Abstract: Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specified ν between 0 and 1. We propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. We provide a theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled data.

Abstract: We study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. We contend that the model displays a normal continuous phase transition with a divergent correlation length as the degree of randomness tends to zero. We propose a real-space renormalization group transformation for the model and demonstrate that the transformation is exact in the limit of large system size. We use this result to calculate the exact value of the single critical exponent for the system, and to derive the scaling form for the average number of "degrees of separation" between two nodes on the network as a function of the three independent variables. We confirm our results by extensive numerical simulation.

Abstract: A family of multivariate representations is introduced to capture the input–output relationships of high‐dimensional physical systems with many input variables. A systematic mapping procedure between the inputs and outputs is prescribed to reveal the hierarchy of correlations amongst the input variables. It is argued that for most well‐defined physical systems, only relatively low‐order correlations of the input variables are expected to have an impact upon the output. The high‐dimensional model representations (HDMR) utilize this property to present an exact hierarchical representation of the physical system. At each new level of HDMR, higher‐order correlated effects of the input variables are introduced. Tests on several systems indicate that the few lowest‐order terms are often sufficient to represent the model in equivalent form to good accuracy. The input variables may be either finite‐dimensional (i.e., a vector of parameters chosen from the Euclidean space $$\mathcal{R}^n$$ ) or may be infinite‐dimensional as in the function space $${\text{C}}^n \left[ {0,1} \right]$$ . Each hierarchical level of HDMR is obtained by applying a suitable projection operator to the output function and each of these levels are orthogonal to each other with respect to an appropriately defined inner product. A family of HDMRs may be generated with each having distinct character by the use of different choices of projection operators. Two types of HDMRs are illustrated in the paper: ANOVA‐HDMR is the same as the analysis of variance (ANOVA) decomposition used in statistics. Another cut‐HDMR will be shown to be computationally more efficient than the ANOVA decomposition. Application of the HDMR tools can dramatically reduce the computational effort needed in representing the input–output relationships of a physical system. In addition, the hierarchy of identified correlation functions can provide valuable insight into the model structure. The notion of a model in the paper also encompasses input–output relationships developed with laboratory experiments, and the HDMR concepts are equally applicable in this domain. HDMRs can be classified as non‐regressive, non‐parametric learning networks. Selected applications of the HDMR concept are presented along with a discussion of its general utility.

Abstract: Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the formWe assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.

Abstract: We propose a new projection algorithm for solving the variational inequality problem, where the underlying function is continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudomonotone). The method is simple and admits a nice geometric interpretation. It consists of two steps. First, we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem. This procedure requires a single projection onto the feasible set and employs an Armijo-type linesearch along a feasible direction. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. Thus, in contrast with most other projection-type methods, only two projection operations per iteration are needed. The method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions. Preliminary computational experience is also reported.

Abstract: Using the Bethe ansatz method, the zero frequency contribution (Drude weight) to the spin current correlations is analyzed for the easy plane antiferromagnetic Heisenberg model. The Drude weight is a monotonically decreasing function of temperature for all $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$; it approaches the zero temperature value with a power law and appears to vanish for all finite temperatures at the isotropic $\ensuremath{\Delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ point.

Abstract: k-particle cumulants λk (for 2⩽k⩽n) corresponding to the k-particle reduced density matrices γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ≡γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3−3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3−3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration self-consistent field wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk co...

Abstract: This work describes schemes for distributing between n servers the evaluation of a function f which is an approximation to a random function, such that only authorized subsets of servers are able to compute the function. A user who wants to compute f(x) should send x to the members of an authorized subset and receive information which enables him to compute f(x). We require that such a scheme is consistent, i.e. that given an input x all authorized subsets compute the same value f(x). The solutions we present enable the operation of many servers, preventing bottlenecks or single points of failure. There are also no single entities which can compromise the security of the entire network. The solutions can be used to distribute the operation of a Key Distribution Center (KDC). They are far better than the known partitioning to domains or replication solutions to this problem, and are especially suited to handle users of multicast groups.

Abstract: Using Mathematica 3.0, the Schrodinger equation for bound states is solved. The method of solution is based on a numerical integration procedure together with convexity arguments and the nodal theorem for wave functions. The interaction potential has to be spherically symmetric. The solving procedure is simply defined as some Mathematica function. The output is the energy eigenvalue and the reduced wave function, which is provided as an interpolated function (and can thus be used for the calculation of, e.g., moments by using any Mathematica built-in function) as well as plotted automatically. The corresponding program schroedinger.nb can be obtained from franz.schoeberl@univie.ac.at.

Abstract: We study the function and show that the poly-Bernoulli numbers introduced in our previous paper are expressed as special values at negative arguments of certain combinations of these functions. As a consequence of our study, we obtain a series of relations among multiple zeta values.

Abstract: Given a function/ ? ?2(0), Q '= (0, l)2 and a real number t > 0, let U(f,t) := infg?BV(g) 11/ ? ??ll/^/) + 'Vgig), where the infimum is taken over all functions g G BV of bounded variation on /. This and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation, as well as in digital image processing. Techniques for finding minimizers g for U(f, t) based on variational calculus and nonlinear partial differential equations have been put forward by several authors (DMS), (RO), (MS), (CL). The main disadvantage of these approaches is that they are numerically intensive. On the other hand, it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems, for example, the above problem with BV replaced by the Besov space B\(L\(I)) (see e.g. (CDLL)). However, since BV has no simple description in terms of wavelet coefficients, it is not clear that minimizers for U(f, t) can be realized in this way. We shall show in this paper that simple methods based on Haar thresholding provide near minimizers for U(f, t). Our analysis of this extremal problem brings forward many interesting relations between Haar decompositions and the space BV. 1. Introduction. Nonlinear approximation has recently played an impor tant role in several problems of image processing including compression, noise removal, and feature extraction. We have in mind techniques such as wavelet compression (DJL), wavelet shrinkage or thresholding (DJKP1), wavelet packets (CW), and greedy algorithms (MZ), (DT). There has also been an impressive contribution of techniques based on variational calculus and nonlinear partial dif ferential equations (see e.g. (DMS), (RO), (MS), (CL)) especially to the problems of noise removal and image segmentation. The common point between these two approaches is their ability to adapt to the composite nature of images: edge, tex tures and smooth regions should be treated adaptively, a requirement which is certainly not fulfilled by the classical linear filtering techniques.

Abstract: An accurate fundamental frequency (F0) estimation method for non-stationary, speech-like sounds is proposed based on the differential properties of the instantaneous frequencies of two sets of filter outputs. A specific type of fixed points of mapping from the filter center frequency to the output instantaneous frequency provides frequencies of the constituent sinusoidal components of the input signal. When the filter is made from an isometric Gabor function convoluted with a cardinal B-spline basis function, the differential properties at the fixed points provide practical estimates of the carrier-to-noise ratio of the corresponding components. These estimates are used to select the fundamental component and to integrate the F0 information distributed among the other harmonic components.

Abstract: In this paper we give a construction of a small approximately min-wise independent family of hash functions, i.e., a family of hash functions such that for any set of arguments X and x?X, the probability that the value of a random function from that family on x will be the smallest among all values of that function on X is roughly 1/|X|. The number of bits needed to represent each function is O(logn·log1/?). This construction gives a solution to the main open problem of A. Broder et al. (in “STOC'98”).

Abstract: We consider the wave equation damped with a boundary nonlinear velocity feedback p(u') . Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function ρ has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction of a special weight function (that depends on the behavior of the function ρ in zero), and on a new nonlinear integral inequality.

Abstract: The statistical features of turbulence can be studied either through spectral quantities, such as the kinetic energy spectrum, or through structure functions, which are statistical moments of the dierence between velocities at two points separated by a variable distance In this paper structure function relations for two-dimensional turbulence are derived and compared with calculations based on wind data from 5754 airplane flights, reported in the MOZAIC data set For the third-order structure function two relations are derived, showing that this function is generally positive in the twodimensional case, contrary to the three-dimensional case In the energy inertial range the third-order structure function grows linearly with separation distance and in the enstrophy inertial range it grows cubically with separation distance A Fourier analysis shows that the linear growth is a reflection of a constant negative spectral energy flux, and the cubic growth is a reflection of a constant positive spectral enstrophy flux Various relations between second-order structure functions and spectral quantities are also derived The measured second-order structure functions can be divided into two dierent types of terms, one of the form r 2=3 , giving a k 5=3 -range and another, including a logarithmic dependence, giving a k 3 -range in the energy spectrum The structure functions agree better with the two-dimensional isotropic relation for larger separations than for smaller separations The flatness factor is found to grow very fast for separations of the order of some kilometres The third-order structure function is accurately measured in the interval [30, 300] km and is found to be positive The average enstrophy flux is measured as ! 1:8 10 13 s 3 and the constant in the k 3 -law is measured asK0:19 It is argued that the k 3 -range can be explained by two-dimensional turbulence and can be interpreted as an enstrophy inertial range, while the k 5=3 -range can probably not be explained by two-dimensional turbulence and should not be interpreted as a two-dimensional energy inertial range

Abstract: There have been many applications of Hilbert curve, such as image processing, image compression, computer holo- gram, etc. The Hilbert curve is a one-to-one mapping between -dimensional space and one-dimensional (1-D) space which preserves point neighborhoods as much as possible. There are several algorithms for -dimensional Hilbert scanning, such as the Butz algorithm and the Quinqueton algorithm. The Butz algorithm is a mapping function using several bit operations such as shifting, exclusive OR, etc. On the other hand, the Quinqueton algorithm computes all addresses of this curve using recursive functions, but takes time to compute a one-to-one mapping correspondence. Both algorithms are complex to compute and both are difficult to implement in hardware. In this paper, we propose a new, simple, nonrecursive algorithm for -dimensional Hilbert scanning using look-up tables. The merit of our algorithm is that the computation is fast and the implementation is much easier than previous ones.

Abstract: The concept of M-convex function, introduced by Murota 1996, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of Dress and Wenzel 1990. In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete optimization problems.

Abstract: In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain \Om in Euclidean n-space, k=1,...,n, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider k-convex functions, not necessarily continuous, and prove the weak continuity of the associated k-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the gradients of k-convex functions.

Abstract: The logic of equality with uninterpreted functions (EUF) provides a means of abstracting the manipulation of data by a processor when verifying the correctness of its control logic. By reducing formulas in this logic to propositional formulas, we can apply Boolean methods such as Ordered Binary Decision Diagrams (BDDs) and Boolean satisfiability checkers to perform the verification. We can exploit characteristics of the formulas describing the verification conditions to greatly simplify the propositional formulas generated. In particular, we exploit the property that many equations appear only in positive form. We can therefore reduce the set of interpretations of the function symbols that must be considered to prove that a formula is universally valid to those that are ``maximally diverse.'' We present experimental results demonstrating the efficiency of this approach when verifying pipelined processors using the method proposed by Burch and Dill.

Abstract: This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials Bn are monotonic in n, as in the classical case.

Abstract: The authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous-time linear systems affected by time-varying memoryless parametric uncertainties. They consider as candidate Lyapunov functions the elements of the class /spl Sigma//sub p//sup z/ which is formed by special homogeneous positive definite functions. They show that this class is universal in the sense that a Lyapunov function exists if and only if there exists a Lyapunov function in /spl Sigma//sub p//sup z/. They prove this result in a constructive way, showing that such a Lyapunov function can always be obtained by "smoothing" a polyhedral function for which construction algorithms are available. The authors show that unlike the polyhedral Lyapunov functions, these functions allow us to derive explicit formulas for the stabilizing controller.

Abstract: In this paper, we introduce the concept of a comparison function, which is a mapping g that assigns numbers to ordered pairs of alternatives (x,y) with the property that g(x,y)=−g(y,x). The paper discusses how some well-known choice correspondences on tournaments such as the uncovered set, the minimal covering set and the bipartisan set can be extended to this general framework. Axiomatic characterizations and properties are studied for these correspondences.

Abstract: Optimization is an important operation in many domains of science and technology. Local optimization techniques typically employ some form of iterative procedure, based on derivatives of the function to be optimized (objective function). These techniques normally involve parameters that must be set by the user, often by trial and error. Those parameters can have a strong influence on the convergence speed of the optimization. In several cases, a significant speed advantage could be gained if one could vary these parameters during the optimization, to reflect the local characteristics of the function being optimized. Some parameter adaptation methods have been proposed for this purpose, for deterministic optimization situations. For stochastic (also called on-line) optimization situations, there appears to be no simple and effective parameter adaptation method. This paper proposes a new method for parameter adaptation in stochastic optimization. The method is applicable to a wide range of objective functions, as well as to a large set of local optimization techniques. We present the derivation of the method, details of its application to gradient descent and to some of its variants, and examples of its use in the gradient optimization of several functions, as well as in the training of a multilayer perceptron by on-line backpropagation. Introduction Optimization is an operation that is often used in several different domains of science and technology. It normally consists of maximizing or minimizing a given function (called objective function ), that is chosen to represent the quality of a given system. The system may be physical, (mechanical, chemical, etc.), a mathematical model, a computer program, etc., or even a mixture of several of these.