Showing papers on "Function (mathematics) published in 1988"
01 Jan 1988
TL;DR: In this article, the authors show that every function of n inputs can be efficiently computed by a complete network of n processors in such a way that if no faults occur, no set of size t can be found.
Abstract: Every function of n inputs can be efficiently computed by a complete network of n processors in such a way that: If no faults occur, no set of size t
2,207 citations
TL;DR: In this article, a method for measuring correlation functions without interpolating in the temporal domain is proposed which provides an assumption-free representation of the correlation measured in the data and allows meaningful error estimates.
Abstract: A method for measuring correlation functions without interpolating in the temporal domain is proposed which provides an assumption-free representation of the correlation measured in the data and allows meaningful error estimates. Physical interpretation of the cross-correlation function of two series believed to be related by a convolution is shown to require knowledge of the input function's fluctuation power spectrum. Application of the method to two systems reveals no correlation for the optical data of Akn 120, but a strong correlation for the UV data of NGC 4151, placing bounds of between 1.2 and 20 light days on the size of the line-emitting region.
1,139 citations
TL;DR: In this paper, the typical fraction of the space of interactions between each pair of N Ising spins which solve the problem of storing a given set of p random patterns as N-bit spin configurations is considered, and a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist.
Abstract: The typical fraction of the space of interactions between each pair of N Ising spins which solve the problem of storing a given set of p random patterns as N-bit spin configurations is considered. The volume is calculated explicitly as a function of the storage ratio, alpha =p/N, of the value kappa (>0) of the product of the spin and the magnetic field at each site and of the magnetisation, m. Here m may vary between 0 (no correlation) and 1 (completely correlated). The capacity increases with the correlation between patterns from alpha =2 for correlated patterns with kappa =0 and tends to infinity as m tends to 1. The calculations use a saddle-point method and the order parameters at the saddle point are assumed to be replica symmetric. This solution is shown to be locally stable. A local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist.
900 citations
TL;DR: The difficulties of extending Zamolodchikov's c-theorem to dimensions d ≠ 2 are discussed in this paper, where it is shown that the existence of such a c-function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.
666 citations
TL;DR: In this article, the authors show that the number of steps per unit length along the trace of major strike-slip fault zones in California and Turkey is a smoothly decreasing function of cumulative geological offset.
Abstract: The mapped traces of strike-slip faults are commonly characterized by discontinuities that appear as steps in map-view. Here I present observations to show that the number of steps per unit length along the trace of major strike-slip fault zones in California and Turkey is a smoothly decreasing function of cumulative geological offset. When coupled with a growing body of evidence that indicates that steps in fault traces work to impede or arrest the propagation of earthquake ruptures, the apparent smoothing of fault traces with displacement is interpreted to suggest that the spatial distribution of strength properties on a fault plane is a function of cumulative geological offset. A consequence of this structural evolution is that faults may undergo a seismological evolution a well, whereby the size and frequency distribution of earthquakes is also a function of cumulative offset.
449 citations
IBM1
TL;DR: An approximation technique, called scaling approximation, is introduced and applied to the analysis of homogeneous fork/join queuing systems consisting of K>or=2 servers to find upper and lower bounds on the mean response time.
Abstract: An approximation technique, called scaling approximation, is introduced and applied to the analysis of homogeneous fork/join queuing systems consisting of K>or=2 servers. The development of the scaling approximation technique is guided by both experimental and theoretical considerations. The approximation is based on the observation that there exist upper and lower bounds on the mean response time that grow at the same rate as a function of K. Simple, closed-form approximate expressions for the mean response time are derived and compared to simulation results. The relative error in the approximation is less than 5% for K >
351 citations
IBM1
TL;DR: The elementary equivalence of thegebra of rational trees and the algebra of infinite trees is obtained and forms the basis of decision procedures for the theories of the corresponding algebras.
Abstract: Complete axiomizations for the algebras of infinite trees and infinite trees are presented. The axiomizations are parameterized by the alphabet of function symbols for both the finite trees and infinite trees. There are two main cases, depending on whether the number of function symbols is finite or infinite. In the former case an extra axiom is necessary to obtain completeness. The method of proof is an elimination of quantifiers. Although a full elimination of quantifiers is not possible, the method forms the basis of decision procedures for the theories of the corresponding algebras. As a corollary to the results in infinite trees, the elementary equivalence of the algebra of rational trees and the algebra of infinite trees is obtained. >
283 citations
01 Jun 1988
TL;DR: A parallel algorithm for the rasterization of polygons is presented that is particularly well suited for 3D Z-buffered graphics implementations and can be interpolated with hardware similar to hardware required to interpolate color and Z pixel values.
Abstract: A parallel algorithm for the rasterization of polygons is presented that is particularly well suited for 3D Z-buffered graphics implementations. The algorithm represents each edge of a polygon by a linear edge function that has a value greater than zero on one side of the edge and less than zero on the opposite side. The value of the function can be interpolated with hardware similar to hardware required to interpolate color and Z pixel values. In addition, the edge function of adjacent pixels may be easily computed in parallel. The coefficients of the "Edge function" can be computed from floating point endpoints in such a way that sub-pixel precision of the endpoints can be retained in an elegant way.
259 citations
TL;DR: It is shown how the selection of two species of equal or almost equal selective value is influenced by its nearest neighbours in sequence space, and how the Hamming distance between the species determines selection.
208 citations
TL;DR: In this paper, an important optimization problem that arises in control is to minimize the largest eigenvalue (in magnitude) of a symmetric matrix function of a given matrix function.
Abstract: An important optimization problem that arises in control is to minimize $\varphi ( x )$, the largest eigenvalue (in magnitude) of a symmetric matrix function of x. If the matrix function is affine,...
203 citations
TL;DR: The electron Green's function in the superconducting state can be solved either at discrete Matsubara frequencies along the imaginary axis, or as a function of a continuous variable along the real frequency axis, and a formally exact analytic continuation of the imaginary-axis solutions to theReal frequency axis is derived.
Abstract: The electron Green's function in the superconducting state can be solved either at discrete Matsubara frequencies along the imaginary axis, or as a function of a continuous variable along the real frequency axis. The former is considerably more convenient to calculate than the latter. We derive a formally exact analytic continuation of the imaginary-axis solutions to the real frequency axis. The procedure is applicable to more general problems involving Green's functions. We apply our method to calculating the tunneling density of states for superconducting lead.
TL;DR: This work presents here an operation for the ‘addition’ of two images, with a physical justification in the context of transmitted light, that permits the construction of the family of ‘positive homothetics' of the grey tone function at hand.
Abstract: SUMMARY
Up to now, image processing and image analysis techniques have borrowed their basic tools from functional analysis: Fourier filtering, differential and integral calculus, and so on. These tools, however, only realize their efficiency when they are put into a well-defined algebraic frame, most of the time of a vectorial nature. Unfortunately, the class of functions modelling ‘images’, commonly referred to as ‘grey tone functions’ does not necessarily present this very type of structure. We present here an operation for the ‘addition’ of two images, with a physical justification in the context of transmitted light. Such an addition permits the construction of the family of ‘positive homothetics' of the grey tone function at hand. The vectorial context sought is well defined: The class of images associated with the class of their grey tone functions naturally becomes the positive cone of an ordered real vector space.
Furthermore, the proposed model holds for logarithmic imaging and is compatible with what is known about the human visual process. This model has been called ‘LIP’ (logarithmic image processing model).
TL;DR: It is proved that the problem of checking local optimality for a feasible point and the problem for checking if a local minimum is strict, are NP-hard problems.
01 Jan 1988
TL;DR: A conventional mathematical programming problem can be described as a maximization of a well-defined objective function subject to well- defined constraints on a set of possible alternatives, if the objective function and the constraints are linear functions and the possible alternatives are vectors of real or integer values.
Abstract: A conventional mathematical programming problem can be described as a maximization of a well-defined objective function subject to well-defined constraints on a set of possible alternatives. Standard software tools (for example MPSX, APEX, etc.) can be used to calculate the optimal solution of the model, if the objective function and the constraints are linear functions and the possible alternatives are vectors of real or integer values. This is a very important advantage, but the possibility of application of mathematical programming is limited by the strict structure of the model. The underlying restricting assumptions are as follows:
The considerations focus on a single objective function.
The set of constraints differentiates exactly between feasible solutions and infeasible ones even if these violate a constraint only to a very small degree.
Constraints are aggregated by intersection. This corresponds to the logical “and”.
The set of feasible solutions is independent of the objective function.
The objective function depends hierarchically on the constraints.
TL;DR: In this article, the authors introduced a two-point cluster function C2(r1,r2) which reflects information about clustering in general continuum-percolation models.
Abstract: We introduce a two‐point cluster function C2(r1,r2) which reflects information about clustering in general continuum–percolation models. Specifically, for any two‐phase disordered medium, C2(r1,r2) gives the probability of finding both points r1 and r2 in the same cluster of one of the phases. For distributions of identical inclusions whose coordiantes are fully specified by center‐of‐mass positions (e.g., disks, spheres, oriented squares, cubes, ellipses, or ellipsoids, etc.), we obtain a series representation of C2 which enables one to compute the two‐point cluster function. Some general asymptotic properties of C2 for such models are discussed. The two‐point cluster function is then computed for the adhesive‐sphere model of Baxter. The two‐point cluster function for arbitrary media provides a better signature of the microstructure than does a commonly employed two‐point correlation function defined in the text.
TL;DR: In this paper, a phased translation function, which takes advantage of prior phase information to determine the position of an oriented molecular replacement model, is examined, which is the coefficient of correlation between the electron density computed with the prior phases and the translated model, evaluated in reciprocal space as a Fourier transform.
Abstract: A phased translation function, which takes advantage of prior phase information to determine the position of an oriented molecular replacement model, is examined. The function is the coefficient of correlation between the electron density computed with the prior phases and the electron density of the translated model, evaluated in reciprocal space as a Fourier transform. The correlation coefficient used in this work is closely related to an overlap function devised by Colman, Fehlhammer & Bartels [in Crystallographic Computing Techniques (1976), edited by F. R. Ahmed, K. Huml & B. Sedlacek, pp. 248–258. Copenhagen: Munksgaard]. Tests with two protein structures, one of which was solved with the help of the phased translation function, show that little phase information is required to resolve the translation problem, and that the function is relatively insensitive to misorientation of the model.
TL;DR: For the positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1, for real α > 0, the Epstein zeta function satisfies the functional equation.
Abstract: Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)
TL;DR: A neural network model is presented in which each neuron performs a threshold logic function that always converges to a stable state when operating in a serial mode and to a cycle of length at most 2 whenoperating in a fully parallel mode.
Abstract: A neural network model is presented in which each neuron performs a threshold logic function. The model always converges to a stable state when operating in a serial mode and to a cycle of length at most 2 when operating in a fully parallel mode. This property is the basis for the potential applications of the model, such as associative memory devices and combinatorial optimization. The two convergence theorems (for serial and fully parallel modes of operation) are reviewed, and a general convergence theorem is presented that unifies the two known cases. New relations between the neural network model and the problem of finding a minimum cut in a graph are obtained. >
Book•
01 Aug 1988
TL;DR: In this paper, general qualitative dynamics of some nonlinear systems are discussed, including a recursive loop system using a subjective Weber function and output Uncertainty, and a generic dynamic Mapping of environment onto.
Abstract: Contents: General Qualitative Dynamics of Some Nonlinear Systems. Choice of a Recursive Core Equation. A Recursive Loop System Using . A Subjective Weber Function and Output Uncertainty. A Generic Dynamic Mapping of Environment onto . Further Variants on Mapping of Inputs. Cascading of the Loop. Elementary Identification of Variants and Parameters. Matching Data Patterns and Theory Patterns. Metric or Nonmetric Scaling: Properties of Outputs. Analogues of SDT and Isocriterion Plots. Range and Transposition Effects. Mixing and Attenuation. R sum .
TL;DR: In this article, it was shown that an additive function on a Cartesian product of connected topological spaces is continuous, whenever the preference relation, represented by this function, is continuous.
TL;DR: In this paper, four discriminant models were compared in a simulation study: Fisher's linear discriminant function, Smith's quadratic discriminant functions, the logistic discriminant model, and a model based on linear programming.
Abstract: Four discriminant models were compared in a simulation study: Fisher's linear discriminant function [14], Smith's quadratic discriminant function [34], the logistic discriminant model, and a model based on linear programming [17]. The study was conducted to estimate expected rates of misclassification for these four procedures when observations were sampled from a variety of normal and nonnormal distributions. In contrast to previous research, data were taken from four types of Kurtotic population distributions. The results indicate the four discriminant procedures are robust toward data from many types of distributions. The misclassification rates for both the logistic discriminant model and the formulation based on linear programming consistently decreased as the kurtosis in the data increased. The decreases, however, were of small magnitude. None of these procedures yielded statistically significant lower rates of misclassification under nonnormality. The quadratic discriminant function produced significantly lower error rates when the variances across groups were heterogeneous.
23 May 1988
TL;DR: A new induction principle based on a constructor model of a data structure is developed that can be used for proving properties by induction for data structures such as integers, finite sets, whose values cannot be freely constructed.
Abstract: Automating proofs of properties of functions defined on inductively constructed data structures is important in many computer science and artificial intelligence applications, in particular in program verification and specification systems. A new induction principle based on a constructor model of a data structure is developed. This principle along with a given function definition as a set of equations is used to construct automatically an induction scheme suitable for proving inductive properties of the function. The proposed induction principle thus gives different induction schema for different function definitions, just as Boyer and Moore's prover does. A novel feature of this approach is that it can also be used for proving properties by induction for data structures such as integers, finite sets, whose values cannot be freely constructed, i.e., constructors for such data structures are related to each other. This method has been implemented in RRL, a rewrite-rule based theorem prover. More than a hundred theorems in number theory including the unique prime factorization theorem, have been proved using the method.
TL;DR: In this paper, a quantitative study of the typical behavior of the simulated annealing algorithm based on a cooling schedule presented previously by the authors is presented based on the analysis of numerical results obtained by systematically applying the algorithm to a 100-city traveling salesman problem.
Abstract: A quantitative study is presented of the typical behavior of the simulated annealing algorithm based on a cooling schedule presented previously by the authors. The study is based on the analysis of numerical results obtained by systematically applying the algorithm to a 100-city traveling salesman problem. The expectation and the variance of the cost are analyzed as a function of the control parameter of the cooling schedule. A semiempirical average-case performance analysis is presented from which estimates are obtained on the expectation of the average final result obtained by the simulated annealing algorithm as a function of the distance parameter, which determines the decrement of the control parameter.
TL;DR: A new class of trial wave functions is introduced to compute variationally the ground-state energy of solid /sup 4/He, which is symmetric under particle exchange, translationally invariant, and does not require the a priori introduction of a crystal lattice.
Abstract: A new class of trial wave functions is introduced to compute variationally the ground-state energy of solid /sup 4/He. This wave function is symmetric under particle exchange, translationally invariant, and does not require the a priori introduction of a crystal lattice. It gives a lower energy than and has properties comparable with those given by previous calculations in which atoms are explicitly localized. The same functional form of the wave function is used to investigate the liquid phase, where a lower energy than those given by a wave function of the Jastrow form is obtained as well.
TL;DR: A real valued function of s vector arguments in Rn is said to be arrangement increasing if the function increases in value as the components of the vector arguments become more similarly arranged.
TL;DR: In this paper, a general production function framework for the analysis of capacity output and long-run factor demands in a world consisting of small open economy (SOE) is presented. But this framework is not applicable to the Irish economy, since the factors of each country are homothetic in world output and weakly separable from the factors in all other countries.
TL;DR: In this paper, the first few moments of a random vector are estimated using a numerical procedure using selected weights and points to estimate integrals, where the points and weights are predetermined in the independent standard normal variable space and the sample points in the space are then obtained by various transformations.
TL;DR: In this article, a factorization ansatz for the triplet direction correlation function c (3) is combined with the exact relation between c(3) and the derivative of the pair function c(2) with respect to density.
Abstract: A factorization ansatz for the triplet direction correlation function c (3) is combined with the exact relation between c (3) and the derivative of the pair function c (2) with respect to density to derive a simple and tractable approximation for c (3) in dense, classical fluids. The predictions of this approximation are found to be in good agreement with the results of ‘exact’ molecular dynamics simulations for the ‘soft sphere’ model near freezing. The approximate c (3) is then used to derive an improved integral equation for the pair correlation function which is found to yield satisfactory results. In another application, the third order term in the density functional theory of freezing is systematically evaluated for the first time and is found to lead to significant improvement over the usual second order theory, particularly for soft interatomic repulsions.
24 Jul 1988
TL;DR: A novel objective function is developed here that requires one weighting factor the value of which is easily determined and in combination with an algorithm combining characteristics of neural networks and simulated annealing allows good, valid solutions to be found.
Abstract: An Ising-like objective function has been used by J. Hopfield (1985) and others for finding the optimal tour in a traveling salesman problem using a neural network. This function contains four terms: one which reflects the length of the tour and three more penalty terms which attempt to maintain a feasible solution. These terms are combined into a weighted sum using four coefficients determined by the user. The quality of the final solution is very sensitive to these weighting factors, and good values for them are difficult to find when even a moderate number of cities are considered. A novel objective function is developed here that requires one weighting factor the value of which is easily determined. The use of this function in combination with an algorithm combining characteristics of neural networks and simulated annealing allows good, valid solutions to be found. >
TL;DR: The Koba-Nielsen amplitude five-point dual function is computed integrating over the p-adic field as discussed by the authors, and the infinite product of all the padic functions and the usual five point functions is not a constant.