Abstract: We Consider the boundary value problem [d] Here λ is a non negative parameter; f is a given real valuede function defined and a class C2 [d] is an arbitrarily specified function of class C1 on [0, n] satisfying [d] = 0. Under suitable hypotheses concerning f, we investigate the existence and stability properties of stationary solutions for (*). Our approach is to interpret (*) as a dynamical system in an appropriately chosen Banach space, and then to apply to (*) certain known results in the theory of Liapunov stability for general dynamical systems

Abstract: Suppose that 1.1 is a transcendental integral function. In this article we develop the theory initiated by Wiman [22, 23] and deepened by other writers including Valiron [18, 19, 20], Saxer [15], Clunie [4, 5] and Kovari [10, 11], which describes the local behaviour of f(z), near a point where | f(z) | is large, in terms of the power seriesf of f(z).

Abstract: This paper develops the properties of the Erlang loss function, B (N, a), used in telephone traffic engineering. The extension to a transcendental function of two complex variables is constructed, thus permitting the methods of complex analysis to be employed for the further study of its properties. Exact representations, Rodrigues formulas, and addition theorems are given both for the loss function and for the related Poisson-Charlier polynomials. Asymptotic formulas and approximations are developed for the loss function and also for its derivatives. A table of coefficients is included which, together with one of the asymptotic formulas, permits computation of B (N, a) by simple means even when the number of trunks, N, is very large. This same table is used to obtain ∂B(x, a)/∂x.

Abstract: A procedure for the synthesis of processes exhibiting temporal and spatial variability is presented The method involves the addition of harmonics of random frequencies that are sampled from the spectral density function or the radial spectral density function The process obtained is asymptotically Gaussian and ergodic An estimate of the error in time or space averages due to the nonergodicity of the process as a function of the number of harmonics is also included

Abstract: Nelder and Mead [2] have developed a simple, robust direct-search procedure for finding the minimum of a function. It consists of evaluating a function of n variables at the (n + 1) vertices of a g...

Abstract: Five special forms for a von Neumann-Morgenstern utility function u on a product set A×X are examined. Each expresses u as a combination of functions defined on the separate factors A and X. The five forms include the additive and multiplicative forms, u=u1+u2 and u=f1f2, respectively, and the additive form with independent product expressed by u=u1+u2+f1f2. Necessary and sufficient preference conditions are identified for each form, along with admissible transformations on the single-argument functions.

Abstract: Let K be a compact set, XT a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and f a continuous real-valued function on K x K. We study the problem of determining for which , e Y1 (if any) the energy integral I(K, A) = IKIKf(X, y)dgXx)dgXy) is maximal, and what this maximum is. The more symmetry K has, the more we can say; our results are best when K is a sphere. In particular, when X is atomic we obtain good upper bounds for the sums of powers of all (2) distances determined by n points on the surface of a sphere. We make use of results from Schoenberg's theory of metric embedding, and of techniques devised by Polya and Szego for the calculation of transfinite diameters. 1. Background and summary of results. In this paper we will investigate a number of extremal problems in distance geometry. Our work is in many ways analogous to the study of energy integrals in classical potential theory. Let K be a compact set in a Euclidean space and 51 be a prescribed family of Borel measures (possibly signed) of total mass one supported by K. Suppose f is a continuous real-valued function on K x K. We consider the family of integrals having the form (1.1) I(K, = | f (x, y) d1(x)d1d(y), it E 1. A number of interesting questions naturally arise concerning 1(K), the supremum of the numbers I(K, pi) with it in Vl: (i) What is the numerical value of I(K)? (ii) Does there exist a y0 in V1I such that I(K, Io) = 1(K)? (iii) If y0 exists, is this measure unique? (iv) Can an extremal measure [t be explicitly produced? Received by the editors March 20, 1973 and, in revised form, April 20, 1973. AMS (MOS) subject classifications (1970). Primary 52A25, 52A40.

Abstract: A predesign cost-estimation function for quite arbitrary buildings is derived utilizing historical cost data. The assumed function is multilinear and its coefficients are determined by means of the least-squares estimation technique. The selection and measurement of the six independent variables of the cost function, the collection of limited but proper historical cost data, and the use of mathematical tools both as a means in arriving at the desired function and as an end in explaining the value and impact of estimation hypotheses are the real contribution of this work. The derived predesign cost-estimation function is subjected to an error analysis and tested with two actual buildings not belonging in the sample data.

Abstract: A simple model of the storage hierarchies is formulated with the assumptions that the effect of the storage management strategy is characterized by the hit ratio fqnction. The hit ratio function and the device technology-cost function are assumed to be representable by power functions (or piece-wise power functions). The optimization of this model is a geometric programming problem. An explicit formula for the minimum hierarchy access time is derived; the capacity and technology of each storage level are determined. The opfimal number of storage levels in a hierarchy is shown to be directly proportional to the logarithm of the systems capacity with the constant of proportionality dependent upon the technolagy and hit ratio characteristics. The optimal cost ratio of adjacent storage levels is constant, as are the ratios of the device access times and storage capacities of the adjacent levels. An illustration of the effect of overhead cost and level-dependent cost, such as the cost per "box" and cos for managing memory faults is given and several generalizations are presented.

Abstract: This paper is concerned with the relationships between Lp differentiability and Sobolev functions. It is shown that if f is a Sobolev function with weak derivatives up to order k in Lp, and 0 s I < k, then f has an Lp derivative of order 1 everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an Lp derivative everywhere except for a set small in capacity and if these derivatives are in Lp, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.

Abstract: Two person zero sum differential games of survival are considered; these terminate as soon as the trajectory enters a given closed set F, at which time a cost or payoff is computed. One controller, or player, chooses his control values to make the payoff as large as possible, the other player chooses his controls to make the payoff as small as possible. A strategy is a function telling a player how to choose his control variable and values of the game are introduced in connection with there being a delay before a player adopts a strategy. It is shown that various values of the differential game satisfy dynamic programming identities or inequalities and these results enable one to show that if the value functions are continuous on the boundary of F then they are continuous everywhere. To discuss continuity of the values on the boundary of F certain comparison theorems for the values of the game are established. In particular if there are suband super-solutions of a related Isaacs-Bellman equation then these provide upper and lower bounds for the appropriate value function. Thus in discussingyalue functions of a game of survival one is studying solutions of a Cauchy problem for the Isaacs-Bellman equation and there are interesting analogies with certain techniques of classical potential theory.

Abstract: The basic setting of nonstandard analysis consists of a set-theoretical structure 4' together with a map * from 4' into another structure *X' of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make *X' into an enlargement of 4' [13]. The structures 4' and *X' may be type-hierarchies as in [11] and [13] or they may be cumulative structures with co levels as in [14]. The assumption that *4' is an enlargement of 4' has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that *4' has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11]. This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number K, *4' satisfies the K-isomorphism property (as an enlargement of 4) if the following condition holds: For each first order language, L with fewer than K nonlogical symbols, if 21 and Z are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to *4' and .X), then 21 and 23 are isomorphic. These properties bring the language and tools of model theory even more firmly into nonstandard analysis than before. In particular, they provide a vehicle for applying such results as the downward Lowenheim-Skolem theorem [16] in topology and analysis. In ?1 it is shown that for each 4' and K there exists an enlargement *4' of 4' which has the K-isomorphism property and that any such enlargement of 4' is necessarily K-saturated. The rest of ?1 is devoted to exploring the effects of the various isomorphism properties on the structure of internal sets and functions in *X.' For example, it is shown that if *4' has the g0-isomorphism property and if A and B are infinite, internal sets in *4', then there are bijectionsf and g of A onto B with the following properties: (1) Cis an internal subset of A if and only if {fx: x E C} is an internal subset of B; (2) C is an internal subset of A if and only if {gx: x E C} or its complement is a *-finite subset of B. In particular, such bijections exist when A is the *-finite set {1, 2,** , co} for some c E *N N and B is *N. (Note that a *-finite set can be infinite.) In ?2 the isomorphism properties are applied in the theory of Banach spaces. Some of the results given here concern the structure of nonstandard hulls of

Abstract: Equations were developed to predict the apparent motion of a physically stationary object resulting from head movement as a function of errors in the perceived distances of the object or of its parts. These equations, which specify the apparent motion in terms of relative and common components, were applied to the results of two experiments. In the experiments, the perceived slant of an object was varied with respect to its physical slant by means of perspective cues. In Experiment I, O reported the apparent motion and apparent distance of each end of the object independently. The results are consistent with the equations in terms of apparent relative motion, but not in terms of apparent common motion. The latter results are attributed to the tendency for apparent relative motion to dominate apparent common motion when both are present simultaneously. In Experiment II, a direct report of apparent relative motion (in this case, apparent rotation) was obtained for illusory slants of a physically frontoparallel object. It was found that apparent rotations in the predicted direction occurred as a result of head motion, even though under these conditions no rotary motion was present on the retina.

Abstract: In this paper a p-adic analytic function of two variables is constructed whose values in some "common" domain coincide with the values of the family of Hecke L-series of an imaginary quadratic field. The functional equation for such a function is obtained. The p-adic Mellin integral transform is the main technique. Bibliography: 13 items.

Abstract: Fritz Oberhettinger New York: Academic Press 1973 pp xi + 64 price $11 This little book lists about 400 examples of real forms of Fourier series, and 49 of exponential and Fourier–Bessel series. The real forms are in four groups according to the function represented and the coefficients are elementary or higher functions.

Abstract: In his book Discourse on Fourier Series, Lanczos deals in some detail with representations of f(x) of the type f(x) = h,-i(x) + gp(x) where h,-I(x) is a polynomial of degree p - 1 and gp(x) has the property that its full range Fourier coefficients converge at the rate r-P In Part I, some properties of h,(x) and of the series Ih,(x) l0 are described These prop- erties are used here to provide criteria for the convergence or divergence of the Euler-Mac- laurin series, in the case when f(x) is an analytic function The similarities and differences between this series and the Lidstone and other two-point series are briefly mentioned In Part II, the Lanczos representation is employed to derive an approximate representation F(x) for an analytic function f(x) on the interval (0, 1) is derived This has the form P-i m12 F(x) = E Xqi1Bq(x)/q! +2 E (Iu cos 2irrx + V, sin 2irrx) toq1 rhO and requires for its determination the values of the derivatives f (a-l )(1) - f (q-l )(O) (q = 1, 2, * * p - 1) and the regularly spaced function values f(j/m) (j = 0, 1, * * *, m) It involves replacing gp(x) by a discrete Fourier expansion based on trapezoidal rule approximations to its Fourier coefficients This representation is a powerful one The drawback is that it requires derivatives Most of Part II is devoted to the effect of using only approximate derivatives It is shown that when these are successively less accurate with increasing order (the sort of behaviour encountered using finite difference formula), then the representation is still powerful and reliable In a computational context the only penalty for using inaccurate derivatives is that a larger value of m may-or may not-be required to attain a specific accuracy PART I** PROPERTIES OF THE SEQUENCE hp(x) 1 The Lanczos Representation In this section, we outline a derivation of what we term Lanczos' representation for a function f(x) We suppose that f(x) is an analytic function of x and is real valued when x is real For convenience, we suppose that f(x) is analytic in a region of the complex plane which contains the unit interval (0, 1), a restriction which we denote by

Abstract: HE method proposed by Almon (1965) has been extensively used in the estimation of distributed lag models. It may be regarded as the least squares method under the linear constraint that the regression coefficients lie on a polynomial of a chosen order. Therefore, the loss or inefficiency of Almon's method (defined as some reasonable function of the mean square error matrix) could be smaller than that of the unconstrained least squares method. Then, an interesting question arises: for what order of polynomial is the loss minimized in a given distributed lag model? The answer depends upon several variables: the true values of the regression coefficients, the number of lags assumed in the model, the sample size, the ratio of the variance of the dependent variable to that of the error term, and the degree of the autocorrelation of the independent variable. In this paper we will evaluate numerically how the optimal order of polynomial is determined by these variables. Because the answer depends on so many variables, it is extremely important to design the study to produce meaningful conclusions. For this purpose we adopt one important simplifying assumption -that the independent variable follows a first-order autoregressive process with a varying correlation coefficient. Such a process is a good approximation of the processes of many economic variables. Given this simplification, we obtain definitive conclusions by judiciously defining the loss function so it depends simply and nicely on the parameters that we allow to change. As a result we can calculate the optimal order of polynomial for a given distributed lag model at a minimal computational cost. The essential part of our definition of the loss function is the trace of the product of the mean square error matrix and the autocovariance matrix of the independent variable. In section II we will offer rationales for this definition, as we believe that this definition has intrinsic merit as well as the advantage of simplifying our computation. Section II defines the model, defines the loss function for Almon's method, and discusses the rationale for and mathematical properties of the loss function. Section III presents and'analyzes the results of the numerical evaluation of the loss function for twelve models. Conclusions are presented in section IV.

Abstract: The stochastic Green's matrix is calculated for a random rough surface with Gaussian statistics and a magnetic boundary condition. The techniques we use are similar to those developed for the scalar and elastic cases. The coupled surface integral equations which are derived are the Green's function version of the Franz formulas. These integral equations are represented in k‐space in a certain gauge and a Feynman‐diagram‐like interpretation is given to each term in the equations. The diagram rules have many formal similarities with the scalar and elastic rules. By using partial summation techniques, the mean and second moment of this Green's function are shown to be solutions to Dyson and Bethe‐Salpeter equations respectively. The Green's function is applied to a scattering problem. Some approximations and simple examples are presented. The lowest order approximations agree with the standard literature results. The main advantage of the diagram method, its systematic presentation of higher order approximations, is stressed.

Abstract: Let S and X be any two sets; then a mapping Γ which assigns to each point t in S a set Γ(t) of points in X is called a multifunction from S into X. A selector for Γ is a function f from S into X such that f(t) ∈ Γ(t) for each t. We introduce here a class of multifunctions which is both well-supplied with measurable selectors and yet is comprehensive enough to include those kinds of multifunction which have been most commonly studied before. Hence in order to show that a multifunction with non-empty values, which may arise naturally in an implicit function problem, has a measurable selector, it is sufficient to show that it is of Souslin type.

Abstract: While Green's functions provide a powerful analytical means of solving the wave equation, they do not give a very realistic description of the mechanism of propagation. Also the adaptation of Green's function techniques to numerical methods of solution is not easy. This paper introduces the assumption that propagation takes place in discrete steps, and it is shown that Huygens's principle then becomes physically much more realistic. The basis for a very simple numerical procedure for the solution of the wave equation is also provided.