Abstract: In this paper, we present a generalization of the Hessian matrix toC 1,1 functions, i.e., to functions whose gradient mapping is locally Lipschitz. This type of function arises quite naturally in nonlinear analysis and optimization. First the properties of the generalized Hessian matrix are investigated and then some calculus rules are given. In particular, a second-order Taylor expansion of aC 1,1 function is derived. This allows us to get second-order optimality conditions for nonlinearly constrained mathematical programming problems withC 1,1 data.

Abstract: A phenomenological renormalisation-group model is proposed for Ising-like spin glasses. The model parameters are a typical interaction strength and a typical energy barrier for block spins. The energy barrier is found to be a function of length scale with the largest barrier at the correlation length. The system equilibration time varies at tau T varies as exp(V/T) or faster depending on dimensionality. For short times t< tau T the system is hysteretic and the noise spectrum is approximately 1/f. A two-level model with distributed level separations and barrier heights is derived.

Abstract: This paper presents two applications of an interesting information theoretic theorem about graphs The first application concerns the derivation of good bounds for the function $Y(b,k,n)$, which is defined to be the minimum size of a family of functions such that for every subset of size k from an n element universe, there exists a perfect hash function in the family mapping the subset into a table of size b The second application concerns the derivation of good bounds for the function $M(i,j,n)$, which is defined to be the minimum size of an $(i,j)$-separating system

Abstract: Modeling the image as a piecewise linear gray-value function of the pixel coordinates considerably improved a change detection test based previously on a piecewise constant gray-value function. These results encouraged investigations into modeling the picture as a mosaic of patches where the gray-value function within each patch is described as a second-order bivariate polynomial of the pixel coordinates. Such a more appropriate model allowed the assumption to be made that the remaining gray-value variation within each patch can be attributed to noise related to the sensing and digitizing devices, independent of the individual image frames in a sequence. This assumption made it possible to relate the likelihood test for change detection to well-known statistical tests ( t test, F test), facilitating the determination of threshold values related to a priori confidence limits.

Abstract: A popular approach to the analysis of mapped planar point patterns is through secondorder methods, where the object of attention is the variance of the number of points falling in a test set of a given size and shape, or the behaviour of all distances between pairs of points in the pattern. Standard references for the analysis of spatial data are works by Ripley (1981, Ch. 8) and Diggle (1983). Many of the methods for point-pattern analysis given in the literature are second-order in nature. The analysis of mapped point patterns is important in a variety of biological applications, ranging from ecology to physiology, and some examples are given in these books by Ripley and by Diggle. Suppose that an observed point pattern can be considered as a realization of a random point-process model which is stationary and isotropic. Of course this is a strong assumption which may not be justified in practice, but it is nevertheless the assumption under which much of the existing point-process methodology has been developed. The property of the underlying random process elicited, directly or indirectly, by second-order methods is the second-moment cumulative function K(t), which satisfies the following properties under suitable regularity conditions (see Ripley, 1977, p. 150): (i) if X is the intensity of the process, then XK(t) is the expected number of further points within distance t of a typical point of the process; (ii) specifying K(t) for all t is equivalent to specifying the variance of the number of points falling in any given set. The function K(t) defined by Ripley (1977) is an edge-corrected version of the empirical distribution function of all interpoint distances in the observed pattern, and provides an approximately unbiased estimator of K(t). The estimate K(t) can be used to construct tests of the hypothesis that an observed pattern is consistent with the 'completely random' Poisson point-process model, and, more importantly, to quantify the apparent deviation

Abstract: A simple and systematic method is described for calculating the integral of a polynomial function over an arbitrary nonconvex polyhedron. First a general formula is presented for direct evaluation of the integral of a polynomial over a 3-D simplex. An integral over a polyhedron can then be easily calculated by using the central projection method and decomposing a polyhedron symmetrically into a set of simplices and accumulating the results from each simplex based on this formula. This method adopts a systematic and automatic decomposition. It is analytically exact, but the practical accuracy of the result is within the accuracy of floating-point arithmetic. Furthermore, the time complexity of this method is linearly proportional to the number of vertices of a polyhedron.

Abstract: A digital phase locked loop is provided by the present invention which includes a digital controlled oscillator (DCO), whereby the frequency of the output signal of the DCO is a function of the value associated with a digital input word. The frequency of the output signal of the DCO is phase compared to a reference signal by a phase comparator. Depending upon which signal is leading or lagging, the phase comparator outputs an increment signal or a decrement signal. These increment and decrement signals are operatively coupled to an up/down counter which provides the digital input word to the DCO, the value of the digital input word being modified by the increment or decrement signal to cause the frequency of the output signal to track the frequency of the input signal to the phase comparator, i.e., the frequency of the reference signal.

Abstract: The problem of measuring distances between discrete frequency distributions is considered. Three conditions are stated, which are believed to reflect basic, intuitive requirements to be met by a distance measure of the above kind with particular reference to genetic frequency distributions. These conditions chiefly concern aspects of maximum distance and linearity. It is shown that exactly one function meets the conditions, and this function, having all properties of a metric, is explicitly given.

Abstract: In this paper, an "energy function" is derived for transient stability analysis of power systems. The main contribution of this paper is the generality of the model for the loads. The loads are modeled as constant real power loads but the reactive part of the loads are modeled as arbitrary function of the load bus voltage. This model accounts for the voltage transients of the load busses and for this general model, it is shown that the system satisfies the law of conservation of energy. An efficient algorithm for determining the stability of the system for a given disturbance is described. Results of simulation of the system studied by El-Abiad and Nagappan [2] are included and it is shown that due to the fluctuations in the system voltages, the commonly used energy function over estimates the energy of the system during system transients.

Abstract: The purpose of this article is to present the approximation theory underlying the p-version of the finite element method. By exploiting the relationship between polynomial approximation and certain weighted Sobolev spaces, which are identified as the domains of positive real powers of the Legendre operator, this one-dimensional result is generalized via a tensor product construction to yield a nonconforming piecewise polynomial approximation result in the usual unweighted Sobolev spaces on triangulated domains of $R^n $. It is then shown that essentially the same result holds for approximation by conforming piecewise polynomials provided that the function being approximated possesses the same degree of conformality across the common boundaries of adjacent simplices and the same homogeneous boundary conditions. Inverse results are given for the special case of approximation in $L_2 $.

Abstract: A practical approach is proposed to the problem of simultaneously computing a function, its partial derivatives with respect to all the variables, and an estimate of the rounding error incurred in the computed value of the function. Theoretically, it has a complexity at most a constant times as large as that of evaluating the function alone, the constant being independent of the number of variables of the function, and it is an alternative graphical interpretation of W. Baur and V. Strassen’s results, with some generalizations. Practically, it is stated in a form easily implementable as a computer program, which enables us to automatically compute the derivatives if we are given only the program for computing the function. Remarks are added also on the cases of several functions, of higher derivatives and of non-straght-line programs, and on application to problems containing differential equations.

Abstract: An exact integral equation for the pair-connectedness function gDouble Dagger (12) is derived on the basis of the author's earlier exact equation for the pair-distribution function g(12). The new equation gives rise to a sequence of approximations for gDouble Dagger (12), starting with the Percus-Yevick approximation. Significant aspects of the first two approximations are noted, as is the relevance of the blocking functions g-gDouble Dagger for ionic and chemically associating particles.

Abstract: This paper examines the use of clue words in argument dialogues. These are special words and phrases directly indicating the structure of the argument to the hearer. Two main conclusions are drawn: 1) clue words can occur in conjunction with coherent transmissions, to reduce processing of the hearer 2) clue words must occur with more complex forms of transmission, to facilitate recognition of the argument structure. Interpretation rules to process clues are proposed. In addition, a relationship between use of clues and complexity of processing is suggested for the case of exceptional transmission strategies.

Abstract: Exhaustive pattern logic testing schemes provide all possible input patterns with respect to an output in the set of test patterns. This paper is concerned with the problem that arises when this is to be done simultaneously with respect to a number of outputs, using a single test set. More specifically, in this paper we describe an iterative procedure for generating a test set consisting of n- dimensional vectors which exhaustively covers all k-subspaces simultaneously, i.e., the projections of n-dimensional vectors in the test set onto any input subset of a specified size k contain all possible patterns of k-tuples. For any given k, we first find an appropriate N (N > k) and generate an efficient N-dimensional test set for exhaustive coverage of all k-subspaces. We next develop a constructive procedure to expand the corresponding test matrix (formed by taking test vectors as its rows) such that a test set of N2-dimensional vectors exhaustively covering the same k-subspaces is obtained. This procedure may be repeated to cover arbitrarily large n (n = N2i after i iterations), while keeping the same k. It is shown that the size of the test set obtained this way grows in size which becomes proportional to (log n) raised to the power of [log (q + 1)], where q is a function of k only, and is approximated (bounded closely below) by k2/4 in binary cases. This approach applies to nonbinary cases as well except that the value of q in an r-ary case is approximated by a number lying between k2/4 and k2/2.

Abstract: In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition x + y, multiplication x v, and two-place exponentiation x '. For a restricted class of such functions, we show that every true identity follows from the natural set of eleven axioms. The rates of growth of such functions, in the case of a single independent variable x, as x - *, are also studied, and we give an algorithm for the Hardy relation of eventual domination, again for a restricted class of functions. Value distribution of analytic functions of one and of several complex variables, especially the Nevanlinna characteristic, plays a major role in our proofs. Introduction. To make precise the questions we consider, it is necessary to introduce a formal language containing the expressions which define the elementary functions to be studied. This language has variables x!, x2,..., a set S of constants (representing a given set of fixed numbers) and function symbols for addition, multiplication and exponentiation. The terms (that is, the function-defining expres- sions) are obtained inductively starting with the variables and constants and continuing according to the rule: if t and s are terms, then so are (t + s), (t * s) and (t'). (We follow some standard conventions for dropping parentheses and below.) Let us refer to these as the exponential terms over S. In most cases considered here the constants in S represent positive real numbers. In such a case, every term t represents a function which is defined for all positive real values of the variables which occur in it. We write t -s to mean that t and s are terms which define exactly the same function (for positive real values of the variables appearing in t or s). It is necessary to distinguish between this function equality relation t s and a formal expression of equality between t and s, which we write in the form t = s. The relation t -s is a mathematical relation between certain elementary functions; the expression t = s is a purely formal expression which is susceptible to formal proof within various axiomatic systems. A closely related topic concerns the orders of growth of functions of one variable defined by exponential terms. If t and s are such terms which contain only the single

Abstract: The identification problem for systems with fuzzy relations is solved by applying a general method of solving finite fuzzy relation equations. It is shown that the identification uncertainty is characterized in terms of the maximum solution and the infimum of all the minimal solutions. The result is independent of the function used to define the uncertainty measure. A method for calculating the infimum of all the minimal solutions without calculating these minimal solutions is also derived and results in an efficient method for calculating the identification uncertainty.

Abstract: The partial derivatives of the squared error loss function for the metric unfolding problem have a unique geometry which can be exploited to produce unfolding methods with very desirable properties. This paper details a simple unidimensional unfolding method which uses the geometry of the partial derivatives to find conditional global minima; i.e., one set of points is held fixed and the global minimum is found for the other set. The two sets are then interchanged. The procedure is very robust. It converges to a minimum very quickly from a random or non-random starting configuration and is particularly useful for the analysis of large data sets with missing entries.

Abstract: We report results for the three-jet cross section to order αs2 using a jet resolution criterion depending on the jet mass and study the sum of three-and four-jet cross sections inO(αs2) as a function of the resolution parameters in order to obtain the limit of infinite resolution.

Abstract: An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x ? 0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is = x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random. NETWORKS; BRANCHING PROCESS

Abstract: In the study of integral solutions to Diophantine equations, many problems can be reduced to that of solving the equationin S-units of the given ring. To accomplish this over number fields, the only known effective method is to use Baker's deep results on linear forms in logarithms, which yield relatively weak upper bounds. For function fields, R. C. Mason [2] has recently given a remarkably strong effective upper bound. In this note we give an independent proof of Mason's bound, relying only on elementary algebraic geometry, principally the Riemann-Hurwitz formula.