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An approximate homomorphism between the Banach Algebras of periodic functions of one variable and that of periodic functions of two variables is studied. Application to device synthesis of nonlinear filters is indicated.

Given the unit cube, the author considers the motion of a point which moves like a billiard ball, i.e. performs a rectilinear and uniform motion that is being reflected in the usual way on striking any one of the 2n faces., See also Part 2, AD-A011 010.

The purpose of the report is to provide a unified and reasonably self-contained source of theoretical techniques for ideal Surface acoustic wave (SAW) bandpass filter design and analysis. Topics discussed include an introduction to sampling theory, transducer analysis from sampling weights, cosine squared-on-a-pedestal transducers, phase reversal transducers, Dolph-Chebyshev and Kaiser weighting for sharp-cutoff filters, optimum (equiripple) filter synthesis plus Butterworth and other no-null frequency response filters. In addition, periodic time and frequency responses are discussed. Synthesis of phase weighted bandpass filters are also considered in detail, as are second order and real life effects such as inaccurate apodization, inaccurate finger placement, electrode resistance effects, and phase distortion due to the finite gap between electrodes and dummy electrodes. In all cases, full computer program descriptions, listings and test runs are provided.

Trans. of Calcul en Flexion de Pales de Giravions S.N.A.C. Rept., Document He3-0.03, 23 Dec 1948.

Various algorithms for generating steady state signals by an all- digital synthesizing system are presented and analyzed. The emphasis is on evaluating the results obtained with synthesis algorithms which employ approximations or short-cuts to decrease implementation costs. Algorithms for generating both broadband and discrete or narrowband signal components using Fast Fourier Transform (FFT) techniques are investigated, and the output spectra for the approximate techniques are derived and compared to the spectra obtained with direct approaches. Finally an algorithm for the generation of arbitrary periodic signals in the time domain is presented and the spectrum of its output derived.

Mathieu's equation ((the second derivative of y with respect to x) + (a-2q cos 2x)y = o) admits of periodic solutions of period pi and 2 pi for four countable sets of characteristic values, a(q), for each assigned value of q. There are extensive tabulations for real values of q but very little is available in the complex plane. The present work provides a tabulation in the complex plane (tables I and II) for q=rho exp(i phi), phi=5 degrees (5 degrees) 90 degrees, rho=0(.5)25, and additional tables on the 90 degree -ray for rho up to 100 (tables III, IV, V).

The primary area of interest of the search is the existence and uniqueness of solution to ordinary differential equations for the following two classes of problems: initial value problems and boundary value problems. Also included are actual construction of solutions. Excluded are stability of solutions and equations with periodic solutions, or equations h periodic coefficients. Arrangement is chronological and the period covered is 1949- October 1962. It is suggested that for literature prior to 1949 the bibliography by da Silva Dias be consulted (Citation No. 18). Search was completed October 1962.

In this Memorandum the author studies the asymptotic behavior of solutions of real difference-differential equations, and the question of the existence and behavior of periodic solutions. The equations considered arise in a wide variety of control problems.

Conventional Fourier series analysis enables one to extract information from a periodic function of time. In practical situations, however, one often deals with finite time functions, where conventional Fourier series techniques are inadequate. Finite functions result from a singular or finite set disturbances. The transient response of a circuit is an example of such a disturbance. The theory incorporated in finite Fourier filters enables one to deal with the type of time functions mentioned above. This technical note attempts to summarize, in a logical development, the basic theory of finite Fourier analysis, and the synthesis which incorporates the theory in the finite Fourier filters.

It is shown that cardinal spline interpolation of degree 2m-1 converges, as m nears infinity, for a certain class of almost periodic functions., Prepared in cooperation with Weizmann Inst. of Science, Rehovoth (Israel).

A new method is presented for finding the characteristic exponents and the Floquet equivalent constant system for periodic linear Hamiltonian systems. This method is based on a detailed study of the phase portrait of periodic Riccati differential equations as well as the study of the stability of solutions of periodic Riccati differential equations subject to algebraic constraints. Questions of structural stability are also considered. (Author)

Contents: Methods of evaluation, atomic weights, fundamental constants, symbols and units, general references, and properties of the elements.

This compilation of energy band gaps in elemental and binary compound semiconductors and insulators is intended for scientists who are concerned with energy states and electronic properties of solids and for materials engineers who are concerned with the application of solid state science to semiconductor device technology. Binary compounds included in this compilation are those involving elements in Groups IIIA, IVA, VA, VIA, and VIIA of the Periodic Table with elements of atomic numbers 3 (lithium) through 92 (uranium). In addition to the band gap, the compilation gives the method of measurement, the form of the sample, and the temperature at which the measurement was made for each material listed. When available, temperature coefficients of band gaps and indications of whether the material exhibits cathodoluminescence, electroluminescence, laser emission, mechanical luminescence, photoluminescence, or thermoluminescence are also given., Reprint no. 22 from Jnl. of Physical and Chemical Reference Data, v2 n1 p163-200 1973.

A method of analysis of the total available kinetic energy of sinusoidal and random vibration programs has been developed, using the common basis of unit mass. Fifteen energy expressions are presented for the sinusoidal domain, including log and liner sweep, and fixed frequency dwell. Twenty-eight energy expressions are presented for the random domain, including broadband, swept narrowband, and harmonic spike series. Two examples are presented, showing application to MIL-STD-810B. Calculated energies for eighteen curves of this standard are tabulated. In order to facilitate energy comparisons, and as a guide to program synthesis, eighty-five energy ratio expressions are presented. (Author)

Contents: Periodic-Fourier series; Aperiodic function - Fourier/Laplace transformation; LaPlace transform; Random signals-Average and variance; Two random variables; Cross correlation and auto correlation; Improved estimation; Fourier transform of R(sub XX); Application of communication concepts-Sampling theorem (Deterministic); Linear systems and optimization; State variable formulation.

The method of differentiation with respect to a parameter is applied to the problem of finding periodic solutions to differential equations., Edited trans. of mono. Prikladnaya Matematika i Kibernetika, n.p., 1973 p61-64, by K. L. Dion.

The scope of the research conducted under the terms of this contract, was centered principally upon the problem of developing physically realizable, as well as unique, acoustic signal characteristic signatures with which passively observed signal samples could be classified probabilistically as to their most likely source of origin.

Several recent theorems provide sufficient conditions for the existence or for the existence and stability of infinitely many periodic trajectories of the the generalized Lienard system dx/dt = muF(x) - y, dy/dt = g(x), mu,x,y real. First by applying a diffeomorphism of the plane two of these theorems which require that g(x) identically equal to x are extended to the case where g is spring-like, i.e. g satisfying the hypothesis xg(x) > 0 for x not = 0. For a third theorem which requires g to be spring-like, the conditions on F are relaxed by using a diffeomorphism of the plane. Next the generalized Lienard system is considered under the assumption that there exists an interval (x(1), x(2)) such that g(x) is negative to the left of x(1) and g(x) is positive to the right of x(2). Sufficient conditions are established which, when is sufficiently small, guarantee that infinitely many periodic trajectories exist and that those trajectories are alternately stable and unstable. A useful asymptotic expansion for a function involved in these sufficient conditions is obtained. The results are applied to the system with F(x) periodic of mean zero and g(x) asymptotic to the identity function. (Author)

The discrete finite Fourier transform can be regarded as a matrix operation, since each element of one member of the pair is a linear combination of all the elements of the other member. A remarkably simple relation between a periodic function of a discrete variable and its discrete finite Fourier transform, namely that the absolute values of their expansion coefficients in these eigenvectors are the same, has been demonstrated. A canonical form for such functions (with respect to the finite Fourier transform) is suggested in which the transform can be done by inspection.

The problem of heat transfer in a layer of disperse material with periodic change in the direction of motion of liquid permeating the bed is formulated, with reference to the thermal resistance of the materials of the layer particles and to the nongradient course of the process., Trans. of Inzhenerno-Fizicheskii Zhurnal (USSR) n1 p21-26 1970.

In the paper the author proves that if a periodic neutral differential equation with uniformly stable D operator (which includes retarded equations) has an uniformly asymptotically stable equilibrium point, and one perturbs the equation slightly then a periodic solution of the same period will exist. Since a generalization of a Browder's asymptotic fixed point theorem is available, the author will not assume that the period is greater than the lag, and this makes possible to extend the results for existence of an equilibrium point if the system and the perturbations are autonomous. (Author), Prepared in cooperation with Universidade de Campinas, Brazil.

Methods of generating and properties of Walsh functions are discussed with emphasis on discrete Walsh functons. Hardware and software versions of two methods of Walsh function generation are shown. The Walsh transform is explored and hardware and software realizations of a fast Walsh transforn algorithm, particularly applicable to hardware, is introduced. Computer programs containing the various algorithms are included. A discussion of the feasibility of the use of Walsh functions to produce a digital filter with a desired frequency response is presented. (Author)

The report describes efforts to extend to several variables the theory of mean periodicity developed in the first year of work. The author obtained a representation of solutions of homogeneous convolution equations through a sum or integral or exponential functions. The coefficient of such a formula would then yield a far reaching generalization of the notion of Fourier transform. In the special case where Z is a manifold a fairly satisfactory solution is provided. In the general case, it was not possible to divide or extrapolate as is required with the special case solution. The partitioning of Z is related to the special synthesis problem. (Author)

Let (S sub n)(x) be the periodic spline of degree (2n-1) that interpolates a given set of k data within the period. The limit of (S sub n) (x) is determined as the data are kept fixed and the author lets n approach infinity. (Author)

The interpolating operator P studied here is the one which produces a trigonometric polynomial of order n taking prescribed values at 2n + 1 equally-spaced nodes (theta sub j) = 2 pi j (2n + 1), j = 0,...,2n. This operator P acts in the space C of 2 pi-periodic continuous real functions, normed with the supremum norm. For any point, phi, other than a node, there is a (2n + 1)-dimensional manifold of projections carried by the point set ((theta sub o),...,(theta sub 2n), phi). The first theorem states that P is the element of least norm in this manifold. Another theorem asserts that if 3 correctly-chosen points are adjoined to the set of nodes, then P is no longer a minimal element in the manifold of projections carried by that point set. Various other related results are given. (Author)

Calculations involving an electrical circuit with one nonlinear element are discussed with piece-linear approximation of the characteristic of this element. The calculation is based on the works of Ye. N. Rozenvasser, dedicated to the application of integral equations for analysis of nonlinear systems. (Author), Trans. of Teoreticheskaya Elektroteknika (USSR) n5 p108-112 1968.

In recent times the Cepstrum technique has been applied with varying degrees of success to problems in seismology and speech analysis. Developed by Bogert, et al, as a technique to detect time delayed signals or echoes, Cepstrum consists of a spectral analysis of the log power spectrum of a time series. In the mathematical development of Cepstrum, it is necessary to approximate a logarithm function by a function which is linear in one parameter, the degree of approximation depending on values of the same parameter. The result of this approximation is to place undue restrictions on the generality of the model. The purpose of the paper is to derive an alternate technique, comparable to Cepstrum, in which a logarithmic approximation is not used and the generality of the model is preserved. (Author)

A neutral functional differential equation as defined below includes the scalar differential-difference equation (1) d/dt(x(t)+ax(t-1)+ epsilon G(t,x(t-1))) = bx(t)+cx(t-1)+ epsilon F(t,x(t),x(t-1)) where epsilon is a parameter, a,b,c are constants and G(t,x), F(t,x,y) are continuous functions of t,x,y. For any continuous function phi defined on (-1,0), a solution of (1) is a continuous function x defined on some interval (-1, alpha), alpha > 0, which coincides with phi on (-1,0) and is such that the expression x(t) + ax(t-1) + G(x(t-1)) (not x(t)) is continuously differentiable and satisfies (1) on (0, alpha). The purpose of this paper is to prove for epsilon small the existence of bounded and periodic solutions of (1). (Author)

The report investigates the system model G(u, u dot, . . ., u sup n, t) = F(t), where F(t) is a periodic excitation. Using an approximate solution of the form u(t) = Summation, R=0 to UH (U sub c)(R) cos R (omega sub o) t + (U sub s) (R) sin R (omega sub o) t), analytical theorems and computer results are obtained that yield information into the nature of the existence and non-existence of subharmonic components in the response. By assuming G(u, u dot, . . . , u sup n, t) to be a polynomial, theorems are developed that determine the conditions for the possible existence and non-existence of subharmonic response. (Author)

The research to be reported developed in the following manner. A study was completed of dissipative retarded functional differential equations, which was then developed in a more general fashion which, however, assumed the strong smoothing of initial data as occurs for retarded functional differential equations. Further generalizations were made for parabolic and hyperbolic partial differential equations and for a wide class of functional differential equations of neutral type. It was then decided to unite the disparate points of view by identifying the general hypothesis which lead to the principal results, which is the subject of this report., Sponsored in part by Contract N00014-67-A-0191-0009, Grants NGL-40-002-015, NSF-GP-15132.

If A is a Lebesque measurable subset of the interval (0,1) and t is any irrational number in that interval, then by a well-known theorem of Weyl, the frequency with which the integer products of t modulo one fall in A converges to the measure of A. This result may be used to evaluate asymptotic error in certain approximations. For a special case, Weyl's theorem is shown to extend to rational numbers t and a lower bound on frequency is derived. (Author)

Considered is the problem of simultaneous approximation on the whole real axis of arbitrary differentiable functions and their derivatives by entire functions of exponential type. S. N. Bernshtein's approximation theorem on functions bounded and uniformly continuous between (minus infinity, plus infinity) is generalized, and an inequality is obtained for the best approximation of derivatives of functions on the whole real axis that is close to the well-known inequality of A. N. Kolmogorov. It is found that upon uniform approximation of arbitrary functions on the whole real axis the constants being considered are appreciably greater in some cases than the corresponding constants in the approximation of 2 pi-periodic functions by trigonometric polynomials. (Author), Trans. of Akademiya Nauk SSSR. Izvestiya. Seriya Matematicheskaya, v24 n3 p421-430 1960, by L. Holtschclag.

Trans. of Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika (USSR) n5(84) p30-34 1969, by L. Holtschlag.

The problem of transport of particles in a slab geometry is considered, the unique feature being that the slab is assumed to have a periodic structure. It is shown that the particle density at any point in the slab can be obtained numerically by integrating a set of differential equations of initial value type over one period, followed by the use of certain relatively easy difference and functional equations. There are strong indications that considerable advantage is gained in time, accuracy, stability, etc. over the standard method of handling such problems. Several rather simple numerical examples are given. Agreement with results obtained in other ways is excellent. It is noted that the algorithm produced is really applicable to a wide class of two-point boundary value problems having little or nothing to do with transport theory per se. (Author)

The objective of this paper is to develop a general and meaningful theory of dissipative periodic systems. For ordinary periodic differential equations one studies the iterates of a map T of a state space into itself where the map T is topological and the space is locally compact (n-dimensional Euclidean space). However, for the applications the authors have in mind, the solutions will be unique only in the forward direction of time and the state spaces are not locally compact. Because of this generalization of the results for ordinary differential equations is by no means trivial. The basic theory of dissipative periodic processes on Banach spaces are developed in Sections 2 and 3 of the paper. How this applies to retarded functional differential equations of retarded type is discussed in the fourth section. Two sufficient conditions for dissipativeness are given in terms of Liapunov functions. They formalize the intuitive notion that many systems for large displacements dissipate energy. Application of these results is illustrated. (Author), Prepared in cooperation with LeMoyne Coll., Syracuse, N. Y. Dept. of Mathematics. Sponsored in part by Grant NGL-40-002-015.

The norms of the inverses and corresponding extremal functions of some differential operators acting on periodic functions are determined. (Author)

Solutions of the Helmholtz equation ((the second partial derivative with respect to x) + (the second partial derivative with respect to y) + k squared + 1)psi = 0 describing a simple scattering process in an L-shaped region are studied. To obtain the correct expansion for psi(x,y) on the square corner of the L-shape when k = 1, it is found necessary to replace terms sin kx sin y and sin x sin ky by sin x sin y and (x cos x sin y - y cos y sin x). Terms similar to the latter have been neglected by previous authors and spurious resonances found in consequence. (Author)

Rounding error bounds are given for the recursion formula: x sub (n+1) = p(x sub n) - x sub (n-1), where p = or < 2. Some statistical bounds are also included. (Author)

It is proved that near a singular point or a periodic solution of a Hamiltonian system with an arbitrary number of degrees of freedom the system is stable for almost all the cases if it is stable in the first approximation. This stability is established in the following strong sense: Near the singular point or periodic solution the entire neighborhood is filled in a smooth way with families of concentric periodic surfaces. The results are for analytic systems only. (Author), See also AD-710 791.

The bulk of the report is concerned with a description of the following two papers: (1) Sets of uniqueness for the vibrating string problem (published in the Transactions of the American Mathematical Society); and, (2) Removable sets for pointwise solutions of the Cauchy-Riemann equations (to be published in the Annals of Mathematics). These two papers represent one of the important concepts developed by the principal investigator during the period of the grant; namely, by using the Laplace and generalized Laplace operators in conjunction with recent sophisticated developments in the theory of multiple trigonometric series, fundamental new results can be achieved in the theory of partial differential equations. (Author)

A short review of recent accomplishments in the general area of nonlinear oscillatory phenomena is presented. Scientists from 17 nations met in the USSR to exchange ideas on the theoretical background and practical tools that appear attractive in the analysis of nonlinear processes. Current status, recent progress, and evolving trends are discussed as reflected by the proceedings of the 5th International Conference on Nonlinear Oscillations held in Kiev, August-September 1969. (Author), Presented at the International Conference on Nonlinear Oscillations (5th) held at Kiev (USSR), 25 Aug-5 Sep 69.

The report contains a necessary and sufficient condition for the existence of solutions of certain difference equations and for multiply periodic solutions of linear partial differential equations. These results deal directly with functional properties and thereby avoid the classical difficulty of 'small divisors' stemming from the use of series. (Author)

The report describes the oscillatory behavior of solutions of the equation (1) x double prime + a(t)x=0, where a(t) is locally integrable on the interval from zero to, but not including, infinity. The main result is an extension of a nonoscillation theorem due to Hartman, the contrapositive of which is a useful criterion for equation (1) to be oscillatory. (Author), Revision of report dated 17 Jun 67. Prepared in cooperation with Wisconsin Univ., Madison. Mathematics Research Center.

In Ch. 1, 'Autonomous Systems', the author considers the system of two equations. The conditions for the existence and stability of a periodic solution are established. The following method is used: we convert to polar coordinates and seek the Lyanunov function V in implicit form. Criteria for the stability or instability of the sought cycle are also established in the form of the function L(subscript 1) (V), and a method for finding an approximate equation of limit cycles is indicated; estmates of the value of the small parameter Mu, for which the theory being developed can be used, are given. The author then considers cases in which studies of more general systems of two equations can be reduced to the study of previously analyzed particular cases. At the end of the chapter the author discusses the feasibility of using Lyapunov functions to find stationary (generally speaking, nonperiodic) oscillations for multidimensional systems as well, but gives no computations. Ch. 2, 'Nonautonomous Systems', briefly reviews the theory of obtaining stationary oscillations., Edited trans. of Universitet Druzhby Narodov, Moscow. Trudy (USSR) v15 n3 p3-55 1966.

The paper discusses the structure of solutions of the linear scaler difference equation x(t + h) = x(t) + f(t), and in part its relationship to the differential equation x dot (t) = f(t). It is shown that the structure of solutions of the difference equation is critically dependent on h. For example, given f periodic, there is a dense set of h's for which there is not even a generalized periodic solution, no matter how smooth f is. Conditions on f for boundedness and uniform continuity of solutions are obtained and an example is presented to show that if f is almost periodic and the solution is bounded, the solution need not be almost periodic. (Author), Report on Vibration and Stability of Military and Other Complex Vehicular Systems.

The report summarizes the research efforts under this project and includes discussions of (1) periodic solutions, (2) a function equation, (3) infinitely differentiable functions, (4) equivalence problems, and (5) operations and distributions. (Author)

The connection between dispersion relations and the solutions of the original partial differential equations in finite systems, involving mixed boundary conditions, is discussed. To illustrate this connection, a generalized wave equation is employed, which can exhibit four common forms of dispersion relations. These four cases are examined to illustrate the modifications arising from finite system, possible misconceptions concerning dispersion relations, and the role or retarded (causal) boundary conditions in classifying instabilities. Some mathematical problems of analysis, which apparently have yet to be solved, are also noted. (Author)

Two empirical methods of approximating nonlinear functions with rational polynomials have already been successfully demonstrated by the author and L. D. Kovach. It was shown that these approximations produce considerably less error than those of polynomials with exponents restricted to integers. In an extension of this work, an iterative algorithm is presented that allows rational polynomial approximations to be computed analytically. Numerical examples are included. (Author)