Abstract: A computational model is a representation of some physical or other system of interest, first expressed mathematically and then implemented in the form of a computer program; it may be viewed as a function of inputs that, when evaluated, produces outputs. Motivation for this article comes from computational models that are deterministic, complicated enough to make classical mathematical analysis impractical and that have a moderate-to-large number of inputs. The problem of designing computational experiments to determine which inputs have important effects on an output is considered. The proposed experimental plans are composed of individually randomized one-factor-at-a-time designs, and data analysis is based on the resulting random sample of observed elementary effects, those changes in an output due solely to changes in a particular input. Advantages of this approach include a lack of reliance on assumptions of relative sparsity of important inputs, monotonicity of outputs with respect to inputs, or ad...

Abstract: Several researchers characterized the activation functions under which multilayer feedforwardnetworks can act as universal approximators. We show that all the characterizationsthat were reported thus far in the literature ark special cases of the following general result:a standard multilayer feedforward network can approximate any continuous functionto any degree of accuracy if and only if the network's activation functions are not polynomial.We also emphasize the important role of the threshold, asserting that without it thelast theorem doesn't hold.

Abstract: The initial state of an unforced linear system is output admissible with respect to a constraint set Y if the resulting output function satisfies the pointwise-in-time condition y(t) in Y, t>or=0. The set of all possible such initial conditions is the maximal output admissible set O/sub infinity /. The properties of O/sub infinity / and its characterization are investigated. In the discrete-time case, it is generally possible to represent O/sub infinity / or a close approximation of it, by a finite number of functional inequalities. Practical algorithms for generating the functions are described. In the continuous-time case simple representations of the maximal output admissible set are not available, however, it is shown that the discrete-time results may be used to obtain approximate representations. >

Abstract: This is a method for determining numerically local minima of differentiable functions of several variables. In the process of locating each minimum, a matrix which characterizes the behavior of the function about the minimum is determined. For a region in which the function depends quadratically on the variables, no more than N iterations are required, where N is the number of variables. By suitable choice of starting values, and without modification of the procedure, linear constraints can be imposed upon the variables.

Abstract: An attempt is made to determine the logically consistent rules for selecting a vector from any feasible set defined by linear constraints, when either all $n$-vectors or those with positive components or the probability vectors are permissible. Some basic postulates are satisfied if and only if the selection rule is to minimize a certain function which, if a "prior guess" is available, is a measure of distance from the prior guess. Two further natural postulates restrict the permissible distances to the author's $f$-divergences and Bregman's divergences, respectively. As corollaries, axiomatic characterizations of the methods of least squares and minimum discrimination information are arrived at. Alternatively, the latter are also characterized by a postulate of composition consistency. As a special case, a derivation of the method of maximum entropy from a small set of natural axioms is obtained.

Abstract: The proximal point algorithm (PPA) for the convex minimization problem $\min _{x \in H} f(x)$, where $f:H \to R \cup \{ \infty \} $ is a proper, lower semicontinuous (lsc) function in a Hilbert space H is considered. Under this minimal assumption on f, it is proved that the PPA, with positive parameters $\{ \lambda _k \} _{k = 1}^\infty $, converges in general if and only if $\sigma _n = \sum_{k = 1}^n {\lambda _k \to \infty } $. Global convergence rate estimates for the residual $f(x_n ) - f(u)$, where $x_n $ is the nth iterate of the PPA and $ u \in H $ is arbitrary are given. An open question of Rockafellar is settled by giving an example of a PPA for which $x_n $ converges weakly but not strongly to a minimizes of f.

Abstract: The spatial Green's function for the open microstrip structure, especially with a thick substrate, is generally represented in the time-consuming Sommerfield integrals. Through the Sommerfield identity, a closed-form spatial Green's function of a few terms is found from the quasi-dynamic images, the complex images, and the surface waves. With the numerical integration of the Sommerfeld integrals thus avoided, this closed-form Green's function is computationally very efficient. Numerical examples show that the closed-form Green's function gives less than 1% error for all substrates and source-to-field distances. >

Abstract: This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums These arise in many problems in analytic number theory It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory The authors show how the method can be applied to problems such as upper bounds for the Riemann-Zeta function the Dirichlet divisor problem, the distribution of square free numbers, and the Piatetski-Shapiro prime number theorem

Abstract: This paper presents a theory of convergence for real coded genetic algorithms GAs that use oating point or other high cardinality codings in their chromosomes The theory is consistent with the theory of schemata and postulates that selection dominates early GA performance and restricts subsequent search to intervals with above average function value dimension by dimension These intervals may be further subdivided on the basis of their attraction under genetic hillclimbing Each of these subintervals is called a virtual character and the collection of characters along a given dimension is called a virtual alphabet It is the virtual alphabet that is searched during the recombinative phase of the genetic algorithm and in many problems this is su cient to ensure that good solutions are found Although the theory helps suggest why many problems have been solved using real coded GAs it also suggests that real coded GAs can be blocked from further progress in those situations when local optima separate the virtual characters from the global optimum

Abstract: A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition. This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leffler function. The retardation function is determined via Laplace transformation and is solely a power law type. The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles. This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value. In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function.

Abstract: A stereo matching method is presented which uses multiple stereo pairs with various baselines to obtain precise depth estimates without suffering from ambiguity. The stereo matching method uses multiple stereo pairs with different baselines generated by a lateral displacement of a camera. Matching is performed by computing the sum of squared-difference (SSD) values. The SSD functions for individual stereo pairs are represented with respect to the inverse depth (rather than the disparity, as is usually done), and then are simply added to produce the sum of SSDs. This resulting function is called the SSSD-in-inverse-depth. The authors define a stereo algorithm, based on the SSSD-in-inverse-depth and then present a mathematical analysis to show how the algorithm can remove ambiguity and increase precision. Experimental results for stereo images are presented to demonstrate the effectiveness of the algorithm. >

Abstract: Linear discrete-time systems affected by both parameter and input uncertainties are considered. The problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned speed of convergence is investigated. It is shown that the problem has a solution if and only if there exists a certain Lyapunov function which does not belong to a pre-assigned class of functions (i.e. the quadratic ones) but it is determined by the target set in which ultimate boundedness is desired. One of the advantages of this approach is that one can handle systems with control constraints. No matching assumptions are made. For systems with linearly constrained uncertainties, it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets. An extension of the technique to continuous-time systems is presented. >

Abstract: In the second‐order reliability method the principal curvatures of the limit‐state surface at the design point are used to construct a paraboloid approximation of the surface, which is then used to compute a second‐order estimate of the failure probability. The principal curvatures are the eigenvalues of the Hessian of the surface. In this paper an efficient algorithm is developed to determine the principal curvatures without computing the Hessian. The curvatures are computed in an iterative manner using the gradient of the limit‐state function, and are obtained in the decreasing order of their absolute magnitudes, which is also the order of their importance in reliability analysis. The computation can be terminated when the last curvature obtained is sufficiently small. The method is efficient for problems with large numbers of random variables, especially when an efficient algorithm for computing the gradient is available. Several numerical examples, including a finite‐element application involving 99 r...

Abstract: A method is presented for nonlinear function identification and application to learning control. The control objective is to identify and compensate for a nonlinear disturbance function. The nonlinear disturbance function is represented as an integral of a predefined kernel function multiplied by an unknown influence function. Sufficient conditions for the existence of such a representation are provided. Similarly, the nonlinear function estimate is generated by an integral of the predefined kernel multiplied by an influence function estimate. Using the time history of the plant, the learning rule indirectly estimates the unknown function by updating the influence function estimate. It is shown that the estimate function converges to the actual disturbance asymptotically. Consequently, the controller achieves the disturbance cancellation asymptotically. The method is extended to repetitive control applications. It is applied to the control of robot manipulators. Simulation and actual real-time implementation results using the Berkeley/NSK robot arm show that the proposed learning algorithm is more robust and converges at a faster rate than conventional repetitive controllers. >

Abstract: The bootstrap procedure provides a way to test the significance of the regression coefficients and the stability of the estimates in response functions generated by regression on principal components A subroutine RESBO, which calculates a bootstrapped response function, has been added to Fritts' program PRECON The principle of the response function is described in Fritts (1976) and discussed in Hughes, et al (1982) To avoid problems with the great number of predictors and their inter correlation, Fritts et al (1971) introduced regression on principal components As with all regression methods, the main problems with this procedure are testing the significance of the coefficients and the stability of the estimates The response function obtained on a sample is considered satisfactory only if it explains the growth over independent years The most straightforward way to assess the stability is to divide climatic and tree -ring data into a dependent calibration set and an independent verification set (Fritts 1976) If the set of tree -ring indices estimated from the verification -set climate data using the regression coefficients that were derived from the calibration data set is close to the observed values, the response function is judged as reliable Gordon et al (1982) clearly set out the problem of verifying the predictive ability of a model calibrated on one data set when applied to another data set Because regression coefficients are validated only to the dependent data, they result in overconfidence in the predictive power of the model We can be convinced of that by simulating tree -ring indices by random numbers and by calculating response functions with real climatic data (Guiot 1981; Cropper 1985) These authors showed that simulated tree -ring series also can produce regression coefficients judged significant by standard Student's tests This result is due mainly to an inadequate number of degrees of freedom To test regression coefficients, Student's test involves n -k -1 degrees of freedom where n is the number of observations and k the number of regressors If k is set to the number of principal components actually introduced into the regression on the basis of their correlation with the predictand (stepwise regression), the significance of the coefficients is overestimated; therefore, the number k must be chosen by a priori considerations independent of the predictand A good practice is to select a relatively large number of principal components taking into account say 90 or 95% of the variance of the climatic data or using the PVP criterion of Guiot (1981, 1985) The number k is then the number of princi-

Abstract: The relation between the Walsh-Hadamard transform and the auto-correlation function of Boolean functions is used to study propagation characteristics of these functions. The Strict Avalanche Criterion and the Perfect Nonlinearity Criterion are generalized in a Propagation Criterion of degree k. New properties and constructions for Boolean bent functions are given and also the extension of the definition to odd values of n is discussed. New properties of functions satisfying higher order SAC are derived. Finally a general framework is established to classify functions according to their propagation characteristics if a number of bits is kept constant.

Abstract: Multiclass learning problems involve finding a definition for an unknown function f(x) whose range is a discrete set containing k < 2 values (i.e., k "classes"). The definition is acquired by studying large collections of training examples of the form [xi, f(xi)]. Existing approaches to this problem include (a) direct application of multiclass algorithms such as the decision-tree algorithms ID3 and CART, (b) application of binary concept learning algorithms to learn individual binary functions for each of the k classes, and (c) application of binary concept learning algorithms with distributed output codes such as those employed by Sejnowski and Rosenberg in the NETtalk system. This paper compares these three approaches to a new technique in which BCH error-correcting codes are employed as a distributed output representation. We show that these output representations improve the performance of ID3 on the NETtalk task and of back propagation on an isolated-letter speech-recognition task. These results demonstrate that error-correcting output codes provide a general-purpose method for improving the performance of inductive learning programs on multiclass problems.

Abstract: Let f be a real differentiate function defined on the open subset A of R n;f is said to be invex if the following inequality is satisfied for a suitable function η The purpose of this paper is to investigate the relationship between invexity and generalized convexity; moreover, we shall give necessary and sufficient conditions for a differentiable function to be η-invex, which are natural extensions of the analogous conditions for convex functions.

Abstract: We investigate the effectiveness of connectionist architectures for predicting the future behavior of nonlinear dynamical systems. We analyze the sunspot series as an example of a real world time series of limited record length. The problem of overfitting, particularly serious for short records of noisy data, is addressed both by using the statistical method of validation and by adding a complexity term to the cost function (weight-elimination). We show why sigmoid units are superior in performance to radial basis functions for high-dimensional input spaces. The ultimate goal is prediction accuracy: we find that sigmoid networks trained with weight-elimination outperform traditional nonlinear statistical approaches. The prediction accuracy does not deteriorate when too many input units are used. Iterated single-step predictions are found to be better than direct multi-step predictions. Furthermore, we compare different sampling times (yearly and monthly), investigate the effect of preprocessing the data (square root and logarithmic transforms) and compare different error functions (corresponding to Gauss and Poisson statistics).

Abstract: Nonlinear function approximation is often solved by finding a set of coefficients for a finite number of fixed nonlinear basis functions. However, if the input data are drawn from a high-dimensional space, the number of required basis functions grows exponentially with dimension, leading many to suggest the use of adaptive nonlinear basis functions whose parameters can be determined by iterative methods. The author proposes a technique based on the idea that for most of the data, only a few dimensions of the input may be necessary to compute the desired output function. Additional input dimensions are incorporated only where needed. The learning procedure grows a tree whose structure depends upon the input data and the function to be approximated. This technique has a fast learning algorithm with no local minima once the network shape is fixed, and it can be used to reduce the number of required measurements in situations where there is a cost associated with sensing. Three examples are given: controlling the dynamics of a simulated planar two-joint robot arm, predicting the dynamics of the chaotic Mackey-Glass equation, and predicting pixel values in real images from pixel values above and to the left. >

Abstract: Given n frames taken at different time instants and m points in each frame, the problem of motion correspondence is to map a point in one frame to another point in the next frame such that no two points map onto the same point. This problem is combinatorially explosive; one needs to introduce constraints to limit the search space. We propose a proximal uniformity constraint to solve the correspondence problem. According to this constraint, most objects in the real world follow smooth paths and cover a small distance in a small time. Therefore, given a location of a point in a frame, its location in the next frame lies in the proximity of its previous location. Further, resulting trajectories are smooth and uniform and do not show abrupt changes in velocity vector over time. An efficient, non-iterative polynomial time approximation algorithm which minimizes the proximal uniformity cost function and establishes correspondence over n frames is proposed. It is argued that any method using smoothness of motion alone cannot operate correctly without assuming correct initial correspondence, the correspondence in the first two frames. Therefore, we propose the use of gradient based optical flow for establishing the initial correspondence. This way the proposed approach combines the qualities of the gradient and token based methos for motion correspondence. The algorithm is then extended to take care of restricted cases of occlusion. A metric called distortion measure for measuring the goodness of solution to this n frame correspondence problem is also proposed. The experimental results for real and synthetic sequences are presented to support our claims.

Abstract: A theoretical analysis of morphological filters for the optimal restoration of noisy binary images is presented. The problem is formulated in a general form, and an optimal solution is obtained by using fundamental tools from mathematical morphology and decision theory. Consideration is given to the set-difference distance function as a measure of comparison between images. This function is used to introduce the mean-difference function as a quantitative measure of the degree of geometrical and topological distortion introduced by morphological filtering. It is proved that the class of alternating sequential filters is a set of parametric, smoothing morphological filters that best preserve the crucial structure of input images in the least-mean-difference sense. >