Kantorovich variant of a new kind of q‐Bernstein–Schurer operators

Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.     

非凸半定规划的增广Lagrangian的微分的计算

迄今为止，还未见出版过有关求解非凸半定规划的算法，但在最近，Chen，et.al(2000)和Sun＆Sun(1999)关于非凸半定规划(SDP)的增广Lagrangian的研究是非常有用的，在本文中，我们证明非凸半定规划的增广Lagrangian是可微的，并且给出它的可微表达式．     

An Unsolved Problem of Fenchel

The Fenchel problem of level sets is solved under the conditions that theboundaries of the nested family of convex sets in R^n>+1 aregiven by C³ n-dimensional differentiable manifolds and theconvex sets determine an open or closed convex set inRⁿ⁺¹.     

Analytic implementation of the duals of some spaces of infinitely differentiable functions

Using the Fourier-Laplace transform for functionals, we describe the duals of some spaces of the infinitely differentiable functions given on convex compact sets or convex domains in ?^N and such that the growth of their derivatives is determined by weight sequences of a general form.     

Differentiability of a class of semi-linear hyperbolic systems with respect to a parameter

If a parameter is contained in a hyperbolic semi-linear equation, then the solution to the equation will depend on the parameter. It is proved by using the theory of semi-groups of bounded operators that the solution continuously depends on the parameter and is Fréchet continuously differentiable, under certain mild assumptions.     

Stiefel-Whitney classes and toral actions

Certain Stiefel-Whitney classes of manifolds with smooth, effective toral actions are shown to be computable in terms of Poincare duals of fixed point sets of isotropy subgroups. As an application the toral degrees of symmetry of certain Dold manifolds are determined.     

A note on solving nonlinear minimax problems via a differentiable penalty function

The note demonstrates that modeling a nonlinear minimax problem as a nonlinear programming problem and applying a classical differentiable penalty function to attempt to solve the problem can lead to convergence to a stationary point of the penalty function which is not a feasible point of the nonlinear programming problem. This occurred naturally in an application from statistical reliability theory. The note resolves the problem through modification of both the problem formulation and the iterative penalty function method.     

Enlarging the region of convergence of Newton's method for constrained optimization

In this paper, we consider Newton's method for solving the system of necessary optimality conditions of optimization problems with equality and inequality constraints. The principal drawbacks of the method are the need for a good starting point, the inability to distinguish between local maxima and local minima, and, when inequality constraints are present, the necessity to solve a quadratic programming problem at each iteration. We show that all these drawbacks can be overcome to a great extent without sacrificing the superlinear convergence rate by making use of exact differentiable penalty functions introduced by Di Pillo and Grippo (Ref. 1). We also show that there is a close relationship between the class of penalty functions of Di Pillo and Grippo and the class of Fletcher (Ref. 2), and that the region of convergence of a variation of Newton's method can be enlarged by making use of one of Fletcher's penalty functions.This work was supported by the National Science Foundation, Grant No. ENG-79-06332.     

A simple proof for the Chernoff-Savage theorem

A proof for the Chernoff-Savage theorem is given based on differentiable statistical functionals. It employs no second order differentiability condition.     

Surjectivity of Linear Differential Operators in a Weighted Space of Infinitely Differentiable Functions

We present a sufficient condition for the surjectivity of a linear differential operator of infinite order with constant coefficients in a weighted space of infinitely differentiable functions on the real line.