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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.

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.

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.

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...

1 Hoffman, W. C, Zvara, J., and Bryson, A. E., Jr., "A Landing Approach Guidance Scheme for Unpowered Lifting Vehicles," Journal of Spacecraft and Rockets, Vol. 7, No. 2, Feb. 1970, pp. 196-202. 2 Duiven, E. M., "Quadratic Cost-Re-Entry Guidance Laws," Internal Memo 2-5482-0100-020, Dec. 1969, Aerospace Group, Boeing Co., Seattle, Wash. 3 Kovatch, G., "Optimal Guidance and Control Synthesis for Maneuverable Lifting Space Vehicles," Transactions of the 9th Symposium on Ballistic Missile and Space Technology, Vol. 1, Air Force Command, San Diego, Calif., 1964, pp. 327-379. 4 Glover, L. A. and Hagan, J. C, "The Motion of Ballistic Missiles," TG1164, July 1971, The Johns Hopkins University, Applied Physics Lab., Baltimore, Md. 5 White, J. S. and Lee, H. Q., "Users Manual for the Variable Dimension Automatic Synthesis Program (VASP)," TM X-2417, Oct. 1971, NASA. 6 Anderson, B. D. O. and Moore, J. B., Linear Optimal Control, Prentice-Hall, Englewood Cliffs, N.J., 1971. 7 Bryson, A. E. and Ho, H. C, Applied Optimal Control, Blaisdell, Waltham, Mass., 1969. 8 Kwakernaak, H. and Si van, R., Linear Optimal Control Systems, Wiley, New York, 1972.

A Bayesian decision theoretic approach is employed to compare sampling schemes designed to estimate the reliability of series and parallel systems by testing individual components. Quadratic loss is assumed and schemes are found which minimize Bayes risk plus sampling cost. Several kinds of initial information concerning the reliability of the individual components, all of which assume the components function independently, are considered. The case for which the initial information is in terms of the system's reliability is briefly considered and related to the aforementioned case

For each cusp singularity of the Hilbert modular variety of a totally real algebraic number field there is associated an L-series which determines the corresponding "parabolic contribution" to the version of Selberg's trace formula worked out by Shimizu. For a real quadratic field Meyer [9, 10] has given an elementary algorithm by Dedekind sums for the computation of the parabolic contribution (see also Siegel [14]) from which a simple formula for the parabolic contribution in terms of the continued fraction associated with the cyclic resolution of the corresponding cusp singularity can be derived ([5, 7, 8]). In fact, the parabolic contribution equals ~2(3r-bo b l . . . . . br-1) where bo .. . . . br-1 are the characteristic numbers of the dual normal bundles of the rational curves which form the exceptional fibre of the cyclic resolution. The first purpose of this note is to give a simple formula for the total parabolic contribution, i.e., the sum of the parabolic contributions for the different cusps. This formula is deduced from a general identity satisfied by certain L-series in any real algebraic number field. In the quadratic case one obtains an expression for the total parabolic contribution in terms of the class numbers of imaginary quadratic fields. Further, in the quadratic case the total parabolic contribution vanishes if and only if the discriminant of the field is the sum of two squares and is negative otherwise. The total parabolic contribution is of additional interest in the quadratic case for two reasons: first, it is one-fourth the signature (adjusted for defects coming from quotient singularities in certain special cases) of the open rational homology 4-manifold obtained as the quotient of the product of two upper-half planes by the non-symmetric Hilbert modular group. This result, announced to some extent in [5], will be proved here. Thus, from [2] the vanishing of this (adjusted) signature is a necessary and sufficient condition for the existence of modular imbeddings. Second, it will be shown that this signature is twice the difference of the arithmetic genera for the ordinary and mixed Hilbert modular groups.

The paper begins with a discussion of the possibility of estimating the parameters of a utility function from the estimated values of the parameters of the demand functions derived from them. The cases of the Klein-Rubin and Quadratic utility functions are analysed in detail. Next, from the assumptions of utility optimization, a method is derived for estimating the parameters of linear and logarithmic utility functions. This method is based on Principal Component Analysis. The method is applied to a sample of 9 countries in all levels of development. It is verified that the results obtained agree with accepted notions about the rankings of goods and services according to preferences.

In an earlier paper, it was shown that the linear and quadratic programming problems in complex space could be unified in the “complex linear complementarity problem” (complex LCP). An existence theory for this problem was provided. The present paper addresses the problem of actually solving the complex LCP and hence complex linear and quadratic programs as well. Two solution procedures are described and an example is given.

A mean-variance portfolio selection model suitable for the small investor is formulated as a sequence of quadratic integer programming problems. The special structure of these quadratic problems is exploited in a partial enumeration algorithm which uses cutting planes to accelerate convergence. Computational experience is reported on problems ranging in size from fifteen to fifty variables.

Potential energy profiles corresponding to libration of 9,10-anthraquinone about its molecular axes were constructed for the five structures derived from data which were collected at - 170, - 112, -72, - 12.5 and 20.5°C [Lonsdale, Milledge & El Sayed (1966). Acta Cryst. 20, 1-13]. These profiles were represented by fourth-degree least-squares polynomials, whereafter r.m.s. libration amplitudes and rigid rotator frequencies of 9,10-anthraquinone were evaluated in the quadratic approximation. The temperature dependence of the calculated quantities is in most cases close to that of the observed ones thus reproducing, by comparison with observed Raman frequencies, the pseudoharmonic behaviour of 9,10-anthraquinone. Calculated r.m.s. libration amplitudes are only qualitatively comparable to the experimental ones and appear to be somewhat too low. The present representation of energy profiles makes it possible to estimate conveniently the contribution of anharmonicity to the profile shape.

A method for the approximation of all the real roots of an n-order polynomial equation is developed. It is assumed that intervals containing the solutions are known. Bilinear transformations are used to approximate the solution. Convergence is achieved.

The problem of the behavior of a pulse when propagating through a dispersive medium has been treated in terms of Fourier transforms. For a linearly dispersive medium the evaluation of the integrals is trivial, but when the dispersion relation contains a significant quadratic term the integrals can be evaluated using the properties of Fresnel integrals. The advantage of this treatment is that it shows that there is a very strong analogy between the pulse shape and the diffraction pattern produced when light passes through a small hole in an opaque screen.

For a bottomless medium characterized by a positive constant gradient, the multiple reflection problem is treated by analysis of the following: (1) The Simulation Problem: Given the relative coordinates x and y between source and receiver, determine the orientations of those rays which succeed in impinging upon the receiver by use of the acquiring‐ray approach. The existence condition for real roots of the acquiring‐ray cubic and quartic equations treated herein facilitate the determination of the caustic equations. When the condition is imposed that the discriminant of the cubic or quartic equation be zero valued, the caustic equation emerges as the boundary line among which repeated roots occur. For the single reflection case, a cubic equation must be solved and the associated caustic curve has the shape of a cusped semicubical parabola. For n reflections (n ⩾ 2), a quartic equation must be treated and the two caustics of the nth order are derived from the discriminants of the quadratic factors composing the quartic equation. Explicit closed‐form solutions for the caustics are obtained by use of reciprocity and reduction‐to‐single‐reflection‐type arguments. The mathematical form and physical significance of the caustics is discussed. (2) The Passive Detection Problem: Given only the ray orientations at the receiver, determine the vertical and horizontal distances between source and receiver. For both problems, no restrictions are imposed on the source and receiver depths.

Recently Britten has proven an analog of Goldie’s theorem for the Jordan ring S S of symmetric elements in a ring with involution of characteristic not 2. In this paper we first extend Britten’s theorem to the situation where R R is an arbitrary ring and the Jordan ring is only an ample subring of the symmetric elements. We apply this result to show that if S S has ACC on quadratic ideals, then the (Jordan) nil radical of S S is nilpotent.

Wheeler [6] proposes simple formulas for quickly determining the sample sizes necessary to achieve a prescribed power. We find Wheeler's evaluation lacking in accuracy because of his application of a linear approximation to quadratic formulas. Therefore, “portable power” is potentially harmful to the experimenter who may not be knowledgeable in the theory.

(i) -2 for n= 1, (ii) 0 for n = N, (iii) 0 for n = 0, and also that (iv) var (ta) = var (tN-f) These are properties that var (tn) must possess on statistical grounds and they provide a partial check on the derivation of (1). Further, such properties can assist recall of the result: the idea is that one writes down Cn(N n) by virtue of (ii) and (iii), or (ii) and (iv), and obtains n(N-n) o2/(N1) by virtue of (i). But now this rather ad hoc reconstruction of (1) seems very close to being a proof of it. The tacit assumption in the above that var (tn) is quadratic in n needs justification, and other weaknesses may be revealed by a closer scrutiny. The object of the present note is to use such methods for higher moments of tn and for the sampling moments of other statistics.

Bayesian estimation and design are considered for a dichoto-mous response surveillance model for defectives. The probability of a defective item after storage time t is assumed to be given by F(t) = l-exp(-At), 0 < t, λ < ∞. Surveillance is carried out by removing k lots from storage and observing the size (ni), storage time (ti) and number of defectives (ri) for each lot. The ri are assumed to be independently and binomially distributed with respective expectations (ni).F(ti) A prior gamma distribution is assumed to be available for x. the posterior distribution of x is Hayes estimate assuming quadratic loss. A formula for the corresponding Bayes risk is derived as are some recursive relationships to aid in computation. A design procedure is given for selecting surveillance times and sample sizes successively one lot at a time. Tables of optimum surveillance times and Bayes risks at these times as functions of the prior parameters and sample size are provided to help with this selection procedure.

Recently, an algorithm for function minimization was presented, based upon an homogeneous, rather than upon a quadratic, model. Numerical experiments with this algorithm indicated that it rapidly minimizes the standard test functions available in the literature. Although it was proved that the algorithm produces function values which continually descend, no proof of convergence was supplied. In this paper, the homogeneous algorithm is modified primarily by replacing the cubic interpolation routine by Armijo's step size rule. Although not quite as fast as the original version on the standard test functions, this modified form has the advantage that a proof of convergence follows from a general theorem of Polak.

The purpose of this paper is to develop a method for evaluation of certain elementary functions on a digital computer by the use of continued fractions. The time required for this evaluation is drastically reduced by using "short" operations like shift and add, instead of multiplications. Functional consistency is the most important factor that aliows the expansion of a function into a continued fraction. Several cases are discussed; in particular the solution of the quadratic equation is discussed in more detail to demonstrate the convergence of the method.

The combination tones of differential type are studied in a non-linear damped mechanical system of two degrees of freedom with quadratic and cubic non-linearities and excited by two external harmonic forces with different frequencies. Approximate steady state solutions and the corresponding Galerkin approximations of high order are obtained and error bounds are given. For a certain frequency the existence of three exact periodic solutions is proved by Urabe's method.

A detailed discussion is given of the quadratic characteristic value problem, x = λPx + ( 1 λ )Qx , where P and Q are compact selfadjoint operators in Hilbert space. Theorems are proven showing that unconditional bases of various types may be constructed from the root vectors of A(λ) = λP + ( 1 λ )Q . These theorems are of interest in the stability theory of linear evolution equations, particularly for application to certain problems in the linearized theory of hydrodynamic stability.

It is known that compliance scheme does not provide simple linear isotopic sum rules in λ −1 similar to the ones in λ in the force constant scheme. The advantage of not having such sum rules in the compliance scheme is discussed in relation to the uniqueness problem of the molecular force field. Incidentally, for the molecules H 2 O, D 2 O, and HDO, two quadratic relations in terms of the inverse frequency parameters λ −1 are obtained. These supplement the three obvious product rule relations.

The classical method of reducing a positive binary quadratic form to a semi-reduced form applies translations alternately left and right to minimize the absolute value of the middle coefficient — and may therefore be called absolute reduction. There is an alternative method which keeps the sign of the middle coefficient constant before the end: we call this method positive reduction. Positive reduction seems to make possible an algorithm for finding the representations of 1 by a binary cubic form with real linear factors, and has various properties somewhat simpler than those of absolute reduction. Some of these properties involve unipositive matrices (with nonnegative integer elements and determinant 1). Certain semigroups of unipositive matrices with unique factorization into primes are described. Two of these semigroups give a neat approach to the reduction of indefinite binary quadratic forms—which may generalize. Some remarks on unimodular automorphs occur in Section 6.

It is well known that in the Gauss-Markov model (Y, Xβ, σ 2 V) with |V| ≠ 0, the BLUE (best linear unbiased estimator) of Xβ is Y 1 , the orthogonal projection of Y on M(X), the space spanned by the columns of X, with inner product defined as (x, y)=x'V −1 y. A quadratic function of Y 2 , the projection of Y on the orthogonal complement of M(X), provides an estimate of σ 2 . It may be seen that Y=Y 1 +Y 2 . When V is singular, the inner product definition as in non-singular case is not possible. In this paper a suitable theory of projection operators is developed for the case |V|=0, and a decomposition Y=Y 1 +Y 2 is obtained such that Y 1 is the BLUE of Xβ and a quadratic function of Y 2 is the MINQUE (Minimum Norm Quadratic Unbiased Estimator) of σ 2 in the sense of Rao (1972).

This paper describes a library of computer field programs available for design investigations during the development of high-voltage switchgear equipment. The emphasis is on recent developments to a general-purpose multidielectric field program suitable for solving practical field problems in two dimensions or in three dimensions with one axis of symmetry. These developments include an improved presentation of printout data, automatic drawings of computed equipotentials, and accurate calculation of gradient values for practical systems. A quadratic fit method and finite-difference techniques to compute gradient values are discussed together with associated accuracies. Finally, examples are given of the application of this numerical technique to high-voltage switchgear design.

The decision-theoretic approach to point estimation involves the choice of an estimate to minimize the expected loss associated with the estimate. The purpose of this paper is to indicate the influence of risk aversion on point estimates for classes of payoff functions including the piecewise linear and quadratic payoff functions. Increased risk aversion results in a point estimate closer to zero for a quadratic payoff function and a lower estimate with a piecewise linear payoff function, for example.

An in-depth analysis of the U. S. apple industry and its problems requires simultaneous consideration of its multiple dimensions of space, time, resources, commodities, production activities, and marketing levels. An empirical model incorporating these dimensions is constructed and its use for policy analysis is demonstrated by measuring the impact of alternative size reductions in regional apple marketings on f.o.b. level industry and regional net returns. The model could also be used in determining the ramifications of changes in consumer demand, transportation costs, and marketing margins on such factors as production, prices, and interregional flows.

and IntroductionQuadratic programming means maximizing or minimizing a quadratic function of one or more variables subject to linear restrictions i.e. linear equations and/or inequalities.Among the numerous insurance problems which can be formulated as quadratic programs we shall only discuss four, namely the Credibility, Retention, IBNR and the Cost Distribution problems.Generally, there is no explicite solution to quadratic optimization problems, only statements about the existence of a solution can be made or some algorithm may be recommended in order to get exact or approximate numerical solutions. Restricting ourselves to typical problems of the above mentioned type, however, enables us to give an explicit solution in terms of general formulae for quite a number of cases, such as the onedimensional credibility problem, the retention problem and—under relatively week assumptions— for the IBNR-problem.The results given here are by no means new. The only goal of this paper is to describe a few fundamental insurance problems from a common mathematical standpoint, namely that of quadratic programming and at the same time, to draw attention to a few special aspects and open questions in this field.

An elementary proof is given of the theorem: If D = −3q or −27q is the discriminant of a cubic field, where q ≡ 1 (mod 4) is a prime, and if p or 4p is represented by c2 + ∥ D ∥ d2, then the fundamental unit in the field Q(q 1 2 ) is a cubic residue of the prime p. In special cases necessary and sufficient conditions are derived.

In constructing the differential equation for stresses and deformations and their rates, it is proposed to use the nonlinear stress-deformation equations in the form of a quadratic parabola, when the loadings are instantaneous or very slow. Relaxation curves and curves of the creep and rate of creep of an elastoplastic material with nonlinear creep are obtained.

The article reviews Ragnar Frisch's contributions to solving the problem of establishing preference functions for macroeconomic decision models, and compares Frisch's approach with other possible approaches to the problem. The article mainly considers linear and quadratic preference functions. In discussing the methodology attention is drawn to the analogy with problems in the analysis of consumer demand.

The solutions of a quadratic partition of primes p ≡ 1 ( mod 7 ) p \equiv 1 \pmod 7 , in terms of which the author and P. A. Leonard have given the cyclotomic numbers of order seven and also necessary and sufficient conditions for 2, 3, 5 and 7 to be seventh powers ( mod p ) \pmod p , are obtained for all such primes > 1000 > 1000 .

In this paper we consider the distribution in the half-plane σ>1/2 of the values of DirichletL-series formed with quadratic characters.

We shall examine the control problem consisting of the system dx dt = f 1 (x, z, u, t, ϵ) ϵ( dz dt ) = f 2 (x, z, u, t, ϵ) on the interval 0 ⩽ t ⩽ 1 with the initial values x(0, ϵ) and z(0, ϵ) prescribed, where the cost functional J(ϵ) = π(x(1, ϵ), z(1, ϵ), ϵ) + ∝01 V(x(t, ϵ), z(t, ϵ), u(t, ϵ), t, ϵ) dt is to be minimized. We shall restrict attention to the special problem where the fi's are linear in z and u, V is quadratic in z and independent of z when ϵ = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ϵ tends to zero. The techniques of constructing such asymptotic expansions will be stressed.

This paper presents an empirical investigation of the importance of risk in decisions. The adaptive expectations geometric lag model is generalized by geometrically including quadratic lag terms indicative of risk. The computation of consistent estimators is described, and the model is applied in the analysis of California field-crop supply response. Results indicate that the effects of stabilization (possibly associated with government programs) might have seriously offset the acreage-reducing effects of voluntary acreage restrictions. Finally, more general implications relating to reduced-form estimation of the standard Nerlovian model are discussed.

This paper discusses the problem of determining which of a large set of possible but improbable malfunctions gave rise to a given set of measurements. The classes of systems under consideration generally lead to underdetermined sets of equations. Three methods of formulating and solving this class of problems are presented: 1) the pseudoinverse method: this leads to an easily-solved computational problem but it is not physically realistic and it tends to give poor results; 2) a pattern recognition approach based on a more realistic problem formulation: unfortunately, the computational problems associated with this formulation may be formoidable; and 3) a quadratic programming approack: this is based on minimization of a physically realistic objective function. A bmaosdification to eliminate discontinuities in the objetive function and a quasilinearization transform the original problem to an inequality-constrained quadratic minimization problem, which is readily solved by Lemke's complementary pivoting method. A sequence of successive quasilinearizations and estimations is defilned which is proved to converge to a minimum of the original objective function. In tests this convergence occurred very fast. Examples are given; very general classes of problems are discussed which can be handled in this way.

The theory of optimal control for time delay systems and the quadratic performance criterion is presented from two points of view: 1) the geometric approach, which yields a maximum principle, and 2) the dynamic programming-Caratheodory approach, which yields a feedback controller synthesis. The relationship of the two approaches is discussed, as well as extensions of the theory to non-linear problems and nonlinear performance criterion.

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?

The problem of obtaining maximum likelihood estimates for the parameters involved in a stationary single-channel, Markovian queuing process is considered. A method of taking observations is presented which simplifies this problem to that of determining a root of a certain quadratic equation. A useful and even simpler rational approximation is also studied.