Abstract: We show how interval analysis can be used to compute the global minimum of a twice continuously differentiable function ofn variables over ann-dimensional parallelopiped with sides parallel to the coordinate axes. Our method provides infallible bounds on both the globally minimum value of the function and the point(s) at which the minimum occurs.

Abstract: EXPLORERS must accept the bad with the good. In the new-found lands gold may lie on the ground for the taking, but pioneers are likely to encounter rattlers and desperadoes. Recently a new territory has been discovered by economists, the intellectual continent we call \"privacy.\" The pioneers are our peerless leaders Posner and Stigler whose golden findings have already dazzled the world. It is high time for rattlers and desperadoes-that's the rest of us-to put in an appearance. Of course, I ought to add parenthetically, \"new\" is relative to one's point of view. Our pioneering economists, like explorers in other places and other times, found aborigines already inhabiting the territory-in this case intellectual primitives, Supreme Court justices and such. Quite properly, our explorers have brushed the natives aside, and I shall follow in that honorable tradition.

Abstract: If the gradient of the function y = f(x/sub 1/,..., x/sub n/) is desired, where f is given by an algoritym Af(x, n, y), most numerical analysts will use numerical differencing. This is a sampling scheme that approximates derivatives by the slope of secants in closely spaced points. Symbolic methods that make full use of the program text of Af should be able to come up with a better way to evaluate the gradient of F. The system Jake described produces gradients significantly faster than numerical differencing. Jake can handle algorithms Af with arbitrary flow of control. Measurements performed on one particular machine suggest that Jake is faster than numerical differencing for n > 8. Somewhat weaker results were obtained for the problem of computing Jacobians of arbitrary shape.

Abstract: We are concerned here with the structural information that it is possible to retrieve from the X-ray scattering study of macromolecules in solu­ tion. The general law governing X-ray scattering phenomena expresses the angular dependence of the scattered intensity as a function of the space distribution of the interatomic distances in the scatterer and of the nature of the atoms involved. When the scatterer is isotropic, the intensity is a function only of the moduli of the interatomic vectors, not of their orientation. Let r]J(r) be this isotropic distribution of the interatomic distances (a precise definition is given below). For a dilute solution of macromolecules the functionp(r) can be expected to display sharp short-range fluctuations (1 to 5 A) corresponding to pairs of neighboring atoms, followed by more damped fluctuations if middle­ range regularities are present-for example in the 5 to 10 A region for a-helical proteins. Beyond approximately 10 A. the number and variety of the interatomic vectors increases very rapidly with increasing r, and the function per) gradually smoothes out and slowly decreases. When r exceeds the maximal dimensions of the macromolecule, per) becomes uniform, since all the vectors involve only the solvent. Because the scattered intensity i(s) and the distribution p(r) are related by a Fourier transformation, such a separation of the fluctua­ tions of per) into two classes entails the presence in ;(s) of two distinct regions. One, at s small, contains the information relevant to the long-range (macromolecular) organization; the other, at s large, mirrors

Abstract: The concept of merged arithmetic is introduced and demonstrated in the context of multiterm multiplication/addition. The merged approach involves synthesizing a composite arithmetic function (such as an inner product) directly instead of decomposing the function into discrete multiplication and addition operations. This approach provides equivalent arithmetic throughput with lower implementation complexity than conventional fast multipliers and carry look-ahead adder trees.

Abstract: We show that any mean-periodic function f can be represented in terms of exponential-polynomial solutions of the same convolution equation f satisfies, ie, ', f 0 (E'(IRn)) This extends to n-variables the work of L Schwartz on mean-periodicity and also extends L Ehrenpreis' work on partial differential equations with constant coefficients to arbitrary convolutors We also answer a number of open questions about mean-periodic functions of one vari- able The basic ingredient is our work on interpolation by entire functions in one and several complex variables

Abstract: We introduce a property of boolean functions, called transitivity which holds of integer, polynomial, and matrix products as well as of many interesting related computational problems. We show that the area of any circuit computing a transitive function grows quadratically with the circuit's maximum data-rate, expressed in bit/second. This result provides a precise analytic expression of an area-time tradeoff for a wide class of V.L.S.I. circuits. Furthermore, (as shown elsewhere), this tradeoff is achievable. Thus we have matching (to within a constant multiplicative factor) upper and lower complexity bounds for the three above products, in the V.L.S.I. circuits computational model.

Abstract: Abstract Suppose A is a bounded set in ℛ d and f a real function defined on A. Suppose m = min{f(x; | x ∈ A} exists. Using a random sample X 1, X 2. …, X n from a uniform distribution over A we construct a confidence interval for m using asymptotic theory. Our results contain some statistical results valid in general extreme value theory (estimation of the main parameter of extreme value distributions).

Abstract: A new class of algorithms for unconstrained optimization has recently been proposed by Davidon [Conic Approximations and Collinear Scalings for Optimers, SIAM J. Num. Anal., to appear.]. This new method called “optimization by collinear scaling” is derived here as a natural extension of existing quasi-Newton methods. The derivation is based upon constructing a collinear scaling of the variables so that a local quadratic model can interpolate both function and gradient values of the transformed objective function at the latest two iterates. Deviation of the function values from quadratic behavior as well as gradient information influences the updating process. A particular member of this algorithm class is shown to have a Q-superlinear rate of convergence under standard assumptions on the objective function. The amount of computation required per update is essentially the same as for existing quasi-Newton methods.

Abstract: On domain C (Rf) we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if f E C '(Rf) and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions f a C '(R f) whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists. In 1917 Radon inverted the first "Radon transform" [18]. This transform, R, maps a function on Rn to a function on the set of hyperplanes in Rn. If f is a continuous function of compact support on R" then Rf evaluated on a hyperplane is the integral of f over that hyperplane in its natural measure. The case n = 2 has many applications in science, engineering, and medicine [2], [3], [15], [21] and the transform on Rn (n arbitrary) has many applications to partial differential equations [13], [14]. Generalizations of this Radon transform to integrations over certain spheres and ellipsoids have been studied by John and others [13], [19] again in connection with partial differential equations. Moreover these examples are all special cases of the generalized Radon transform: given smooth manifolds X, Y, and a class of submanifolds of X, { Hy I y E Y), one specifies smooth measures on each Hy. The generalized Radon transform R from Co'(X) to functions on Y takes f E C0?(X) to the integrals of f over the manifolds Hy in the measures /, [7]. In many cases restrictions on the support of Rf imply restrictions on the support of f [10]; this fact is useful in applications to partial differential equations [11], [14]. In this article we define a Radon transform over spheres passing through the origin in Rn. 1ff E C(Rn), the transform f evaluated on a sphere containing 0 is the mean value of f over that sphere in its natural measure. Our main result, Theorem 1, is an inversion formula for this transform: if f E C (R n) then f(x) is determined by the values of f on spheres that lie inside the disc of radius Ixi Received by the editors August 17, 1979; presented to the Society, October 19, 1979 at Howard University. AMS (MOS) subject classifications (1970). Primary 44A05, 35Q05.

Abstract: SUMMARY The concept of the inverse correlation function of a stationary process xt was first introduced by Cleveland (1972), who also introduced the autoregressive and the window methods for estimating this function. The asymptotic distribution of the estimates provided by these two methods is derived and their asymptotic covariance structure is shown to be in accordance with a remark of Parzen (1974). The results are extended to show that the two procedures suggested by Durbin (1959, 1961) for estimating the parameters of a moving average model are asymptotically efficient, relative to maximum likelihood in the Gaussian case. Some key word8: Akaike's information criterion; Autoregressive spectral estimate; Inverse correlation function; Inverse covariance function; Moving average model; Window spectral estimate.

Abstract: A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configurationX relative to the proximity dataP and the loss ofX relative to a pseudo-data matrixR. The matrixR is set up such that the side constraints to be imposed onX's distances are expressed by the relations amongR's numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solutionX. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matricesP andR. Further generalizations are introduced by substitutingR by a set ofR matrices,Ri,i=1, ...r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.

Abstract: Multiple reflections are filtered from seismograms by transforming them into an f-k array representing amplitude as a function of frequency and wave number The inverse of the f-k transform of the multiple reflections is generated The f-k array of the seismograms is filtered by weighting all samples with the inverse of the f-k transform of the multiple reflections

Abstract: Publisher Summary This chapter discusses the existence of 2π-periodic weak solutions for the equation XPrime; + g(x(t)) = p(t), where p: ℝ→ ℝ is a 2π-periodic measurable function with ∫2 < ∞ and g: ℝ → ℝ is a continuous T-periodic function such that (g(u) − g(v))/(u − v) < 1 for all u, v, ∈ ℝ, u ≠ v. The chapter presents a theorem that assumes that the above equation holds. If = (1/T) ∫02π g(s) ds and if P0: ℝ → ℝ is a 2π-periodic measurable function such that ∫0 = 0 and ∫02 < ∞, then there exist two real numbers d(p0) and D(p0), withmin t∈ℝ g(t) < d(p0) ≤ < D(P0) < max t∈ℝ g(t).

Abstract: An understanding of the behavior of vegetation canopy reflectance as a function of solar zenith angle is important to several remote sensing applications. Spectral hemispherical-conical reflectances of a nadir looking sensor were taken throughout the day of a lodgepole pine and two grass canopies. Mathematical simulations of both a spectral hemispherical-conical reflectance factor and a spectral bi-hemispherical reflectance were performed for two theoretical canopies of contrasting geometric structure. These results and results from literature studies showed a great amount of variability of vegetation canopy reflectances as a function of solar zenith angle. Explanations for this variability are discussed and recommendations for future measurements are proposed.

Abstract: The theory is presented of spin glasses for which the condition n m 1/3 l « dl is valid, l being the range of interaction forces between spins and n m the volume concentration of spins. Various examples where such an interaction appears are considered. Thermodynamic and kinetic characteristics of such systems are calculated at high and low temperatures. Particular attention is paid to the transition region; for its analysis percolation theory methods are applied. The theory of one-dimensional spin glasses of several types is constructed. A ‘kinetic’ equation is derived for the cluster distribution function in the transition region and the behaviour of this function is studied.

Abstract: : Consider the nonlinear Volterra equation u(t) + (b*Au) not an element of f(t). This paper discusses existing and recent results for the following problems concerning this equation the global existence and uniqueness of solutions and their continuous dependence on the data; the boundedness and asymptotic behavior as t approaches infinity in th special cases when X = H is a real Hilbert space and A is either a maximal monotone operator on H or A is a subdifferential of a proper, convex, lower semicontinuous function; and the existence, boundedness, and asymptotic behavior of positive solutions in the general settting. The theory is used to study one possible model problem for heat flow in a material with 'memory' which can be transformed to the equivalent from of the equation under physically reasonable assumptions; the latter provide a motivation for the natural setting of much of the theory developed here.

Abstract: This note presents an extension of a well-known synthesis technique for linear systems to bilinear plants. A classical quadratic Lyapunov function candidate is first selected. Then the controller is designed so as to make the derivative of that function along the trajectories of the bilinear system as negative as possible. This leads to a bang-bang feedback control law using a nonlinear switching function. It is shown that the closed-loop system is guaranteed to be globally asymptotically stable.

Abstract: Real cyclic cubic fields.- Real cyclic sextic fields.- The function ? and the structure of UR.- Bergstrom's product formula.- Bergstrom's product formula in the case of real cyclic sextic fields.- Formulas for computing ?A.- The class number of K6.- The signature rank of U6.- The computer program.- Numerical results.

Abstract: Let g(x) be the spectral density of a Gaussian stationary process. Then, for each continuous function 4u(x) we shall give an estimator of f$", (x)4t(g(x))dx whose asymptotic variance is O(n-), where 1((-) is an appropriate known function. Also we shall investigate the asymptotic properties of its estimator.