Explicit Functions Of Time Research Articles

A new boundary element technique for solving linear viscoelastic problems and its application.

Viscoelastic analysis is one of the most important subjects in engineering. Several attempts have been so far made for the integral equation approach to viscoelastic problems. To authors' knowlege, however, there is no available literature for the application of a more direct boundary element method to the time dependent problems of this kind using a step-wise time integration scheme. In this paper, we propose a direct boundary integral formulation. This formulation needs only Kelvin's fundamental solution of isotropic elastostatics with material constants prescribed as explicit functions of time. Successive application of this solution scheme can determine the whole behavior of viscoelastic problems. Finally, some sample problems of viscoelastic materials are computed by means of the proposed method of solution to demonstrate its potential usefulness.

On the eigenvalue problem associated with Feynman path integration

When the lagrangian is not explicit function of time, the Nth approximation to the propagator may be viewed as the Nth power of anitary operator — the infinitesimal time propagator. We solve the eigenvalue problem associated with the operator for some special cases. In the limit of large N the eigenfunctions are shown to be identical to those of the finite time propagator. We also present an elementary method to evaluate the propagator corresponding to an action function encountered in the study of electron gas in a random potential. The evaluation of this propagator within Feynman's polygonal approach was not possible until recently.

The rotational Brownian motion of a nonspherical body and its application to the theory of dielectric relaxation

The conditional probability density function in angular velocity space for a symmetrical body is obtained exactly both from the stochastic differential equations governing the rotational Brownian motion and from the Fokker–Planck equation. Furthermore, it is shown how the Laplace transform of the angular velocity correlation function for an asymmetrical body may be calculated from the Fokker–Planck equation. At the same time, the Laplace transform of the dipole correlation function for the symmetrical body is calculated both from a stochastic integrodifferential equation and Fokker–Planck–Kramers equation. Also, the Laplace transform of the dipole correlation function for the asymmetrical body is obtained using the Fokker–Planck–Kramers equation. Also, the dipole correlation function as an explicit function of time is calculated numerically for a symmetrical body based on the Fokker–Plank–Kramers equation, and is compared to that obtained from the free rotational model and to a single exponential decay function with the Debye relaxation time.

Optimization of the energy output in pulsed lasers

The time dependence of the reflection of the output mirror for a two-level pulsed laser is determined for an arbitrary time-dependent pumping rate. The result derived using the methods of optimal control theory is given as an explicit function of time. The optimal reflectivity eliminates relaxation oscillation spikes so that the output is a smooth function of time. Moreover, the peak instantaneous intracavity photon density of the optimal solution is significantly reduced relative to the photon density using constant reflectivity, thereby increasing the fluence without reaching the damage threshold. Numerical examples for a Nd+3/YAG system is provided. A practical feedback mechanism for controlling the output mirror reflectivity as a function of time that well approximates the optimal solution is proposed.

Approximate dynamics using Hamilton's principle, including applications to non-conservative and constrained systems

There has been recent interest in simulating the dynamical behavior of mechanical systems using approximate techniques based on Hamilton's principle. A method is presented which does not require the user to form partial derivatives of the energy function and produces a solution which is an explicit function of time, not a numerical solution. The method is shown to be suitable for both multidegree of freedom and constrained systems; in the latter case being markedly superior to existing techniques.

Note—On the Relationship of Adaptive Filtering Forecasting Models to Simple Brown Smoothing

This Note establishes two linkages between adaptive estimation procedures used in the analysis and forecasting of multivariate time series and the simple Brown smoothing technique. First, for a simple model it is shown that AEP and LMS are identical to the Brown technique. Thus they are accorded the theoretical properties possessed by the Brown estimator for the simple model. This linkage establishes that the Brown technique has been extended in two complementary directions: to higher order smoothing models which use explicit functions of time and implicitly include effects of other variables and to adaptive time-varying parameter models which explicitly include the influences of other variables and implicitly incorporate time. Second, for the same simple model, Stochastic Approximation, Kalman Filtering, and Bayesian Analysis reduce to a form similar to the Brown technique except that they have a time-varying smoothing constant which approaches zero as the sample size grows large. Hence, while they provide optimal estimates which converge in mean square error, they are not responsive to change.

Source extraction in a random medium

A general formulation of the problem of extracting information about nontransient sources immersed in a randomly fluctuating medium is presented. The formulation is based upon a Green’s function approach in which the medium is treated as a random linear system. It is shown that the problem may be cast into two possible forms, each involving an integral equation the solution of which describes the locations and spectral content of all sources. One of these integral equations is time independent and consists exclusively of deterministic quantities. The other equation is time dependent, consists of stochastic as well as deterministic quantities, and contains considerably more detail. Approximate solutions to the second equation enable one to study image undulations as explicit functions of time. Examples are presented which illustrate some of the difficulties induced by the medium fluctuations when one attempts to solve the basic integral equations.

Source extraction in a random medium

A general formulation of the problem of extracting information about nontransient sources immersed in a randomly fluctuating medium is presented. The formulation is based upon a Green function approach in which the medium is treated as a random linear system. It is shown that the problem may be cast into two possible forms, each involving an integral equation the solution of which describes the locations and spectral content of all sources. One of these integral equations is time independent and consists exclusively of deterministic quantities. The other equation is time dependent, consists of stochastic as well as deterministic quantities, and contains considerably more detail. Approximate solutions to the second equation enable one to study image undulations as explicit functions of time. Examples are presented which illustrate some of the difficulties induced by the medium fluctuations when one attempts to solve the basic integral equations.

A Compartment Model Approach to the Estimation of Tumor Incidence and Growth: Investigation of a Model of Cancer Latency

Consideration is made of the problems involved in determining the effects of a chronic disease process, such as stomach cancer, on the observed mortality of the U.S. population. Specifically, since the time of initiation of tumor growth is unknown and the tumor becomes clinically manifest only after reaching considerable size, the early rate and pattern of tumor growth is unobserved. As a possible solution to the analysis of such problems, it is proposed to use stochastic compartment modelling techniques which deal with the problems of estimating the transition probabilities of a partially observed stochastic process. Implementation of the stochastic compartment techniques in this case depends on the selection of certain mathematical expressions from theories of carcinogenesis, epidemiologic studies and animal studies which allow the calculation of transition probabilities to unobserved states by making them explicit functions of time or age. Though the selection of the specific functions might be subject to debate, the general strategy of explicitly selecting such functions, and thereby exposing them for review in terms of biologic reasonableness and consistency with the data, seems to be a valid and useful methodology. Furthermore, various ways of viewing the model results (say from its internal behavior, e.g., from implied distributions of waiting times in various disease states) yield different insights into the various factors in carcinogenesis. The model, with parameters representing tumor incidence, time to tumor death given onset, genetic susceptibility to tumor growth and the effects of competing forces of mortality, is fitted to data on deaths due to stomach cancer for male U.S. residents age 25 and over in 1969. Two basic forms of the model, one with a waiting time distribution for occupants of the latent state and another with a single latency time, achieved excellent fits to the data. Examination of parameter estimates and compartment waiting time distributions are consistent with theoretical expectations and intuition. It is concluded that such strategies, involving the integration of clinical, experimental and vital statistics data into a comprehensive model of population carcinogenesis, are potentially powerful tools for investigation of the temporal dimensions of disease development in a human population.

A MODAL APPROACH TO THE TRANSIENT ANALYSIS OF SYNCHRONOUS MACHINES

ABSTRACT A novel method for calculating the transient currents, flux linkages and electromagnetic torque of a synchronous machine following a sudden voltage disturbance is presented. The solution is obtained for a sudden change from an arbitrary balanced to an arbitrary unbalanced voltage condition. Since minimum constraints are placed on the disturbance, many typical types of transients such as three-phase and line-to-line short-circuit faults can be accommodated. Complete solutions for all machine currents are obtained as explicit functions of time. Any number of rotor circuits can be included and full allowance is inherently made for coupling between all stator and rotor circuits. Constant speed is assumed but no additional approximations are made other than those implicit in the derivation of Park's Equations. The technique incorporates the state space concept of modal analysis in its implementation. Closed form solutions obtained by the modal theory are compared with standard solution techniques

Volterra Series Synthesis of Nonlinear Stochastic Tracking Systems

One of the most important objectives of a radar angle-tracking loop is to keep the target within the beamwidth of the radar antenna. Thus, the behavior of the antenna pointing error is of vital interest in determination of tracking performance. For a tracker with a general polynomial linearity (representing nonlinear receiver characteristics), subjected to constant line-of-sight rate inputs, random initial antenna pointing errors, and white Gaussian receiver noise, a method to obtain approximations to the transient mean and variance of the antenna pointing error as explicit functions of time is presented.

Transient Load Model of an Induction Motor

The power invariant, nonlinear differential equations, that describe the behavior of a two-phase equivalent of a balanced three-phase induction motor, are linearized about an arbitrary nominal point. Under the assumption that terminal voltage and system frequency can be approximated by a straight line segment over any interval of time, solutions of these small perturbation equations are used to give expressions for active and reactive power delivered tb the motor during transient conditions. These newly developed expressions for power, which turn out to be implicit functions of voltage and frequency, and explicit functions of time and the rates of change of voltage and frequency, are proposed as a load model represeptation of an induction motor for use in transient stability studies. An intuitively appealing, corrective technique, which reduces the error introduced by the use of a linearized model, is presented. Eigenvalues of the linearized system of equations for a group of typical induction motors are given. For a small low voltage motor, a sensitivity study of the dominant eigenvalue to changes in nominal point quantities and parameters is made. Values of active and reactive power calculated by a numericzal solution of the motor nonlinear differential equations are compared with those values of power predicted by the model for certain specific rates of change of voltage and frequency.

Approximations of Interplanetary Trajectories by Chebyshev Series

A METHOD to obtain rapidly convergent series approximations of trajectories with close planetary encounters is presented. The expansions are obtained as finite Chebyshev polynomial series with numerical coefficients. A high-order solution is obtained by the Picard method of successive approximations. The series solution method may be used to solve either initial value or two-point boundary value problems, and may include nongravitational forces. A modification of the Picard method is introduced to reduce considerably the number of iterates. The method yields highly accurate and strongly convergent series solutions as explicit functions of time. The method requires less time of calculation than a more conventional method of numerical integration and provides more definitive and bounded error properties.

The hall mobility of a small polaron in a square lattice

The two-dimensional molecular crystal model of Friedman and Holstein describing the motion of a small polaron in the presence of a magnetic field is utilized as the basis of a study of the small polaron Hall mobility in a square lattice. This work is concerned solely with the high temperature regime where the polaron motion is predominately by means of thermally activated jumps between lattice sites. In addition, the calculation employs the Holstein classical occurrence-probability approach (valid at sufficiently high temperature) which considers the vibrational coordinates to be explicit functions of time. The magnetic field manifests itself solely through the interference of the transition amplitudes associated with the carrier moving between two sites via different paths. Hence in the square lattice (ignoring direct diagonal hops) a minimum of four site are involved in processes associated with the Hall effect. Furthermore, the contributions to the Hall mobility arise from processes in which the initial, final and one intermediate site experience a momentary equality of the electronic energy levels associated with the carrier occupation of these sites, i.e., a triple coincidence event. The distortion of the fourth (noncoincident) site at the time of the triple coincidence event is shown to play an important role in the calculation. It is found that (for reasonable choices of the physical parameters) the temperature dependence of the Hall mobility in a square lattice differs considerably from that found by Friedman and Holstein for the hexagonal lattice structure. In particular, the Hall mobility is shown to be a slowly varying function of temperature which decreases with increasing temperature at temperatures above 2ϵ 2 9k , ϵ 2 being the drift mobility activation energy. The associated Hall coefficient is greater than its “usual” value ( R = −1 nec ) and varies with temperature as exp[ 2ϵ 2 3kT ] .

Segmental flexibility in an antibody molecule

The rotational motions of an antibody molecule were investigated by nanosecond fluorescence polarization techniques. The fluorescent chromophore, ϵ-dansyl- l-lysine, was specifically bound to the combining sites of antidansyl antibody. The emission anisotropy of the bound dansyl group was measured as an explicit function of time following excitation by a nanosecond light pulse. The F( (ab′)2) and F ab fragments of the antibody were studied in the same way. The striking finding is that the antibody molecule displays a discrete mode of flexibility in the nanosecond time range. The F ab portions of the intact antibody molecule are free to rotate over an angular range of the order of 33 degrees in times of nanoseconds. The rotational correlation time for this restricted segmental motion is 33 nsec. The over-all rotational motion of the antibody molecule has a correlation time of 168 nsec, indicating that the antibody has a markedly non-spherical shape. Our findings are consistent with an important feature of the model proposed by Valentine & Green (1967), namely that there is a flexible joint at the junction of the F ab segments. This joint may be biologically significant in facilitating the formation of antibody-antigen complexes. The facile segmental flexibility exhibited by the antibody exemplifies a structural design which is probably used in other large proteins and in organized macromolecular assemblies.

Classical Electrodynamic Equations of Motion with Radiative Reaction

The current status of classical description of radiating bodies is summarized and classical equations of motion are presented for the motion of charged bodies. The general physical solutions for the case of force as an explicit function of time and the exact solutions for many different force fields are given for relativistic, nonrelativistic, one-dimensional, and three- dimensional motion. It is concluded that these equations provide an exact classical description of a radiating body. (D.L.C.)

Classical Electrodynamic Equations of Motion with Radiative Reaction

The current status of classical description of radiating bodies is summarized and classical equations of motion are presented for the motion of charged bodies. The general physical solutions for the case of force as an explicit function of time and the exact solutions for many different force fields are given for relativistic, nonrelativistic, one-dimensional, and three- dimensional motion. It is concluded that these equations provide an exact classical description of a radiating body. (D.L.C.)

Optimization of nonlinear control systems by means of nonlinear feedbacks

The first portion of this paper is devoted to the logical selection of a category of nonlinear systems - systems having: 1. controllers with any number of time lags or leads, and usually operating in their saturated region (velocity saturation, force saturation, etc.), and 2. loads consisting of combinations of nonlinear masses, nonlinear dampers, and nonlinear springs. (Nonlinearities which involve explicit functions of time, such as mass and inertia of a controlled rocket motor missile, introduce special considerations not included here.) In the second part of the paper, characteristics of an intentionally nonlinear element added to the feedback are considered. In the final part, the nonlinear element is shaped to provide optimum response in the saturated region of operation.

The Inadequacy of Testing Dynamic Theory by Comparing Theoretical Solutions and Observed Cycles

IN MODERN business-cycle research the following proceeding is being commonly used: First, a mathematical model (a determinate dynamic system) is set up, as an attempt to describe approximately the interconnections between a set of economic variables, their time derivatives, lagged values, and so on, in terms of strict functional relations. By some statistical procedure the constants in the system are estimated from the corresponding observed time series. Then the system is solved, i.e., the variables are expressed as explicit functions of time, involving the estimated parameters. The degree of conformity between these theoretical solutions and the corresponding observed time series is used as a test of the validity of the model. In particular, since most economic time series show cyclical movements, one is led to consider only mathematical models the solutions of which are cycles corresponding approximately to those appearing in the data.' This means that one restricts the class of admissible hypotheses by inspecting the apparent form of the observed time series. This condition for a good theory is of course not a sufficient one, since there are in general many different a priori setups of theory which are capable of reproducing approximately the observed cycles. But, what is more important, it may not even be a necessary condition, and its application may result in a dangerous and misleading discrimination between theories. The whole question is connected with the type of errors we have to introduce as a bridge between pure theory and actual observations. Compared with actual observations, each equation in a dynamic model splits the observed variations into two parts, one part which is explained by the equation, and another part which is not accounted for, and which is ascribed to external factors. This kind of splitting is common to all theory. We usually consider such equations as good and useful theories if, in order to get full agreement between theory and observations, it is and continues to be sufficient to allow for only relatively small and random external factors. There are two main ways in which such external factors may be