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In finite mixture of location-scale distributions maximum likelihood estimator does not exist because of the unboundedness of the likelihood function when the scale parameter of some mixture component approaches zero. In order to study the strong consistency of maximum likelihood estimator, we consider the case that the scale parameters of the component distributions are restricted from below by c n , where is a sequence of positive real numbers which tends to zero as the sample size n increases. We prove that under mild regularity conditions maximum likelihood estimator is strongly consistent if the scale parameters are restricted from below by c n = exp(-n ), 0 < d < 1. 1

We show that there is a Lyapunov functional for the neural field equation, a neural network model which represents highly dense cortical neurons as a spatially continuous field, and that the system necessarily converges to an equilibrium point as far as the length of the field is finite. We also show that the Lyapunov functional is a natural extension of the Lyapunov function of the Hopfield model. The results suggest that the two models have generally common global dynamics characterized by the intimately related Lyapunov functional/function.

Trees are useful data structures, but to design e#cient parallel programs over trees is known to be more di#cult than to do over lists. Although several important tree skeletons have been proposed to simplify parallel programming on trees, few studies have been reported on how to systematically use them in solving practical problems; it is neither clear how to make a good combination of skeletons to solve a given problem, nor obvious how to find suitable operators used in a single skeleton. In this paper, we report our first attempt to resolve these problems, proposing two important transformations, the tree di#usion transformation and the tree context preservation transformation. The tree di#usion transformation allows one to use familiar recursive definitions to develop his parallel programs, while the tree context preservation transformation shows how to derive associative operators that are required when using tree skeletons. We illustrate our approach by deriving an e#cient parallel program for solving a nontrivial problem called the party planning problem, the tree version of the famous maximum-weight-sum problem.

Abstract—We consider a network formation game where a finite number of nodes wish to send traffic to each other. Nodes contract bilaterally with each other to form communication links; once the network is formed, traffic is routed along shortest paths (if possible). Cost is incurred to a node from four sources: (1) routing traffic; (2) maintaining links to other nodes; (3) disconnection from destinations the node wishes to reach; and (4) payments made to other nodes. We assume that a network is stable if no single node wishes to unilaterally deviate, and no pair of nodes can profitably deviate together. We characterize stable networks, and study the efficiency of those networks. We also consider myopic best response dynamics in the case where links are bidirectional. Under certain assumptions, these myopic dynamics converge to a stable network; further, they naturally select an efficient equilibrium out of the set of possible equilibria. I.

are applied in case studies from the design of ontologies for engineering mathematics and bibliographic data. Selected design decisions are discussed, and alternative representation choices and evaluated against the design criteria.

Recent calls for instructional innovation in undergraduate science, mathematics, engineering, and technology (SMET) courses and pro-grams highlight the need for a solid foundation of education research at the undergraduate level on which to base policy and practice. We report herein the results

Partial differential equations provide mathematical models of many important problems in the physical sciences and engineering. This book treats one class of such equations, concentrating on methods involving the use of surface potentials. It provides the first detailed exposition

engineering, computational fluid dynamics, model reduction, electromagnetics, semiconductor devices, thermodynamics, materials, acoustics, computer graphics/vision, robotics/kinematics, and other discretizations) and those that typically do not have such geometry (optimization, circuit simulation, networks

genetic operations such as crossover (sexual recombination) and mutation. John Holland's pioneering Adaptation in Natural and Artificial Systems (1975) described how an analog of the evolutionary process can be applied to solving mathematical problems and engineering optimization problems using what