Showing papers on "Constant (mathematics) published in 1968"

The stochastic computer simulation of ion motion in a gas subjected to a constant electric field

Abstract: The stochastic computer simulation of ion motion in a gas subjected to an electric field involves the problem of statistically choosing free times between the collisions, when the velocity cannot be assumed constant during a free time. It is shown how free times can be chosen, without any approximations or numerical integrations, by the use of a piecewise velocity-independent mean collision frequency chosen to be greater than the actual collision frequency, combined with the assumption that an appropriate part of the collisions results in zero momentum transfer.

Optimal control of the linear regulator with constant disturbances

Abstract: An optimal control problem for a linear regulator with constant external disturbance is formulated. It is shown that, for a suitably selected quadratic-type performance index, the optimal control is not an explicit function of the external disturbance. Moreover, the optimal control can be synthesized as a time-invariant linear function of the state plus the first time integral of a certain other time-invariant linear function of the state.

Stark Ladder in Solids

Abstract: It is shown that the Stark ladder for conduction electrons in a constant electric field does not follow from the existing theory. The former proofs of the ladder are shown to be based on inconsistent assumptions. A discussion is given of a tight-binding model which leads to a ladder when overlap is completely neglected.

Finite State Continuous Time Markov Decision Processes with a Finite Planning Horizon

Abstract: The system we consider may be in one of n states at any point in time and its probability law is a Markov process which depends on the policy (control) chosen. The return to the system over a given planning horizon is the integral (over that horizon) of a return rate which depends on both the policy and the sample path of the process. Our objective is to find a policy which maximizes the expected return over the given planning horizon. A necessary and sufficient condition for optimality is obtained, and a constructive proof is given that there is a piecewise constant policy which is optimal. A bound on the number of switches (points where the piecewise constant policy jumps) is obtained for the case where there are two states.

Quasi-separable utility functions†

Abstract: This paper is concerned with a method for the assessment of utility functions of multi-numeraire consequences. It is proven that given von Neumann and Morgenstern's axioms of “rational behavior” and two additional assumptions, the utility function for (x, y) consequences can be written as U(x, y) = Ux(x) + Uy(y) + KUx(x) Uy(y). K is a constant that must be evaluated empirically. This form shall be designated as a quasi-separable utility function. It is more general than the separable utility function and is shown to be nearly as easy to use. Implications and ramifications of such a utility function and its requisite assumptions are discussed. A technique for practical application of this work is presented.

Plant Competition and Crop Yield

Abstract: Shinozaki and Kira1 by combining a logistic equation of population growth (equation (1)) with the law of constant final yield (equation (2)) derived equation (3) to describe the relationship between plant density and yield. or where at a given time t, w is the dry weight of each plant and W is the asymptote of w, λ is the coefficient of growth which is independent of plant density ρ, k is the integration constant, determined by the initial conditions and independent of plant density, Y is the final yield/unit area, and independent of density, and W0 is w at t = 0. The law of constant final yield, however, may not always be satisfied (data to be published elsewhere) and the yield in higher densities may decline. In these conditions, equation (3) cannot describe the yield–density relationship. The validity of the logistic equation (equation (1)), as tested for different crops1,2, has proved to be justified so far, and it seems that it is the law of constant final yield which needs further modification.

An optimal policy for lay-up decisions

Abstract: Summary The paper is concerned with the problem of deciding when a ship should be laid up and when it should be put back into operation, when revenue is a stationary random walk and cost parameters constant over time. The solution is given in terms of a decision rule of the form: If revenue falls to y, lay up; if it rises to z, start operation. Optimal values for z and y are derived, and some interesting properties of the solution discussed.

Transverse Vibrations of Tapered Cantilever Beams with End Support

Abstract: Free vibrations of nonuniform cantilever beams with an end support have been investigated, using the equations of Bernoulli‐Euler. Two configurations of interest are treated: (a) constant width and linearly variable thickness, and (b) constant thickness and linearly variable width. Charts have been plotted for each case from which the fundamental frequency, the second harmonic, and the third harmonic can be easily determined for various taper ratios. The Tables from which these charts were plotted are also included.

Invalid results from the method of constant stimuli

Abstract: The Method of Constant Stimuli was shown to produce constant errors in the direction expected by E. In an experiment, 89 Ss compared variable lines with a standard, each S producing a psychometric function and Point of Subjective Equality (PSE). Four groups differed in the particular range of variable stimuli used; whether the stimuli averaged longer or shorter than the standard, and whether they had a wide or narrow range. All sets of variable stimuli encompassed the standard. PSE was consistently between the standard and the mean of the variables, as predicted by Adaptation Level Theory. In many experiments, the set of variables is chosen on the basis of theory or pilot studies; the present study shows that PSE will be spuriously drawn toward the center of the variables, hence toward E’s pre-experimental expectations.

Extreme Points and Linear Isometries of the Banach Space of Lipschitz Functions

Abstract: Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such that The smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,

Optimal control of unstable linear plants with inaccessible states

Abstract: A theory is presented for the optimal design of linear time-varying systems with a quadratic cost functional when not all the states of the plant are accessible for measurement. The procedure involves choosing a constant set of eigenvalues and ensures closed-loop stability, even when the plant is originally unstable. The performance of the resulting feedback system can be made arbitrarily close to the optimum which results when all states of the plant can be measured.

Analytic solutions of limit cycles in a feedback-regulated converter system with hysteresis

Abstract: A mathematical model is derived for the voltage step-down dc-to-dc converter in which a hysteretic bistable trigger circuit is used to regulate the output voltage. Normalized second-order differential equations are derived for the output-voltage error, or output-voltage ripple, measured with respect to a constant reference. The method of successor functions is applied to the piecewise analytic phase plane trajectory for the errors and the conditions leading to a limit cycle are initially formulated in two transcendental equations. With the objective of obtaining an analytic solution for the period of the limit cycles these two equations are then replaced by approximate algebraic equations which are solvable in general terms. The approximations are carefully based on properties that are typical of all converters of this type, and lead to quite simple but accurate expressions for the period of the limit cycle in terms of arbitrary system parameters. Expressions for the amplitude of the limit cycle are also given, and its stability is tested. A numerical example based on an actual representive system is given. Certain unusual characteristics of the limit cycle as a function of certain system parameters are pointed out.

Minimum-weight design with piecewise constant specific stiffness

Abstract: Much of the literature on minimum-weight design is concerned with structures with continuously varying dimensions. Structures of this kind achieve an absolute minimum of weight and thereby provide a useful standard of comparison, but their manufacturing cost is prohibitive in most applications. As a step toward the establishment of optimality conditions for more practical structures, the present paper discusses optimal design of sandwich beams and frames with segmentwise constant cross sections. Optimality is shown to require that the mean square stress in the face sheets has the same value for all segments. The use of the optimality condition is illustrated by examples.

On grey absorption coefficients in radiative transfer

Abstract: The radiative heat flux and several frequency-averaged absorption coefficients were calculated for a plane-parallel radiating gas slab, for a variety of absorption spectra. Both constant temperature and variable temperature are considered. None of the assumptions concerning these coefficients of several recent non-grey models turn out to be verifiable, except at best under more restrictive conditions. With constant temperature, the heat flux is to a good approximation the same function of a single optical depth as in a grey gas. With variable temperature, a radiative transfer equation with a common coefficient for both emission and absorption terms becomes meaningless; a separate coefficient should be used for the absorption term.

Self-diffusion in liquid sodium at constant volume and constant pressure

Abstract: The self-diffusion coefficient of liquid sodium has been determined in the temperature range melting point to 300°C, under both constant pressure and constant volume conditions. The experimental technique is described in detail and current theories are reviewed in the light of the results obtained. None of the theoretical approaches is entirely correct in predicting the experimentally observed data. This is probably because there is more than one diffusion process operative in the liquid. It is, however, concluded that models of the fluctuation type most accurately describe observed conditions. The critical fluctuation concept is thought not to be applicable.

Time-Translation Invariance for Certain Dissipative Classical Systems

Abstract: Time-translation invariance of the equation of motion for an arbitrary one-dimensional classical system has been shown to generate a conservation law. For conservative systems, this conserved quantity is shown here to be the usual energy. For the linearly damped harmonic oscillator, the explicit form of this constant is found, as well as for quadratically damped systems. In both cases, as the damping approaches zero, the conserved quantity is shown to approach a function of the usual energy. For both types of dissipation, Lagrangians are given which yield the equations of motion.

On solutions for two waves with periodic coupling

Abstract: An exact solution for the coupling effects between two waves with a particular complex periodic coupling function is presented; the particular coupling function gives the same wave interactions as constant coupling but at a translated value of differential phase constant. A transformation is given which permits known theory for constant coupling to be applied to the periodic coupling case. Approximate solutions are given for periodically reversed coupling (sinusoidal or square wave) between two waves, and calculations are presented which indicate the solutions are valid for arbitrarily long coupling regions or arbitrarily large integrated coupling strengths. The region of validity for earlier perturbation theory is defined and proved to include the cases of interest for multimode circular electric waveguides.

Friction in a vacuum

Abstract: : The effect of vacuum on the friction behavior of steel has been investigated. It was found that the weight loss of the specimens due to friction increases rapidly with decreased pressure until the pressure reaches a certain value below which the weight also remains constant. (Author)

Optimal unsupervised learning multicategory dependent hypotheses pattern recognition

Abstract: A Bayesian decision theory approach is applied to the solution of the problem of unsupervised parametric pattern recognition. The parametric model for this investigation includes the cases where both constant and time-varying unknown parameters are present, and, most significantly, the unknown hypotheses do not constitute a statistically independent sequence. They are restricted only to be from a source with finite-order Markov dependence. The resulting optimal learning system is found and shown to grow initially in size and memory until the N th observation (where N is the highest Markov order), and subsequently to remain of fixed size and memory. It can, therefore, operate indefinitely and continue to improve its ability to recognize patterns utilizing only a fixed size memory. In summary, the main contributions of this paper are the following: \begin{enumerate} \item the extension of previous investigations of the unsupervised parametric pattern recognition problem to include cases where both constant and time-varying unknown parameter vectors are simultaneously present; \item that the a priori probabilities of the hypotheses, the time-varying parameters, and their transition laws may, if constant, be expressed as functions of the constant unknown parameter and, thus, also be learned; and \item the removal of the assumption of statistical independence between hypotheses for the sequence of observations. \end{enumerate}

A new determination of the constant of gravitation

Abstract: Summary A new determination of the constant of gravitation is being prepared, more than 30 years after the previous one. The reasons for undertaking a new determination are outlined and the methods that have been adopted in the past are reviewed. The design of the new determination, which incorporates a number of refinements of principle and execution, is discussed.

An economic lot-sizing echnique: II mathematical analysis of the part-period algorithm

Abstract: The Part Period Algorithm discussed in Part I is compared with optimal solutions, determining the maximum deviations possible. Optimality of the algorithm is established for the case of constant demand. Performance characteristics are compared with those of the Least Unit Cost algorithm.