Trigonometric Operations Research Articles

Design of precision switched-capacitor generalised integrators and their applications to the synthesis of nonlinear networks

The paper presents a novel approach to the design of generalised integrators using switched-capacitor techniques. These generalised integrators, which integrate with respect to a dependent variable, can easily be constructed in IC form using MOS technology and possess other attractive features, for instance low cost and precision. Potential applications of the generalised integrators to the synthesis of nonlinear devices and circuits are also considered in the paper. It is shown that the switched-capacitor generalised integrators can perform many useful basic nonlinear operations such as multiplication, division, trigonometric and logarithmic operations etc. Finally, experimental results are presented which are in close agreement with the theoretical results.

Reformulation of the Brouwer geopotential theory for improved computational efficiency

The theory, as derived by Brouwer and later modified by Lyddane, of the motion of an artificial Earth satellite perturbed by the first five zonal harmonics is reformulated in terms of an alternate set of variables. This alternate set of variables produces an equivalent solution, has no small eccentricity or small inclination restrictions, and allows calculation of position and velocity with considerably fewer algebraic and trigonometric operations. In addition, the alternate set of variables avoids one solution of Kepler's equation.

Saturation of trigonometric polynomial operators

Let C,, denote the Banach space of all 2r-periodic continuous functions on the real line with the supremum norm II . Jo . Let {L,} be a sequence of bounded linear operators of C,, into itself. Suppose that there exists a sequence {&} of positive numbers converging to zero such that every f in C,, for which (1 L,(f) -fij = ~(4~) is a constant function, and there exists a nonconstant function f0 in C,, such that 11 L,(f,) f0 11 = U(+,). Then the sequence {L,} is said to be saturated with the order { I#+J and the class Y(L:,), consisting of all fin C,, for which jj L,(f) fi’ = 0(J an excellent source for references and a systematic treatment of theorems of saturation can be found in Butzer and Nessel [2] and DeVore [5]. Saturation theory in an arbitrary Banach space setting is treated by Butzer, Nessel, and Trebels [3,4]. Here we are concerned with trigonometric polynomial operators which can be defined as follows. Let (h(n; I?)),,~>~ b e a lower triangular matrix, that is, an infinite real matrix satisfying h(n; k) = 0 whenever k > n. Let f~ C,, , let its Fourier series be

Zur Minimalit�t trigonometrischer Polynomoperatoren

Cheney and others [2] have shown that, with respect to the norm of uniform convergence, the Fourier operator Fn:C2π→pn is the only minimal projection. For a more detailed study of operators A:C2π→Pn we investigate the evaluation functionals\(\hat xAf: = (Af)(x)\) respectively the mean of their norms\(\frac{1}{{2\pi }}\int\limits_0^{2\pi } {\left| {\hat xA} \right|dx} \). We give a complete characterization of polynomial operators which minimize this quantity. As an application we can simplify the proof in [2]. Moreover, we show that the trigonometric interpolation operators, having the above minimal-property,are exactly those with equidistant nodes.

On a Sequence of Linear Trigonometric Polynomial Operators

A sequence of linear trigonometric polynomial operators is constructed with the help of the operators $L_{np - p} $, which are generalizations of Jackson operators, and is shown to have the order of approximation given by Jackson's theorems. With the help of these operators, Jackson’s theorems may be given a direct proof, without resorting to the relation $E_n (f) \leqq CE_n (f')/ n$.

The solution of the problem of integration in finite terms

Introduction. The problem of integration in finite terms asks for an algorithm for deciding whether an elementary function has an elementary indefinite integral and for finding the integral if it does. Elementary is used here to denote those functions built up from the rational functions using only exponentiation, logarithms, trigonometric, inverse trigonometric and algebraic operations. This vaguely worded question has several precise, but inequivalent formulations. The writer has devised an algorithm which solves the classical problem of Liouville. A complete account is planned for a future publication. The present note is intended to indicate some of the ideas and techniques involved.

On a trigonometric convolution operator with kernel having two zeros of simple multiplicity

On a trigonometric convolution operator with kernel having two zeros of simple multiplicity

Atmospheric Refraction

I PERCEIVE on further consideration that my suggestion (NATURE, January 5, p. 8) of a spherical wave-front in place of a plane one to account for the discrepancy between the coefficient of terrestrial refraction as derived by Dr. Hunter from Mr. Mallock's proposition and the coefficient deduced from trigonometric levelling operations under ordinary conditions is inapplicable, for the reason that in practice we have to deal, not with a single point-source of light, but with an assemblage of point-sources the wave-front from which is sensibly plane.

The Slide Rule

Having accepted with some mental reservation the assurance of the Editor of this Gazette that the Slide Rule, although invented by Gunter as far back as 1624, is still unknown to many mathematicians, the writer has ventured to make this article explanatory and descriptive. To such then as are unenlightened he offers the good news that half-a-guinea will purchase an instrument capable of multiplying and dividing, of squaring and cubing, of extracting square and cube roots, of solving triangles and performing rapidly many similar combinations of arithmetic and trigonometric operations, all without any mental effort and to an accuracy of from ⅕ to ⅓ %. To examiners having to reduce marks to percentages the Slide Rule is invaluable, to physicists it is indispensable, to engineers it is both invaluable and indispensable.