Showing papers on "Function (mathematics) published in 1987"
TL;DR: In this article, bounds on asymptotic efficiency are derived for a class of nonparametric models, where the data are independent and identically distributed according to some unknown distribution F. This bound turns out to depend only upon certain conditional moments, and not upon the support of the distribution.
908 citations
TL;DR: The problem of finding a linear time invariant complete system which models an observed time series will be continued, however, in this part it will be assumed that the model is required to explain the observations only approximately, and what is meant by an approximate model is raised.
366 citations
1 Jan 1987
TL;DR: This paper gives approximation algorithms of solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles.
Abstract: This paper gives approximation algorithms of solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path. Approximation algorithms will be given for versions of this problem in the plane and in three-dimensional space. The algorithms return an e-short path, that is, a path with length within (1 + e) of shortest. Let n be the total number of faces of the polyhedral obstacles, and e a given value satisfying O e ≤ p. The algorithm for the planar case requires O(n log n)/e time to build a data structure of size O(n/e). Given points s and t, and e-short path from s to t can be found with the use of the data structure in time O(n/e + n log n). The data structure is associated with a new variety of Voronoi diagram. Given obstacles S ⊂ E3 and points s, t e E3, an e-short path between s and t can be found in O(n2l(n) log(n/e)/e4 + n2 lognp log(n logp)) time, where p is the ratio of the length of the longest obstacle edge to the distance between s to t. The function l(n) = a(n)O(a(n)O(1)), where the a(n) is a form of inverse of Ackermann's function. For log(1/e) and log p that are O(log n), this bound is O(log n2(n)l(n)/e4).
324 citations
264 citations
TL;DR: In this article, a generalized full-wave Green's function completely defining the field inside a multilayer dielectric structure due to a current element arbitrarily placed between any two layers is derived in two-dimensional spectral-domain form.
Abstract: A generalized full-wave Green's function completely defining the field inside a multilayer dielectric structure due to a current element arbitrarily placed between any two layers is derived in two-dimensional spectral-domain form. It is derived by solving a "standard" form containing the current element with two substrates on either side of it, and using an iterative algorithm to take care of additional layers. Another iterative algorithm is then used to find the field in any layer in terms of the field expressions in the two layers of the "standard" form. The locations of the poles of the Green's function are predicted, and an asymptotic form is derived along with the asymptotic limit, by use of which the multilayer Green's function can be used in numerical methods as efficiently as the single-layer grounded-dielectric-substrate Green's function. This Green's function is then applied to a few multilayer transmission lines for which data are not found in the literature to date.
238 citations
TL;DR: In this article, the Gutzwiller variational wave function of correlated fermions with a short-range interaction was investigated by using a new, analytically tractable approach.
Abstract: The properties of the Gutzwiller variational wave function, which is frequently used to study ground-state properties of correlated fermions with a short-range interaction, are investigated by use of a new, analytically tractable approach As a first application several ground-state quantities are evaluated exactly in dimensionality d=1 for arbitrary band-filling and interaction strengths The results allow for the first approximation-free assessment of the wave function The method itself is applicable to arbitrary space dimensions
214 citations
TL;DR: In this article, an eigenvector decomposition technique is used to extract the mode amplitudes for data received on a long (1.km) vertical array, which is successfully estimated both for simulated data and for data from the 1982 FRAM IV experiment in the Arctic Ocean.
Abstract: Normal mode amplitudes are definite functions of depth and have a characteristic phase as a function of source range rs [i.e., exp(−ikirs), where ki is the ith mode wavenumber]. The range and depth of an acoustic source in the ocean can then be determined by decomposing array data and beamforming on the mode amplitudes. In particular, the product of the normal mode amplitudes with the steering vector Ui [where Ui=exp(ikir) ] is maximum for the true source range (r=rs). Similarly, the correlation of the measured (decomposed) mode amplitudes with the theoretically calculated mode amplitudes is maximum at the source depth. Accurate range and depth estimation with this approach, however, requires reliable estimates of the mode amplitudes. In this article, an eigenvector decomposition technique is used to extract the mode amplitudes for data received on a long (1‐km) vertical array. Range and depth are successfully estimated both for simulated data and for data from the 1982 FRAM IV experiment in the Arctic Oc...
186 citations
Patent•
8 Apr 1987TL;DR: In this paper, the authors present a method for optimizing resource allocations in commercial enterprises that are characterized by a linear set of constraint relationships of the controllable parameters, where the cost function to be minimized is piecewise linear and convex.
Abstract: A method for optimizing resource allocations in commercial enterprises that are characterized by a linear set of constraint relationships of the controllable parameters. The method is particularly useful in enterprises where the cost function to be minimized is piece-wise linear and convex, or where the cost function is linear and the constraint relationships are such that the enterprise can conveniently be decomposed into a plurality of relatively independent subentities and a correlating portion that expresses the interrelationships among the subentities. More specifically, in accordance with the disclosed method, an enterprise with a linear system of constraint equations that can be decomposed into a master problem and a plurality of subproblems leads to a need to solve a system of equations (corresponding to the master problem) where the cost function is piece-wise linear and convex. The piece-wise linear problem is solved by a method that is a modification of the algorithm invented by Karmarkar. The method moves iteratively within the polytope of feasible solutions (as in Karmarkar's invention) in a move direction that improves the value of the cost function. The move direction is ascertained scaling of the problem and determining a descent direction through a sensitivity vector evaluation, and the step size is determined through a line search along chosen direction to the best set of allocation values along that direction. Whenever possible, that direction corresponds to the steepest descent direction. The iterations are repeated until an optimal solution is found, whereupon the last evaluated set of allocations (controllable parameters of the system) is communicated to the system.
176 citations
TL;DR: In this article, it is shown that the influence function of a locally asymptotically linear estimator can be estimated consistently and conversely, that, given a consistent estimator of the influence functions, estimators can be constructed which are locally asmptotic linear in that influence function.
Abstract: Consider estimators which behave locally asymptotically like an average of some function taken at the observations. This function is called the influence function and one calls such estimators locally asymptotically linear. It is shown that the influence function of a locally asymptotically linear estimator can be estimated consistently and conversely, that, given a consistent estimator of the influence function, estimators can be constructed which are locally asymptotically linear in that influence function. With the help of these results an adaptive estimator is constructed for a partially irregular model.
173 citations
TL;DR: In this paper, the H∞-optimal sensitivity problem for delay systems was considered and a new explicit solution was derived for pure delay case where φ = e−sh, h > 0.
132 citations
TL;DR: The associative-commutative matching problem is shown to be NP-complete; more precisely, the matching problem for terms in which some function symbols are uninterpreted and others are both associative and commutative, isNP-complete.
TL;DR: In this paper, the problem of approximating a given stable rational matrix function G(s) of McMillan degree n by a function Ĝ(s)+F(s), where ǫ is antistable, is reduced to a symmetric Wiener-Hopf factorization problem, which can be solved by applying a result of Kaashoek and Ran.
Abstract: We consider the problem of approximating a given stable rational matrix function G(s) of McMillan degree n by a function Ĝ(s)+F(s), where Ĝ has McMillan degree l
TL;DR: In this article, the L multi-soliton solution corresponding to poles of order L of the function r(k), and their asymptotic behaviors are obtained. But the weakly bound states of the two-and three-solomon solutions are explicitly given (i.e. double and triple pole solutions).
TL;DR: It is shown that the detectors' optimal decisions are characterized by thresholds which are coupled and whose computation requires the solution of two coupled sets of dynamic programming equations.
Abstract: Two detectors making independent observations must decide which one of two hypotheses is true. The decisions are coupled through a common cost function. It is shown that the detectors' optimal decisions are characterized by thresholds which are coupled and whose computation requires the solution of two coupled sets of dynamic programming equations. An approximate computation of the thresholds is proposed and numerical results are presented.
TL;DR: In this article, an adaptive procedure which involves a Venter-type estimate of the Jacobian of the response function is proposed and shown to be asymptotically efficient from both the estimation and the control points of view.
Abstract: Herein we study a multivariate version of the adaptive stochastic approximation developed recently by Lai and Robbins. An adaptive procedure which involves a Venter-type estimate of the Jacobian of the response function is proposed and shown to be asymptotically efficient from both the estimation and the control points of view.
TL;DR: In this paper, it was shown that the policy function obtained by optimizing a discounted infinite sum of stationary return functions is Lipschitz continuous when the instantaneous function is strongly concave.
1 Jun 1987
TL;DR: This paper shows that the problem of determining whether a query is safe when terms involving function symbols are represented as distinct occurrences of uninterpreted infinite predicates over which certain finiteness dependencies hold.
Abstract: A database query is said to be safe if its result consists of a finite set of tuples If a query is expressed using a set of pure Horn Clauses, the problem of determining whether it is safe is in general undecidable In this paper, we show that the problem is decidable when terms involving function symbols (including arithmetic) are represented as distinct occurrences of uninterpreted infinite predicates over which certain finiteness dependencies hold. We present a sufficient condition for safety when some monotonicity constraints also hold.
TL;DR: A computational procedure is proposed for MRCI energy gradients that does not require the solution to any “coupled-perturbed MCSCF” equations, it doesNot require any expensive direct-CI matrix-vector products involving derivative integrals, and it does not requirement any derivative integral to be transformed from the AO basis to the MO basis.
Abstract: The theory of MCSCF and CI energy derivatives with respect to geometrical variations is briefly reviewed with special attention given to the MCSCF and MRCI energy gradients. A computational procedure is proposed for MRCI energy gradients that does not require the solution to any “coupled-perturbed MCSCF” equations, it does not require any expensive direct-CI matrix-vector products involving derivative integrals, and it does not require any derivative integrals to be transformed from the AO basis to the MO basis. An additional feature is that it does not require any changes to existing MCSCF gradient evaluation programs in order to compute MRCI gradients. The only difference in the two cases is the exact nature of the data passed to the gradient evaluation program from the previous steps in the computational procedure. The additional effort required to compute the entire MRCI energy gradient vector is approximately that required for one additional iteration of the MRCI diagonalization procedure and for one additional MCSCF iteration. For large scale MRCI wave functions, the MRCI energy gradient evaluation should only require about 10% of the effort of computing the wave function itself. This computational procedure removes a major computational botleneck of potential energy surface evaluation.
TL;DR: In this paper, a new empirical function was proposed to describe the relationship between cutting speed and cutting temperature, and applied to non-metallic coat formation in the case of intermittent cutting.
Abstract: With increasing cutting speed, the cutting temperature increases only to a certain limit, and in the case of intermittent cutting, it even reduces after a certain maximum. A new empirical function is suggested, which describes the relationship between cutting speed and cutting temperature even at high cutting speeds and in the case of intermittent cutting appropriately. The new function has been applied to non-metallic coat formation in the case of intermittent cutting.
Patent•
3M1
TL;DR: In this paper, a motor driven centrifugal blood pump is measured indirectly and non-invasively by sensing a first parameter representative of pump RPM and a second parameter representing motor torque.
Abstract: Blood flow rate in a motor driven centrifugal blood pump is measured indirectly and non-invasively by sensing a first parameter representative of pump RPM and a second parameter representative of motor torque. The blood flow rate is computed as a function of the first and second parameters. By measuring the second parameter in a zero flow condition while the motor is being driven at a known RPM, a viscosity calibration factor is determined which is used, together with the first and second parameters in the calculation of flow rate.
TL;DR: In this article, a multilevel optimization scheme for large laminated composite structures is proposed, and a suitable element/lower-level optimization scheme using a multicriteria objective function is developed.
Abstract: A multilevel optimization scheme for large laminated composite structures is proposed, and a suitable element/lower-level optimization scheme using a multicriteria objective function is developed. The objective function combines a weight function and a strain energy change function into a utility function which is minimized and in which the relative importance of each part is reflected by weighting coefficients. Minimizing the change in strain energy ensures load path continuity in the overall structure when switching between upper and lower levels of optimization, and so decouples the problems at the two levels. Continuous lamina thickness and ply-angle variation is used to minimize the objective function while satisfying strain, buckling, and gauge constraints. Numerical applications are given to illustrate the effect of the weighting coefficients in the objective function on the final result, and to demonstrate the algorithm's effectiveness as a pure weight minimization routine.
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TL;DR: In this article, the authors introduce the characteristic symmetry function, based on the characteristic function of the underlying distribution, whose behaviour is indicative of symmetry or its absence, and propose a statistic for testing symmetry about an unspecified center, derived from the empirical characteristic symmetry functions.
Abstract: SUMMARY We introduce the characteristic symmetry function, based on the characteristic function of the underlying distribution, whose behaviour is indicative of symmetry or its absence. A statistic is proposed for testing symmetry about an unspecified centre, derived from the empirical characteristic symmetry function. The statistic is readily computible, it utilizes information in the empirical characteristic function over an interval, and does not require the estimation of the centre of symmetry. Under general symmetry the asymptotic null distribution of the statistic is folded normal. The empirical power for selected alternatives is studied by a small-scale simulation and a numerical illustration is given.
TL;DR: Another efficient algorithm for this problem is presented here, based on characterization of the MFP's of a regular function in terms of its MTP's, which is used to derive a new upper bound on the number of MFP’s of aregular function.
TL;DR: In this article, a boundary value problem describing the onset of linear instability in a Benard layer is considered, and the solutions of the sixth-order differential equation arising are expressed as Laplace integrals whose integrands involve a function satisfying a second-order equation with six transition points.
Abstract: A boundary-value problem describing the onset of linear instability in a Benard layer, is considered. The solutions of the sixth-order differential equation arising are expressed as Laplace integrals whose integrands involve a function satisfying a second-order equation with six transition points. W .K.B. approximations to this function, valid in regions associated with each transition point, are related by using global phase-integral methods. This allows solutions of the sixth-order problem to be estimated by steepest descents, and leads to an eigenvalue condition. The eigenvalue estimates are confirmed numerically by using the compound matrix method.
TL;DR: In this article, the problem of choosing the Lyapunov function from the class of quadratic forms for differential inclusions is reduced to that of searching saddle points of a suitable function.
TL;DR: The general aim of this review is to critically examine the power function within the theoretical framework of dimensional analysis and principles of similitude to analyze the interpretations proposed for the metabolic power function.
Abstract: Since its introduction by Snell (40), the power function has become a basic mathematical tool for analyzing form, structure, and function in animals of different size. To date about 750 power functions or allometric equations have been reported (7). The power function can be derived from exponential growth curves (19), from dimensional analysis (16), from principles of similitude (10), and from fractal geometry (29, 37, 38). The general aim of this review is to critically examine the power function within the theoretical framework of dimensional analysis and principles of sim,ilitude. The specific aim is to analyze the interpretations proposed for the metabolic power function. A seminal theoretical approach was put forth by Sarrus & Rameaux (33, 35) and has been continued by von Hoesslin (18), Lambert & Teissier (25), Kayser & Heusner (20), and Heusner ( 13-17). A more empirical approach can be traced back to Rubner (34) through the works of Brody (5), Kleiber (2 1-23), and McMahon (26, 27). Both approaches are based on concepts of dimensional analysis and similitude; they differ with respect to their aims and conclusions. In Heusner's homomorphism ( 13, 16, 17), the concept of similitude is defined in terms of qualitative sameness (i.e. same intensive properties). Criteria for mass-independent qualitative changes are defined that focus on
1 Jun 1987
TL;DR: In this article, it is shown that the roots of a polynomial over C vary continuously as a function of the coefficients and that the coefficients can be given as a symmetric mapping of the roots.
Abstract: We present an elementary topological proof that the roots of a polynomial vary continuously as a function of the coefficients. It is, or should be, common knowledge that the roots of a polynomial over C vary continuously as a function of the coefficients. Here we present an elementary topological proof of this important fact. Of course, the coefficients vary continuously as a function of the roots; indeed, the coefficients can be given as a symmetric polynomial mapping of the roots. It is natural to ask if this mapping is continuously invertible, and this is the question we answer in Theorem A. Before becoming more precise, we should mention that proofs, probably in large numbers, exist in the literature in differing contexts. For example, for polynomials of degree 2, 3, or 4, Galois theory tells us the roots can be found by explicit formulas of the coefficients involving radicals. ([4] is a good source.) However, a general study of continuity can still be tricky. For polynomials without multiple roots, the Complex Implicit Function Theorem can be used to show the roots vary analytically with the coefficients [1]. For general polynomials our Theorem B follows from an application of Rouché's theorem [3]. As stated above, our purpose is to give a topological proof that the roots of a polynomial are given as a continuous function of the coefficients. We may assume all polynomials P of degree n are normalized so P(z) = zn + axzn-1 + --+ an. We will identify P with the vector o = (ai,... ,an) in CTM. By the Fundamental Theorem of Algebra, we know P can be factored as
1 Nov 1987
TL;DR: There is a deep and useful connection between statistical mechanics and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters), and a detailed analogy with annealing in solids provides a framework for optimization of very large and complex systems as mentioned in this paper.
Abstract: There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and provides an unfamiliar perspective on traditional optimization problems and methods.
TL;DR: In this article, the oscillatory and nonoscillatory behavior of the solutions of second-order difference equations is studied for the form A:x, = A,(AtIx) for k > 1.