Abstract: The LambertW function is defined to be the multivalued inverse of the functionw →we w . It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.

Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Abstract: We derive and discuss the equations of motion for the condensate and its fluctuations for a dilute, weakly interacting Bose gas in an external potential within the self-consistent Hartree-Fock-Bogoliubov (HFB) approximation. Account is taken of the depletion of the condensate and the anomalous Bose correlations, which are important at finite temperatures. We give a critical analysis of the self-consistent HFB approximation in terms of the Hohenberg-Martin classification of approximations (conserving vs gapless) and point out that the Popov approximation to the full HFB gives a gapless single-particle spectrum at all temperatures. The Beliaev second-order approximation is discussed as the spectrum generated by functional differentiation of the HFB single-particle Green's function. We emphasize that the problem of determining the excitation spectrum of a Bose-condensed gas (homogeneous or inhomogeneous) is difficult because of the need to satisfy several different constraints.

Abstract: This paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms. Unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a (possibly infinite) basis function expansion with random coefficients. The unknown parameter function is related to the measurements used for estimation by a ''forward operator'' which describes the measurement process. In the particular case considered here, the parameter of interest is the large-scale log hydraulic conductivity, the measurements are point values of log conductivity and piezometric head, and the forward operator is derived from an upscaled groundwaterflow equation. The inverse algorithm seeks the ''most probable'' or maximum a posteriori estimate of the unknown parameter function. When the measurement errors and parameter function are Gaussian and independent, the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness-of- fit and prior terms. When the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate. This constraint tends to make the problem well- posed by limiting the range of admissible solutions. The Gaussian maximum a posteriori problem may be solved with variational methods, using functional generalizations of Gauss-Newton or gradient-based search techniques. Several popular groundwater inverse algorithms are either special cases of, or variants on, the functional maximum a posteriori algorithm. These algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use (linear versus nonlinear). The accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a Bayesian extension of the Cramer-Rao lower bound on the estimation error covariance. This bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements.

Abstract: SUMMARY This paper explores the inconsistency of common scale estimators when output is proxied by deflated sales, based on a common output deflator across firms. The problem arises when firms operate in an imperfectly competitive environment and prices differ between them. In particular, we show that this problem reveals itself as a downward bias in the scale estimates obtained from production function regressions, under a variety of assumptions about the pattern of technology, demand and factor price shocks. The result also holds for scale estimates obtained from cost functions. The analysis is carried one step further by adding a model of product demand. Within this augmented model we examine the probability limit of the scale estimate obtained from an ordinary production function regression. This analysis reveals that the OLS estimate will be biased towards a value below one, and how this bias is affected by the magnitude of the parameters and the amount of variation in the various shocks. We have included an empirical section which illustrates the issues. The empirical analysis presents a tentative approach to solve the problem discussed in the theoretical part of this paper.

Abstract: The subject of this paper is the simulation of multi-dimensional, homogeneous, Gaussian stochastic fields using the spectral representation method. Following this methodology, sample functions of the stochastic field can be generated using a cosine series formula. These sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic field when the number of terms in the cosine series is large. The ensemble-averaged power spectral density or autocorrelation function approaches the corresponding target function as the sample size increases. In addition, the generated sample functions possess ergodic characteristics in the sense that the spatially-averaged mean value, autocorrelation function and power spectral density function are identical with the corresponding targets, when the averaging takes place over the multi-dimensional domain associated with the fundamental period of the cosine series. Another property of the simulated stochastic field is that it is asymptotically Gaussian as the number of terms in the cosine series approaches infinity. The most important feature of the method is that the cosine series formula can be numerically computed very efficiently using the Fast Fourier Transform technique. The main area of application of this method is the Monte Carlo solution of stochastic problems in structural engineering, engineering mechanics and physics. Specifically, the method has been applied to problems involving random loading (random vibration theory) and random material and geometric properties (response variability due to system stochasticity).

Abstract: We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation function evaluated by each principal element satisfies certain technical conditions. Under these conditions, it is also possible to construct networks that provide a geometric order of approximation for analytic target functions. The permissible activation functions include the squashing function (1-e-x)-1 as well as a variety of radial basis functions. Our proofs are constructive. The weights and thresholds of our networks are chosen independently of the target function; we give explicit formulas for the coefficients as simple, continuous, linear functionals of the target function.

Abstract: We study the approximation problem ofE f(X T ) byE f(X ), where (X t ) is the solution of a stochastic differential equation, (X ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X T ) −f(X ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hormander type for the infinitesimal generator of (X t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX and compare it to the density of the law ofX T .

Abstract: The strength and range of interpoint interactions in a spatial point process can be quantified by the function J = (1 - G)/(1 - F), where G is the nearest-neighbour distance distribution function and F the empty space function of the process. J(r) is identically equal to 1 for a Poisson process; values of J(r) smaller or larger than 1 indicate clustering or regularity, respectively. We show that, for a large class of point processes, J(r) is constant for distances r greater than the range of spatial interaction. Hence both the range and type of interaction can be inferred from J without parametric model assumptions. It is also possible to evaluate J(r) explicitly for many point process models, so that J is also useful for parameter estimation. Various properties are derived, including the fact that the J function of the superposition of independent point processes is a weighted mean of the J functions of the individual processes. Estimators of J can be constructed from standard estimators of F and G. We compute estimates of J for several standard point pattern datasets and implement a Monte Carlo test for complete spatial randomness.

Abstract: Searching for a goal is a central and extensively studied problem in computer science. In classical searching problems, the cost of a search function is simply the number of queries made to an oracle that knows the position of the goal. In many robotics problems, as well as in problems from other areas, we want to charge a cost proportional to the distance between queries (e.g., the time required to travel between two query points). With this cost function in mind, the abstract problem known as thew-lane cow-path problem was designed. There are known optimal deterministic algorithms for the cow-path problem; we give the first randomized algorithm in this paper. We show that our algorithm is optimal for two paths (w=2) and give evidence that it is optimal for larger values ofw. Subsequent to the preliminary version of this paper, Kaoet al.(in“Proceedings, 5th ACM?SIAM Symposium on Discrete Algorithm,” pp. 372?381, 1994) have shown that our algorithm is indeed optimal for allw?2. Our randomized algorithm gives expected performance that is almost twice as good as is possible with a deterministic algorithm. For the performance of our algorithm, we also derive the asymptotic growth with respect tow?despite similar complexity results for related problems, it appears that this growth has never been analyzed.

Abstract: We show that the class of two-layer neural networks with bounded fan-in is efficiently learnable in a realistic extension to the probably approximately correct (PAC) learning model. In this model, a joint probability distribution is assumed to exist on the observations and the learner is required to approximate the neural network which minimizes the expected quadratic error. As special cases, the model allows learning real-valued functions with bounded noise, learning probabilistic concepts, and learning the best approximation to a target function that cannot be well approximated by the neural network. The networks we consider have real-valued inputs and outputs, an unlimited number of threshold hidden units with bounded fan-in, and a bound on the sum of the absolute values of the output weights. The number of computation steps of the learning algorithm is bounded by a polynomial in 1//spl epsiv/, 1//spl delta/, n and B where /spl epsiv/ is the desired accuracy, /spl delta/ is the probability that the algorithm fails, n is the input dimension, and B is the bound on both the absolute value of the target (which may be a random variable) and the sum of the absolute values of the output weights. In obtaining the result, we also extended some results on iterative approximation of functions in the closure of the convex hull of a function class and on the sample complexity of agnostic learning with the quadratic loss function.

Abstract: We present a zero-knowledge proof system [19] for any NP language L, which allows showing that x in L with error probability less than 2^−k using communication corresponding to O(|x|^c) + k bit commitments, where c is a constant depending only on L. The proof can be based on any bit commitment scheme with a particular set of properties. We suggest an efficient implementation based on factoring. We also present a 4-move perfect zero-knowledge interactive argument for any NP-language L. On input x in L, the communication complexity is O(|x|^c) max(k; l) bits, where l is the security parameter for the prover. Again, the protocol can be based on any bit commitment scheme with a particular set of properties. We suggest efficient implementations based on discrete logarithms or factoring. We present an application of our techniques to multiparty computations, allowing for example t committed oblivious transfers with error probability 2^−k to be done simultaneously using O(t+k) commitments. Results for general computations follow from this. As a function of the security parameters, our protocols have the smallest known asymptotic communication complexity among general proofs or arguments for NP. Moreover, the constants involved are small enough for the protocols to be practical in a realistic situation: both protocols are based on a Boolean formula Phi containing and- , or- and not-operators which verifies an NP-witness of membership in L. Let n be the number of times this formula reads an input variable. Then the communication complexity of the protocols when using our concrete commitment schemes can be more precisely stated as at most 4n + k + 1 commitments for the interactive proof and at most 5nl +5l bits for the argument (assuming k the number of commitments required for the proof is linear in n. Both protocols are also proofs of knowledge of an NP-witness of membership in the language involved.

Abstract: We make three related contributions. First, we propose a new technique for solving prediction problems under asymmetric loss using piecewise-linear approximations to the loss function, and we establish existence and uniqueness of the optimal predictor. Second, we provide a detailed application to optimal prediction of a conditionally heteroscedastic process under asymmetric loss, the insights gained from which are broadly applicable. Finally, we incorporate our results into a general framework for recursive prediction-based model selection under the relevant loss function.

Abstract: An efficient implementation of a multidimensional data aggregation operator that generates all aggregates and super-aggregates for all available values in a results set by first generating a minimal number of aggregates at the lowest possible system level using a minimal number of function calls, and second categorizing the aggregate function being applied and applying the aggregate function with the fewest possible function calls. The aggregates are generated from a union of roll-ups of the n attributes to the GROUP BY clause of the SELECT statement. The number of roll-ups are minimized by including a barrel shift of the attributes being rolled up. A scoreboard array of 2n bits is updated during the roll-up and barrel shifting process to keep track of which roll-ups are complete and with are not yet complete. Generating super-aggregates is further optimized by identifying the type of aggregate function being applied and facilitating the most efficient application of the aggregate function. A lter-- super() function is implemented to facilitate the most efficient application of algebraic aggregate functions that require access to intermediate aggregate data that heretofore was not available to any algebraic aggregation operator.

Abstract: We have constructed fully self-consistent models of triaxial galaxies with central density cusps. The triaxial generalizations of Dehnen’s (1993) spherical mass models are presented, which have densities that vary as r near the center and r at large radii. We computed libraries of ∼ 7000 orbits in each of two triaxial models with γ = 1 (“weak cusp”) and γ = 2 (“strong cusp”); these two models have density profiles similar to those of the “core” and “power-law” galaxies observed by HST. Both mass models have short-to-long axis ratios of 1:2 and are maximally triaxial. The major orbit families and their associated periodic orbits were mapped as a function of energy. A large fraction of the orbits in both model potentials are stochastic, as evidenced by their non-zero Liapunov exponents. We show that most of the stochastic orbits in the strong-cusp potential diffuse relatively quickly through their allowed phase-space volumes, on time scales of 10 − 10 dynamical times. Stochastic orbits in the weak-cusp potential diffuse more slowly, often retaining their box-like shapes for 10 dynamical times or longer. Attempts to construct self-consistent solutions using just the regular orbits failed for both mass models. Quasi-equilibrium solutions that include the stochastic orbits exist for both models; however, real galaxies constructed in this way would evolve near the center due to the continued mixing of the stochastic orbits. We attempted to construct more nearly stationary models in which stochastic phase space was uniformly populated at low energies. These “fully mixed” solutions were found to exist only for the weak-cusp potential; as much as ∼ 1/3 of the mass near the center of these models could be placed on stochastic orbits without destroying the self-consistency. No significant fraction of the mass could be placed on fully-mixed stochastic orbits in the strong-cusp model, demonstrating that strong triaxiality can be inconsistent with a high central density. Our results suggest that chaos is a generic feature of the motion in realistic triaxial potentials, but that the presence of chaos is not necessarily inconsistent with the existence of stationary triaxial configurations. Triaxial galaxies with cusps 2 Merritt & Fridman

Abstract: In any steady-state optimization problem, the output being optimized could be a nonmonotonic function of the controlled variable. Often the output is dependent on the temperature, the load impedance, and other unknown and variable quantities. Thus, it is very useful to have an automatic tuning scheme that will take the system to the desired operating point using only input and output information. The present invention is a generalized tuning scheme that uses the correlation between changes in the input and corresponding changes in the output to tune the operating point. In general terms, the present invention utilizes a correlation function between the controlled variable and a perturbed waveform. When the system reaches the desired operating point, the correlation goes to zero and the system converges. This corresponds to the point at which the derivative of the output with respect to the input is zero. This tuning scheme is appropriate for any tuning problem which has a single maximum or minimum. A variety of tuning problems in power electronics and other areas fall into this category. A tuning scheme based on correlation usually requires an excitation input. The switching action in the controlled circuit perturbs all the states and provides this excitation. Thus, this tuning scheme is appropriate for switching power applications. A preferred embodiment of the present application is used to control a power converter in a solar array application.

Abstract: In this paper I present a new two-dimensional decomposition technique, which models the surface photometry of a galaxy with an exponential light profile for both bulge and disk and, when necessary, with a Freeman bar. The new technique was tested for systematic errors on both artificial and real data and compared with widely used one-dimensional decomposition techniques, where the luminosity profile of the galaxy is used. The comparisons indicate that a decomposition of the two-dimensional image of the galaxy with an exponential light profile for both bulge and disk yields the most reproducible and representative bulge and disk parameters. An extensive error analysis was made to determine the reliability of the model parameters. If the model with an exponential bulge profile is a reasonable description of a galaxy, the maximum errors in the derived model parameters are of order 20%. The uncertainties in the model parameters will increase, if the exponential bulge function is replaced by other often used bulge functions as the de Vaucouleurs law. All decomposition methods were applied to the optical and near-infrared data set presented by de Jong & van der Kruit (1994), which comprises 86 galaxies in six passbands.

Abstract: Two upward directed sets of sequences of zeroes and ones are positively correlated. We provide a lower bound on the correlation, in function of how much the two sets simultaneously depend on the same coordinates.

Abstract: We characterize a firm's optimal factor adjustment when any number of factors face "kinked" linear adjustment costs so that all factor accumulation is costly to reverse We first consider a general non-stationary case with a concave operating profit function, unrestricted form of uncertainty and a horizon of arbitrary length We show that the optimal investment strategy follows a control limit policy at each point in time The state space of the firm's problem is partitioned into various domains, including a continuation region where no adjustment should optimally be made to factor levels We then consider two specific model classes and exploit their special structure to derive expressions for their continuation regions

Abstract: A constraint can be thought of intuitively as a restriction on a space of possibilities. Mathematical constraints are precisely specifiable relations among several unknowns (or variables), each taking a value in a given domain. Constraints restrict the possible values that variables can take, representing some (partial) information about the variables of interest. For instance, “The second side of a sheet of a paper must be imaged 9000 milliseconds after the time at which the first side is imaged,” relates two variables without precisely specifying the values they must take. One can think of such a constraint as standing for (a possibly infinite) set of values, in this case the set {^0, 9000&, ^1500, 10500&, . . .}. Constraints arise naturally in most areas of human endeavor. They are the natural medium of expression for formalizing regularities that underlie the computational and (natural or designed) physical worlds and their mathematical abstractions, with a rich tradition going back to the days of Euclidean geometry, if not earlier. For instance, the three angles of a triangle sum to 180 degrees; the four bases that make up DNA strands can only combine in particular orders; the sum of the currents flowing into a node must equal zero; the trusses supporting a bridge can only carry a certain static and dynamic load; the pressure, volume, and temperature of an enclosed gas must obey the “gas law”; Mary, John, and Susan must have different offices; the relative position of the scroller in the window scroll-bar must reflect the position of the current text in the underlying document; the derivative of a function is positive at zero; the function is monotone in its first argument, and so on. Indeed, whole subfields of mathematics (e.g., theory of Diophantine equations, group theory) and many celebrated conjectures of mathematics (e.g., Fermat’s Last Theorem) deal with whether certain constraints are satisfiable. Constraints naturally enjoy several interesting properties. First, as previously remarked, constraints may specify partial information—a constraint need not uniquely specify the value of its variables. Second, they are additive: given a constraint c1, say, X 1 Y $ Z, another constraint c2 can be added, say, X 1 Y # Z. The order of imposition of constraints does not matter; all that matters at the end is that the conjunction of constraints is in effect. Third,

Abstract: Apparatus for determining the location of a signal-generating source (e.g., a conferee in a telephone conference) includes at least three sensors (e.g., microphones) arranged in a plurality of sets, each having two or more sensors. A surface-finding element responds to receipt at each sensor set of signals (e.g., speech) from the source for identifying a geometric surface (e.g., the surface of a hyperboloid or cone) representing potential locations of the source as a function of sensor locations and time difference of arrival of the signals. A location-approximating element coupled to two or more of the sets identifies a line that further defines potential source locations at the intersection of the surfaces. A location signal representing those potential locations is generated in accord with parameters of that line. Further functionality generates generating the location signal as a function of closest intersections the plural ones of the aforementioned lines.

Abstract: In this work a symmetric representation of the three-body Coulomb continuum wave function is constructed that represents an exact asymptotic solution of the many-body Schr\"odinger equation on a five-dimensional hypersphere of large hyperradius. Consequently, the wave function is shown to satisfy the Kato cusp conditions at all three two-body collision points. At finite distances the proposed solution is designed to account for properties of the total potential surface. In particular, dynamical stabilization due to the presence of ridge structure in the total potential (Wannier ridge) is encompassed in the present treatment. The behavior of the wave function at the total dissociation threshold is investigated. In order to allow for three-body interactions we linearly expand each two-body Coulomb potential in terms of all three two-body potentials. The expansion coefficients determine the amount of distortion of each two-body subsystem by the presence of the third particle and thus give direct information on the strength of three-body interactions. \textcopyright{} 1996 The American Physical Society.

Abstract: A neural network system is provided that models the system in a system model (12) with the output thereof providing a predicted output. This predicted output is modified or controlled by an output control (14). Input data is processed in a data preprocess step (10) to reconcile the data for input to the system model (12). Additionally, the error resulted from the reconciliation is input to an uncertainty model to predict the uncertainty in the predicted output. This is input to a decision processor (20) which is utilized to control the output control (14). The output control (14) is controlled to either vary the predicted output or to inhibit the predicted output whenever the output of the uncertainty model (18) exceeds a predetermined decision threshold, input by a decision threshold block (22). Additionally, a validity model (16) is also provided which represents the reliability or validity of the output as a function of the number of data points in a given data region during training of the system model (12). This predicts the confidence in the predicted output which is also input to the decision processor (20). The decision processor (20) therefore bases its decision on the predicted confidence and the predicted uncertainty. Additionally, the uncertainty output by the data preprocess block (10) can be utilized to train the system model (12).

Abstract: We consider the NJL model as an effective quark theory to describe the interaction which is responsible for the quark flavor dynamics at intermediate energies. In addition to the usual ultraviolet cut-off which is necessary since the model is non-renormalizable, we also introduce an infrared cut-off which drops off the unknown confinement part of the quark interaction, which is believed to be less important for the flavor dynamics. The infrared cut-off eliminates all q-qbar thresholds, which plague the application of the usual NJL model beyond low-energy pion physics. We apply this two-cut-off prescription to the extended NJL model with chiral and heavy quark symmetries proposed recently by us. We find a satisfactoring description even of the heavy mesons with spin/parity J/P = (0+, 1+). Furthermore, the shape-parameters of the Isgur-Wise function are studied as a function of the residual heavy meson mass.