Abstract: 1. Internal set theory. We present here a new approach to Abraham Robinson's nonstandard analysis [10] with the aim of making these powerful methods readily available to the working mathematician. This approach to nonstandard analysis is based on a theory which we call internal set theory (1ST). We start with axiomatic set theory, say ZFC (Zermelo-Fraenkel set theory with the axiom of choice [1]). In addition to the usual undefined binary predicate E of set theory we adjoin a new undefined unary predicate standard. The axioms of 1ST are the usual axioms of ZFC plus three others, which we will state below. All theorems of conventional mathematics remain valid. No change in terminology is required. What is new in internal set theory is only an addition, not a change. We choose to call certain sets standard (and we recall that in ZFC every mathematical object-a real number, a function, etc.-is a set), but the theorems of conventional mathematics apply to all sets, nonstandard as well as standard. In writing formulas we use A for and, V for or, ~ for not, =* for implies, and for is equivalent to. We call a formula of 1ST internal in case it does not involve the new predicate "standard" (that is, in case it is a formula of ZFC); otherwise we call it external. Thus "x standard" is the simplest example of an external formula. To assert that x is a standard set has no meaning within conventional mathematics-it is a new undefined notion. The fact that we have adjoined "standard" as an undefined predicate (rather than defining it in terms of E as is the case with all of the predicates of conventional mathematics) requires a readjustment of an engrained habit. We are used to defining subsets by means of predicates. In fact, it follows from the axioms of ZFC that if A(z) is an internal formula then for all sets x there is a set y = {z E x: A(z)} such that for all sets z we have z&y n E N A n standard. We may not use external predicates to define subsets. We call the violation of this rule illegal set formation. We adopt the following abbreviations:

Abstract: OVER the last 20 years a large literature has developed concerning evolution equations which for certain initial data possess solutions that do not exist for all time. The bulk of this literature relates to problems arising from partial differential equations. To establish nonexistence it is customary to argue by contradiction. One supposes that for given UQ and t0 a solution u(t) with u(ro)= u0 exists for all times t^t0; typically u takes values in some Banach space X and we will assume that this is the case. A function p : X —» R is then constructed, and by use of differential inequalities it is shown that lim p(u(()) = ° for some tle(t0, °°). This usually

Abstract: The task of finding global optima to general classes of nonconvex optimization problem is attracting increasing attention. McCormick points out that many such problems can conveniently be expressed in separable form, when they can be tackled by the special methods of Falk and Soland or Soland, or by Special Ordered Sets. Special Ordered Sets, introduced by Beale and Tomlin, have lived up to their early promise of being useful for a wide range of practical problems. Forrest, Hirst and Tomlin show how they have benefitted from the last few years, as a result of being incorporated in a general mathematical programming system. Nevertheless, Special Ordered Sets in their original form require that any continuous functions arising in the problem be approximated by piecewise linear functions at the start of the analysis. The motivation for the new work described in this paper is the relaxation of this requirement by allowing automatic interpolation of additional relevant points in the course of the analysis. This is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward. Two bt-products of the work are of interest. One is an improved branching strategy for general special-ordered-set problems. The other is a method for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval. The paper describes these methods, their implementation in the UMPIRE system, and preliminary computational experience.

Abstract: A nonlinear time series system is considered. The system has the property that the output series corresponding to a given input series is the sum of a noise series and the result of applying in turn the operations of linear filtering, instantaneous functional composition and linear filtering to the input series. Given a stretch of Gaussian input series and corresponding output series, estimates are constructed of the transfer functions of the linear filters, up to constant multipliers. The investigation discloses that for such a system, the best linear predictor of the output given Gaussian input, has a broader interpretation than might be suspected. The result is derived from a simple expression for the covariance function of a normal variate with a function of a jointly normal variate.

Abstract: The self-tuning controller is extended to include rational transfer function (as opposed to polynomial) terms in the associated cost function. Two interpretations of the self-tuning controller are examined in some detail: a model reference adaptive control, and a self-tuning least-squares predictor in conjunction with conventional compensation. The former version is shown to give not only prespecified set point response, but also a closed-loop disturbance with largely prespecified spectral density. The latter version is compared with the method of O.J.M. Smith, and is shown to be less sensitive to uncertainty in some system parameters. Examples are given which illustrate the continuous-time performance of these discrete-time control laws.

Abstract: A simple technique for reasoning about equalities that is fast and complete for ground formulas with function symbols and equality is presented. A proof of correctness is given as well.

Abstract: In recent years numerous results of piecewise-linear analysis of nonlinear resistive networks have been derived. The applicability of the method relies on the fact that every nonlinear device is modeled by a piecewise-linear continuous function. In order to extend the applicability of piecewise-linear analysis to treat more general nonlinear networks, three steps need to be carried out: i) the subdivision of the domain of the multi-dimensional nonlinear network function; ii) the interpolation of a piecewise-linear continuous function on the subdivided domain; and iii) the application of piecewise-linear analysis. It turns out that the above three steps can be accomplished effectively by using simplicial subdivision. In addition, the difficulties encountered in the conventional piecewise-linear analysis are simplified. The memory space needed for the analysis is also greatly reduced. The complete analysis has been implemented in a program on CDC 6400.

Abstract: In the simplest possible description of an n-electron system, one one-electron function (spin-orbital) is associated with each electron and the n -electron wave function is a Slater determinant built up from these spin-orbitals. The one-to-one correspondence between electrons and spin-orbitals gives an acceptable first-order description only for closed-shell and certain open-shell states. A one-electron theory that is applicable in general to open-shell states as well is characterized by assigning sets of electrons to sets of degenerate spin-orbitals, where the number of electrons within one set can be equal to or smaller than the dimension of the irreducible representation spanned by the degenerate set of spin-orbitals. An example is the well-known characterization of an atomic state by its configuration,(1) e.g., for the carbon ground state 1s22s22p2, without specifying the ms and ml values. (For a general discussion of closed- and open-shell states in the framework of rigorous quantum mechanics, see Refs. 2 and 3.)

Abstract: SUMMARY The paper develops in a more general setting the methods used' by Paulson, Holcomb & Leitch (1975) to estimate the parameters of a stable law The statistic considered minimizes a distance function determined, by the empirical characteristic function Consistency is established under the condition of differentiability of the characteristio function and the existence of bounded second derivatives is required to obtain a central limit theorem for the estimators of one or more parameters Questions concerning efficiency and robustness are discussed

Abstract: THE CHOICE MODEL in which alternatives are ordered in terms of the mean and variance of their return has been utilized in a variety of fields with the best developed, both theoretically and empirically, being that of portfolio selection. Since the pioneering works of Markowitz [22] and Tobin [32], a series of results has been obtained regarding the utility theoretic foundations of this model. While these results are widely known, they continue to be the source of discussion.' For example, Markowitz demonstrated that if the ordering of alternatives is to satisfy the von Neumann-Morgenstern (NM) [35] axioms of rational behavior, only a quadratic (NM) utility function is consistent with an ordinal expected utility function that depends solely on the mean and variance of the return. Consequently, even if the return for each alternative has a normal distribution, the mean-variance framework cannot be used to rank alternatives consistently with the NM axioms unless a quadratic NM utility function is specified. The implications of this restriction are disturbing not only because of the undesirable properties of a quadratic utility function but also because, for example, the indifference curves in the mean-standard deviation plane are concentric circles with the center on the mean axis. Furthermore, a quadratic utility function leads to a rather disquieting result in portfolio theory, since it implies that in equilibrium each investor holds an equal percentage of every security (Mossin [23, p. 69]). Also, Ekern and Wilson [11, p. 179] have shown that in a mean-variance portfolio model all shareholders prefer that a firm characterized by stochastic constant returns operate so that its market value is zero. Another disquieting result is that of Borch [7] who demonstrates that if preferences satisfy a monotonicity condition, an indifference curve in the ( ,a)-plane consists of a single point. A number of authors, including Samuelson [27] [29] and Tsiang [33], have argued that mean-variance analysis may be viewed as an approximation to a more general choice model. The appropriateness of such an approximation has been considered by Borch [8], Bierwag [6], Levy [20], and Tsiang [34], for example, and will not be considered further here. The purpose of this paper is instead to consider the foundations of mean-variance analysis and the consistency of the ordering of alternatives according to means and variances with the NM axioms. More specifically, given alternatives a, and a2 and their corresponding random returns X, and X2 with distribution functions F,(x) and F2(x), respectively, preferences satisfying the NM axioms imply the existence of a measurable, continuous utility function

Abstract: A dynamic central place theory is formulated as a simulation model in which retail activities, described by cost equations, and consumers, described by spatial interaction equations, interact to generate a central place system. The behavior of the model is then examined. Simulation results show that the basic character of the system—whether it is agglomerated or dispersed—depends primarily on a single parameter in the interaction equation, and only secondarily on the specification of the cost function. The results are highly robust in that they are largely independent of the initial sizes and locations of centers, and even independent of the type of interaction equation used. The patterns generated appear plausible.

Abstract: A model for finding the local optima of a multimodal function defined in a region A ? Rn is proposed. The method uses a local optimizer which is started from a number of points sampled in A. In order to reduce the number of function evaluations needed to reach the local optima, the parallel local search processes are stopped repeatedly, the working points clustered, and a reduced number of processes from each cluster resumed. A direct nonhierarchical cluster analysis technique is presented. The dissimilarity measure used is the Euclidean distance between points. Clusters are grown from seed points. The number of required distance evaluations is less than or equal to c(n-1), where n is the number of points and c is the number of clusters arrived at. Thresholds are determined by the point density in a body which in turn is determined by the given points. The covariance matrix is diagonalized, and a decision on the dimensionality of the space containing the points can be made. The volume of the body is proportional to the square root of the product of the corresponding eigenvalues. The performance of the clustering analysis technique is illustrated. It is demonstrated that there exist classes of global optimization problems for which the probability of obtaining a solution is greater for the proposed model than for multiple local optimizations. Some experiences gained from using the model are reported.

Abstract: Bearing impedance vectors are introduced for plain journal bearings which define the bearing reaction force components as a function of the bearing motion. Impedance descriptions are developed directly for the approximate Ocvirk (short) and Sommerfeld (long) bearing solutions. The impedance vector magnitude and the mobility vector magnitude of Booker are shown to be reciprocals. The transformation relationships between mobilities and impedance are derived and used to define impedance vectors for a number of existing mobility vectors including the finite-length mobility vectors developed by Moes. The attractiveness and utility of the impedance-vector formulation for transient simulation work is demonstrated by numerical examples for the Ocvirk "p", and "2p" bearing impedances and the cavitating finite-length-bearing impedance. The examples presented demonstrate both bearing and squeeze-film damper application. A direct analytic method for deriving a complete set of (analytic) stiffness and damping coefficients from impedance descriptions is developed and demonstrated for the cavitating finite length-bearing impedances. Analytic expressions are provided for all direct and cross-coupled stiffness and damping coefficients, and compared to previously developed numerical results. These coefficients are used for stability analysis of a rotor, supported in finite-length cavitating bearings. Onset-speed-of-instability results are presented as a function of the L/D ratio for a range of bearing numbers. Damping coefficients are also presented for finite-length squeeze-film dampers.

Abstract: The first part of this article integrates the concept of (relative) risk aversion with respect to income (r) with the static analysis of demand for many commodities Alternative representations of preferences and demand functions, using duality, give rise to many alternative representations and interpretations of r, and to theorems regarding attitudes towards risk in bundles of quantities and in prices In the second part, a previous analysis by Deschamps is corrected and completed by specifying the general form of preferences and demands such that r is a function of the utility level only, independent of relative prices Finally, preferences and demand functions associated with constant r (previously analyzed by Stiglitz and Deschamps) are specified more explicitly and completely A general conclusion emerging is that demand behavior under certainty can hardly throw any light on the nature of attitudes towards risk THE CONCEPT OF THE relative risk aversion function as a unit-free measure of individual aversion to income risk under expected utility maximization, was defined by Arrow [1] and Pratt [15], and has proved useful in various applications Stiglitz [21] studied relations between an individual's aversion to income risk, and his indirect utility and demand functions for many commodities, obtained under certainty in competitive markets In particular, he analyzed the cases of risk indifference and constant relative risk aversion (r) Deschamps [4] extended this analysis to study implications of alternative assumptions: (i) That absolute risk aversion R = rly is independent of prices, nominal or relative, for given income This is equivalent, however, to constant r (which is the case analyzed by Stiglitz) for both cases (ii) That R or r are constant on each indifference surface This is shown to be impossible for R, but meaningful and interesting for r = r(u) Unfortunately, however, Deschamps could not show the utility and demand functions for this case, and conducted an indirect analysis based on the second-order differential equation implied, failing to note that any such r(u) is compatible with homothetic preferences In addition, his analysis contains errors (eg, the case r = 1 constant) and may be subject to misleading interpretations The purpose of this article is twofold: (i) To complete the analysis of risk aversion with many commodities, by using various alternative formulations of the relative risk aversion function to study general relations between income risk aversion and attitudes towards risk with respect to quantities (eg, when both relative prices and income are subject to risk), or with respect to prices (ii) To complete the analysis of Deschamps by showing the general forms of utility and demand functions when r = r(u), and to correct some errors of analysis and interpretation

Abstract: Publisher Summary Equilibrium problems in two-dimensional, and higher, continua give rise to elliptic partial differential equations. An alternative argument employs the maximum (minimum) modulus theorem. When a partial differential equation has accompanying auxiliary conditions, which select among all possible solutions, a uniquely determined function, the data is called properly posed, provided that the solution depends continuously on this data. Methods of solution for general computational problems fall into two categories—the direct and iterative procedures. Direct methods, of which the solution of a tridiagonal system is typical, give the exact answer in a finite number of steps, if there were no round-off error. The algorithm for such a procedure is complicated and non-repetitive. Many direct methods for linear systems are available. Iterative methods consist of repeated application of a simple algorithm. They yield the answer as a limit of a sequence, even without consideration of round-off errors.

Abstract: The sum of a linear and linear-fractional function is investigated in terms of quasi-convexity and quasi-concavity From this we obtain some insight into the nature of local optima of these functions useful in algorithms

Abstract: Computation of an optimal policy for ordering a perishable commodity with a fixed lifetime of m periods requires the solution of a dynamic program whose state variable has dimension m − 1. Unless m is small, the computations quickly become unreasonable. By bounding the expected outdating function and using an approximate transfer function, we obtain an approximation of the original problem that can be computed as if the lifetime were r < m periods. When r = 2, it is demonstrated under reasonable assumptions that the appropriate function to be minimized each period is quasi-convex. We include computations that compare this approximation to both the optimal policy and a critical number approximation.

Abstract: It is proposed to describe the temporal characteristics of a wave propagating in a random medium in terms of its temporal moments. The first two moments are related to the mean arrival time and the mean pulse width. It is shown that the one-position two-frequency mutual coherence function enters in the formulation naturally. The form of the expression suggests expanding the mutual coherence function in a narrow-band expansion whose coefficients can be solved exactly from the parabolic equation that takes into account all multiple scattering effects except the backscattering. A brief survey of the literature shows that the irregularity spectrum, under various conditions, has a power-law dependence. In order to conform to this observation a Bessel function spectrum proposed by Shkarofsky is found convenient to use since it not only reduces to the desired power-law form in the proper range of wavenumber space, but also has all the finite moments. Exact expressions for the mean arrival time and mean square pulse width are obtained; some numerical examples are given. Finally, the effect of noise on these moments is discussed.

Abstract: Two classical foms of the time of flight equation for the two body, two point boundary value problem, known as "Lambert's problem," are combined to produce an elegant formulation which may serve as the nucleus of an extremely efficient cornputat.- ion algorithm. (i.e., includes elliptic, parabolic and hyperbolic orbits), is a well-behaved function of a single convenient independent variable, and requires the evaluation of a single hypergeometric function. New recursive identities f or hypergeometric functions are developed and efFectively exploited to enhance computation speed. "top-down" continued fraction algorithm, used for efficient evaluation of the hypergeometric function, which avoids the necessity of repeated scaling. This formulation represents a considerable improve- ment over other methods known to the author, The time equation is universal