A Family of Algebraically Closed Fields Containing Polynomials in Several Variables

We introduce a family of fields of series with support in strongly convex rational cones. All these fields contain polynomials in several variables. We prove that they are algebraically closed with a construction that is analogous to the Newton polygon for algebraic curves. As a corollary, we show the existence of fractional power solutions with support in cones for systems of equations.     

Convergence analysis of a proximal newton method 1

Recently Fukushima and Qi proposed a proximal Newton method for minimizating a nonsmooth convex function. An alternative global convergence proof for that method is presented in this paper. Global convergence was established without any additional assumption on the objective function. We also show that the infimum of a convex function is always equal to the infimun of its Moreau—Yosida regularization     

迹与行列式的相互表示

应用牛顿恒等式,得到矩阵的特征值的对称多项式与等幂和之间的关系,以此为基础给出行列式的迹表示,另由克莱姆法则导出迹的行列式表示。     

Smooth and Semismooth Newton Methods for Constrained Approximation and Estimation

In the article, we show that the constrained L ₂ approximation problem, the positive polynomial interpolation, and the density estimation problems can all be reformulated as a system of smooth or semismooth equations by using Lagrange duality theory. The obtained equations contain integral functions of the same form. The differentiability or (strong) semismoothness of the integral functions and the Hölder continuity of the Jacobian of the integral function were investigated. Then a globalized Newton-type method for solving these problems was introduced. Global convergence and numerical tests for estimating probability density functions with wavelet basis were also given. The research in this article not only strengthened the theoretical results in literatures but also provided a possibility for solving the probability density function estimation problem by Newton-type method.     

吉尔伯特·牛顿·路易斯——加州大学伯克利分校化学系的缔造者*

吉尔伯特·牛顿·路易斯是20世纪最伟大的化学家之一。在加州大学伯克利分校,路易斯通过大力引进人才、提倡学术民主和学术合作、改革人才培养模式,逐步建立起了世界上最强大、最具创造力的化学系之一,培养了一批日后在化学领域做出了重大贡献的博士研究生。介绍了路易斯担任加州大学伯克利分校化学学院院长兼化学系主任期间的教育改革措施。     

-functions of twisted diagonal exponential sums over finite fields

Let be the finite field of elements with characteristic and its extension of degree . Fix a nontrivial additive character and let be multiplicative characters of For

one can form the twisted exponential sum . The corresponding -function is defined by

In this paper, by using the -adic gamma function and the Gross-Koblitz formula on Gauss sums, we give an explicit formula for the -function if is a Laurent diagonal polynomial. We also determine its -adic Newton polygon.

     

Polynomials with the half-plane property and matroid theory

A polynomial f is said to have the half-plane property if there is an open half-plane H⊂C, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions relating multivariate polynomials with the half-plane property to matroid theory.

(1)

We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous.

(2)

We prove that a multivariate multi-affine polynomial f∈R[z₁,…,zn] has the half-plane property (with respect to the upper half-plane) if and only if

     

Sixth order derivative free family of iterative methods

In this study, we develop a four-parameter family of sixth order convergent iterative methods for solving nonlinear scalar equations. Methods of the family require evaluation of four functions per iteration. These methods are totally free of derivatives. Convergence analysis shows that the family is sixth order convergent, which is also verified through the numerical work. Though the methods are independent of derivatives, computational results demonstrate that family of methods are efficient and demonstrate equal or better performance as compared with other six order methods, and the classical Newton method.     

A new smoothing Newton method for solving constrained nonlinear equations

In this paper, a new smoothing Newton method is proposed for solving constrained nonlinear equations. We first transform the constrained nonlinear equations to a system of semismooth equations by using the so-called absolute value function of the slack variables, and then present a new smoothing Newton method for solving the semismooth equations by constructing a new smoothing approximation function. This new method is globally and quadratically convergent. It needs to solve only one system of unconstrained equations and to perform one line search at each iteration. Numerical results show that the new algorithm works quite well.     

无约束优化问题的修正PRP共轭梯度法

本文通过结合牛顿法与PRP共轭梯度法提出一修正PRP方法,新方法中包含了二阶导数信息,在适当的假设下算法全局收敛,数值算例表明了算法的有效性.