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Showing papers on "Piecewise published in 1969"


Journal ArticleDOI
TL;DR: In this article, a new method was developed for integrating coupled differential equations arising in bound state and scattering problems in quantum mechanics, and wavefunctions were easily constructed in piecewise analytic form, to any prescribed accuracy.
Abstract: A new method is developed for integrating coupled differential equations arising in bound state and scattering problems in quantum mechanics. The wavefunctions are easily constructed in piecewise analytic form, to any prescribed accuracy.

531 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the Fourier transform of the indicator function of the set C is of class LP on Sn1, for some p>2, where p is the largest order of contact which can occur between AC and its tangent line, at which the exterior normal is either 00 or- o
Abstract: Suppose C is a compact subset of the plane having a piecewise smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates, of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. In ?1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -x- 00. In ?2, we shall give applications of the results of ?1 to some questions in the geometry of numbers. 1. If AC is sufficiently smooth, and has everywhere positive Gaussian curvature, it is known that the function 'D(6) = sup, r312IF(r, 0)1 is bounded on S' (cf. [1]). If AC has points of zero curvature, this need no longer be true (cf. [3]). The following, however, remains true: THEOREM 1. If AC is of class Cn + 3, for some integer n 1, and if the Gaussian curvature of AC is nonzero at all points of AC, with the possible exception of a finite set, at each point of which the tangent line has contact of order 1. Moreover, 'D(6) is always bounded, except in neighborhoods of those points of S' which, regarded as vectors, correspond to exterior or interior normals to AC at points of zero curvature. In a neighborhood of such a point 006 'D(6) is bounded by a multiple of [dist (6, 00)] -n - 1)/2n , where dist (6, 00) is the length of the smaller arc of S' connecting 0 and 6o, and nj is the largest order of contact which can occur between AC and its tangent line, at those points of AC at which the exterior normal is either 00 or- o REMARK. Theorem 1 has analogues in higher dimensions. I shall show in another paper, by different methods, that if C is a compact convex subset of Rn, whose boundary is analytic, and if F(r, 0) is the Fourier transform, in polar coordinates, of the indicator function of the set C, then supr r(n + 1)/21 F(r, 0)1 is of class LP on Sn1, for some p>2. If C is a polygon, the estimates are of a quite different character. THEOREM 2. Suppose C is a polygon. Then

97 citations


Journal ArticleDOI
TL;DR: One such function, an exponential function, is discussed and is shown to provide a better fit to several sets of empirical data than the traditional power function.
Abstract: The startup or learning curve literature has in the past concentrated mainly on the algebraic power function or on versions based on this function. This concentration is not unusual because the power function has proven, in numerous studies, to fit empirical data quite well. However, other easy-to-apply algebraic functions should also be analyzed and considered. One such function, an exponential function, is discussed and is shown to provide a better fit to several sets of empirical data than the traditional power function. In addition to providing a better fit, this exponential function is easy to apply in practice.

80 citations


Journal ArticleDOI
TL;DR: In this paper, a comprehensive set of H-mode contour plots are presented for four well-known guide sections for four different types of waveguide cross-sections, including convex and convex-guides.
Abstract: Computer programs have been developed to investigate the construction of polynomial approximations to H-modes in empty guide of arbitrary shape. Mathematically this corresponds to the approximate solution of the Helmholtz equation with homogeneous Neumann boundary conditions. The method used is the well-known Rayleigh-Ritz. Both polynomial and piecewise polynomial function spaces have been investigated for several types of waveguide cross section. Particularly good results have been obtained for convexguides.A comprehensive set of H-mode contour plots are presented for four well-known guide sections.

41 citations


Journal ArticleDOI
TL;DR: In this article, the authors re-examine the original problem in the light of the modern theory of optimal plastic design and apply general optimality criteria to full circular as well as annular plates.
Abstract: Hopkins and Prager (1955) used an intuitive approach based on the concept of competing yield mechanisms to discuss plastic minimum-weight design of a circular plate with piecewise constant cross section. Since this paper was written, the theory of optimal plastic design has progressed considerably, but subsequent papers on optimal plate design were exclusively concerned with plates of continuously varying cross section. In view of the lesser manufacturing cost of plates with piecewise constant cross section, the present paper re-examines the original problem in the light of the modern theory of optimal plastic design. General optimality criteria are established and applied to full circular as well as annular plates. The relation of the designs obtained for annular plates to the singular designs investigated by Mpegarefs (1966, 1967) is discussed.

31 citations


Journal ArticleDOI
01 Jun 1969-Calcolo
TL;DR: In this article, the authors constructed approximations of the So bolev spaces W m,p (Ω) by piecewise polynomial functions on sets of simplexes of the Euclidean space R fixme n.
Abstract: The purpose of the present paper is to construct approximations of the So bolev spacesW m,p (Ω) by piecewise polynomial functions on sets of simplexes of the Euclidean spaceR n. These approximations are obtained by repeated convolution of the characteristic function of the unit hypercube ofR n with a positive measure. Although they involve polynomials of lower degree, they yield the same accuracy as the approximations on rectaugular nets usually adopted. Moreover when used for the approximate solution of variational boundary value problems, they lead to systems of linear equations with matrices where the diagonals with non-vanishing elements are less numerous; their number is reduced from (2m+1) n , for approximations on rectangular nets, to2(2m) n−(2m−1)n for approximations on simplexes nets. The gain thus realised cannot be improved when one requires that these approximations verify a commutation property with differentiation operators.

29 citations


Journal ArticleDOI
TL;DR: In this article, the authors obtained characteristic properties of piecewise-polynomial functions (spline functions) that have least deviation from zero in the metric of C. This has allowed them to obtain quadrature formulas with least estimate of the remainder on a number of classes of differentiable functions.
Abstract: In this article we obtain characteristic properties of piecewise-polynomial functions (spline functions) that have least deviation from zero in the metric of C. This has allowed us to obtain quadrature formulas with least estimate of the remainder on a number of classes of differentiable functions.

24 citations


Journal ArticleDOI
TL;DR: A branch-and-bound algorithm that will find a global solution to multi-level fixed-charge problems, where the separable portion of the objective function is the sum of piecewise continuous functions of a single variable.
Abstract: Multi-level fixed-charge problems are mathematical optimization problems in which the separable portion of the objective function is the sum of piecewise continuous functions of a single variable. This paper describes a branch-and-bound algorithm that will find a global solution to this type of problem. The algorithm has the feature that a good feasible solution is generated at the start. Moreover, at each step of the algorithm an additional feasible solution may be generated for comparison with the best solution found previously.

21 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that if two invertible measure-preserving point transformations commute, in what sense is one a function of the other, and if the first admits of approximation by periodic transformations, then the second transformation is a piecewise power of the first.
Abstract: The question considered is the following: If two invertible measure preserving point transformations commute, in what sense is one a function of the other? The main theorem is the following: If two invertible measure preserving transformations commute, and if the first admits of approximation by periodic transformations, then the second transformation is a piecewise power of the first, where we say that σ is a piecewise power of Τ if there exists a sequence [j(n)] of positive integers such that for each measurable set A the limit of the measure of the symmetric difference of Τ (A) and σ j(n) (A) tends to zero.

20 citations


Journal ArticleDOI
TL;DR: In this paper, the minimum-weight design of an elastic sandwich beam with a prescribed deflection constraint at a given point is investigated based on geometrical considerations using then-dimensional space of discretized specific bending stiffness.
Abstract: In this paper, minimum-weight design of an elastic sandwich beam with a prescribed deflection constraint at a given point is investigated. The analysis is based on geometrical considerations using then-dimensional space of discretized specific bending stiffness. Since the present method of analysis is different from the method based on the calculus of variations, the conditions of piecewise continuity and differentiability on specific bending stiffness can be relaxed. Necessary and sufficient conditions for optimality are derived for both statically determinate and statically indeterminate beams. Beams subject to a single loading and beams subject to multiple loadings are analyzed. The degree to which the optimality condition renders the solution unique is discussed. To illustrate the method of solution, two examples are presented for minimum-weight designs under dual loading of a simply supported beam and a beam built in at both ends. The present analysis is also extended to the following problems: (a) optimal design of a beam built in at both ends with piecewise specific stiffness and a prescribed deflection constraint and (b) minimum-cost design of a sandwich beam with prescribed deflection constraints.

19 citations


Journal ArticleDOI
TL;DR: Superposition is used to obtain expressions for the field of a line source with piecewise-sinusoidal current distribution which are rigorous even in the near zone, despite their simplicity.
Abstract: Superposition is used to obtain expressions for the field of a line source with piecewise-sinusoidal current distribution which are rigorous even in the near zone, despite their simplicity

Journal ArticleDOI
TL;DR: In this paper, a Lyapunov function for w' = 2Bw was used to obtain new instability results for the Hill equation which extend the classical results of Lyapinov and Haupt.
Abstract: the same These results are used to obtain new instability results for the Hill equation which extend the classical results of Lyapunov and Haupt Finally, we show that a Lyapunov function for w' = 2Bw can be used to algorithmically obtain a Lyapunov function for y' = By along with certain verifiable conditions from which stability properties of y' = By (and hence of x' = Ax) may be obtained No such algorithm may be found in the present literature 2 Preliminary results The matrix A can be written as a sum of piecewise

Journal ArticleDOI
TL;DR: In this paper, necessary and sufficient conditions for the existence of Lyapunov functions of the piecewise-quadratic type for relay-control systems are given. But these conditions are restricted to a special case of the problem.
Abstract: This paper develops necessary and sufficient conditions for the existence of Lyapunov functions of tho Lur'e type (‘ piecewise-quadratic ’) for relay-control systems and gives explicit formulae- for ‘optimum ’functions of this form for estimating regions of asymptotic stability in special cases, comparing these with results of numerical optimization. Certain similarities and advantages of ‘piecewise-linear ’Lyapunov functions are discussed. Among their benefits is algebraic simplification which allows one to compute ranges of time varying and non-linear parameters which preserve asymptotic stability in a given region in the state space; an example of this type is given.

Journal ArticleDOI
TL;DR: In this article, the successive sweep method is applied to the computation of optimal solutions to control problems which are characterized by terminal constraints and discontinuities in the control functions, such as attitude control of an orbiting vehicle and transfer of a low thrust vehicle in a gravitational field.
Abstract: The "successive sweep method," a Newton-Raphson algorithm in function space, based upon [1]-[3], is applied to the computation of optimal solutions to control problems which are characterized by terminal constraints and discontinuities in the control functions. The computational technique of using piecewise constant gains is introduced for the successive sweep method in order to reduce computer storage. Two examples are considered. The first example concerns the three-dimensional attitude control of an orbiting vehicle. The second example is the two-dimensional transfer of a low thrust vehicle in a gravitational field.


Journal ArticleDOI
TL;DR: In this article, a theory of quadratic and linear programming theory is developed, where structures are discretizised in finite elements, constitutive laws piecewise linearized, the problem is split in a preliminary linear elastic solution and a nonlinear subproblem.
Abstract: The theory developed exhibits the following peculiar features: structures are discretizised in finite elements, the constitutive laws piecewise linearized, the problem is split in a preliminary linear elastic solution and a “corrective” nonlinear subproblem; concepts and techniques of quadratic and linear programming theory are utilized. The main results are: for the analysis under given loads and dislocations, a pair of extremum theorems for locking stresses, corresponding to dual quadratic programs; for the limit analysis with respect to locking situations two already known theorems, which are here deduced from the solvability conditions of the above quadratic programs and formulated as dual linear programs. The extension of the results to imperfectly locking behavior is carried out. Some examples illustrate the solution techniques based on the theory expounded.

Journal ArticleDOI
TL;DR: In this article, the problem of optimal control of systems whose control signals belong to a pre-specified class of functions is re-examined for a broader class of systems and the results are formulated in a maximum principle of the Pontryagin type.
Abstract: A previous paper treating the problem of optimal control of systems whoso control signals belong to a pre-specified class of functions is discussed. The problem is re-examined for a broader class of systems and the results are formulated in a maximum principle of the Pontryagin type. The maximum principle states that a necessary condition for the optimal control is that the first variation of the time integral of the Hamiltonian is zero if the control function can vary in two opposite directions, is non-positive if the control function can vary in only one direction. The free end-point and the fixed end-point problems as well as the variable end-point problems are considered. The admissible controls are assumed to be piecewise continuous and bounded. The requirement of operative convexity as imposed in the related paper is lifted.

01 Aug 1969
TL;DR: In this paper, the authors developed a new class of solution concepts in n-person game theory as optimal solutions to specially constructed linear programming problems whose constraint matrices and hence optimal solutions depend on a certain parameter, c. The authors obtained asymptotic results for the limiting payoff configuration as c approaches infinity.
Abstract: : In a previous paper the authors developed a new class of solution concepts in n-person game theory as optimal solutions to specially constructed linear programming problems whose constraint matrices and hence optimal solutions depend on a certain parameter, c. In this paper asymptotic results are obtained for the limiting payoff configuration as c approaches infinity. It is shown that the limiting payoff configuration in general shares some properties with Schmeidler's solution concept of the nucleolus and under additional assumptions does converge to the nucleolus. By using recent results of Kohlberg, a new proof is obtained for the author's theorem of the piecewise linearity of the nucleolus as a function of the characteristic function of n-person games. (Author)


01 Jan 1969
TL;DR: In this article, a piecewise cross correlation technique has been developed to analyze the outputs of remote detection devices, and the purpose of this technique is to eliminate the noise from optical background fluctuations, from transmission fluctuations and from detectors by calculating the instantaneous product of the detector output and a reference signal.
Abstract: A piecewise cross correlation technique has been developed to analyze the outputs of remote detection devices. The purpose of this technique is to eliminate the noise from optical background fluctuations, from transmission fluctuations and from detectors by calculating the instantaneous product of the detector output and a reference signal. Each noise component causes positive and negative oscillations of the instantaneous product and may thus be cancelled by an integration of the instantaneous product. The resultant product mean values will then contain the desired information on the spatial and temporal variation of emission, absorption and scattering processes in the atmosphere.


Journal ArticleDOI
TL;DR: In this paper, an approximate approximation to a function by a polynomial in another function is presented, where the polynomials are represented by a Polynomial and the function is a function.
Abstract: (1969). Approximation to a Function by a Polynomial in Another Function. The American Mathematical Monthly: Vol. 76, No. 9, pp. 1049-1050.

Journal ArticleDOI
TL;DR: The Jordan Curve Theorem for Piecewise Smooth Curves as discussed by the authors is a well-known piecewise smooth curve theorem, which is based on the Jordan Curve theorem for smooth curves.
Abstract: (1969). The Jordan Curve Theorem for Piecewise Smooth Curves. The American Mathematical Monthly: Vol. 76, No. 6, pp. 605-610.

Patent
03 Feb 1969
TL;DR: A hybrid digital-to-analogue function generator for generating analogue voltages which are nonlinear functions of an input digital number was proposed in this article, where the function generator uses digital to analogue converters which have two reference voltage inputs, Eo greater than Ei, the output voltage of each converter being defined as:
Abstract: A hybrid digital-to-analogue function generator for generating analogue voltages which are nonlinear functions of an input digital number. The function generator uses digital-to-analogue converters which have two reference voltage inputs, Eo greater than Ei, the output voltage of each converter being defined as:


Journal ArticleDOI
TL;DR: For a class of piecewise linear correlation functions, it is shown that optimal linear mean-square filtering is achieved with a finite number of samples of the process for any finite observation interval.
Abstract: For a class of piecewise linear correlation functions, it is shown that optimal linear mean-square filtering is achieved with a finite number of samples of the process for any finite observation interval. The class of correlation functions is defined by a particular property of the points at which they change slope. Conditions are discussed under which an arbitrary piecewise linear function is a correlation function. An example demonstrating various aspects of the theory is given, and applications of the theory are considered.

Journal ArticleDOI
TL;DR: A novel hybrid function generator that replaces a servo multiplier previously used in an instrument to measure the length distribution function of a population of cotton fibers is described.
Abstract: A novel hybrid function generator that replaces a servo multiplier previously used in an instrument to measure the length distribution function of a population of cotton fibers is described. The function generator linearizes the output of a photocell by extracting the square root of this signal and displaying the result in digital form as well as producing an analog signal. It has general application as a replacement for analog function generation and multiplication.


01 Jan 1969
TL;DR: For a class of piecewise linear correlation functions, it is shown that optimal linear mean-square filtering is achieved with a finite number of samples of the process for any 6nite observation interval.
Abstract: Absfracf-For a class of piecewise linear correlation functions, it is shown that optimal linear mean-square filtering is achieved with a finite number of samples of the process for any 6nite observation interval. The class of correlation functions is defined by a particular property of the points at which they change slope. Conditions are discussed under which an arbitrary piecewise linear function is a correlation function. An example demonstrating various aspects of the theory is given, and applications of the theory are considered.