THE BERNSTEIN TYPE INEQUALITY AND SIMULTANEOUS APPROXIMATION BY INTERPOLATION POLYNOMIALS IN COMPLEX DOMAIN

In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.

Complex Function Approximation Based on Multi-ANN Approach

This paper presents a multi-ANN approximation approach to approximate complex non-linear function. Comparing with single-ANN methods the proposed approach improves and increases the approximation and generalization ability, and adaptability greatly in learning processes of networks. The simulation results have been shown that the method can be applied to the modeling and identification of complex dynamic control systems.

A new complex variable meshless method for transient heat conduction problems

In this paper, based on the improved complex variable moving least-square （ICVMLS） approximation, a new complex variable meshless method （CVMM） for two-dimensional （2D） transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. As the transient heat conduction problems are related to time, the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization. Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained. In order to demonstrate the applicability of the proposed method, numerical examples are given to show the high convergence rate, good accuracy, and high efficiency of the CVMM presented in this paper.

A new complex variable element-free Galerkin method for two-dimensional potential problems

In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-free Galerkin （ICVEFG） method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square （CVMLS） approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.

Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method

In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin（ICVEFG） method is presented for two-dimensional（2D） elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.

Complex variable element-free Galerkin method for viscoelasticity problems

Based on the complex variable moving least-square （CVMLS） approximation, the complex variable element-free Galerkin （CVEFG） method for two-dimensional viscoelasticity problems under the creep condition is presented in this paper. The Galerkin weak form is employed to obtain the equation system, and the penalty method is used to apply the essential boundary conditions, then the corresponding formulae of the CVEFG method for two-dimensional viscoelasticity problems under the creep condition are obtained. Compared with the element-free Galerkin （EFG） method, with the same node distribution, the CVEFG method has higher precision, and to obtain the similar precision, the CVEFG method has greater computational efficiency. Some numerical examples are given to demonstrate the validity and the efficiency of the method.

An improved complex variable element-free Galerkin method for two-dimensional elasticity problems

In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares （CVMLS） approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin （EFG） method and the ICVMLS approximation, the improved complex variable element-free Galerkin （ICVEFG） method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFC method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.

New complex variable meshless method for advection-diffusion problems

In this paper,an improved complex variable meshless method（ICVMM） for two-dimensional advection-diffusion problems is developed based on improved complex variable moving least-square（ICVMLS） approximation.The equivalent functional of two-dimensional advection-diffusion problems is formed,the variation method is used to obtain the equation system,and the penalty method is employed to impose the essential boundary conditions.The difference method for twopoint boundary value problems is used to obtain the discrete equations.Then the corresponding formulas of the ICVMM for advection-diffusion problems are presented.Two numerical examples with different node distributions are used to validate and investigate the accuracy and efficiency of the new method in this paper.It is shown that ICVMM is very effective for advection-diffusion problems,and has a good convergent character,accuracy,and computational efficiency.

Changes in Electroencephalogram Approximate Entropy Reflect Auditory Processing and Functional Complexity in Frogs

Brain systems engage in what are generally considered to be among the most complex forms of information processing. In the present study, we investigated the functional complexity of anuran auditory processing using the approximate entropy（Ap En） protocol for electroencephalogram（EEG） recordings from the forebrain and midbrain while male and female music frogs（Babina daunchina） listened to acoustic stimuli whose biological significance varied. The stimuli used were synthesized white noise（reflecting a novel signal）, conspecific male advertisement calls with either high or low sexual attractiveness（reflecting sexual selection） and silence（reflecting a baseline）. The results showed that 1） Ap En evoked by conspecific calls exceeded Ap En evoked by synthesized white noise in the left mesencephalon indicating this structure plays a critical role in processing acoustic signals with biological significance; 2） Ap En in the mesencephalon was significantly higher than for the telencephalon, consistent with the fact that the anuran midbrain contains a large well-organized auditory nucleus（torus semicircularis） while the forebrain does not; 3） for females Ap En in the mesencephalon was significantly different than that of males, suggesting that males and females process biological stimuli related to mate choice differently.

Complex effective depth approximation for soft solid bottom

In this paper, the approximate equation of Chapman's (real) effective depth for Pekeris guide is extended to the complex effective depth approximation whose real and imaginary parts explicate respectively the phase change and energy loss on reflection. It is shown that the homogeneous acoustic field, which comprises the complex effective depth approximation,closely reproduces the character of low modes at small grazing angles, and calculates effectively the acoustic field at longer ranges in shallow water. Application of the complex effective depth approximation can be extended to bottoms having two soft solid layers.

Meshless acoustic analysis using a weakly singular Burton-Miller boundary integral formulation

The Burton-Miller boundary integral formulation is solved by a complex variable boundary element-free method(CVBEFM)for the boundary-only meshless analysis of acoustic problems with arbitrary wavenumbers.To regularize both strongly singular and hypersingular integrals and to avoid the computation of the solid angle and its normal derivative,a weakly singular Burton-Miller formulation is derived by considering the normal derivative of the solid angle and adopting the singularity subtraction procedures.To facilitate the implementation of the CVBEFM and the approximation of gradients of the boundary variables,a stabilized complex variable moving least-square approximation is selected in the meshless discretization procedure.The results show the accuracy and efficiency of the present CVBEFM and reveal that the method can produce satisfactory results for all wavenumbers,even for extremely large wavenumbers such as k=10000.

Randomized Primal–Dual Proximal Block Coordinate Updates

In this paper,we propose a randomized primal–dual proximal block coordinate updating framework for a general multi-block convex optimization model with coupled objective function and linear constraints.Assuming mere convexity,we establish its O(1/t)convergence rate in terms of the objective value and feasibility measure.The framework includes several existing algorithms as special cases such as a primal–dual method for bilinear saddle-point problems(PD-S),the proximal Jacobian alternating direction method of multipliers(Prox-JADMM)and a randomized variant of the ADMM for multi-block convex optimization.Our analysis recovers and/or strengthens the convergence properties of several existing algorithms.For example,for PD-S our result leads to the same order of convergence rate without the previously assumed boundedness condition on the constraint sets,and for Prox-JADMM the new result provides convergence rate in terms of the objective value and the feasibility violation.It is well known that the original ADMM may fail to converge when the number of blocks exceeds two.Our result shows that if an appropriate randomization procedure is invoked to select the updating blocks,then a sublinear rate of convergence in expectation can be guaranteed for multi-block ADMM,without assuming any strong convexity.The new approach is also extended to solve problems where only a stochastic approximation of the subgradient of the objective is available,and we establish an O(1/√t)convergence rate of the extended approach for solving stochastic programming.