithms where each one can be further divided into vector data, matrix data and tensor data from the simple to the complex. Related Articles | Metrics

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The Research and Simulation for Chaotic Behavior and Synchronization of the Earth's Magnetic Field System WANG He-yuan, KAN Meng, DUAN Wen-yuan 2017, 34 (6):  591-598.  doi: 10.3969/j.issn.1005-3085.2017.06.002 Abstract ( 180 )   PDF (743KB) ( 284 )   Save In order to find the essential features of magnetohydrodynamics, this paper investigates the dynamical behavior and its synchronization of the earth's magnetic field system. We study the stability of stationary solutions to the earth's magnetic field system by using the linear stability analysis method. The chaotic behavior of the system is simulated, the simulation results show that the system can reach the chaos through the subcritical Hopf bifurcation, and there exists the periodic window in the chaotic region, a quasi-periodic regime could be formed at the high values of the parameter. The globally attractive set and positively invariant set of the earth's magnetic field system are discussed via constructing a generalized Lyapnov function. Further, an adaptive control approach is proposed to the synchronization between two chaotic systems. Numerical simulations show the effectiveness of the method. Related Articles | Metrics

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A Branch and Bound Reduction Algorithm for Solving a Class of Linear Multiplicative Programming Problems with Constant Coefficients JING Xia, GAO Lei 2017, 34 (6):  599-608.  doi: 10.3969/j.issn.1005-3085.2017.06.003 Abstract ( 79 )   PDF (165KB) ( 192 )   Save We propose a new branch and bound reduction algorithm for solving a class of linear multiplicative programming problems with constant coefficients. Firstly, by utilizing the convex envelopes of the products of two variables, the lower and upper bounds for the multiplications in the objective and constraint functions are determined. And we use these bounds to construct the corresponding convex programming relaxation for the original problem. Then, resorting to the hyper-rectangular reduction strategy, a new branch and bound reduction algorithm is designed. It is shown that this new algorithm is globally convergent. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm. Related Articles | Metrics

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Algorithm Based on Primal-dual Splitting Method for Solving a Class of Constrained Separable Convex Optimization Problem and its Application TANG Yu-chao, CHEN Bao, ZHU Chuan-xi, YU Hui 2017, 34 (6):  609-621.  doi: 10.3969/j.issn.1005-3085.2017.06.004 Abstract ( 142 )   PDF (272KB) ( 407 )   Save In this paper, we study a constrained separable convex optimization model, in which the data error term in the objective function is differentiable. Many problems arising in image restoration and image reconstruction are the special cases of this convex optimization problem. In order to overcome the shortcomings of existing methods for solving this model, we transform the original problem into an unconstrained convex optimization problem by using an indicator function. Then, we propose a new iterative algorithm, which is based on the idea of the primal-dual splitting method. The proposed algorithm has a simple structure and is easy for parameter selection. At the same time, we prove the convergence of the new iterative algorithm. Finally, to verify its effectiveness, we apply the algorithm to CT image reconstruction. Numerical experiments show that the proposed algorithm outperforms existing iterative algorithms in terms of reconstruction time and reconstruction image quality. Related Articles | Metrics

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Existence and Uniqueness of Solutions to a Class of Neumann Problems for the Semilinear Elliptic Equation XING Hui, CHEN Hong-bin 2017, 34 (6):  622-628.  doi: 10.3969/j.issn.1005-3085.2017.06.005 Abstract ( 127 )   PDF (151KB) ( 389 )   Save The solutions to semilinear elliptic partial differential equations contain rich information about the equations, which is very important for describing the development of various phenomena. The existence of equilibrium solutions of multi-species mutual aid model and the economic equilibrium point can be transformed into the existence of the solutions to Neumann boundary value problems. In this paper, we study the existence and uniqueness of the solutions for a class of semilinear elliptic equations with Neumann boundary value conditions. Using the topological degree theory and the eigenvalue comparison principle, we obtain the existence of the solutions under the assumption that the nonlinear terms satisfy the asymptotic nonuniform conditions. Using the eigenvalue comparison principle, we prove the uniqueness of the solutions. The obtained results extend and complement some relevant existing works. As an application, an example is given to verify the obtained results. Related Articles | Metrics

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Quenching Phenomena for Second-order Nonlinear Parabolic Equation with Nonlinear Source NIU Yi, PENG Xiu-yan, ZHANG Ming-you, SHEN Ji-hong 2017, 34 (6):  629-636.  doi: 10.3969/j.issn.1005-3085.2017.06.006 Abstract ( 107 )   PDF (812KB) ( 301 )   Save In this paper, we investigate a class of second-order nonlinear parabolic equations. Under some conditions about the nonlinear source term, we obtain the quenching phenomena of the Cauchy problem. It is shown that, with more generally nonlinear absorption, the solution quenches in finite time under some restrictions on the exponents of the source term and the initial data. When the structure of the nonlinear absorption is changed, the solution of the Cauchy problem for the second-order nonlinear parabolic equation may exist globally. In the end, we illustrate the behavior of quenching phenomena through simulation experiments. The larger the source term exponents are, the shorter the quenching time is. Our main tools are the comparison principle, the maximum principle and the eigenfunction method. Related Articles | Metrics

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Infinitely Many Sign-changing Solutions for a Class of Fourth-order Elliptic Equations GAO Min, WU Ying 2017, 34 (6):  637-645.  doi: 10.3969/j.issn.1005-3085.2017.06.007 Abstract ( 148 )   PDF (155KB) ( 208 )   Save In engineering practice, the fourth-order elliptic equation with the biharmonic operator $\Delta^2 u + c \Delta u = f(x,u), x \in \Omega$, can be used to describe the deformation of an suspension bridge. When the bridge is in equilibrium and there are no external forces, the corresponding equation satisfies the boundary condition $u|_{\partial \Omega} = \Delta u|_{\partial \Omega} = 0$. In this paper, a class of fourth-order elliptic boundary value problems is examined under the assumption that the nonlinear term $f$ is asymptotically linear at $0$ and superquadric at $\infty$ with respect to $u$. The proof method is the descending flow invariant set method. The main results are two theorems which establish the existence of one sign-changing solution and infinitely many sign-changing solutions, respectively. The main results and the proofs are different from those presented in current literature. Related Articles | Metrics

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High Order Energy Preserving Method for the Ito-type Coupled KdV Equations WANG Yi-fan, SUN Jian-qiang, CHEN Xiao-wei 2017, 34 (6):  646-654.  doi: 10.3969/j.issn.1005-3085.2017.06.008 Abstract ( 101 )   PDF (322KB) ( 297 )   Save Constructing the high order energy preserving scheme for the Ito-type coupled KdV equation with the energy conservation property has the important application in simulating the motion of the equation. In this paper, the high order energy preserving scheme for the Ito-type coupled KdV equation is obtained by applying the fourth order average vector field method and the Fourier pseudo spectral method. The new high order energy preserving scheme is applied to simulate the solitary wave behaviors  of the equation. Numerical results show the new high order energy preserving scheme can well simulate the solitary wave evolution behaviors of the Ito-type coupled KdV equation and preserve the discrete energy conservation exactly. Related Articles | Metrics

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Enumeration Technique for Some Paths and Generalized Spectrum of a Graph LIU Fen-jin, WANG Wei 2017, 34 (6):  655-671.  doi: 10.3969/j.issn.1005-3085.2017.06.009 Abstract ( 127 )   PDF (153KB) ( 209 )   Save A graph $X$ is said to be determined by its generalized spectrum if for any graph $H$, $H$ and $X$ being cospectral with cospectral complements implies that $H$ is isomorphic to $X$. This paper determines formulas for enumerating the number of paths with length no more than $5$ in any graph and proves that if $k\not\equiv2 (mod 3)$, then the grid graph $P_k\square P_2$ is determined by its generalized spectrum. Related Articles | Metrics

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Integrated Pest Management of One-predator Two-prey System with Mutual Interference WANG Zhen, SHAO Yuan-fu, FANG Xian-jia 2017, 34 (6):  672-692.  doi: 10.3969/j.issn.1005-3085.2017.06.010 Abstract ( 77 )   PDF (10796KB) ( 123 )   Save Taking into account chemical control, biological control for pest management at different fixed moments, and the mutual interference of the predator, we propose in this paper a three-species predator-prey system with chemical control, biological control and mutual interference of predator. Based on the theory of impulsive equation, small amplitude perturbation and Floquet theory, we investigate the existence and globally asymptotic stability of the prey-eradication periodic solution for this system. By using comparison methods involving multiple Lyapunov functions, some sufficient conditions assuring the permanence of this system are obtained. Some examples and numerical simulations are given to show the complex behaviors of this system. Further, the biological implications of our main results are analyzed and some suggestions for feasible control strategies are given. Related Articles | Metrics