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A change of source skin distance results in a significant change in percentage depth dose. The factors P(f 2 ) / P(f 1 ) by which percentage depth dose at SSD f 1 is to be multiplied to find new percentage depth dose at SSD f 2 are tabulated and presented in a graphical form, for f 1 = 100 cm and f 2 = 110–140 cm for 8 MV x rays and for f 1 = 80 cm and f 2 = 90–140 cm for 60Co gamma rays. The use of quadratic and linear equations to calculate this factor have been suggested. The ratios calculated using a quadratic equation agree very well with values from experimental data.

THE object of this paper is to propose for discussion the following topic: b1, b2, ... are unbiased estimates of P1, ,2, ... , distributed normally with variances and covariances jointly estimated, with f degrees of freedom and independently of bl, b2, . .. , as vll, v12, V22, . .. , and the functions F,(cc) do not involve the parameters P3, What can we say about the roots of the equation in F(r, M) = rlFl (a) + P2 F2 (a) + . . . = 0? Numerical examples are discussed in detail to illustrate the problems of determining the fiducial distributions of (i) the root of a simple equation (i.e., a ratio), (ii) the roots of a quadratic equation with variable coefficients. The solutions proposed are based on a consideration of the region of the (aC, t2) plane lying above the curve

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.

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.

— Lower bound reliabilities for the various parameters of two previously suggested models for the serial colour word test—one linear and one quadratic —are derived and estimated. The quadratic model gave mostly very low reliabilities, and the stronger, linear model also gave only a few high reliabilities. Validity was studied by means of discriminant functions (4 groups being used; 3 clinical and 1 normal). The validity of the test was low. The linear model gave the best results under cross validation.

Edge displacement of assumed-stress hybrid plate elements are commonly chosen as either linear, quadratic or cubic functions of the edge co-ordinate. The present paper examines the effect of this choice on the acceptability, accuracy, and possible zero energy modes of elements based on a linear stress field. It is concluded that triangular elements may be preferable to quadrilaterals, and that linear edge displacements are not desirable.

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 provides a method for solving mixed integer quadratic programs with the help of cutting-plane technique.

The development of a general quadratic multilayer plate element is presented for the analysis of arbitrarily layered curved plates. In the formulation, each layer of the multilayer plate can have different orthotropic properties and can deform locally. Examples of bending problems are presented which demonstrate the applicability of the formulation.

A new class of algorithms for the suboptimal feedback control of parabolic systems is presented. Three specific techniques within this class are developed and tested numerically using a linear example with three different types of boundary conditions. Performance of two of these techniques is found to compare favorably with that obtained by use of a rigorous open-loop optimal control computed by the gradient method. It is also shown for linear systems having quadratic performance criteria that the new algorithms are superior to the technique presented by Vermeychuk and Lapidus (1973) in cases where state deviation has a weighting coefficient much greater than that for control energy.

Some elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n-dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three-dimensional spaces (e.g. space–time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third-degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher-order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.

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.

Budgets are established and decisions in a firm are made on the basis of the sales forecast. In the majority of firms this forecast is a single value estimate of sales. The decisions made on the basis of the single value estimate frequently fall short of optimal strategies because of costs that occur when actual sales differ from forecast sales. These costs, measured in terms of opportunity losses, can be minimized when considered in conjunction with a probability distribution of sales. Three methods for determining the probability distribution of sales are considered in this paper. These are regression analysis Schlaifer's approach for establishing a subjective distribution, and Bayesian regression. The use of the distribution in conjunction with linear and quadratic loss functions is illustrated.

A technique which calculates approximate solutions to optimal control problems is developed in this work. The technique, which uses an empirically found weighted quadratic Liapunov function to transform the original n-dimensional optimization problem into a scalar optimization problem, is applied to a number of optimal control problems typical of those encountered in chemical engineering practice. The problems considered include both linear and nonlinear, lumped and distributed systems with minimum time, and quadratic and final value type performance indices. The solutions of these control problems show that the Liapunov-like suboptimal method, in general, requires less computer time and storage than that required by iterative and noniterative optimal methods currently used to solve optimal control problems. The reduction in computer storage and time enables the Liapunov-like technique to handle large dimensional control problems with relatively little effort.

A particular linear estimate with three terms approximation using ordered observations is discussed which seems to be simple to calculate. This estimate is applicable for any sample size for some special distributions without considerable loss in efficiency and is, therefore, of practical value.

This paper describes an approximate solution procedure for quadratic programming problems using parametric linear programming. Limited computational experience suggests that the approximation can be expected to be “good”.

This paper deals with the various ways in which the requirements of a certain strategic material could be met in a period of limited war. Besides a small stock of ores to draw on at the beginning of the period, high-quality foreign ores have to be imported or low-quality domestic ores made suitable for the production of alloys. In the latter case, however, the plants for upgrading and processing the ores first have to be constructed. The objective is to meet the requirements, assumed here to be given for the years of limited war, at minimum cost. The cost function contains, besides linear, also quadratic terms, positive as well as negative ones. Two recently developed computational methods have been used to solve the problem.