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Using Newton-type corrections, a simultaneous improvement of a complete set of approximate factors of arbitrary degree of a polynomial is developed, which can form the basis for an iterative method of factoring the polynomial. The simultaneous corrections for a given step are a generalization of a method introduced by Weierstrass [5] in the case of all linear factors. The present paper includes, therefore, an independent derivation of the Weierstrass formulas. From a different point of view the resulting iterative method is a generalization of a method of Samelson [3], recently analyzed by Stewart [4]. Basic properties of the iteration include the fact that the correction terms are related to a certain partial fraction decomposition. Emphasis is placed on the case of approximate quadratic factors, where formulas for the corrections are derived suitable for use in computation with the practically important case of a polynomial with real coefficients whose zeros are not necessarily all real. The formulas so...

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.

This paper investigates the proper selection of a performance index to satisfy a word statement of desired effect. A performance index formed as the time integral of Un is solved for various values of n. When viewed in this light, the classical quadratic index (n = 2) gives a seemingly arbitrary solution. From this investigation the true optimal control solution can be ascertained. For the different values of n, solutions for totally different performance can be obtained. Performance indices combining two effects of minimum control effort and minimum response error are also investigated. The differences of a summed combination and a multiplicative combination are studied.

A quadratic performance index is considered as a generalized criterion for designing linear multivariable control systems. A method is developed for expressing the characteristic polynomial of the optimal system as an explicit function of the weighting matrix elements of the performance index. Since these elements are actually the design parameters it then becomes possible to obtain some general properties of the optimal systems as well as to select the weighting matrix elements, such that more conventional criteria on the system transient response are satisfied. The results are applied to the design of two multivariable systems; in one case, the plant is initially unstable and, in the second case, it is desired to have the system response follow that of a linear model.

Properties of the algebraic equation A^TX+XA-XBQ"2^-^1B^TX+Q"1=0 are studied for arbitrary nonnegative definite and positive definite matrices Q"1 and Q"2. The results are used to study the possible number of stationary solutions of the Riccati equation. The theory for linear systems with quadratic loss is then generalized, and numerical consequences are studied.

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.

This bibliography contains over 900 references arranged in four sections: books and chapters in books, journal papers, proceedings papers, and contractor reports and Ph.D. dissertations. All references are indexed into categories and subcategories. The main categories are: Deterministic Optimal Control of Linear Systems with Respect to Quadratic Criteria, State Estimation, Stochastic Control, Computational Aspects, and Applications.

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.

The equilibrium of a dislocation distribution is considered for the case where the applied stress field is a rational function. A function of a complex variable is introduced and it is shown that for equilibrium this function satisfies a quadratic equation. The stress field of the distribution and the distribution itself are different aspects of this complex function and are determined together. The known prevalence of square roots in distributions is a consequence of finding the roots of the quadratic equation. The connection between this complex representation of equilibrium and that used in Part IV for dislocation dynamics is discussed.

I propose to prove the following theorem.With n > 2, the (n + 2) equations derived from the matrixwhereby equating to zero all (n + 1)-rowed determinants from the matrix ‖Δ‖ are equivalent to only two, one of which is linear in li (i = 1, 2, …, n) and the other is homogeneous and quadratic in a certain n – 1 of li (i = 1, 2, …, n); the elements of the matrix are real; (r, s = 1, 2, …, n) and d is arbitrary.

A general method of solving the inverse secular problem is described which makes it possible to arrive conveniently at the various mathematically permissible solutions. The calculation of multiple solutions for the quadratic force constants from vibrational spectra and for the chemical shifts and spin-spin coupling constants from NMR data is considered in detail.

The thermal conductivity of liquid and gaseous ammonia has been determined between 20° and 177°C and at pressures from 1 to 500 atm using a vertical, co-axial cylinder apparatus. Measurements in the vicinity of the critical point ( pc c = 111.5 atm , t c = 132.4° C ) have been carried out with special care along the reduced isotherms, 1.016 and 1.061. An anomalous increase of thermal conductivity in this regime has been found, similar to that for carbon dioxide previously reported by two other observers. A tentative explanation of this phenomenon, based on transport of energy by clusters of molecules, is given. Experimental results for the liquid-like fluid state, both below and above the critical temperature, have been correlated by a quadratic equation between density and thermal conductivity. With the aid of this equation, values have been computed for conditions outside those of this study, and the table containing the smoothed, experimentally verified figures at even increments of pressure and temperature has been complemented by computed values ranging up to 1000 atm and 480°K. Computed values are also presented in tabular form for the technically important temperature range from room temperature to the normal fusion point (195.5°K). To provide a general survey, and for rapid interpolation, experimental results are presented also in the form of two diagrams, in which isobars and isotherms of the thermal conductivity are shown, using temperature and pressure, respectively, as the independent variables. Low pressure, gas phase data of previous observers and those of this research have been reduced to 1 atm with the aid of pressure coefficients derived from this study. Using the relation for the thermal conductivity of polar gases, proposed by Mason and Monchick, the results of all sources have been satisfactorily correlated between 300° and 500°K. The results of this work agree well with the few dense gas phase data reported by Keyes, and are in perfect agreement with tentative data along a short section of the saturation line given by Sellschopp in the only reference found on the thermal conductivity of liquid ammonia. Considering all known causes of error, the accuracy of this study is estimated to be within ±1.5 per cent for the liquid and dense gas phases, and within ±2 per cent for the low-pressure gas phase and the near-critical region.

An important parameter in the design of countercurrent dialyzers is the permeability of the membrane which separates the blood and dialyzate. In this paper, methods of optimization theory are applied to specify membrane permeability along the length of a countercurrent dialyzer operating under quasi-static conditions. Two optimization problems are considered. In the first the performance functional is a quadratic measure of membrane complexity; in the second it includes also a quadratic measure of the deviation of the blood output concentration from a desired (low) level. For both situations it is shown that the optimal membrane is one of constant permeability along its length, a result which is intuitively appealing and has an obvious economic consequence. Formulas for computing the optimal permeabilities are given, and some problems for future research in this area are outlined. A generalization of the principal result is also included, and some deficiencies in the present analysis are discussed.

This paper is a brief description of the development of simple hybrid sampled-data controllers. The control function generated is amplitude-constrained and is approximately optimum with respect to a given quadratic performance index. The plant controlled is linear with accessible state variables and is sampled with constant period. An optimum controller for a free-terminal-state plant is designed. Quasi-optimum controllers for a fixed-terminal-state plant are designed and tested. The third-order plant 1/[s(s+0.2)(s+1)] is simulated on an analog computer and controlled by each of the quasi-optimum controllers.

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.

In a recent paper on a divisor problem the author showed incidentally that there is a certain regularity in the distribution of the roots of the congruencefor variable k, where D is a fixed integer that is not a perfect square. In fact, to be more precise, it was shown that the ratios v/k, when arranged in the obvious way, are uniformly distributed in the sense of Weyl. In this paper we shall prove that a similar result is true when the special quadratic congruence above is replaced by the general polynomial congruencewhere f(u) is any irreducible primitive polynomial of degree greater than one. An entirely different procedure is adopted, since the method used in the former paper is only applicable to quadratic congruences.

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.

The following method is suggested for the treatment of the quadratic equation in the teaching of algebra. The method makes use of factoring and substitution in their simplest form. If the quadratic is incomplete it can be factored as the difference of two squares, hence given the equation.

It is shown that the occurrence of two satellite components in the quadratic Zeeman effect of the absorption spectrum of caesium vapour as has been obtained by Harting and Klinkenberg 1 ) can be understood as an effect due to spin-orbit coupling.

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.

This note considers the question of why some convex quadratic programming algorithms fail and others succeed when applied to nonconvex quasiconvex quadratic programs. Several algorithms are identified as being capable of solving quasiconvex quadratic programs using only a finite number of arithmetic and logical operations. These algorithms are all primal feasible, pivot algorithms.

This paper deals with the attachment of the Frequency Response Slide Rule. The slide rule which has been published in "Automatic Control" Vol.1 (1953) No.3 by K.IZAWA, is used for frequency response computation of real linear factors : (1+Ts), Ks and K/s and exponential factor : e-Ls of such a transfer function as [numerical formula] The attachment is a transparent sheet upon which a logarithmic scale of hyperbolic sine function is graduated with two reference lines. The frequency response of the quadratic factor (1+2ζTs+T2s2) of the above transfer function is easily computed by the slide rule and the attachment, when the input frequency ω, 0.001aωa1000, the time constant T, 0.01aTa100, and the relative damping ζ, 0.005aζa2, are given. Three steps are necessary for the computation ; the first step is the setting of the cursor and the slide, the second and third steps are the settings of the attachment.

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.

A wide class of linear one dimensional propagation phenomena in a homogeneous medium can be described by a vector equation containing a time derivative and a spatial operator, which after a Fourier transformation amounts into a multiplication by a holomorphic matrix. This class of wave equations admits a general treatment concerning a representation of the solution of the initial value problem, mode-decomposition, quadratic conservation laws, group velocity and stability. This first paper provides the mathematical preliminaries for such a treatment. A theory of holomorphic matrices is developed which includes a.o. a discussion of the analytic behaviour of eigenvalues and eigenvectors and a commutator theory. The "differentiation-like operator" concept is introduced.

The calculus of variations is applied to the problem of finding the optimal control for linear systems with quadratic costs which are not necessarily positive definite. The conjugate point condition is transformed to an explicit condition upon the parameters of the problem, in the one-dimensional case for arbitrary costs and in the n -dimensional case for final costs only. The expressions for the optimal controls are given. A numerical example of a second-order system is solved.

A rapid method to evaluate the classical turning points of vibrational levels for the electronic states of diatomic molecules whose energy terms can be represented adequately by a quadratic in (v+1/2) is discussed. The method is based on the fact that the internuclear distance "r" corresponding to energy "U" can be represented by an equation of the form y(r) =mx(U)+c, the equation of a straight line. The method is also extended to electronic states whose we,ye is not very large. To check the validity of these procedures potential energy curves are constructed for X2-, A 2II3/2, and A 2II3/2 states of CN; X 1 g+ and A 3u+ states of N2; and 3II0u+ state of 79Br81Br. All these values are compared with those obtained by RKRV method. In all cases the agreement is found to be satisfactory, with a deviation of less than 0.1% to RKRV values. Further, a method to check the validity of a Morse function for a particular electronic state is discussed, and a method to adjust the Morse function to represent the true curve is illustrated by considering the A 2II3/2 state of CN.

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 Stark effect, arising from the interaction of a uniform electric field with a permanent electric dipole that is arbitrarily oriented within a rigid asymmetric rotor, and with a dipole induced in the rotor by the field, has been evaluated by perturbation methods. Tables are given for the perturbation of the energy levels so that, for J≤2, the rotational energies of an asymmetric molecule in an electric field may be readily approximated to terms quadratic in the electric field.The effect of accidental degeneracy upon the Stark effect and line intensities has been considered. A qualitative discussion of certain features of Stark patterns is given that may be useful in the identification of rotational spectral lines.

Relations between parameters of the nucleon-nucleon triplet-even potentials consistent with the n-p data are investigated. Strength of a triplet-even spin-orbit potential cannot be determined without close reference to the strength of a triplet-even quadratic spin-orbit potential. For a strong quadratic spin-orbit potential, strong positive spin-orbit potentials are allowable. For a weak quadratic spin-orbit potential, tail-part of spin-orbit potentials should be negative and positive inner part is favoured by the deuteron magnetic moment. Central potentials are strongly attractive inside and tensor potentials are weaker than the OPEP. Reliable estimation of quadratic spin-orbit potentials is important for the determi­ nation of spin-orbit potentials in the triplet-even state. § I. Introduction In order to understand the origin of the spin-orbit potential indispensable for the explanation of the high energy p-p scattering, it is important to know the isotopic spin-dependence of the spin-orbit potential. In this paper, by analy­ sing the scattering up to 310 MeV, we attempt to determine allowable region of parameters of triplet-even spin-orbit potentials, 3 VL"s.**) Several authors have discussed the possibility that bosons such as p and w are responsible for the

The quadratic-error regulator problem with a control-energy constraint has been investigated from the viewpoint of having limited control energy available or as an indirect method of limiting the maximum control amplitude. The solution of this problem requires iteration on a parameter @l"n" "+" "1 with the required value of @l"n" "+" "1 being a function of the initial state. Thus the result is not as straightforward to apply as the well known Kalman result where no constraints are imposed; however, it is much simpler than the problem where control amplitude constraints are applied directly for which case one must iterate on an n dimensional vector. Also, for the systems studied it was found that @l"n" "+" "1 had a regular behavior as a function of initial state and that this behavior could be approximated by empirical equations. Therefore, an approximate control law could be computed for each initial state without iterating yielding real-time solutions. Three examples were considered in the numerical evaluation portion of the study. In addition to revealing the orderly behavior of the required value of @l"n" "+" "1 as a function of initial state, the numerical results also indicated that constraining control energy was quite effective in constraining maximum control amplitude.

A criterion of stability given by [1] for the linear systems of ordinary differential equations with variable coefficients, is extended to systems with lag. The explicit form of the quadratic functionals developed in [2] for the systems of differential equations with lag is used. These functionals play a role analogous to that of the Liapunov quadratic forms for the systems of ordinary differential equations. A criterion of stability of a nonlinear system of differential equations with lag is obtained in the manner analogous to that of [1].

The atomic distances and the mean-square amplitudes of silicon tetrachloride have been determined by the sector-microphotometer method. The possible sources of experimental errors have been thoroughly examined. The quadratic force constants of the potential function have been determined by the use of the mean-square amplitudes thus obtained, by combining them with the vibrational frequencies obtained from the Raman spectrum in the gaseous state. It has been shown that the errors of the force constants originate from those of the mean-square amplitudes by the same order of magnitude as from those of the frequencies.

The paper gives a comparison of Beale's method for quadratic programming and the Simplex Method for quadratic programming as developed by Dantzig and Van de Panne and Whinston. After summary expositions of both methods and a reformulation of Beale's method in the framework of quadratic Simplex tableaux, the relation between the two methods is analyzed. A numerical example is used as illustration.

The search for a linear Λ-minimax estimate (see, e.g., [19]) of a multivariate location parameter (with quadratic loss) is reduced to a problem of maximizing a continuous function on a compact, convex set—a problem which can be solved numerically with little difficulty. This reduction is accomplished by showing that a minimax theorem applies, thus changing a difficult minimax calculation to a simpler maximin problem. The technique is observed to be applicable to other, related problems. Also considered is the choice of experiment under the assumption that the decision maker can, at some cost, improve the specification of his prior distribution.

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.

The role of linear control theory as an aid to the integral control of hydrologic systems is investigated for the case of a combined lake and aquifer storage system that supplies either a deterministic or stochastic water demand. Only lumped time-invariant systems are considered but both deterministic and stochastic inflows to storage are allowed. The computational example allows for recharge of lake water into the aquifer as well as for the subsequent diversion of pumped groundwater back to the lake. Stability criteria are presented for the closed-loop features of the overall control system. Under a quadratic loss criterion, a calculus of variations problem, subject to constraints imposed by the system equations can be solved for the optimal release policy from the lake and aquifer and optimal feedback policy from aquifer to lake.

This paper considers the design of linear time-varying networks and of nonlinear time-invariant networks, the latter being operated in the small-signal mode. In the first part, the design example considered is a network whose time delay is a prescribed function of time. A quadratic performance criterion 'is formulated and the design is obtained iteratively by steepest descent. The second part of the paper considers the design of a nonlinear timeinvariant network with variable bias sources whose small-signal equivalent network is identical with a given linear time-varying network. Explicit conditions are given under which this can be done.

When the substrate of a 2 n–q factorial or fractional factorial experiment may be expected to show trends representable by linear and quadratic terms in time, then certain orderings spaced at equal time-intervals permit better estimation of the effects than do others. Some of these ordered plans are given for p – q = 2, 3, 4, 5. Simple methods are given for computing effects, trends, and efficiencies.

Introduction. The form f=ax2+by2+cz2 +2ryz+2sxz+2txy is properly primitive when (a, b, c, r, s, t) = 1 and (a, b, c) is odd. With f there is associated a determinant d. The greatest common divisor of the cofactors of the elements of d is designated by Q. Then A is defined by d=-2A. For an indefinite form dA must be positive, where Q is chosen so that QA is negative. Also Q' and A' are defined by Q-= 'Q" and A = A'A" where Q" and A" are the greatest powers of 2 dividing Q and A respectively, so that Q' and A' are odd. With f there is associated a reciprocal form F=AX2+B Y2+ CZ2+2R YZ +2SXZ+2TXY, where QA, QB, ***, QT are the cofactors of the elements of d. The purpose of this paper is to show genera containing more than onel class of properly primitive indefinite forms. The method is that which C. L. Siegel used in a specific example communicated to the first of the authors by letter and which is here generalized to include many forms. By employing this method in Lemma 3 of this paper, many genera are explicitly shown containing at least two classes.