Abstract: when certain parameters in the problem tend to limiting values (for example, when the sample size increases indefinitely, the intensity of the noise ap proaches zero, etc.) To address the problem of asymptotically optimal estimators consider the following important case. Let X 1, X 2, ..., X n be independent observations with the joint probability density !(x, O) (with respect to the Lebesgue measure on the real line) which depends on the unknown patameter o e 9 c R1. It is required to derive the best (asymptotically) estimator 0: ( X b ..., X n) of the parameter O. The first question which arises in connection with this problem is how to compare different estimators or, equivalently, how to assess their quality, in terms of the mean square deviation from the parameter or perhaps in some other way. The presently accepted approach to this problem, resulting from A. Wald's contributions, is as follows: introduce a nonnegative function w(0l> ( ), Ob Oe 9 (the loss function) and given two estimators Of and O! n 2 2 the estimator for which the expected loss (risk) Eown(Oj, 0), j = 1 or 2, is smallest is called the better with respect to Wn at point 0 (here EoO is the expectation evaluated under the assumption that the true value of the parameter is 0). Obviously, such a method of comparison is not without its defects."

Abstract: NL2SOL is a modular program for solving nonlinear least-squares problems that incorporate a number of novel features. It maintains a secant approximation S to the second-order part of the least-squares Hessian and adaptively decides when to use this approximation. S is "sized" before updating, something which is similar to Oren-Luenberger scaling. The step choice algorithm is based on minimizing a local quadratic model of the sum of squares function constrained to an elliptical trust region centered at the current approximate minimizer. This is accomplished using ideas discussed by More'', together with a special module for assessing the quality of the step thus computed. These and other ideas behind NL2SOL are discussed and its evolution and current implemetation are also described briefly.

Abstract: In 1959 A. Renyi proposed a set of axioms for a measure of dependence for pairs of random variables. In the same year A. Sklar introduced the general notion of a copula. This is a function which links an $n$-dimensional distribution function to its one-dimensional margins and is itself a continuous distribution function on the unit $n$-cube, with uniform margins. We show that the copula of a pair of random variables $X, Y$ is invariant under a.s. strictly increasing transformations of $X$ and $Y$, and that any property of the joint distribution function of $X$ and $Y$ which is invariant under such transformations is solely a function of their copula. Exploiting these facts, we use copulas to define several natural nonparametric measures of dependence for pairs of random variables. We show that these measures satisfy reasonable modifications of Renyi's conditions and compare them to various known measures of dependence, e.g., the correlation coefficient and Spearman's $\rho$.

Abstract: An algorithm is developed for the simultaneous optimization of several response functions that depend on the same set of controllable variables and are adequately represented by polynomial regression models of the same degree. The data are first checked for linear dependencies among the responses. If such dependencies exist, a basic set of responses among which no linear functional relationships exist is chosen and used in developing a function that measures the distance of the vector of estimated responses from the estimated “ideal” optimum. This distance function permits the user to account for the variances and covariances of the estimated responses and for the random error variation associated with the estimated ideal optimum. Suitable operating conditions for the simultaneous optimization of the responses are specified by minimizing the prescribed distance function over the experimental region. An extension of the optimization procedure to mixture experiments is also given and the method is illustrat...

Abstract: An important component of automated theorem-proving systems are unification algorithms which fred most general substitutions which, when apphed to two expressions, make them equivalent Functions which are associative and commutative (such as the arithmetic addition and multiphcatton functions) are often the subject of automated theorem proving An algorithm which unifies terms whose function is associative and commutauve is presented here Termmaaon, soundness, and completeness of the algorithm have been proved for a subclass of the general case. The algorithm is an efficient alternative to other methods of handling associative-commutative functions

Abstract: A computer controlled interactive instruction system for teaching an individual comprising a peripheral having at least one sensor for producing a signal representing a function which has been or is to be manually performed by the individual, a first device for storing a sequence of signals representing instructions relating to the correct manner of performing the function, a first display coupled to the storing device for displaying the instructions, a second device for storing video and associated audio signals representing both pictorial and oral representations of the correct manner of performing the function, a second display for displaying the pictorial representation, and a computer coupled to the peripheral, the first and second storage devices and the first and second displays for causing an ordered sequence of the instruction signals in the first signal storage device to be displayed by the first display whereby the individual may learn to recognize and to perform the function, for receiving the signals from the peripheral and detecting correct or incorrect performance of the function by the individual and for causing appropriate ones of the stored video signals to be displayed by the second display and the associated audio signals to be produced to illustrate the correct performance of the function if incorrectly performed by the individual whereby telling, showing and coaching of the individual in the recognition and performance of the function may be accomplished.

Abstract: It is well known that the classical difference formulas for evaluating high derivatives of a real function f(ζ) are very ill-conditioned. However, if the function f(ζ) is analytic and can be evaluated for complex values of ζ, the problem can be shown to be perfectly well-conditioned. An algorithm that performs this evaluation for an arbitrary analytic function f(~) is described. A short FORTRAN program for generating up to 50 leading derivatives is to be found in the algorithm section of this issue. To use this program, no knowledge is required either of the method or of the analytical nature (e.g., position of nearest singularity, its type) of the function.

Abstract: Any string-matching algorithm requires at least linear time and a constant number of local storage locations. We design and analyze an algorithm which realizes both asymptotic bounds simultaneously. This can be viewed as completely eliminating the need for the tabulated “failure function” in the linear-time algorithm of Knuth, Morris, and Pratt. It makes possible a completely general implementation as a Fortran subroutine or even as a six-head finite automaton.

Abstract: We give a characterization of Tyhonov well-posedness for the problem of minimising a convex lower-semieontinuous function f on a closed convex set K. To get this result, we use Ekeland's theorem on the“approximate variational principle”[21; when f is differentiable, the condition is and diam , where . We prove also some results relating the diameter of level sets of a sub- homogeneous function g with a function which characterizes the minimum increase of g above the minimum value. This result allows us to relate our characterization with a former one due to Zolezzi [10] Then we use the condition on diam(T∊) to give a definition of“well-posedness”for variational inequalities and prove some related results.

Abstract: In this paper we give a systematic treatment of the exact and approximate realization of a positive real matrix-valued function on the open unit disc by means of a lossless circuit connected to a passive load. We discuss the mathematical properties of the chain scattering matrix which describes the lossless circuit and rederive a form of the classical Darlington synthesis theorem generalized to roomy matrix-valued transmission functions. We then develop a matrix version of an algorithm due to Schur for the construction of approximate realizations which produces (minimal degree) Nevanlinna–Pick approximants to the original positive real matrix. We further identify the normalized inverse of one of the outer factors of the approximant to the positive real matrix as the orthogonal projection of the identity onto a suitably defined subspace, give its interpretation as a reproducing kernel, and establish strong convergence under mild conditions on the growth of the order of the approximation. Finally we interpret and apply the mathematical theory developed in the body of the paper to the theory of prediction for vector-valued second order stationary stochastic sequences and briefly discuss connections with the theory of maximum entropy extensions and of inverse scattering.

Abstract: Given a ring (cycle) of n processes it is required to design the processes so that they will be able to choose a leader (a uniquely designated process) by sending messages along the ring. If the processes are indistiguishable there is no deterministic algorithm, and therefore probabilistic algorithms are proposed. These algorithms need not terminate, but their expected complexity (time or number of bits of communication) is bounded by a function of n. If the processes work asynchronously then on the average O(n log2n) bits are transmitted. In the above cases the size n of the ring was assumed to be known. If n is not known it is suggested first to determine the value of n and then use the above algorithm. However, n may only be determined probabilistically and any algorithm may yield an incorrect value. In addition, it is shown that the size of the ring cannot be calculated by any probabilistic algorithm in which the processes can sense termination.

Abstract: Suppose A is a bounded set in ℛ d and f a real function defined on A. Suppose m = min{f(x; | x ∈ A} exists. Using a random sample X 1, X 2. …, X n from a uniform distribution over A we construct a confidence interval for m using asymptotic theory. Our results contain some statistical results valid in general extreme value theory (estimation of the main parameter of extreme value distributions).

Abstract: In a previous paper, it was shown that parameter-effects nonlinearities of a nonlinear regression model-experimental design-parameterization combination can be quantified by means of a parameter-effects curvature array $A$ based on second derivatives of the model function. In this paper, the individual terms of $A$ are interpreted and local compensation methods are suggested. A method of computing the parameter-effects array corresponding to a transformed set of parameters is given and we discuss how this result could be used to determine reparameterizations which have zero local parameter-effects nonlinearity.

Abstract: This paper contains a complete class theorem (Theorem 3.2) which applies to most statistical estimation problems having a finite sample space. This theorem also applies to many other statistical problems with finite sample spaces. The description of this complete class involves a stepwise algorithm. At each step of the process it is necessary to construct the Bayes procedures in a suitably modified version of the original problem. The complete class is a minimal complete class if the loss function is strictly convex. Some examples are given to illustrate the application of this complete class theorem. Among these is a new result concerning the estimation of the parameters of a multinomial distribution under a normalized quadratic loss function. (See Example 4.5).

Abstract: : THE OXIDATION AND REDUCTION POTENTIALS OF MAIN GROUP METALLOPHTHALOCYANINES ARE SHOWN TO FOLLOW A LINEAR RELATIONSHIP WITH THE FUNCTION R/(ZE) (RADIUS/CHARGE). This relationship may be used to facilitate the design of photocatalysts with specific redox potentials. (Author)

Abstract: Information is not transferred instantaneously; there is always a propagation delay before an output is available as an input to the next computational step. Propagation delay is a function of wire length, so we study the length of edges in planar graphs. We prove matching (to within a constant factor) upper and lower bounds on minimax edge length for four planar embedding problems for complete binary trees. (The results are summarized in Table 1.) Because trees are often subcircuits of larger circuits, these results imply general performance limits due to propagation delay. The results give important information for the popular technique of pipelining.

Abstract: Just as the concept of number is fundamental to arithmetic, the concept of variable is fundamental to algebra and all higher mathematics. Just as numbers provide a concise means of describing operations on sets, variables provide a concise means of describing relations between sets. It is the study of relations, from simple equations and functions to complex patterns and structures, that constitutes, to a very large extent, the content of higher mathematics.

Abstract: Let the space curveL be defined implicitly by the (n, n+1) nonlinear systemH(u)=0. A new direct Newton-like method for computing turning points ofL is described that requires per step only the evaluation of one Jacobian and 5 function values ofH. Moreover, a linear system of dimensionn+1 with 4 different right hand sides has to be solved per step. Under suitable conditions the method is shown to converge locally withQ-order two if a certain discretization stepsize is appropriately chosen. Two numerical examples confirm the theoretical results.

Abstract: In this paper we present the derivation of the bivariate distribution function for median filtered sequences of independent, arbitrary, second-order random variables. This result is then used to qualitatively analyze second moment properties of median filtered data. The results hint towards a low-pass characteristic of the median filters and a low dependency of the output spectrum on input alphabet size and distribution.

Abstract: G. H. Hardy in this paper [3] and his book [5] considered a class L of logarithmic-exponential functions (L-functions for short). These are real, singlevalued functions, defined for all x greater than some definite value. By d~finition, L is the minimal class of such functions with respect to the following conditions: (C1) This class contains real constants and the identity function x. (C2) This class is closed under ring operations (pointwise addition and multiplication). (C3) If the function f(x) belongs to this class and is ultimately either positive or negative (for large x), then 1/[(x) also belongs. (C4) If f(x) belongs to the class, then also e j(x) does. If, moreover, f(x) does not vanish for large x, then ln[f(x)[ also belongs to the class. (C5) If a real continuous function f(x) is algebraic over this class, then f(x) itself belongs to the class. For example, the equation

Abstract: An efficient method for representing multivalued functions is described. The method employs an algorithm which generates an efficient cover for a given function "directly," i.e., without resorting to the intermediate step of creating a table of prime implicants. Data are presented to show that the covers generated are as efficient in terms of cover size as prime implicant based methods. More importantly, however, the direct cover method is shown to require much less computation time than prime implicant based methods, thus making it practical for functions with a large number of input variables and/or as the radix of implementation increases. The algorithm is introduced by applying it to functional representations employing the traditional max and min operation. Next, a modified form of the algorithm is presented for use with the sum and product operators more appropriate to I2L and other current summation technologies.

Abstract: 1* Introduction* In a recent paper (2), Cameron and Storvick treat a Banach algebra S of functions on Wiener space which are a kind of stochastic Fourier transform of Borel measures on L2(a, &). (Precise definitions will be given in the next section.) For such functions they show that the analytic Feynman integral, defined by analytic continuation of the Wiener integral, exists, and they give a formula for this Feynman integral. The work in (2) is related to Albeverio and Hώegh-Krohn's beautiful theory (1) of infinite dimensional oscillatory integrals ("Fresnel integrals") as well as to (5). Cameron and Storvick's work is highly promising and has some appealing features. For example, as we will show in a later note, the existence of the Feynman integral for certain qudratic potentials can be established without having to construct special spaces, quad- ratic forms, etc. to fit the particular problem of interest. The main purpose of this note is to show that a crucial part of (2) can be substantially simplified. Let R, C denote the real and complex numbers respectively. Let θ map (α, b) x R to C. Let C(a, b) denote Wiener space; that is, the space of R-valued continuous functions on (α, b) which vanish at α. Let m denote Wiener measure on C(a, &). Under certain hypotheses on θ, Cameron and Storvick show that the function

Abstract: The far infrared collision-induced spectrum of N2 gas at 300 and 124 K is analysed using an empirical lineshape function. The theory of the collision-induced spectrum of N2 is developed and express...