A feed forward architecture is suggested that increases the complexity of conventional neural network components through the implementation of a more complex scheme of interconnection. This is done with a view to increasing the range of application of the feed forward paradigm. The uniqueness of this new network design is illustrated by developing an extended taxonomy of accepted published constructs specific and similar to the higher order, product kernel approximations achievable using "parallel paths". Network topologies from this taxonomy are then compared to each other and the architectures containing parallel paths. In attempting this comparison, the context of the term "network topology" is reconsidered. The output of "channels" in these parallel paths are the products of a conventional connection as observed facilitating interconnection between two layers in a multilayered perceptron and the output of a network processing unit, a "control element", that can assume the identity of a number of pre-existing processing paradigms. The inherent property of universal approximation is tested by existence proof and the method found to be inconclusive. In so doing an argument is suggested to indicate that the parametric nature of the functions as determined by conditions upon initialization may only lead to conditional approximations. The property of universal approximation is neither, confirmed or denied. Universal approximation cannot be conclusively determined by the application of Stone Weierstrass Theorem, as adopted from real analysis. This novel implementation requires modifications to component concepts and the training algorithm. The inspiration for these modifications is related back to previously published work that also provides the basis of "proof of concept". By achieving proof of concept the appropriateness of considering network topology without assessing the impact of the method of training on this topology is considered and discussed in some detail. Results of limited testing are discussed with an emphasis on visualising component contributions to the global network output.

We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a time-indexed formulation, the other on a completiontime formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent a-points, respectively. The analysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853, and we provide instances showing that it is at least e/(e - 1) 1.5819. Both algorithms may be derandomized, their deterministic versions running in O(n2 ) time. The randomized algorithms also apply to the on-line setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards.

Networks can be considered as approximation schemes. Multilayer networks of the backpropagation type can approximate arbitrarily well continuous functions (Cybenko, 1989; Funahashi, 1989; Stinchcombe and White, 1989). We prove that networks derived from regularization theory and including Radial Basis Function (Poggio and Girosi, 1989), have a similar property. From the point of view of approximation theory, however, the property of approximating continous functions arbitrarily well is not sufficient for characterizing good approximation schemes. More critical is the property of best approximation. The main result of this paper is that multilayer networks, of the type used in backpropagation, are not best approximation. For regularization networks (in particular Radial Basis Function networks) we prove existence and uniqueness of best approximation.

Recent developments in the area of reinforcement learning have yielded a number of new algorithms for the prediction and control of Markovian environments. These algorithms, including the TD(lambda) algorithm of Sutton (1988) and the Q-learning algorithm of Watkins (1989), can be motivated heuristically as approximations to dynamic programming (DP). In this paper we provide a rigorous proof of convergence of these DP-based learning algorithms by relating them to the powerful techniques of stochastic approximation theory via a new convergence theorem. The theorem establishes a general class of convergent algorithms to which both TD(lambda) and Q-learning belong.

by Maurice F. Aburdene, Ram-Nandan P. Singh. / Bibliography: leaf 9. / "February, 1979." / Partial support by NASA Ames Research Center under Grant NGL-22-009-124

In this paper we consider the hypo-ellipticity of differential forms on a closed manifold.The main results show that there are some topological obstruct for the existence of the differential forms with hypoellipticity.

We give an explicit construction of a fundamental solution for an arbitrary non-degenerate partial differential equation with smooth coefficients.

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To differ from the usually applied Eulerian method for describing the motion of fluid, the governing equations complete in the Lagrangian form for describing three-dimensional progressive and short-crested waves system are derived in this paper. A systematical ordering expansion by an appropriate perturbation approximation is developed, and the exactly satisfactory solutions in a form of functional, up to third-order progressive waves and up to second-order short-crested waves, are obtained. The kinematic properties of the waves, including the surface profile, pressure, the paths of fluid particles, and the mass transport velocity, are then described directly. The obtained solution for the short-crested waves system is successfully verified by reducing to two special cases, one is the two-dimensional simple progressive waves, and the other is the two-dimensional standing waves. Also, the analytical results are compared with experimental data including the surface profiles, the pressures and the paths of fluid particles for validation.

The focus of this dissertation is the interpolation and approximation of multivariate bandlimited functions via sampled (function) values. The first set of results investigates polynomial interpolation in connection with multivariate bandlimited functions. To this end, the concept of a uniformly invertible Riesz basis is developed (with examples), and is used to construct Lagrangian polynomial interpolants for particular classes of sampled square-summable data. These interpolants are used to derive two asymptotic recovery and approximation formulas. The first recovery formula is theoretically straightforward, with global convergence in the appropriate metrics; however, it becomes computationally complicated in the limit. This complexity is sidestepped in the second recovery formula, at the cost of requiring a more local form of convergence. The second set of results uses oversampling of data to establish a multivariate recovery formula. Under additional restrictions on the sampling sites and the frequency band, this formula demonstrates a certain stability with respect to sampling errors. Computational simplifications of this formula are also given.