1. Helly's [8] theorem on intersections of convex sets ("If every k-\-l members of a family of compact, convex subsets of Ek have a nonempty intersection, then the intersection of all the members of the family is not empty") has been generalized in various directions. Helly himself gave (in [9]) a generalization to families of not necessarily convex sets, in which the intersections of any 2, 3,...

(1.1) det(D2u) = 1 in Rn must be a quadratic polynomial. For n = 2, a classical solution is either convex or concave; the result holds without the convexity hypothesis. A simpler and more analytical proof, along the lines of affine geometry, of the theorem was later given by Cheng and Yau [9]. The first author extended the result for classical solutions to viscosity solutions [4]. It was proven...

The purpose of this paper is to propose a compositeiterative scheme for approximating a common solution for a finitefamily of m-accretive operators in a strictly convex Banach spacehaving a uniformly Gateaux differentiable norm. As a consequence,the strong convergence of the scheme for a common fixed point ofa finite family of pseudocontractive mappings is also obtained.

In this paper, we introduce a new iterative algorithm for approximating a common solution of certain class of multiple-sets split variational inequality problems. The sequence of the proposed iterative algorithm is proved to converge strongly in Hilbert spaces. As application, we obtain some strong convergence results for some classes of multiple-sets split convex minimization problems.

We utilize radial basis functions (RBFs) to construct numerical schemes for Hamilton–Jacobi (HJ) equations on unstructured data sets in arbitrary dimensions. The computational setup is a meshless discretization of the physical domain. We derive monotone schemes on unstructured data sets to compute the viscosity solutions. The essentially nonoscillatory (ENO) mechanism is combined with radial ba...

This paper deals with convex optimization problems in the face of data uncertainty within the framework of robust optimization. It provides various properties and characterizations of the set of all robust optimal solutions of the problems. In particular, it provides generalizations of the constant subdifferential property as well as the constant Lagrangian property for solution sets of convex ...

in this paper, an effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (rbfs). we present an algorithm based on interpolation by radial basis functions including multiquadratics (mqs), using legendre-gauss-lobatto nodes and weights. also a theorem is proved for convergence of the algorithm. some numerical examples are presented...

Several characterizations of solution sets of a class of convex infinite programs are given using Lagrange multiplier conditions. The results are then applied to some classes of optimization problems: cone-constrained convex programs and fractional programs. A class of semi-convex problems with convex constraints are also examined. Optimality conditions are obtained and several characterization...

We consider learning formulations with non-convex objective functions that often occur in practical applications. There are two approaches to this problem: • Heuristic methods such as gradient descent that only find a local minimum. A drawback of this approach is the lack of theoretical guarantee showing that the local minimum gives a good solution. • Convex relaxation such as L1-regularization...

In cylindrical continua, hoop stresses are induced due to the circumferential failure. This mainly happens when the cylinder is subjected to mechanical loads which vary in the circumferential directions. On the other hand, radial stress is stress in the direction of or opposite to the central axis of a cylindrical body. In the present study, the influence of curvilinear anisotropy on the radial...