A fast iterative method to find the matrix geometric mean of two HPD matrices

The purpose of this research is to present a novel scheme based on a quick iterative scheme for calculating the matrix geometric mean of two Hermitian positive definite (HPD) matrices. To do this, an iterative scheme with global convergence is constructed for the sign function using a novel three‐step root‐solver. It is proved that the new scheme is convergent and shown to have global convergence behavior for this target, when square matrices having no pure imaginary eigenvalues. Next, the constructed scheme is used and extended through a well‐known identity for the calculation of the matrix geometric mean of two HPD matrices. Ultimately, several experiments are collected to show its usefulness.     

Mean curvature flow with free boundary – Type 2 singularities

In this paper we give sufficient conditions that guarantee the mean curvature flow with free boundary on a pinched cylinder develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.     

On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

Lipschitz properties of the scalarization function and applications

The scalarization functions were used in vector optimization for a long period. Similar functions were introduced and used in economics under the name of shortage function or in mathematical finance under the name of (convex or coherent) measures of risk. The main aim of this article is to study Lipschitz continuity properties of such functions and to give some applications for deriving necessary optimality conditions for vector optimization problems using the Mordukhovich subdifferential.     

实现均匀照度光伏聚光镜设计

为满足聚光光伏系统的聚光需求,解决传统点聚焦式聚光光伏系统中聚焦光斑不均匀、径长比过大和聚光比较小的缺点,在不增加二次匀光器件的前提下,设计了径长比小、聚焦光斑相对均匀、聚光比高的聚光光伏系统。根据几何光学柯勒照明原理、等光程原理和反射定律,通过数值求解等光程方程组获得聚光镜各个面型的轮廓曲线。利用TracePro软件对所设计的聚光系统进行光线追迹模拟,结果表明:在聚光比为725的情况下,聚焦光斑最大照度仅为太阳照度的2300倍,是点聚焦情况下的1/10左右,系统的径长比为0.3,接收角为0.72°。系统设计实现了结构紧凑,聚光性能高的设计目标,为高倍聚光光伏系统的小型化,简单化提供了一种有效的解决途径。     

Approximation by walsh-kaczmarz-fejér means on the hardy space

The main aim of this paper is to find necessary and sufficient conditions for the convergence of Walsh-Kaczmarz-Fejér means in the terms of the modulus of continuity on the Hardy spaces H_p, when 0<p≤ 1/2.     

On finiteness and continuity of shot noise processes

For a rather general class of stochastic processes induced by time-stationary and by event-stationary random marked point processes, respectively, conditions are given for the almost sure finiteness of these processes and for their continuous dependence on the underlying random marked point process.     

Crystal planes and reciprocal space in Clifford geometric algebra

This paper discusses the geometry of kD crystal cells given by (k+ 1) points in a projective space ?^{n+ 1}. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual representation are related (and geometrically interpreted) in the projective geometric algebra Cl(?^{n+ 1}) (see (Die Ausdehnungslehre von 1844 und die Geom. Anal. Teubner: Leipzig, 1894)) and in the conformal algebra Cl(?^{n+ 1, 1}). The crystallographic notions of d‐spacing, phase angle, structure factors, conditions for Bragg reflections, and the interfacial angles of crystal planes are obtained in the same context. Copyright © 2011 John Wiley & Sons, Ltd.     

Resonant Leading Term Geometric Optics Expansions with Boundary Layers for Quasilinear Hyperbolic Boundary Problems

We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinear hyperbolic initial value problems, including the compressible Euler equations. The technique of simultaneous Picard iteration is employed to show approximate solutions tend to the exact solutions in the small wavelength limit. Recent work [2 Coulombel, J.-F., Gues, O., and Williams, M., 2011. Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems, Comm. Part. Diff. Eqs. 36 (2011), pp. 1797–1859.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] by Coulombel et al. studied the case of reflecting wave trains whose expansions involve only real phases. We treat generic boundary frequencies by incorporating into our expansions both real and nonreal phases. Nonreal phases introduce difficulties such as approximately solving complex transport equations and result in the addition of boundary layers with exponential decay. This also prevents us from doing an error analysis based on almost periodic profiles as in [2 Coulombel, J.-F., Gues, O., and Williams, M., 2011. Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems, Comm. Part. Diff. Eqs. 36 (2011), pp. 1797–1859.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

The construction and approximation of feedforward neural network with hyp erb olic tangent function

In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tangent activation functionFirstly, an equation of partitions of unity for the hyperbolic tangent function is givenThen, two kinds of quasi-interpolation type neural network operators are constructed to approximate univariate and bivariate functions, respectivelyAlso, the errors of the approximation are estimated by means of the modulus of continuity of functionMoreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.