Journal of Advances in Applied Mathematics

8

Authors: Fei Yang, Yuanjian Lin

In this paper, by the use of a new fixed point theorem and the Boundary Value Problem’s Green function. the existence of at least one positive solutions for the fourth-order two point boundary value problem with all order derivatives  u (4)(t) + u 00 (t) = λf(t, u(t), u 0 (t), u 00 (t), u 000 (t)), t ∈ [0, 1], u(0) = u(1) = u 00 (0) = u 00 (1) = 0. is considered, where f is a nonnegative continuous function and λ > 0, 0 < A < π2 .

8

Authors: Yan Feng, Luping Xu, Minghui Wang

The nonlinear matrix equation ( ) t X AI X C A Q m − ∗ + ⊗− = is considered in this paper. A sufficient and necessary condition for Hermite positive definite (HPD) solutions of this equation is presented by using properties of the Kronecker product. Two iterative methods for solving the equation are constructed by making use of the principle of the bounded sequence. The feasibility and effectiveness of the iterative methods are verified by numerical example.

14

Authors: John Guenther, Herbert Lee

The black box functions found in computer experiments often have multiple optima. When there are multiple optima, the goal of a computer experiment should be to determine the best optimum for the purposes of the experiment. This brings up the concept of the utility of an optimum: The degree to which the optimum fulfills the purposes of the experiment. The utility of an optima is based on measures of the variability of the surface in the tolerance region (region of interest around the optima) defined by the accuracy with which influential variables can be specified. This tolerance region can be symmetric or asymmetric. For symmetric optima the tolerance region that provides maximum utility is centered at the optimum point. For asymmetric optima, it may be advantageous to displace the center of the tolerance region from the optimum point. For example, a very skewed optimum (minimum in this case) can increase significantly in one direction but only slightly in the opposite direction from the minimum point. For such an optimum the center point of the tolerance region that provides maximum utility for the optimum is displaced from the optimum point along the direction that only slightly increases. The algorithm discussed in this paper locates the center point for the tolerance region to achieve the best utility for any optimum, symmetric or asymmetric. Making use of the surface predictions of the emulator around the minimum point, it employs pattern search to find the best center point for any optimum and for any dimensionality.