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A Bayesian treatment is given of a two-phase sampling problem when sampling is from a double dichotomy, losses are quadratic, and sampling costs are linear. The motivating question is: how does one determine an optimal first phase sample from a dichotomous population given that an optimal follow-up sample of nonrespondents in the first phase is to be taken?

This work considers the regulator problem for single input-output linear plants in which only the plant output is available for control. A method is presented to specify an optimal nonlinear controller characteristic, constrained to lie within a given Popov sector, that yields improved performance over that obtained by proportional controllers. System performance is measured by a quadratic performance index, and performance is compared with that of both proportional and optimum linear state regulators.

Two types of kinetic hysteresis curves are modeled in this paper: hysteresis in batch enzyme kinetic reactions and growth rate hysteresis in open fermentation systems. It is shown that a sigmoidal shape can be developed by including two features in a kinetic model without resorting to allosteric arguments. These are parallel pathways branching from an intermediate to the product, and a bimolecular terminal step. Parallel pathways are needed to cause a localized convexity change and the bimolecular step is required for the quadratic initial product rate rise. The counter-clockwise hysteresis effect in open fermentation vessels can be developed starting from a limiting substrate-lumped cell machinery kinetic model. The formation of an intermediate complex, which in turn leads to more cells, makes the fermentation model analogous to an enzyme-substrate model. Relating the enzyme and fermentation models with a common approach makes it possible to directly extend the extensive theory of enzyme systems to the more complex fermentation process.

A particle code was written using weighted particles and a quiet start technique to simulate the nonlinear propagation of whistler waves parallel to an external magnetic field. The algorithm is derived from the Daewin Lagrangian by a variational method and two versions of the code are considered, using linear and quadratic splines. Numerical effects due to the discrete initial loading of the particles in parallel velocities, perpendicular velocity phase angles and positions are analyzed; and the performance of the linear and quadratic versions of the code are compared.

A nonlinear analysis of xenon-temperature controlled nuclear reactor dynamics is presented. The set of equations in question belongs to a general class of rate equations with quadratic nonlinearities. Boundedness of the solutions is examined. The mean value of periodic solutions for the flux is shown to be always less than the equilibrium value. The Bogoliubov's method of averaging as extended by Case is applied to obtain approximate solutions. The mechanism of the existence of relaxation oscillations in the linear stability region is analyzed. Computer calculations are performed and found in good agreement with the approximate solutions obtained by means of the method of averaging.

We let , where the xi are positive integers and i ∈ N, the set of all integers. We define , where where . We let = [0; x1, x2, … ] whereWe letand defineThe range of L(ξ) is known as the Lagrange spectrum and the range of M(ξ) as the Markov spectrum. It is known that both are closed and that the Markov spectrum includes the Lagrange spectrum.

The analysis of time series data can be significantly eased by the use of orthogonal polynomials. This technique is based on traditional regression analysis, but the necessity for solving the associated normal equations is eliminated. Also, the orthogonal polynomial technique includes a means to statistically test the level of fit of the independent terms in a forecast model (e.g., constant, linear, or quadratic). This paper shows the application of orthogonal polynomial technique to the analysis of time series data, but more importantly develops a mathematical means for updating the forecast model from one period to the next without completely recomputing the model from stored historical data. This result considerably improves the ease of application of the orthogonal polynomial technique and increases the value of the technique in practice.

The problem of designing an optimal discrete-time feedback controller is discussed for a linear system governed by coupled partial and ordinary differential equations. For the case of spatially concentrated controls and a quadratic performance index, the minimum cost and the optimal discrete-time controls are determined. Evaluation of computational requirements demonstrate the applicability to on-line computer control.

Generalized variable-metric algorithms presented in Ref. 1 produce a unique search direction independently of parameters in the algorithms under several conditions. They generate a unique sequence of minimizing points for the given initial conditions if the objective function is quadratic.

Techniques for the design of feedback controllers for systems having linear time delay models are considered and reappraised. Design methods based on the quadratic performance index are presented. These lead to a controller in the form of a linear functional operating on a segment of the past trajectory. Simplified feedback controllers involving only “current” state variables are then considered. The Nyquist technique of synthesis and analysis is generalized to cover this time delay feedback structure and a design approach based on triangularization of the forward loop transfer function by elementary transformation type compensation matrices and application of the single input-single output theory is proposed.

The paper shows how the parameter plane technique can be applied to the characteristic equation of the optimal linear control system chosen to minimize a quadratic performance criterion. The technique, as applied to a general polynomial equation, is described paying particular attention to the plotting of constant relative damping, constant undamped natural frequency and constant real pole position characteristic curves in the parameter plane which has two variable parameters as co-ordinates. The method is then applied to an optimal linear control system characteristic equation which is written in a convenient form for determining the structure of the characteristic equation coefficients in terms of the variable parameters. Particular attention is given to two parameters contained within a diagonal control weighting matrix and by observing the characteristic curves in the parameter plane the effect of different values of these two parameters on the optimal transient performance can be investigated without...

This paper examines the transmission of binary data signals over channels which contain quadratic nonlinearities and additive Gaussian noise. We consider the case where the channel is nonlinear with memory and where the signal is passed through an input receiver filter and sampled once every signaling interval. The samples are represented by a discrete Volterra series and a special case where the received sample contains a single quadratic distortion term is examined. The optimum (maximum-likelihood) receiver (processor) is derived and upper and lower performance bounds obtained. The performance of a practical, suboptimum receiver is examined by means of computer simulation and is shown to be very close to the lower bound of the optimum receiver. Next we examine the case where the received sample contains two quadratic distortion terms. Again, upper and lower performance bounds are obtained. The performance of a suboptimum receiver which uses nonlinear decision feedback is evaluated by computer simulation. Its performance is shown to be superior to an optimum linear receiver.

By the introduction of the time integral of the output variable of a second-order system, the optimal feedback control law with respect to the quadratic performance criterion gives the conventional proportional-integral-derivative (PID) control. Some comparisons between the weighting factors of the quadratic criterion and the parameters of the PID controller are given. The values of the weighting factors corresponding to the classical methods of Cohen-Coon and Ziegler-Nichols are calculated ; negative values of the weighting factors are obtained.

A generalized theory for depolarized (VH) light scattering from fluids of nonspherical rigid molecules at low frequencies is presented. The dispersion equation for the part of the spectrum which can have fine structures (the autocorrelation function of αk x z) is reduced to a simple quadratic equation given the condition that the internal angular momentum of the molecules decays much faster than the anisotropic polarizability and the transverse linear momentum. The solutions of the quadratic equation and the spectrum of the autocorrelation function of αk x z are similar to those of a linear damped harmonic oscillator. Thus, the depolarized spectrum has different line shapes depending on the experimental situation (e.g., temperature, wave vector, etc.).

The problem of obtaining output feedback gains to minimize a quadratic performance criterion which includes sensitivity variables is discussed. The equivalent stochastic problems are also studied and when a Kalman filter is allowed, the minimum variance estimate of the state is employed to implement the control. A set of non-linear matrix equations are obtained which constitute the necessary conditions that must be satisfied for an optimal solution. For the deterministic case it is shown that when no sensitivity is considered and the matrix which relates output and input is invertible, the usual Riccati formulation is obtained. The problem is also reformulated using a compensator and it is shown that it can be cast into an instantaneous feedback form.

A subroutine is described, allowing the computation of equilibrium concentrations for an enzyme system, labeled “Elementary Bimolecular Cycle”, or also “The Quadratic Scheme”. The cycle consists of four bimolecular steps with free enzyme, two different bimolecular enzyme complexes and one termolecular enzyme complex. The system is related to a wide variety of enzyme reactions. If the analytical concentrations of the three principal components are in the range of the dissociation constants and also are comparable with each other, the originally cubic equation cannot be reduced to a linear equation or even a quadratic equation. The numerical solution is presented here together with a variety of test results.

This paper presents some results obtained from the computer simulation of a two-dimensional linear vehicle model with coupled bounce and pitch modes. The effects of variations in the ratio of front to rear spring stiffnesses and the inertia coupling ratio on various performance indices for ride quality and road holding were investigated. Two forms of input to the system were considered, ( a) isolated smooth bumps of varying length and ( b) continuous random type excitations. Allowance was made for the time delay between the inputs at the front and rear wheels due to the forward speed of the vehicle.

New data for the site percolation problem on the simple quadratic lattice are given. It is concluded that the critical concentration pc does not correspond to the radius of convergence of the series expansion for the mean cluster size.

This paper establishes conditions of optimality for linear systems whoso control functions are quantized either in magnitude or in switching instants, or both. Optimization of such systems hitherto has been limited primarily to dynamic programming techniques and direct enumeration. The results are derived from the total variations of the cost functional. The main theorem provides sufficient conditions of optimality for linear systems with quadratic and positive semi-definite performance index. The results are significant for digital control systems, pulse-code-modulated control systems and some types of pulse-frequency-modulated systems which are subject to quantized variations only. For the special case where the control functions are not quantized, the optimality conditions provided by the extended maximum principle are shown to be necessary and sufficient.

The connection between the anisotropic Ising model and the second-neighbour Ising model is discussed. Series results are consistent with the prediction that the second-neighbour Ising model on a simple quadratic lattice has critical behaviour described by scaling with respect to the nearest-neighbour coupling strength.

Modern control theory is applied to the problem of maneuvering a large tanker through a narrow passage when it is disturbed by stochastic current. The problem consists of two parts, estimation and control. Kalman theory is used to derive optimal estimates from measurements contaminated by measurement errors. The estimates are used in the control and the control law derived minimizes a quadratic performance index. The algorithms obtained are applied to a tanker of 210 000 tdw and the numerical results show that an accuracy of a few meters can be achieved.

The quadratic Stark effect for the seventeen levels of krypton in the 4s2p5ns-, 4s2p5np- and 4s2p5nd -configurations is calculated in the Coulomb approximation. The calculation is conducted for j-l and j-j coupling. The results obtained are found to agree satisfactorily with experiment. The Stark effect constants are given.

In this article, we consider the problem of Nash points for quadratic functionals in Hilbert spaces. Necessary and sufficient conditions of optimality are given. We apply the results to N-person differential games. Open-loop and closed-loop solutions are considered. When there are no constraints, two types of Riccati equations are obtained. We completely develop the parabolic case, generalizing results of A. Friedman [2] who considered finite-dimensional systems. The infinite-dimensional systems that are modeled are the ones which represent distributed parameter systems as in Lions [4].

A study of the one-dimensional compressible Ising model, in zero field, with quadratic potential energy and linear exchange parameter, is presented, using a canonical transformation of the hamiltonian. Earlier solutions are corrected and generalized, the role of the four-spin interaction in different ensembles being made explicit. The results agree with the solutions of the Baker-Essam model.

This paper describes a numerical method for the spectrum factorization procedure. This numerical method enables the solution of the optimal control problem for linear distributed systems with quadratic performance indices. The numerical method is demonstrated on an optimal feed forward—feed back control system for a heat exchanger.

This paper gives counterexamples to: (1) Ritter's algorithm for the global maximization of a quadratic subject to linear inequality constraints, and (2) Tui's algorithm for the global minimization of a concave function subject to linear inequality constraints.

Explicit expressions are derived for evaluation of the general time-weighted quadratic performance index associated "with a discrete-time constant linear system. The method used relies on formulating the problem in terms of Kronecker matrix products, which enable a formula for the sum of a scalar series to be applied. Similarities with the corresponding problem for continuous-time systems are demonstrated.

The situation in which two groups of people have conflicts of interest is considered as a two-team zero-sum game problem. Two special cases of this problem are solved to illustrate that communication among members of a team may not be worth-while and extra information need not always be desired by decision makers. In the appendix, it is shown that the optimal saddle-point solution exists and is still affine for the general problem with quadratic Gaussian performance index.

An optimal control problem with linear dynamics and quadratic criterion is imbedded in a family of problems characterized by both initial and terminal points. The optimal value function is jointly quadratic in initial and terminal points, and the optimal control is jointly linear. Recursive formulas for the coefficients of these functions are developed. This generalized procedure can be used to solve several versions of the problem not solvable by the standard one-ended imbedding technique. In particular, a procedure doubling the number of stages at each iteration is given for problems with time-invariant coefficients.

This article investigates some theoretical and computational properties dealing with the selection of an “optimum” experiment in a constrained environment. The squared Euclidean norm of the information matrix is developed as a measure of optimality for any experimental factorial arrangement. The problem of experimental layout is structured as a mathematical optimization which can be transformed into a quadratic integer program. Computational results are given.

The tables of Pitzer et al. for the compressibility factor of nonpolar substances have been extended to polar fluids for the gaseous and liquid regions for reduced temperatures from 0.8 to 1.15 and reduced pressures from 0.2 to 6.0. Available experimental PVT data for polar fluids were related to the acentric factor and fourth parameter at constant reduced temperature and pressure by a quadratic least squares procedure. The resulting relationships reproduced the data utilized with good accuracy for the entire region considered.

The capital budgeting problem is treated as a part of the general theory of choice, where utility is to be maximized subject to the opportunities and constraints. The utility function is assumed to be quadratic to take care of any risk-averse behavior of the shareholders and a quadratic mixed zero-one integer programming model is developed. Special solution techniques for the resulting mixed zero-one integer quadratic programming model are discussed.

In this article we give a partial solution to Bernstein's problem concerning quadratic mappings of a simplex into itself: the description of all those mappings V satisfying the condition V2 = V. The formulation of this question is connected with mathematical genetics. In this article, the selected class of mappings is characterized by the existence of linear preservation laws sufficient for the parametrization of a mapping. In genetic terms, this means the existence of genes (but, generally speaking, with a more complex behavior than in classical genetics). This article describes all these mappings. Bibliography: 8 items.

This paper studies how the solution of the problem of minimizingQ(x) = 1/2x T Kx − k T x subject toGx ≦ g andDx = d behaves whenK, k, G, g, D andd are perturbed, say by terms of size∈, assuming thatK is positive definite. It is shown that in general the solution moves by roughly∈ ifG, g, D andd are not perturbed; whenG, g, D andd are in fact perturbed, much stronger hypotheses allow one to show that the solution moves by roughly∈. Many of these results can be extended to more general, nonquadratic, functionals.

Equations of motion are generated for a classical particle in a Riemannian space with arbitrary metric, moving in a velocity-independent potential, and acted upon by a linear frictional force. Only the Lagrangian formalism is used, but the Lagrangian function is not of standard form. As an example, a satellite moving in a linearly-viscous atmosphere is considered. This approach yields immediately a constant of the motion which is a generalization of the usual angular momentum. A particle moving on the viscous surface of a sphere is also discussed. Finally, the equation of motion for a particle moving in one dimension subject to a frictional force quadratic in the velocity is generated geometrically by choosing an appropriate nonanalytic metric.

Quadratic approximation is a three-dimensional analogue of the two-dimensional Pade approximation. The advantages of employing quadratic approximation are demonstrated by several examples. With automatic computers having a relatively inexpensive square root instruction, quadratic approximation is an attractive alternative to Pade approximation if convergence of the Pade approximation is not rapid.

A form of artificial viscosity, which is second-order and quadratic, is developed for elastic solids. This new form and the standard linear artificial viscosity are compared. It is observed that the artificial viscosity works best when applied only to expanding zones.

An appropriate path differential measure is employed for obtaining the path integral representation of the propagator of a particle with a time dependent 'mass'. The evaluations are restricted to quadratic Lagrangians and the propagator for the damped harmonic oscillator is given explicitly.

In application of the previously developed modified structure method to the study of the nonlinear behavior of continuous elastic structures, including bifurcation and snap-through, one is led to an eigenvalue problem in which the eigenparameter appears quadratically in the coefficients of a linear differential equation. In this present study, the elementary example of a fixed-free column loaded both axially and laterally is considered as a continuum problem, an approach possible within the scope of the modified structure method. Earlier studies of this problem in finite element formulation showed nonexistence of an eigenvalue for very slender columns. The present paper investigates the question of existence of a positive eigenvalue in a hierarchy of linearized quadratic eigenvalue problems which result from the retention of higher order terms in the expansion of the potential energy functional. Nonexistence of eigenvalues for columns of sufficiently high slenderness ratio $\rho $ is shown, and the Raylei...

A queueing problem is considered with one server, independent discrete inter-arrival times, and independent discrete service times. The steady state distribution of the waiting time is approximated by means of the solution of certain linear equations and applying a generalization of the Lin-Bairstow method for finding quadratic factors of polynomials.