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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.

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.

It is proposed to use a viscoelastic layer to protect equipment against vibration. The principal quadratic theory of hereditary viscoelasticity is used as the physical relation between the forces and displacements. The solutions obtained for the integrodifferential vibration equation make it possible to minimize the displacements and accelerations of the protected equipment.

A method of calculating a viscoelastic shock-absorbing device or structure by the method of averaging systems of integro-differential equations is considered. The principal quadratic theory of hereditary viscoelasticity is used as the physical relationship between forces and displacements of the shock-absorbing base or structure.

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.

The theory of gravitation developed in a previous paper rests upon the hypothesis that any permissible metric field whatsoever must satisfy some self-contained set of field equations which, it is proposed, are provided by a quadratic action principle. The set of exact, spherically symmetric, static solutions of these equations is now extended to include all those which belong to Takeno’s «class 2». The surface area of a sphere in such a space-time has a value which does not depend upon the radial co-ordinate.

The Euler-Lagrange equation derived from Schwinger's action principle (1951) has been shown by Kianget al. (1969) and Linet al. (1970) to lead to inconsistencies for quadratic lagrangians of the form $$\bar L(\dot q,q) = \tfrac{1}{2}\dot q^j g_{jk} (q)\dot q^k - V(q)$$ except in the Euclidean caseg jk =δ jk . This inadequacy is linked to Schwinger's specification that the variations of operators bec-numbers. We reformulate the action principle by introducing the concept of ‘proper’ Gauteaux variation of operators to find the most general class of admissible variation consistent with the postulated quantisation rules. This new action principle, applied to the LagrangianL, yields a quantum Euler equation consistent with the Hamilton-Heisenberg equations.

The influence of the physical properties of crystals on the noncollinear interaction of light waves in nonlinear media is examined in an example of LiIO3 and K2S2O6. Peculiarities in the vector interaction are noted in crystals with quadratic and cubic nonlinearities. Experimental results are presented.

A principal pivoting algorithm is given for finding local minimizing points for general quadratic minimization problems. The method is a generalization of algorithms of Dantzig, and Van de Panne and Whinston for convex quadratic minimization problems.

In this note we demonstrate the inadmissibility of an extensive class of polynomial estimates of the shift parameter in the case of a quadratic loss function.

In this note we prove the “quadratic property” of the boundary mapping in the exact sequence of cohomologies. We present calculations for the action of the Galois group on the Mumford group.

This paper provides a method for solving quadratic programs in integers with the help of a branch and bound technique.

The projectional properties and associated global and local error bounds for quadratic interpolatory splines are studied, along with applications to the numerical solution of two-point boundary value problems via collocation.

In the present paper, we continue the study of the so-called model systems with quasispin Hamiltonians of Thirring type [i], which allow an exact solution in the thermodynamic limit, (i.e. as V ~ ~o, N -~ ~, V/N = const). Such model problems with both positive and negative interaction introduced byterms quadratic and quartic in Pauli operators have been considered in a number of papers, in which asymptotically exact expressions were obtained for the free energies and the time correlation functions [2-8] on the basis of methods developed earlier for models with four-fermion binary interaction [9-12]. For models with positive interaction it was shown [4-7] that in the thermodynamic limit the free energies and the correlation functions for a system without interaction and the original model system coincide.

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.

Using the methods of group theory, we obtained parametric expressions for the laws of dispersion in a quadratic approximation of the resolution along components of the wave vector, the possible multiplicity of degeneration, and the zero slopes of the energy surfaces in the neighborhood of typical points of the Brillouin zone for crystals having a monoclinic lattice Γ m b . The spin—orbital interaction and the operation of time reversal were considered.

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

The linear complementarity problem is the problem of finding solutionsw, z tow = q + Mz, w≥0,z≥0, andw T z=0, whereq is ann-dimensional constant column, andM is a given square matrix of dimensionn. In this paper, the author introduces a class of matrices such that for anyM in this class a solution to the above problem exists for all feasibleq, and such that Lemke's algorithm will yield a solution or demonstrate infeasibility. This class is a refinement of that introduced and characterized by Eaves. It is also shown that for someM in this class, there is an even number of solutions for all nondegenerateq, and that matrices for general quadratic programs and matrices for polymatrix games nicely relate to these matrices.

In this paper, the problem of minimizing a functionf(x) subject to a constraint ϕ(x)=0 is considered. Here,f is a scalar,x ann-vector, and ϕ aq-vector, withq

The optimal control of time-invariant, uncontrollable linear systems with respect to an integral quadratic performance index over an infinite time interval is considered. It is proved that the asymptotic stability of the uncontrollable part of the system is a necessary and sufficient condition for the existence of the solution.

Explicit forms for the concomitants which are bilinear in two spinors of type [3/2, 1/2] and the concomitants which are quadratic in a single spinor of type [3/2, 1/2] are obtained. The dual-tensors, where they exist, are also given.

We consider the self-mass of a lepton resulting from gravitational interactions, electromagnetic interactions and weak interactions mediated by an intermediate vector boson. If the quadratic weak and gravitational self-mass divergences, and the logarithmic weak, electromagnetic and gravitational self-mass divergences are, respectively, assumed to cancel, the weak-coupling constantg and the mass of the intermediate vector bosonMB can be predicted.

This paper discusses the dynamics of rigid-viscoplastic bodies, bearing in mind large displacement effects. The extremal properties of the incremental solution of the dynamic problem are demonstrated. It is shown that the true instantaneous acceleration field minimizes some quadratic functionals in a class of functions constrained by linear equations and inequalities.

In this paper, the problem of minimizing a functionf(x) subject to a constraint ϕ(x)=0 is considered, wheref is a scalar,x ann-vector, and ϕ aq-vector, withq

An explicit formula is found for the number of solutions in a finite field of the system of polynomial equations(11), when the coefficients satisfy certain conditions.

Definitions of and relations among quadratic plasma response functions are established from Maxwell’s equations and nonlinear constitutive relations. Then from statistical mechanical perturbation theory, equilibrium triple correlations of the microscopic charge density are expressed in terms of these response functions.

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.

We have performed precise measurements of the low temperature resistivity of aluminium, indium, potassium, and sodium, and have obtained evidence of aT2 temperature dependence in the resistivity of aluminium and indium.

The numerical solution of nonlinear programming problems by first-order techniques, such as gradient projection techniques, is difficult when ridges are encountered. Ridges are described and analyzed. Second-order information about the augmented objective function, properly used, can yield a search direction that moves along a ridge, even in the constrained case. The analysis applies to situations in which the objective function (maximization) is not necessarily concave and the constraint functions (G j ⩽ 0) are not necessarily convex.

Altogether a total of almost 200,000 totally positive numbers from different fields were decomposed into squares; in no case were more than five squares required, although in many cases no number of squares sufficed. For some quadratic fields, from our evidence it would seem safe to completely characterize the couples for whichQ=5 or 0, particularly whenm?13; and furthermore, form=17 or 33 it seems possible to characterize all cases whereQ=4, 5, or 0. The whole calculation seems to be pointed toward the result that three squares are sufficient except for "special" cases. Incidentally, the analytic methods ofSiegel andMaass run parallel to the calculation in that these methods involve the third, fourth, and fifth power of a theta-function. The numberical evidence would therefore suggest that their methods point to an analytic (or even a purely algebraic) proof of the futility of using more than five squares in any case. The work was supported in part by the U. S. National Science Foundation Grant G-4222 and the computer services were contributed by the Argonne National Laboratory of the U. S. Atomic Energy Commission during the summer of 1958. The coding was performed by Mr.Alan V. Lemmon with remarkable economy of length of program and running time. The deepest debt of gratitude is owed to the lateDonald A. (Moll) Flanders whose contributions to the logical design ofGeorge had made the rapid execution of the program possible and whose personal interest made possible the availability of the computer for this work.

A matrix formalism is developed for the purpose of facilitating the Liapunov stability analysis of discrete, holonomic, mechanical systems with cyclic coordinates and with the Hamiltonian free of explicit time dependence. Matrix expressions are developmed for the kinetic energy, the Routhian, the Hamiltonian, and the quadratic approximation of the dynamic potential energy, with cyclic coordinates, cyclic-coordinate velocities, and cyclic-coordinate generalized momenta not explicitly involved in the last of these functions. The final result is an expression for the quadratic approximation of the dynamic potential energy that is calculated much more readily than by scalar analysis. From the condition for positive-definiteness of this function, Liapunov stability conditions are available. The method is applied to a dual-spin satellite to illustrate the procedure.

By rearranging the Margules equations and postulating that the coefficients of these are inversely proportional to absolute temperature, it is shown that these equations may be used to deduce the relation established by Turkdogan and Darken (T.M.S.-AIME, 1968, vol. 242, pp. 1997–2005) between the parameters of terminal solution behavior, according to their quadratic formalism, for about thirty binary metal systems.

The effective spin Hamiltonian is obtained in the Dirac-van Vleck form by the method of nonorthogonal atomic orbitals with an estimate of virtual interatomic transitions. The equivalence of the orthogonal- and nonorthogonal-orbital methods is demonstrated for a bounded configurational interaction, with an accuracy to quadratic terms in the overlap integral.

Hestenes' method of multipliers is used to approximate a quadratic optimal control problem. The global existence of a family of unconstrained problems is established. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the original differentially constrained problem.

The nucleon-nucleon potential for the isotopic triplet state is computed as a superposition of Yukawa-type potentials. It contains the central, spin, spin-orbit, quadratic spin-orbit and tensor term and is dependent on the total angular momentumJ. The parameters of the individual terms are fitted to the experimental data.

Cubic-spline and discrete-quadratic polynomial techniques are presented for reliably computing up to third-order derivatives of experimental information. The concept is demonstrated by stress analyzing from measured displacements a transversely loaded plate and a beam under four-point bending. The respective displacement fields were recorded using holography and moire. The accuracy of the employed numerical-differentiation techniques is indicated.

A method is introduced to determine the HamiltoniansH of the quantum-mechanical systems having a given Lie algebra as their spectrum-generating algebra (SGA). Application of this method to the particular case in which the SGA is assumed to be the Lie algebraSO2,1 allows under several assumptions the determination of a general solutionH, for a system of two spinless particles, having this algebra as its SGA, and possessing both a discrete and a continuous spectrum. Velocity-dependent potentials have been included, the only restrictions imposed onH being that it be at most quadratic in the momentum, and rotationally and time-reversal invariant. The potential obtained as the general solution contains as an essential part a term which asymptotically is Coulomb-like and is more general than those which have been obtained previously in the same context. The requirement that the potential be angular-momentum independent reduces the possible Hamiltonian to that of the hydrogen atom, with an extra cubic force, which has already been solved. In the case in which a purely discrete spectrum is allowed, the general answer is not given, but still a sufficiently general Hamiltonian is obtained withSO2,1 as its SGA, that includes as a particular case the known harmonic oscillator with an extra cubic force.

A gas of two Boson systems coexisting inR3, and interacting only mutually, is analyzed. The interaction is quadratic, so that the dynamical problem may be solved completely and exactly.

A new quasilinearization algorithm is presented which essentially eliminates the necessity for computer storage. The representation theorem for the standard quasilinearization procedure is reformulated in terms of the initial value of the solution to a final-value problem, leading to a modification of the successive approximations. Several theorems establishing the convergence properties are proved; as in the original procedure, these convergence properties are both quadratic and monotonic. Finally, the modified approximation scheme is illustrated through several examples.

We analyze the errors in determining the rate constants of elementary reactions that arise when quadratic rather than compound (simultaneous bi- and monomolecular) chain termination is assumed. We show that the value of the chain propagation constant calculated from nonsteady-state kinetic data is independent of the method of calculation and is determined completely by the instrumental experimental error and the accuracy in determining the initiation rate.