Abstract: which is the precision to which F approximates K. An instance of classical interest is that in which L is C([O, 1]), K is AO-i.e. those functions bounded by one, and whose modulus of continuity is dominated by the modulus of continuity function w-and F is the family P,, of polynomials of degree n-1. For this case it is known that there are positive constants A and B, independent of w and n, such that

Abstract: This is a straightforward continuation of Hajek (1968). We provide a further extension of the Chernoff-Savage (1958) limit theorem. The requirements concerning the scores-generating function are relaxed to a minimum: we assume that this function is a difference of two non-decreasing and square integrable functions. Thus, in contradistinction to Hajek (1968), we dropped the assumption of absolute continuity. The main results are accumulated in Section 2 without proofs. The proofs are given in Sections 4 through 7. Section 3 contains auxiliary results.

Abstract: New techniques for the model simplification problem are presented. If the given system is expressed by a high-order transfer function, the technique is to expand the function into a continued fraction and then ignore some quotients. If the system is in the state equations form, the method is realized by partitioning the matrix and discarding some parts. The new approach not only offers a simple procedure and a good approximation but also gives a unified viewpoint of the general analysis of linear systems.

Abstract: It has been suggested that the power law J = an, describing the relationship between numerical magnitude judgments and physical magnitudes, confounds a sensory or input function with an output function flawing to do with O’s use of numbers. Judged magnitudes of differences between stimuli offer some opportunity for separating these functions. We obtained magnitude judgments of differences between paired weights, as well as magnitude judgments of the weights making up the pairs. From the former we calculated simultaneously an input exponent and an output exponent, working upon Attneave’s assumption that both transformations are describable as power functions. The inferred input and output functions, in combination, closely predict the judgments of individual weights by the same Os. Although pooled data (geometric means of judgments) conform fairly well to a linear output function, individual data do not; i.e., individual Os deviate quite significantly fromlinearity and from one another in their use of numbers. Individual values of the inferred sensory exponent, k, show significantly better uniformity over Os than do values of the phenotypica! magnitude exponent previously found to describe interval judgments of weight.

Abstract: The Brueckner-Goldstone many-body perturbation method, previously utilized for calculations of atomic correlation energies and polarizabilities, has been extended to the study of the hyperfine structure. The correlation energy as well as the hyperfine coupling constant of the lithium atom are calculated and compared with the results of some earlier methods. The present method makes use of Feynman-like diagrams which facilitate the evaluation of the importance of various physical effects. Analysis of the hyperfine diagrams shows that the difference between the experimental and the Hartree-Fock values is mainly accounted for by spin polarization, although correlation effects are by no means negligible. Our result of 2.887 a. u. agrees very well with the experimental value of 2.9096 a. u. The excellent result for the total energy of -7.478 a. u., comparing with the corresponding experimental value of -7.47807 a. u., shows that the wave function is good over-all, as well as in the region near the nucleus.

Abstract: A formula is given for the coordinates of the point that maximizes a given function F(x1, …, xn) over the closure of a bounded domain S in n-dimensional Euclidean space. The principal assumption made in deriving the formula is that F attains a global maximum at exactly one point of S. In certain cases the formula may be used to discuss the maximization problem as a function of the parameters involved. Some simple examples are given.

Abstract: Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function o:I1 → I2 such that f(α) ≤ g(o(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.

Abstract: wheref is a complex-valued function on a group G, for all x, y in G. On the line this functional equation is obviously satisfied by the cosine function and may be called the cosine equation. Of course this equation has a meaning on any group. One obvious way to solve the functional equation (A) on any group is by means of a homomorphism of G, say g, into the multiplicative group of nonzero complex numbers, K. If g is such a homomorphism, then the function defined by

Abstract: It is shown that the coherent-state representation of a many-boson wave function may be identified with the order-parameter function conventionally used to describe a superfluid. The statistical mechanics of the many-boson system is reformulated in terms of the coherent states, and a theory of the Ginzburg-Landau form is recovered in an obvious approximation. The formalism is particularly useful for describing metastable states of finite superflow and the fluctuations which may cause spontaneous decay of such states.

Abstract: In a previous paper (Lyness and Moler (11), several closely related formulas of use for obtaining a derivative of an analytic function numerically are derived. Each of these formulas consists of a convergent series, each term being a sum of function evaluations in the complex plane. In this paper we introduce a simple generalization of the previous methods; we investigate the "truncation error" associated with truncating the infinite series. Finally we recommend a particular differentiation rule, not given in the previous paper.

Abstract: A device is presented for scoring peripheral acuity Values are expressed in percent analagous to what the Snellen scale does for central acuity Like the recently published scale for the tangent-screen field, it is based on function The grid consists of 100 units whose unequal size and distribution reflect the unequal functional value of different parts of the field—in effect a weighted or relative-value scale Because each unit equals 1%, a simple count of units yields the functional score in percent The grid improves on the American Medical Association method of scoring, although it is based on the same AMA standard isopter for the normal (100%) peripheral field The device, tested on 1,000 fields by 20 experienced ophthalmologists, yielded a 959% correspondence between their estimates and the grid scores It is simple, quick, inexpensive, consistent, and can, after brief instruction, be delegated by the ophthalmologist to a nonprofessional aide

Abstract: This paper is concerned with a method for the assessment of utility functions of multi-numeraire consequences. It is proven that given von Neumann and Morgenstern's axioms of “rational behavior” and two additional assumptions, the utility function for (x, y) consequences can be written as U(x, y) = Ux(x) + Uy(y) + KUx(x) Uy(y). K is a constant that must be evaluated empirically. This form shall be designated as a quasi-separable utility function. It is more general than the separable utility function and is shown to be nearly as easy to use. Implications and ramifications of such a utility function and its requisite assumptions are discussed. A technique for practical application of this work is presented.

Abstract: It is shown that an extended Hartree‐Fock (EHF) function, obtained by applying a projection operator O to a Hartree product φ with subsequent optimization, has vanishing matrix elements of the operator H‐EI with all its singly excited states. A method for obtaining the EHF function using this property is described. Calculations for the Li atom are reported.

Abstract: Sufficient conditions on the modulus of continuity ω(δ) are found for an imbedding of function classes Hpω(δ) ⊂ (L). It is shown that in a number of cases these conditions are also necessary.

Abstract: A closed form has been derived for the function αl(NML | a,r) introduced by Lowdin for two‐center integrals in molecules and solid states. This expression is general and applied to all values of l, L, and M.

Abstract: THE lactation curve in cattle has been represented by the function where yn is the average daily yield in the nth week and A, b and c are constants. I have previously1 suggested a measure of persistency (the extent to which peak yield is maintained) given by which was related to expected total yield (Y) by Y = AeS.

Abstract: Given a nonlinear device described by a power series and having an input consisting of a sum of any number of cosine waves, a formula is derived which enables one to express the amplitude of any chosen intermodulation product in the output as a function of the power series coefficients and the amplitudes of the input frequency components.

Abstract: This paper contains an original investigation of the conditions which are necessary and sufficient for a function of many variables to have a stationary local minimum when its second variation is only semi-definite. The approach is to determine the order of contact which the function makes with the space of the variables upon which it depends. The analysis is slanted towards potential energy functions in order to get conditions for the colloquial stability of a conservative system. The sequence of governing equations are shown to be specializations of equations defining equilibrium paths which are explored in part I. They are characterized by variational principles. The classical notions of neutral equilibrium and buckling modes are re-examined in the light of this nonlinear analysis.

Abstract: A previous paper [2] was concerned with the determination of optimal policies for restocking an inventory which is continuous!y depleted by a random process of demands. The purpose of the present paper is to develop a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process. This reversal of the roles of input and output does not, by itself, change the character of the problem. But the consideration of set-up costs for ordering replacements, which leads to inventory policies of the (s,S) type, has no counterpart here. It is natural to regard the dam as a device for smoothing out random fluctuations in a flow of water and, under utility assumptions which reflect this attitude, it follows that the optimal output rate is a continuous function of the level of water in the reservoir. Our main object is to determine this function.

Abstract: A simple model has been assumed for the anharmonic intramolecular potential function of the methyl halides; a quadratic potential in instantaneous curvilinear coordinates has been supplemented with Morse functions for the bond‐stretching coordinates. All parameters in this potential function are known from other work, and the model could therefore be used to estimate the anharmonic spectroscopic parameters of the methyl halides. These are discussed and compared with the experimental values where the latter are known. Systematic differences in the calculated anharmonic constants Xss′ for the symmetric and asymmetric CH stretching vibrations are noted, and an explanation is offered. The concept of group anharmonicities has also been investigated.

Abstract: New techniques for the model simplification problem are presented. If the given system is expressed by a high-order transfer function, the technique is to expand the function into a continued fraction and then ignore some quotients. If the system is in the state equations form, the method is realized by partitioning the matrix and discarding some parts. The new approach not only offers a simple procedure and a good approximation but also gives a unified viewpoint of the general analysis of linear systems.

Abstract: A computationally convenient formulation of perturbation theory is developed. In this formulation either the SCF wavefunction or the SCF function plus important corrections may be used as a trial function. Corrections to the trial function are expanded in powers of the residual error. The reduced density matrices may be expanded in a series in this error. If the SCF function is used as the trial function, parts of the density matrix are needed to second order in the error to determine the natural orbitals and geminals to zeroth order.

Abstract: The variation of frequency F as a function of wave number K and the associated spectral transfer function are computed for different modes in a complex oceanic wave-guide. The model consists of a fluid layer resting upon a three-layer elastic half-space. The layers and the half-space are homogeneous. The comparison of theoretical results with measured power spectra for two records taken in the Pacific Ocean shows qualitative agreement stressing strongly the role of the leaking compressional organ-pipe modes which are not continuations of normal modes beyond cutoff frequency. The mathematical procedure consists in the integration of the second-minor propagator equation of Gilbert and Backus (1966). The determinant representing the secular function is computed directly rather than by summing the products of its elements. This improves both accuracy and computing time. The integration can be reduced to that of a third-order nonlinear differential system which, for K = 0, splits into two Riccati equations. The ( F, K )-diagram corresponding to every mode is obtained by a technique based on properties of similar diagrams for simple oceanic and continental structures.

Abstract: Several problems in one‐speed neutron transport theory for spherically symmetrical systems are discussed. The singular eigenfunction expansion technique is used to construct a solution for a specific finite‐slab Green's function. This slab solution is then used to construct the finite‐medium spherical Green's function by extending the point‐to‐plane transformation concept. For the general case, the expansion coefficients are shown to obey a Fredholm equation, and first‐order solutions are obtained; however, in the infinite‐medium limit the solution is represented in closed form. In addition, the solution for the angular density in an infinite‐medium due to an isotropic point source is developed directly from the set of normal modes of the transport equation. A proof that the result so obtained obeys the proper source condition at the origin is given.