Abstract: Informally, an obfuscator O is an (efficient, probabilistic) "compiler" that takes as input a program (or circuit) P and produces a new program O(P) that has the same functionality as P yet is "unintelligible" in some sense. Obfuscators, if they exist, would have a wide variety of cryptographic and complexity-theoretic applications, ranging from software protection to homomorphic encryption to complexity-theoretic analogues of Rice's theorem. Most of these applications are based on an interpretation of the "unintelligibility" condition in obfuscation as meaning that O(P) is a "virtual black box," in the sense that anything one can efficiently compute given O(P), one could also efficiently compute given oracle access to P. In this work, we initiate a theoretical investigation of obfuscation. Our main result is that, even under very weak formalizations of the above intuition, obfuscation is impossible. We prove this by constructing a family of functions F that are inherently unobfuscatable in the following sense: there is a property π : F → {0, 1} such that (a) given any program that computes a function f ∈ F, the value π(f) can be efficiently computed, yet (b) given oracle access to a (randomly selected) function f ∈ F, no efficient algorithm can compute π(f) much better than random guessing. We extend our impossibility result in a number of ways, including even obfuscators that (a) are not necessarily computable in polynomial time, (b) only approximately preserve the functionality, and (c) only need to work for very restricted models of computation (TC0). We also rule out several potential applications of obfuscators, by constructing "unobfuscatable" signature schemes, encryption schemes, and pseudorandom function families.

Abstract: We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of \dRd It is assumed that function evaluations are expensive and that no additional information is available Radial basis function interpolation is used to define a utility function The maximizer of this function is the next point where the objective function is evaluated We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm A general framework for both methods is presented Finally, a few numerical examples show that on the set of Dixon-Szego test functions our method yields favourable results in comparison to other global optimization methods

Abstract: This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift-invariant subspaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p-spaces.

Abstract: We introduce and analyze a simulation-based approximate dynamic programming method for pricing complex American-style options, with a possibly high-dimensional underlying state space. We work within a finitely parameterized family of approximate value functions, and introduce a variant of value iteration, adapted to this parametric setting. We also introduce a related method which uses a single (parameterized) value function, which is a function of the time-state pair, as opposed to using a separate (independently parameterized) value function for each time. Our methods involve the evaluation of value functions at a finite set, consisting of "representative" elements of the state space. We show that with an arbitrary choice of this set, the approximation error can grow exponentially with the time horizon (time to expiration). On the other hand, if representative states are chosen by simulating the state process using the underlying risk-neutral probability distribution, then the approximation error remains bounded.

Abstract: This work introduces a continuous smooth permittivity function into Poisson–Boltzmann techniques for continuum approaches to modeling the solvation of small molecules and proteins. The permittivity function is derived using a Gaussian method to describe volume exclusion. The new method allows a rigorous determination of solvent forces within a grid-based technology. The generality of approach is demonstrated by considering a range of applications for small molecules and macromolecules. We also present a very complete statistical analysis of grid errors, and show that the accuracy of our Gaussian-based method is improved over standard techniques. The method has been implemented in a new code called ZAP, which is freely available to academic institutions.1 © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 608–640, 2001

Abstract: This paper is concerned with mean-variance portfolio selection problems in continuous-time under the constraint that short-selling of stocks is prohibited. The problem is formulated as a stochastic optimal linear-quadratic (LQ) control problem. However, this LQ problem is not a conventional one in that the control (portfolio) is constrained to take nonnegative values due to the no-shorting restriction, and thereby the usual Riccati equation approach (involving a "completion of squares") does not apply directly. In addition, the corresponding Hamilton--Jacobi--Bellman (HJB) equation inherently has no smooth solution. To tackle these difficulties, a continuous function is constructed via two Riccati equations, and then it is shown that this function is a viscosity solution to the HJB equation. Solving these Riccati equations enables one to explicitly obtain the efficient frontier and efficient investment strategies for the original mean-variance problem. An example illustrating these results is also presented.

Abstract: Suppose that under a semiparametric setting an estimator of a vector of parameters of interest is obtained by optimising an objective function which has a U-process structure. The covariance matrix of the estimator is generally a function of the underlying density function, which may be difficult to estimate well by conventional methods. In this paper, we present a simple resampling method by perturbing the objective function repeatedly. Inferences of the parameters can then be made based on a large collection of the resulting optimisers. We illustrate our proposal by three examples with a heteroscedastic regression model.

Abstract: Physical models of various phenomena are often represented by a mathematical model where the output(s) of interest have a multivariate dependence on the inputs Frequently, the underlying laws governing this dependence are not known and one has to interpolate the mathematical model from a finite number of output samples Multivariate approximation is normally viewed as suffering from the curse of dimensionality as the number of sample points needed to learn the function to a sufficient accuracy increases exponentially with the dimensionality of the function However, the outputs of most physical systems are mathematically well behaved and the scarcity of the data is usually compensated for by additional assumptions on the function (ie, imposition of smoothness conditions or confinement to a specific function space) High dimensional model representations (HDMR) are a particular family of representations where each term in the representation reflects the individual or cooperative contributions of the inputs upon the output The main assumption of this paper is that for most well defined physical systems the output can be approximated by the sum of these hierarchical functions whose dimensionality is much smaller than the dimensionality of the output This ansatz can dramatically reduce the sampling effort in representing the multivariate function HDMR has a variety of applications where an efficient representation of multivariate functions arise with scarce data The formulation of HDMR in this paper assumes that the data is randomly scattered throughout the domain of the output Under these conditions and the assumptions underlying the HDMR it is argued that the number of samples needed for representation to a given tolerance is invariant to the dimensionality of the function, thereby providing for a very efficient means to perform high dimensional interpolation Selected applications of HDMR's are presented from sensitivity analysis and time-series analysis

Abstract: In this paper we study the existence and stability of asymp- totically large stationary multi-pulse solutions in a family of singularly perturbed reaction-diVusion equations. This family includes the gen- eralized Gierer-Meinhardt equation. The existence of N-pulse homo- clinic orbits (N 1) is established by the methods of geometric singular perturbation theory. A theory, called the NLEP (=NonLocal Eigenvalue Problem) approach, is developed, by which the stability of these patterns can be studied explicitly. This theory is based on the ideas developed in our earlier work on the Gray-Scott model. It is known that the Evans function of the linear eigenvalue problem associated to the stability of the pattern can be decomposed into the product of a slow and a fast transmission function. The NLEP approach determines explicit leading order approximations of these transmission functions. It is shown that the zero/pole cancellation in the decomposition of the Evans function, called the NLEP paradox, is a phenomenon that occurs naturally in sin- gularly perturbed eigenvalue problems. It follows that the zeroes of the Evans function, and thus the spectrum of the stability problem, can be studied by the slow transmission function. The key ingredient of the analysis of this expression is a transformation of the associated nonlocal eigenvalue problem into an inhomogeneous hypergeometric diVerential equation. By this transformation it is possible to determine both the number and the position of all elements in the discrete spectrum of the linear eigenvalue problem. The method is applied to a special case that corresponds to the classical model proposed by Gierer and Meinhardt. It is shown that the one-pulse pattern can gain (or lose) stability through a Hopf bifurcation at a certain valueHopf of the main parameter. The NLEP approach not only yields a leading order approximation ofHopf , but it also shows that there is another bifurcation value,edge, at which a new (stable) eigenvalue bifurcates from the edge of the essential spec- trum. Finally, it is shown that theN-pulse patterns are always unstable whenN 2.

Abstract: An important aspect of the physical sciences is to make inferences about physical parameters from data. In general, the laws of physics provide the means for computing the data values given a model. This is called the “forward problem”, see figure 1. In the inverse problem, the aim is to reconstruct the model from a set of measurements. In the ideal case, an exact theory exists that prescribes how the data should be transformed in order to reproduce the model. For some selected examples such a theory exists assuming that the required infinite and noise-free data sets would be available. A quantum mechanical potential in one spatial dimension can be reconstructed when the reflection coefficient is known for all energies [Marchenko, 1955; Burridge, 1980]. This technique can be generalized for the reconstruction of a quantum mechanical potential in three dimensions [Newton, 1989], but in that case a redundant data set is required for reasons that are not well understood. The mass-density in a one-dimensional string can be constructed from the measurements of all eigenfrequencies of that string [Borg,1946], but due to the symmetry of this problem only the even part of the mass-density can be determined. If the seismic velocity in the earth depends only on depth, the velocity can be constructed exactly from the measurement of the arrival time as a function of distance of seismic waves using an Abel transform [Herglotz, 1907; Wiechert, 1907]. Mathematically this problem is identical to the construction of a spherically symmetric quantum mechanical potential in three dimensions [Keller et al., 1956]. However, the construction method of Herglotz-Wiechert only gives an unique result when the velocity increases monotonically with depth [Gerver and Markushevitch, 1966]. This situation is similar in quantum mechanics where a radially symmetric potential can only be constructed uniquely when the potential does not have local minima [Sabatier, 1973].

Abstract: A system for locating a number of devices (112-130) by measuring signals transmitted between known location devices (112-118, 134-138, 214-218, 224-228) and unknown location devices (120-130, 222), and signals transmitted between pairs of unknown location devices (120-130,222), entering signal measurements into a graph function that includes a number of first sub-expressions, a number of which include signal measurement prediction sub-expressions, and have extrema when a predicted signal measurement is equal to an actual signal measurement, and optimizing the graph function.

Abstract: A technique entitled robust baseline estimation is introduced, which uses techniques of robust local regression to estimate baselines in spectra that consist of sharp features superimposed upon a continuous, slowly varying baseline. The technique is applied to synthetic spectra, to evaluate its capabilities, and to laser-induced fluorescence spectra of OH (produced from the reaction of ozone with hydrogen atoms). The latter example is a particularly challenging case for baseline estimation because the experimental noise varies as a function of frequency.

Abstract: Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the computationally hard ~NP-hard! graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of run time and system size. The method has a single free parameter, which we determine numerically and justify using a simple argument. On random graphs, our numerical results demonstrate that extremal optimization maintains consistent accuracy for increasing system sizes, with an approximation error decreasing over run time roughly as a power law t 20.4 . On geometrically structured graphs, the scaling of results from the average run suggests that these are far from optimal with large fluctuations between individual trials. But when only the best runs are considered, results consistent with theoretical arguments are recovered.

Abstract: Preface 1. Introduction 2. Basic theory 3. Theory development 4. The SL-integral 5. Generalized AC function 6. Integration in several dimensions 7. Some applications 8. List of symbols Appendices.

Abstract: A new noniterative frequency domain parameter estimation technique is proposed. It is based on a weighted total least squares approach, starting from multiple input multiple output frequency response functions. One of thespecific advantages of the technique lies in the very stable identification of the system poles as a function of the specified system order leading to easy-to-interpret stabilization diagrams. This implies a potential for automating the method and to apply it to "difficult" estimation cases. Several real-life case studies are discussed, one related to holographic modal analysis in the medium frequency range, one to the modal testing of a fully trimmed vehicle.

Abstract: We use the Sampson and Guttorp approach to model the non-stationary correlation function r(x, x′) of a Gaussian spatial process through a bijective space deformation, f, so that in the deformed space the spatial correlation function can be considered isotropic, namely r(x, x′) = ρ(∣ f(x)−f(x′)∣), where ρ belongs to a known parametric family. Given the locations in the deformed space of a number of geographic sites at which data are available, we smoothly extrapolate the deformation to the whole region of interest. Using a Bayesian framework, we estimate jointly these locations, as well as the parameters of the correlation function and the variance parameters. The advantage of our Bayesian approach is that it allows us to obtain measures of uncertainty of all these parameters. As the parameter space is of a very high dimension, we implement an MCMC method for obtaining samples from the posterior distributions of interest. We demonstrate our method through a simulation study, and show an application to a real data set. Copyright © 2001 John Wiley & Sons, Ltd.

Abstract: An important issue in the design and implementation of a neural network is the sensitivity of its output to input and weight perturbations. In this paper, we discuss the sensitivity of the most popular and general feedforward neural networks-multilayer perceptron (MLP). The sensitivity is defined as the mathematical expectation of the output errors of the MLP due to input and weight perturbations with respect to all input and weight values in a given continuous interval. The sensitivity for a single neuron is discussed first and an analytical expression that is a function of the absolute values of input and weight perturbations is approximately derived. Then an algorithm is given to compute the sensitivity for the entire MLP. As intuitively expected, the sensitivity increases with input and weight perturbations, but the increase has an upper bound that is determined by the structural configuration of the MLP, namely the number of neurons per layer and the number of layers. There exists an optimal value for the number of neurons in a layer, which yields the highest sensitivity value. The effect caused by the number of layers is quite unexpected. The sensitivity of a neural network may decrease at first and then almost keeps constant while the number increases.

Abstract: In this paper we propose a novel schedulability analysis for verifying the feasibility of large periodic task sets under the rate monotonic algorithm, when the exact test cannot be applied on line due to prohibitively long execution times. The proposed test has the same complexity as the original Liu and Layland bound but it is less pessimistic, so allowing to accept task sets that would be rejected using the original approach. The performance of the proposed approach is evaluated with respect to the classical Liu and Layland method, and theoretical bounds are derived as a function of n (the number of tasks) and for the limit case of n tending to infinity. The analysis is also extended to include aperiodic servers and blocking times due to concurrency control protocols. Extensive simulations on synthetic tasks sets are presented to compare the effectiveness of the proposed test with respect to the Liu and Layland method and the exact response time analysis.

Abstract: This paper is concerned with filled function techniques for unconstrained global minimization of a continuous function of several variables. More general forms of filled functions are presented for smooth and non-smooth optimization problems. These functions have either one or two adjustable parameters. Conditions on functions and on the values of parameters are given so that the constructed functions have the desired properties of filled functions.

Abstract: Let ℜ be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f: [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: If G is a proper nontrivial subgroup of (R, *) definable in ℜ, then either G is dense and codense in R, or G contains an element u such that (R, <, *, e, u, G) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), where e denotes the identity element of (R, *) (Theorem 2.3).Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).

Abstract: This paper develops a unified approach to the analysis and design of adaptive filters with error nonlinearities. In particular, the paper performs stability and steady-state analysis of this class of filters under weaker conditions than what is usually encountered in the literature, and without imposing any restriction on the color or statistics of the input. The analysis results are subsequently used to derive an expression for the optimum nonlinearity, which turns out to be a function of the probability density function of the estimation error. Some common nonlinearities are shown to be approximations to the optimum nonlinearity. The framework pursued here is based on energy conservation arguments.

Abstract: The robust nonlinear output regulation problem was first solved under a polynomial condition on an input feedforward function. Another condition was given later which appears less restrictive than the first one. In this note, we will show that both these two conditions lead to the same sufficient condition that the input feedforward function along the trajectories of the exosystem is a sum of finitely many harmonics, or what is called trigonometric polynomial.

Abstract: We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height ${M>0$ cite{butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite{lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with~$n$ sides centered in the disc, and numerical experiments indicate that the natural number $ngeq2$ is a non-decreasing function of $M$. The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height~$M$.

Abstract: Recently, the beta function has been shown to be an accurate mathematical model of the human dental arch. In this research, we tried to find the equation of a curve that would be similar to the generalized beta function curve and at the same time could represent tapered, ovoid, and square dental arches. A total of 23 sets of naturally well-aligned Class I casts were selected, and the depths and widths of the dental arches were measured at the canine and second molar regions. Using the mean depths and widths, functions in the form of Y 5 AX m 1 BX n were calculated that would pass through the central incisors, canines, and second molars. Each function was compared with the generalized beta function with the use of root mean square values. It was shown that the polynomial function Y 5 AX 6 1 BX 2 was the nearest to the generalized beta function. Then the coordinates of the midincisal edges and buccal cusp tips of each dental arch were measured, and the correlation coefficient of each dental arch with its corresponding sixth order polynomial function was calculated. The results showed that the function Y 5 AX 6 1 BX 2 could be an accurate substitute for the beta function in less common forms of the human dental arch. (Angle Orthod 2001;71:386-389.)