Showing papers on "Bounded function published in 1998"

Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry

Abstract: The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of $\mathrm{PT}$ symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These $\mathrm{PT}$ symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories.

Stability of Functional Equations in Several Variables

Abstract: Approximately additive and approximately linear mappings stability of the quadratic functional equation generalizations - the method of invariant means approximately multiplicative mappings - superstability stability of functional equations for trigonometric and similar functions functions with bounded nth differences approximately convex functions stability of the generalized orthogonality functional equation stability and set-valued mappings stability of stationary and minimum points functional congruences quasi-additive functions and related topics.

The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network

Abstract: Sample complexity results from computational learning theory, when applied to neural network learning for pattern classification problems, suggest that for good generalization performance the number of training examples should grow at least linearly with the number of adjustable parameters in the network. Results in this paper show that if a large neural network is used for a pattern classification problem and the learning algorithm finds a network with small weights that has small squared error on the training patterns, then the generalization performance depends on the size of the weights rather than the number of weights. For example, consider a two-layer feedforward network of sigmoid units, in which the sum of the magnitudes of the weights associated with each unit is bounded by A and the input dimension is n. We show that the misclassification probability is no more than a certain error estimate (that is related to squared error on the training set) plus A/sup 3/ /spl radic/((log n)/m) (ignoring log A and log m factors), where m is the number of training patterns. This may explain the generalization performance of neural networks, particularly when the number of training examples is considerably smaller than the number of weights. It also supports heuristics (such as weight decay and early stopping) that attempt to keep the weights small during training. The proof techniques appear to be useful for the analysis of other pattern classifiers: when the input domain is a totally bounded metric space, we use the same approach to give upper bounds on misclassification probability for classifiers with decision boundaries that are far from the training examples.

Identification of genetic networks from a small number of gene expression patterns under the Boolean network model.

Abstract: Liang, Fuhrman and Somogyi (PSB98, 18-29, 1998) have described an algorithm for inferring genetic network architectures from state transition tables which correspond to time series of gene expression patterns, using the Boolean network model. Their results of computational experiments suggested that a small number of state transition (INPUT/OUTPUT) pairs are sufficient in order to infer the original Boolean network correctly. This paper gives a mathematical proof for their observation. Precisely, this paper devises a much simpler algorithm for the same problem and proves that, if the indegree of each node (i.e., the number of input nodes to each node) is bounded by a constant, only O(log n) state transition pairs (from 2n pairs) are necessary and sufficient to identify the original Boolean network of n nodes correctly with high probability. We made computational experiments in order to expose the constant factor involved in O(log n) notation. The computational results show that the Boolean network of size 100,000 can be identified by our algorithm from about 100 INPUT/OUTPUT pairs if the maximum indegree is bounded by 2. It is also a merit of our algorithm that the algorithm is conceptually so simple that it is extensible for more realistic network models.

Perfect Incompressible Fluids

Abstract: Introduction 1. Presentation of the equations 2. Littlewood-Paley theory 3. Around Biot-Savart's law 4. The case of a smooth initial data 5. When the vorticity is bounded 6. Vortex sheets 7. The wave front and the product 8. Analyticity and Gevrey regularity 9. Singular vortex patches References

Upper bounds on the number of hidden neurons in feedforward networks with arbitrary bounded nonlinear activation functions

Abstract: It is well known that standard single-hidden layer feedforward networks (SLFNs) with at most N hidden neurons (including biases) can learn N distinct samples (x/sub i/,t/sub i/) with zero error, and the weights connecting the input neurons and the hidden neurons can be chosen "almost" arbitrarily. However, these results have been obtained for the case when the activation function for the hidden neurons is the signum function. This paper rigorously proves that standard single-hidden layer feedforward networks (SLFNs) with at most N hidden neurons and with any bounded nonlinear activation function which has a limit at one infinity can learn N distinct samples (x/sub i/,t/sub i/) with zero error. The previous method of arbitrarily choosing weights is not feasible for any SLFN. The proof of our result is constructive and thus gives a method to directly find the weights of the standard SLFNs with any such bounded nonlinear activation function as opposed to iterative training algorithms in the literature.

Conjunctive-query containment and constraint satisfaction

Abstract: Conjunctive-query containment is recognized as a fundamental problem in database query evaluation and optimization. At the same time, constraint satisfaction is recognized as a fundamental problem in artificial intelligence. What do conjunctive-query containment and constraint satisfaction have in common? Our main conceptual contribution in this paper is to point out that, despite their very different formulation, conjunctive-query containment and constraint satisfaction are essentially the same problem. The reason is that they can be recast as the following fundamental algebraic problem: given two finite relational structures A and B, is there a homomorphism h:A?B? As formulated above, the homomorphism problem is uniform in the sense that both relational structures A and B are part of the input. By fixing the structure B, one obtains the following nonuniform problem: given a finite relational structure A, is there a homomorphism h:A?B? In general, nonuniform tractability results do not uniformize. Thus, it is natural to ask: which tractable cases of nonuniform tractability results for constraint satisfaction and conjunctive-query containment do uniformize? Our main technical contribution in this paper is to show that several cases of tractable nonuniform constraint-satisfaction problems do indeed uniformize. We exhibit three nonuniform tractability results that uniformize and, thus, give rise to polynomial-time solvable cases of constraint satisfaction and conjunctive-query containment. We begin by examining the tractable cases of Boolean constraint-satisfaction problems and show that they do uniformize. This can be applied to conjunctive-query containment via Booleanization; in particular, it yields one of the known tractable cases of conjunctive-query containment. After this, we show that tractability results for constraint-satisfaction problems that can be expressed using Datalog programs with bounded number of distinct variables also uniformize. Finally, we provide a new proof for the fact that tractability results for queries with bounded treewidth uniformize as well, via a connection with first-order logic with a bounded number of distinct variables.

Decoupling: From Dependence to Independence

Abstract: 1 Sums of Independent Random Variables.- 1.1 Levy-Type Maximal Inequalities.- 1.2 Hoffmann-J?rgensen Type Inequalities.- 1.3 The Khinchin-Kahane Inequalities.- 1.4 Moment Bounds.- 1.4.1 Maxima.- 1.4.2 Estimating La-Norms in Hilbert Space.- 1.4.3 K-Function Bounds.- 1.4.4 A General Decoupling for Sums of Arbitrary Positive Random Variables and Martingales.- 1.5 Estimates with Sharp Constants for the La-Norms of Sums of Independent Random Variables: The L-Function.- 1.6 References for Chapter 1.- 2 Randomly Stopped Processes With Independent Increments.- 2.1 Wald's Equations.- 2.2 Good-Lambda Inequalities.- 2.3 Randomly Stopped Sums of Independent Banach-Valued Variables.- 2.4 Proof of the Lower Bound of Theorem 2.3.1.- 2.5 Continuous Time Processes.- 2.6 Burkholder-Gundy Type Inequalities in Banach Spaces.- 2.7 From Boundary Crossing of Nonrandom Functions to First Passage Times of Processes with Independent Increments.- 2.8 References for Chapter 2.- 3 Decoupling of U-Statistics and U-Processes.- 3.1 Decoupling of U-Processes: Convex Functions.- 3.2 Hypercontractivity of Rademacher Chaos Variables.- 3.3 Minorization of Tail Probabilities: The Paley-Zygmund Argument and a Conditional Jensen's Inequality.- 3.4 Decoupling of U-processes: Tail Probabilities.- 3.5 Randomization136.- 3.5.1 Moment Inequalities for Randomized U-Statistics139.- 3.5.2 Randomization of Tail Probabilities for U-Statistics and Processes.- 3.6 References for Chapter 3.- 4 Limit Theorems for U-Statistics.- 4.1 Some Inequalities the Law of Large Numbers.- 4.1.1 Hoffmann-J?rgensen's Inequality for U-Processes.- 4.1.2 An Application to the Law of Large Numbers.- 4.1.3 Exponential Inequalities for Canonical U-Statistics.- 4.2 Gaussian Chaos and the Central Limit Theorem for Canonical U-Statistics.- 4.3 The Law of the Iterated Logarithm for Canonical U-Statistics.- 4.4 References for Chapter 4.- 5 Limit Theorems for U-Processes.- 5.1 Some Background on Asymptotics of Processes, Metric Entropy, and Vapnik-?ervonenkis Classes of Functions: Maximal Inequalities.- 5.1.1 Convergence in Law of Sample Bounded Processes.- 5.1.2 Maximal Inequalities Based on Metric Entropy.- 5.1.3 Vapnik-?ervonenkis Classes of Functions.- 5.2 The Law of Large Numbers for U-Processes.- 5.3 The Central Limit Theorem for U-Processes.- 5.4 The Law of the Iterated Logarithm for Canonical U-Processes.- 5.4.1 The Bounded LIL.- 5.4.2 The Compact LIL.- 5.5 Statistical Applications.- 5.5.1 The Law of Large Numbers for the Simplicial Median.- 5.5.2 The Central Limit Theorem for the Simplicial Median.- 5.5.3 Truncated Data.- 5.6 References for Chapter 5.- 6 General Decoupling Inequalities for Tangent Sequences.- 6.1 Some Definitions and Examples.- 6.2 Exponential Decoupling Inequalities for Sums.- 6.3 Tail Probability andLpInequalities for Tangent Sequences I.- 6.4 Tail Probability and Moment Inequalities for Tangent Sequences II: Good-Lambda Inequalities.- 6.5 Differential Subordination and Applications.- 6.6 Decoupling Inequalities Compared to Martingale Inequalities.- 6.7 References for Chapter 6323.- 7 Conditionally Independent Sequences.- 7.1 The Principle of Conditioning and Related Results.- 7.2 Analysis of a Sequence of Two-by-Two Tables.- 7.3 SharpLpComparison of Sums of Arbitrarily Dependent Variables to Sums of CI Variables.- 7.4 References for Chapter 7.- 8 Further Applications of Decoupling.- 8.1 Randomly Stopped Canonical U-Statistics.- 8.1.1 Wald's Equation for Canonical U-Statistics.- 8.1.2 Moment Bounds for Regular and Randomly StoppedU-Statistics.- 8.1.3 Moment Convergence in Anscombe's Theorem forU-Statistics.- 8.2 A General Class of Exponential Inequalities for Martingales and Ratios.- 8.3 References for Chapter 8.- References.

Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem

Abstract: A new technique to solve elliptic linear PDEs, called ultra weak variational formulation (UWVF) in this paper, is introduced in [B. Despres, C. R. Acad. Sci. Paris, 318 (1994), pp. 939--944]. This paper is devoted to an evaluation of the potentialities of this technique. It is applied to a model wave problem, the two-dimensional Helmholtz problem. The new method is presented in three parts following the same style of presentation as the classical one of the finite elements method, even though they are definitely conceptually different methods. The first part is committed to the variational formulation and to the continuous problem. The second part defines the discretization process using a Galerkin procedure. The third part actually studies the efficiency of the technique from the order of convergence point of view. This is achieved using theoretical proofs and a series of numerical experiments. In particular, it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom. An application to scattering problems is presented in a fourth part.

Set-membership filtering and a set-membership normalized LMS algorithm with an adaptive step size

Abstract: Set-membership identification (SMI) theory is extended to the more general problem of linear-in-parameters filtering by defining a set-membership specification, as opposed to a bounded noise assumption. This sets the framework for several important filtering problems that are not modeled by a "true" unknown system with bounded noise, such as adaptive equalization, to exploit the unique advantages of SMI algorithms. A recursive solution for set membership filtering is derived that resembles a variable step size normalized least mean squares (NLMS) algorithm. Interesting properties of the algorithm, such as asymptotic cessation of updates and monotonically non-increasing parameter error, are established. Simulations show significant performance improvement in varied environments with a greatly reduced number of updates.

Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results

Abstract: We prove some weak and strong comparison theorems for solutions of differential inequalities involving a class of elliptic operators that includes the p-laplacian operator. We then use these theorems together with the method of moving planes and the sliding method to get symmetry and monotonicity properties of solutions to quasilinear elliptic equations in bounded domains.

Multi-peak bound states for nonlinear Schrödinger equations

Abstract: In this paper we consider the study of standing wave solutions for a nonlinear Schrodinger equation. This problem reduces to that of finding nonnegative solutions of ɛ 2 Δ u - V ( x ) u + f ( u ) = 0 in Ω , with finite energy. Here ɛ is a small parameter, ω is a smooth, possibly unbounded domain, f is an appropriate superlinear function, and V is a positive potential, bounded away from zero. It is the purpose of this article to obtain multi-peak solutions in the “multiple well case”. We find solutions exhibiting concentration at any prescribed finite set of local minima, possibly degenerate, of the potential. The proof relies on variational arguments, where a penalization-type method is developed for the identification of the desired solutions.

Global Behaviour of a Reaction‐Diffusion System Modelling Chemotaxis

Abstract: Using Lyapunov functionals the global behaviour of the solutions of a reaction-diffusion system modelling chemotaxis is studied for bounded piecewise smooth domains in the plane. Geometric criteria can be given so that this dynamical system tends to a (not necessarily trivial) stationary state.

One-dimensional transport equations with discontinuous coefficients

Abstract: We consider one-dimensional linear transport equations with bounded but possibly discontinuous coefficient a. The Cauchy problem is studied from two different points of view. In the first case we assume that a is piecewise continuous. We give an existence result and a precise description of the solutions on the lines of discontinuity. In the second case, we assume that a satisfies a one-sided Lipschitz condition. We give existence, uniqueness and general stability results for backward Lipschitz solutions and forward measure solutions, by using a duality method. We prove that the flux associated to these measure solutions is a product by some canonical representative â of a. Key-words. Linear transport equations, discontinuous coefficients, weak stability, duality, product of a measure by a discontinuous function, nonnegative solutions. 1991 Mathematics Subject Classification. Primary 35F10, 35B35, 34A12. To appear in Nonlinear Analysis, TMA ∗Departement de Mathematiques et Applications, UMR CNRS 8553, Ecole Normale Superieure et CNRS, 45 rue d’Ulm, 75230 Paris Cedex 05, France, francois.bouchut@ens.fr †Mathematiques, Applications et Physique Mathematique d’Orleans, UMR CNRS 6628, Universite d’Orleans, 45067 Orleans Cedex 2, France, james@math.cnrs.fr

Lipschitz stability in inverse parabolic problems by the Carleman estimate

Abstract: We consider a system with a suitable boundary condition, where is a bounded domain, is a uniformly elliptic operator of the second order whose coefficients are suitably regular for , is fixed, and a function satisfies Our inverse problems are determinations of g using overdetermining data or , where and . Our main result is the Lipschitz stability in these inverse problems. We also consider the determination of , in the case of with given R satisfying on . Finally, we discuss an upper estimation of our overdetermining data by means of f.

On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions

Abstract: The vanishing viscosity limit is considered for the incompressible 2D Navier-Stokes equations in a bounded domain. Motivated by studies of turbulent flow we suppose Navier's friction condition in the tangential direction, i.e. the creation of a vorticity proportional to the tangential velocity. We prove the existence of the regular solutions for the Navier-Stokes equations with smooth compatible data and of the solutions with bounded vorticity for initial vorticity being only bounded. Finally, we establish a uniform -bound for the vorticity and convergence to the incompressible 2D Euler equations in the inviscid limit.

On Some Tighter Inapproximability Results

Abstract: We give a number of improved inapproximability results, including the best up to date explicit approximation thresholds for bounded occurrence satisfiability problems like MAX-2SAT and E2-LIN-2, and the bounded degree graph problems, like MIS, NodeCover, and MAX CUT. We prove also for the first time inapproximability of the problem of Sorting by Reversals and display an explicit approximation threshold.

On Some Tighter Inapproximability Results

Abstract: We give a number of improved inapproximability results, including the best up to date explicit approximation thresholds for bounded occurrence satisfiability problems like MAX-2SAT and E2-LIN-2, and the bounded degree graph problems, like MIS, NodeCover, and MAX CUT We prove also for the first time inapproximability of the problem of Sorting by Reversals and display an explicit approximation threshold

Description of the lack of compactness for the Sobolev imbedding

Abstract: We prove that any bounded sequence in a Hilbert homogeneous Sobolev space has a subsequence which can be decomposed as an almost-orthogonal sum of a sequence going strongly to zero in the corresponding Lebesgue space, and of a superposition of terms obtained from fixed profiles by applying sequences of translations and dilations. This decomposition contains in particular the various versions of the concentration-compactness principle.

Multivalued perturbations ofm-accretive differential inclusions

Abstract: Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2 X , we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x 0. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L 1([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ4 in which no integral solution exists.

Quasi-exactly solvable quartic potential

Abstract: A new two-parameter family of quasi-exactly solvable quartic polynomial potentials is introduced Heretofore, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial potential is sextic This belief is based on the assumption that the Hamiltonian must be Hermitian However, it has recently been discovered that there are huge classes of non-Hermitian, -symmetric Hamiltonians whose spectra are real, discrete, and bounded below Replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi-exactly solvable theories The spectra of this family of quartic potentials discussed here are also real, discrete and bounded below and the quasi-exact portion of the spectra consists of the lowest J eigenvalues These eigenvalues are the roots of a Jth-degree polynomial

Long-wave instabilities and saturation in thin film equations

Abstract: Hocherman and Rosenau conjectured that long-wave unstable Cahn-Hilliard type interface models develop nite-time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term (Phys. D 67:113-125, 1993). We consider this conjecture for a class of equations, often used to model thin lms in a lubrication context, by showing that if the solutions are uniformly bounded above or below (e.g. are nonnegative) then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded. For example, they conjecture that for the long-wave unstable equation ht = (h n hxxx)x (h m hx)x; m>n leads to blow-up. Using a conservation of volume constraint for the case m> n> 0, we conjecture a dierent critical exponent for possible singularities of nonnegative solutions. We prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions. Specically, for the above equation, solutions are bounded if m

How to learn an unknown environment. I: the rectilinear case

Abstract: We consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it. We seek algorithms that achieve a bounded ratio of the worst-case distance traversed in order to see all visible points of the environment (thus creating a map), divided by the optimum distance needed to verify the map, if we had it in the beginning. The situation is complicated by the fact that the latter off-line problem (the problem of optimally verifying a map) is NP-hard. Although we show that there is no such “competitive” algorithm for general obstacle courses, we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it. We restrict ourselves to the rectilinear case, where each side of the obstacles and the room is parallel to one of the coordinates, and the robot must also move either parallel or perpendicular to the sides. (In a subsequent paper, we will discuss the extension to polygons of general shapes.)We also discuss the off-line problem for simple rectilinear polygons and find an optimal solution (in the L1 metric) in polynomial time, in the case where the entry and the exit are different points.

An Incremental Gradient(-Projection) Method with Momentum Term and Adaptive Stepsize Rule

Abstract: We consider an incremental gradient method with momentum term for minimizing the sum of continuously differentiable functions. This method uses a new adaptive stepsize rule that decreases the stepsize whenever sufficient progress is not made. We show that if the gradients of the functions are bounded and Lipschitz continuous over a certain level set, then every cluster point of the iterates generated by the method is a stationary point. In addition, if the gradient of the functions have a certain growth property, then the method is either linearly convergent in some sense or the stepsizes are bounded away from zero. The new stepsize rule is much in the spirit of heuristic learning rules used in practice for training neural networks via backpropagation. As such, the new stepsize rule may suggest improvements on existing learning rules. Finally, extension of the method and the convergence results to constrained minimization is discussed, as are some implementation issues and numerical experience.

Analysis of Concentration and Oscillation Effects Generated by Gradients

Abstract: A general theorem characterizing the interaction of concentrations and oscillations effects associated with sequences of gradients bounded in Lp, t p > 1, is proved. The oscillations are recorded in the Young measure while the concentrations are encoded in the varifold.

Local and global results for wave maps I

Abstract: We consider the initial value problem for wave-maps corresponding to constant coefficient second order hyperbolic equations in dimensions, . We prove that this problem is globally well-posed for initial data which is small in the homogeneous Besov space . Our second result deals with more regular solutions; it essentially says that if in addition the initial data is in then the solutions stay bounded in the same space. In part II of this work we shall prove that the same result holds in dimensions n = 2,3.

Parameter Estimation in the Presence of Bounded Data Uncertainties

Abstract: We formulate and solve a new parameter estimation problem in the presence of data uncertainties. The new method is suitable when a priori bounds on the uncertain data are available, and its solution leads to more meaningful results, especially when compared with other methods such as total least-squares and robust estimation. Its superior performance is due to the fact that the new method guarantees that the effect of the uncertainties will never be unnecessarily over-estimated, beyond what is reasonably assumed by the a priori bounds. A geometric interpretation of the solution is provided, along with a closed form expression for it. We also consider the case in which only selected columns of the coefficient matrix are subject to perturbations.

Unrectictifiable 1-sets have vanishing analytic capacity

Abstract: We complete the proof of a conjecture of Vitushkin that says that if E is a compact set in the complex plane with finite 1-dimensional Hausdorff measure, then E has vanishing analytic capacity (i.e., all bounded anlytic functions on the complement of E are constant) if and only if E is purely unrectifiable (i.e., the intersection of E with any curve of finite length has zero 1-dimensional Hausdorff measure). As in a previous paper with P. Mattila, the proof relies on a rectifiability criterion using Menger curvature, and an extension of a construction of M. Christ. The main new part is a generalization of the T(b)-theorem to some spaces that are non necessarily of homogeneous type.