Showing papers on "Function (mathematics) published in 1985"

The Dip Test of Unimodality

Abstract: The dip test measures multimodality in a sample by the maximum difference, over all sample points, between the empirical distribution function, and the unimodal distribution function that minimizes that maximum difference. The uniform distribution is the asymptotically least favorable unimodal distribution, and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform.

An efficient integrator that uses Gauss-Radau spacings

Abstract: This describes our integrator RADAU, which has been used by several groups in the U.S.A., in Italy, and in the U.S.S.R, over the past 10 years in the numerical integration of orbits and other problems involving numerical solution of systems of ordinary differential equations. First- and second-order equations are solved directly, including the general second-order case. A self-starting integrator, RADAU proceeds by sequences within which the substeps are taken at Gauss-Radau spacings. This allows rather high orders of accuracy with relatively few function evaluations. After the first sequence the information from previous sequences is used to improve the accuracy. The integrator itself chooses the next sequence size. When a 64-bit double word is available in double precision, a 15th-order version is often appropriate, and the FORTRAN code for this case is included here. RADAU is at least comparable with the best of other integrators in speed and accuracy, and it is often superior, particularly at high accuracies.

Lambda lifting: transforming programs to recursive equations

Abstract: Lambda lifting is a technique for transforming a functional program with local function definitions, possibly with free variables in the function definitions, into a program consisting only of global function (combinator) definitions which will be used as rewrite rules. Different ways of doing lambda lifting are presented, as well as reasons for rejecting or selecting the method used in our Lazy ML compiler. A functional program implementing the chosen algorithm is given.

Neighborhood Models of Plant Population Dynamics. I. Single-Species Models of Annuals

Abstract: We present tractable formulations for neighborhood models of annual plant population dynamic processes. These models are constructed from submodels, termed predictors, of individual plants. Fecundity and survivorship predictors give the fecundity and survivorship of an individual as a function of local population density. Dispersal predictors predict the dispersal pattern of a plant's maternal progeny and the survivorship of plants from seed to seedlings. We develop both computer models and analytically tractable models. Our computer models are designed to determine the population dynamic consequences of specific fecundity, survivorship and dispersal predictors. The analytical models are valid when dispersal is sufficiently large, and are used to explain the predictions of analogous computer models. We show through examples that the predictions of corresponding computer and analytical models may be virtually identical. Empirical tests of these models are practical because all model parameters and function...

On Approximation Algorithms for # P

Abstract: The theme of this paper is to investigate to what extent approximation, possibly together with randomization, can reduce the complexity of problems in Valiant’s class # P. In general, any function in # P can be approximated to within any constant factor by a function in the class $\Delta _3^p $ of the polynomial-time, hierarchy. Relative to a particular oracle, $\Delta _3^p $ cannot be replaced by $\Delta _2^p $ in this result. Another part of the paper introduces a model of random sampling where the size of a set X is estimated by checking, for various “sample sets” S, whether or not S intersects X For various classes of sample sets, upper and lower bounds on the number of samples required to estimate the size of X are discussed. This type of sampling is motivated by particular problems in # P such as computing the size of a backtrack search tree. In the case of backtrack search trees, a sample amounts to checking whether a certain path exists in the tree. One of the lower bounds suggests that such tests...

Extremal Splittings of Point Processes

Abstract: The sequence with nth term defined by [(n + 1)p] − [np] is an extremal zero-one valued sequence of asymptotic mean p in the following sense (for example): if a fraction p of customers from a point process with iid interarrival times is sent to an exponential server queue according to a prespecified splitting sequence, then the long-term average queue size is minimized when the above sequence is used. The proof involves consideration of the lower convex envelope J (which is a function on Rm) of a function J on Zm. An explicit representation is given for J in terms of J, for J in a broad class of functions, which we call “multimodular.” The expected queue size just before an arrival, considered as a function of the zero-one splitting sequence, is shown to belong to this class.

Slope Reliability and Response Surface Method

Abstract: A method to analyze reliability of soil slopes using the response surface method is described. The soil slope is modelled and analyzed by a finite element code as in prevalent deterministic studies. The simulation which is usually expensive, is repeated a limited number of times to give point estimates of the response corresponding to uncertainties in the model parameters. A graduating function is then fit to these point estimates so that the response given by the finite element code can be reasonably approximated by the graduating function within the region of interest. The approximating function, called the response surface, is used to replace the code in subsequent repetitive computations required in a statistical reliability analysis. The procedure is applied to a sample problem in slope stability involving uncertain soil properties. It is shown that the slope stability statistics from the response surface is within 1–9% of the statistics based on a direct Monte Carlo simulation using the finite eleme...

Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals

Abstract: We study semicontinuity of multiple integrals ~ where the vector-valued function u is defined for xE ~C N with values in ~ . The function f(x,s,~) is assumed to be Carath@odory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology 1 N of the Sobolev space H 'P(~;~ ). The proofs are based on some approximation results for f. In particular we can approximate f by a nondecreasing sequence of quasiconvex functions, each of them being convex and independent of (x,s) for large values of ~. In the special polyconvex case, for example if n: N and f(Du) is equal to a convex function of the Jacobian detDu, then we obtain semicontinuity in the weak topology of 1,p(~ n H ;~ ) for small p, in particular for some p smaller than n.

A multivariate Weierstrass-Mandelbrot function

Abstract: The univariate Weierstrass-Mandelbrot function is generalized to many variables to model higher dimensional stochastic processes such as undersea topography. Because this topography is difficult to measure at small length scales over the many large regions that affect long-ranged acoustic propagation in the ocean, one needs a stochastic description that can be extrapolated to both large and small features. Fractal surfaces are a convenient framework for such a description. Computer-generated plots for the two-variable case are presented. It is shown that in the continuum limit the multivariate function is equivalent to the multivariate fractional Brownian motion.

Global optimization and stochastic differential equations

Abstract: Let ℝ n be then-dimensional real Euclidean space,x=(x 1,x 2, ,x n)T ∈ ℝ n , and letf:ℝ n → R be a real-valued function We consider the problem of finding the global minimizers off A new method to compute numerically the global minimizers by following the paths of a system of stochastic differential equations is proposed This method is motivated by quantum mechanics Some numerical experience on a set of test problems is presented The method compares favorably with other existing methods for global optimization

Selection from alphabetic and numeric menu trees using a touch screen: breadth, depth, and width

Abstract: Goal items were selected by a series of touch-menu choices among sequentially subdivided ranges of integers or alphabetically ordered words The number of alternatives at each step, b, was varied, and, inversely, the size of the target area for the touch Mean response time for each screen was well described by T = k+clogb, in agreement with the Hick-Hyman and Fitts' laws for decision and movement components in series It is shown that this function favors breadth over depth in menus, whereas others might not Speculations are offered as to when various functions could be expected

Choice Functions over a Finite Set: A Summary

Abstract: A choice function picks some outcome(s) from every issue (subset of a fixed set A of outcomes). When is this function derived from one preference relation on A (the choice set being then made up of the best preferred outcomes within the issue), or from several preference relations (the choice set being then the Pareto optimal outcome within the issue, or the union of the best preferred outcomes for each preference relation)? A complete and unified treatment of these problems is given based on three functional properties of the choice function. None of the main results is original.

Uncertainty principle and uncertainty relations

Abstract: It is generally believed that the uncertainty relation Δq Δp≥1/2ħ, where Δq and Δp are standard deviations, is the precise mathematical expression of the uncertainty principle for position and momentum in quantum mechanics. We show that actually it is not possible to derive from this relation two central claims of the uncertainty principle, namely, the impossibility of an arbitrarily sharp specification of both position and momentum (as in the single-slit diffraction experiment), and the impossibility of the determination of the path of a particle in an interference experiment (such as the double-slit experiment). The failure of the uncertainty relation to produce these results is not a question of the interpretation of the formalism; it is a mathematical fact which follows from general considerations about the widths of wave functions. To express the uncertainty principle, one must distinguish two aspects of the spread of a wave function: its extent and its fine structure. We define the overall widthW Ψ and the mean peak width wψ of a general wave function ψ and show that the productW Ψ w φ is bounded from below if φ is the Fourier transform of ψ. It is shown that this relation expresses the uncertainty principle as it is used in the single- and double-slit experiments.

Local basis-function approach to computed tomography.

Abstract: In the local basis-function approach, a reconstruction is represented as a linear expansion of basis functions, which are arranged on a rectangular grid and possess a local region of support. The basis functions considered here are positive and may overlap. It is found that basis functions based on cubic B-splines offer significant improvements in the calculational accuracy that can be achieved with iterative tomographic reconstruction algorithms. By employing repetitive basis functions, the computational effort involved in these algorithms can be minimized through the use of tabulated values for the line or strip integrals over a single-basis function. The local nature of the basis functions reduces the difficulties associated with applying local constraints on reconstruction values, such as upper and lower limits. Since a reconstruction is specified everywhere by a set of coefficients, display of a coarsely represented image does not require an arbitrary choice of an interpolation function.

Variance Reduction Analysis

Abstract: This paper presents an algorithm for optimal data collection in random fields, the so-called variance reduction analysis, which is an extension of kriging. The basis of variance reduction analysis is an information response function (i.e., the amount of information gain at an arbitrary point due to a measurement at another site). The ranking of potential sites is conducted using an information ranking function. The optimal number of new points is then identified by an economic gain function. The selected sequence of sites for further sampling shows a high degree of stability with respect to noisy inputs.

Extremal domains associated with an analytic function I

Abstract: In this paper we investigate the following two extremal problems: A) Let F be a continuum in the extended complex plane that does not divide and let f(z) be a function analytic on F By D we denote domains in such that f(z) has a single-valued analytic continuation in D. Does there exist a domain D 0 with minimal condenser capacity B) Let f(z) be a function analytic in a neighborhood of infinity. By D we denote domains in , such that f{z) has a single-valued analytic continuation in D. Does there exist a domain D 0 with minimal logarithmic capacity It is proved that there exist extremal domains D 0 in both problems. In a second part of the paper it will be shown that these domains are uniquely determined up to a boundary set of capacity zero.

A linear algorithm for finding dominators in flow graphs and related problems

Abstract: In the first part of the paper we show how to extend recent methods for solving a special case of the union-find problem in linear time, to a special case of the eval-link-update problem for computing the minimum function defined on paths of trees. In the cases where our approach is applicable, we give a way to perform m eval, link, and update operations on n elements in O(m + n) time and O(n) space, improved from O(m a(m + n, n) + n) time and O(n) space in the more general case, where a is a functional inverse of Ackermans function. The technique gives similar improvements in the efficiency of algorithms for solving several network optimization problems in the case where all the keys involved are integers in some suitable range. In the second part of the paper we show how to use the new technique for speeding up the fastest known algorithm for finding dominators in flow graphs so that it runs in linear time. We introduce the notions of pseudo and external dominators which are both computable in linear time and make the technique introduced in the first part applicable for finding immediate dominators. We first give an algorithm for a limited class of graphs which include cycle free graphs, and thus can be used to find dominators in reducible flow graphs. We then show how to extend our technique for computing dominators on any flow graph. All the algorithms we describe run on a Random Access Machine.

On submodular function minimization

Abstract: Earlier work of Bixby, Cunningham, and Topkis is extended to give a combinatorial algorithm for the problem of minimizing a submodular function, for which the amount of work is bounded by a polynomial in the size of the underlying set and the largest function value (not its length).

First integrals for the modified Emden equation q̈+α(t) q̇+qn =0

Abstract: It is shown that the modified Emden equation q+α(t)q+qn=0 possesses first integrals for functions α(t) other than kt−1. The function α(t) is obtained explicitly in the case n=3 and parametrically for other n(≠2). The case n=2 is seen to be particularly difficult to solve.

Critique of the replica trick

Abstract: It is shown that the replica trick fails to give the correct non-perturbative result for the two-point function S2 of the Gaussian unitary ensemble of N*N random matrices. The failure arises from an incorrect description of the symmetries of the random-matrix system in the limit N to infinity . The correct description, which involves integration over both non-compact and compact degrees of freedom, is obtained by using the method of superfields. Some implications for the localisation transition in disordered electronic systems and the theory of the quantised Hall effect are suggested.

Model of cortical organization embodying a basis for a theory of information processing and memory recall

Abstract: Motivated by V. B. Mountcastle's organizational principle for neocortical function, and by M. E. Fisher's model of physical spin systems, we introduce a cooperative model of the cortical column incorporating an idealized substructure, the trion, which represents a localized group of neurons. Computer studies reveal that typical networks composed of a small number of trions (with symmetric interactions) exhibit striking behavior--e.g., hundreds to thousands of quasistable, periodic firing patterns, any of which can be selected out and enhanced with only small changes in interaction strengths by using a Hebb-type algorithm.

A dynamization of the all pairs least cost path problem

Abstract: Given a digraph and a cost function of the edges we update on-line, i.e., between two successive modifications of the cost function, the solution of the All Pairs Least Cost Path Problem (APLCPP). We derive lower and upper bounds for time complexity and show that the gaps between two corresponding bounds are small. Space complexity is quadratic and therefore optimal in our model.

The extinction time of a birth, death and catastrophe process and of a related diffusion model

Abstract: The distribution of the extinction time for a linear birth and death process subject to catastrophes is determined. The catastrophes occur at a rate proportional to the population size and their magnitudes are random variables having an arbitrary distribution with generating function d(·). The asymptotic behaviour (for large initial population size) of the expected time to extinction is found under the assumption that d(.) has radius of convergence greater than 1. Corresponding results are derived for a related class of diffusion processes interrupted by catastrophes with sizes having an arbitrary distribution function.