Showing papers on "Bounded function published in 1999"

Symbolic Model Checking without BDDs

Abstract: Symbolic Model Checking [3, 14] has proven to be a powerful technique for the verification of reactive systems. BDDs [2] have traditionally been used as a symbolic representation of the system. In this paper we show how boolean decision procedures, like Stalmarck's Method [16] or the Davis & Putnam Procedure [7], can replace BDDs. This new technique avoids the space blow up of BDDs, generates counterexamples much faster, and sometimes speeds up the verification. In addition, it produces counterexamples of minimal length. We introduce a bounded model checking procedure for LTL which reduces model checking to propositional satisfiability. We show that bounded LTL model checking can be done without a tableau construction. We have implemented a model checker BMC, based on bounded model checking, and preliminary results are presented.

On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN

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.

A separation principle for the stabilization of a class of nonlinear systems

Abstract: It is shown that the performance of a globally bounded partial state feedback control of a certain class of nonlinear systems can be recovered by a sufficiently fast high-gain observer. The performance recovery includes recovery of asymptotic stability of the origin, the region of attraction, and trajectories.

High frequency approximation of solutions to critical nonlinear wave equations

Abstract: This work is devoted to the description of bounded energy sequences of solutions to the equation (1) □ u + | u |4 = 0 in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], up to remainder terms small in energy norm and in every Strichartz norm. The proof relies on scattering theory for (1) and on a structure theorem for bounded energy sequences of solutions to the linear wave equation. In particular, we infer the existence of an a priori estimate of Strichartz norms of solutions to (1) in terms of their energy.

Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay

Abstract: This paper studies the problem of control for discrete time delay linear systems with Markovian jump parameters. The system under consideration is subjected to both time-varying norm-bounded parameter uncertainty and unknown time delay in the state, and Markovian jump parameters in all system matrices. We address the problem of robust state feedback control in which both robust stochastic stability and a prescribed H/sub /spl infin// performance are required to be achieved irrespective of the uncertainty and time delay. It is shown that the above problem can be solved if a set of coupled linear matrix inequalities has a solution.

Backstepping Controller Design for Nonlinear Stochastic Systems Under a Risk-Sensitive Cost Criterion

Abstract: This paper develops a methodology for recursive construction of optimal and near-optimal controllers for strict-feedback stochastic nonlinear systems under a risk-sensitive cost function criterion. The design procedure follows the integrator backstepping methodology, and the controllers obtained guarantee any desired achievable level of long-term average cost for a given risk-sensitivity parameter $\theta$. Furthermore, they lead to closed-loop system trajectories that are bounded in probability, and in some cases asymptotically stable in the large. These results also generalize to nonlinear systems with strongly stabilizable zero dynamics. A numerical example included in the paper illustrates the analytical results.

Robust principal component analysis for functional data

Abstract: A method for exploring the structure of populations of complex objects, such as images, is considered. The objects are summarized by feature vectors. The statistical backbone is Principal Component Analysis in the space of feature vectors. Visual insights come from representing the results in the original data space. In an ophthalmological example, endemic outliers motivate the development of a bounded influence approach to PCA.

Model Checking of Message Sequence Charts

Abstract: Scenario-based specifications such as message sequence charts (MSC) offer an intuitive and visual way of describing design requirements Such specifications focus on message exchanges among communicating entities in distributed software systems Structured specifications such as MSC-graphs and Hierarchical MSC-graphs (HMSC) allow convenient expression of multiple scenarios, and can be viewed as an early model of the system In this paper, we present a comprehensive study of the problem of verifying whether this model satisfies a temporal requirement given by an automaton, by developing algorithms for the different cases along with matching lower bounds When the model is given as an MSC, model checking can be done by constructing a suitable automaton for the linearizations of the partial order specified by the MSC, and the problem is coNP-complete When the model is given by an MSC-graph, we consider two possible semantics depending on the synchronous or asynchronous interpretation of concatenating two MSCs For synchronous model checking of MSC-graphs and HMSCs, we present algorithms whose time complexity is proportional to the product of the size of the description and the cost of processing MSCs at individual vertices Under the asynchronous interpretation, we prove undecidability of the model checking problem We, then, identify a natural requirement of boundedness, give algorithms to check boundedness, and establish asynchronous model checking to be Pspace-complete for bounded MSC-graphs and Expspace-complete for bounded HMSCs

Optimal linear filtering under parameter uncertainty

Abstract: This paper addresses the problem of designing a guaranteed minimum error variance robust filter for convex bounded parameter uncertainty in the state, output, and input matrices. The design procedure is valid for linear filters that are obtained from the minimization of an upper bound of the error variance holding for all admissible parameter uncertainty. The results provided generalize the ones available in the literature to date in several directions. First, all system matrices can be corrupted by parameter uncertainty, and the admissible uncertainty may be structured. Assuming the order of the uncertain system is known, the optimal robust linear filter is proved to be of the same order as the order of the system. In the present case of convex bounded parameter uncertainty, the basic numerical design tools are linear matrix inequality (LMI) solvers instead of the Riccati equation solvers used for the design of robust filters available in the literature to date. The paper that contains the most important and very recent results on robust filtering is used for comparison purposes. In particular, it is shown that under the same assumptions, our results are generally better as far as the minimization of a guaranteed error variance is considered. Some numerical examples illustrate the theoretical results.

Generalized derivations of left faithful rings

Abstract: Let R be a left faithful ringU its right Utumi quotient ring and ρ a dense right ideal of R. An additive map g: ρ → U is called a generalized derivation if there exists a derivation δ of ρ into U such that for all x,y∈ρ. In this note, we prove that there exists an element a∈ U such that for all x ∈ ρ. From this characterization, it is proved that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0. Analogous results are also obtained for the case of generalized derivations with nilpotent values on Lie ideals or one-sided ideals.

Singular integral operators with non-smooth kernels on irregular domains

Abstract: Let ? be a space of homogeneous type. The aims of this paper are as follows. i) Assuming that T is a bounded linear operator on L2(?), we give a sufficient condition on the kernel of T such that T is of weak type (1,1), hence bounded on Lp(?) for 1 < p = 2; our condition is weaker then the usual Hormander integral condition. ii) Assuming that T is a bounded linear operator on L2(O) where O is a measurable subset of ?, we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on Lp(O) for 1 < p = 2. iii) We establish sufficient conditions for the maximal truncated operator T* which is defined by T*u(x) = supe>0 |Teu(x)|, to be Lp bounded, 1 < p < 8. Applications include weak (1,1) estimates of certain Riesz transforms, and Lp boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.

Holder Regularity and Dimension Bounds for Random Curves

Abstract: Random systems of curves exhibiting fluctuating features on arbitrarily small scales (δ) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the probabilities of crossing events imply that typically all the realized curves admit Holder continuous parametrizations with a common exponent and a common random prefactor, which in the scaling limit (δ → 0) remains stochastically bounded. The regularity is used for the construction of scaling limits, formulated in terms of probability measures on the space of closed sets of curves. Under the hypotheses presented here the limiting measures are supported on sets of curves which are Holder continuous but not rectifiable, and have Hausdorff dimensions strictly greater than one. The hypotheses are known to be satisfied in certain two dimensional percolation models. Other potential applications are also mentioned.

On Some Tighter Inapproximability Results (Extended Abstract)

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

Modulation spaces and pseudodifferential operators

Abstract: We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR 2d , which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol σ lies in the modulation spaceM ∞,1 (R 2d ), then the corresponding pseudodifferential operator is bounded onL 2(R d ) and, more generally, on the modulation spacesM p,p (R d ) for 1≤p≤∞. If σ lies in the modulation spaceM 2,2 (R 2d )=L /2 (R 2d )∩H s (R 2d ), i.e., the intersection of a weightedL 2-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderon-Vaillancourt boundedness theorem and on Daubechies' trace-class results.

Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses

Abstract: We establish hardness versus randomness trade-offs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round Arthur-Merlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with access to satisfiability. We show that every language with a bounded round Arthur-Merlin game has subexponential size membership proofs for infinitely many input lengths unless the polynomial-time hierarchy collapses. This provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We set up a general framework for derandomization which encompasses more than the traditional model of randomized computation. For a randomized procedure to fit within this framework, we only require that for any fixed input the complexity of checking whether the procedure succeeds on a given random bit sequence is not too high. We then apply our derandomization technique to four fundamental complexity theoretic constructions: The Valiant-Vazirani random hashing technique which prunes the number of satisfying assignments of a Boolean formula to one, and related procedures like computing satisfying assignments to Boolean formulas non-adaptively given access to an oracle for satisfiability. The algorithm of Bshouty et al. for learning Boolean circuits. Constructing matrices with high rigidity. Constructing polynomial-size universal traversal sequences. We also show that if linear space requires exponential size circuits, then space bounded randomized computations can be simulated deterministically with only a constant factor overhead in space.

Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture.

Abstract: We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and "hairiness" of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N > 1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results we supply the space of quadratic-like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension-one foliation of the connectedness locus.

An analytical approach to the Schrodinger-Newton equations

Abstract: In this paper we present the second part of a study of spherically-symmetric solutions of the Schrodinger-Newton equations for a single particle (Penrose R 1998 Quantum computation, entanglement and state reduction Phil. Trans. R. Soc. 356 1-13). We show that there exists an infinite family of normalizable, finite energy solutions which are characterized by being smooth and bounded for all values of the radial coordinate. We therefore provide analytical support for our earlier numerical integrations (Moroz I M et al 1998 Spherically-symmetric solutions of the Schrodinger-Newton equations Classical and Quantum Gravity 1998 15 2733-42).

Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems

Abstract: We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths problem (EDP), we are given a network G with source-sink pairs (si; ti), 1 i k, and the goal is to nd a largest subset of source-sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any > 0, EDP is NP-hard to approximate within m 1=2 . We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any > 0, bounded length EDP is hard to approximate within m 1=2 even in undirected networks, and give an O( p m)-approximation algorithm for it. For directed networks, we show that even the single source-sink pair case (i.e. nd the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1=2 , for any > 0.

Robustness Properties of k Means and Trimmed k Means

Abstract: The generalized k means method is based on the minimization of the discrepancy between a random variable (or a sample of this random variable) and a set with k points measured through a penalty function Φ. As in the M estimators setting (k = 1), a penalty function, Φ, with unbounded derivative, Ψ, naturally leads to nonrobust generalized k means. However, surprisingly the lack of robustness extends also to the case of bounded Ψ; that is, generalized k means do not inherit the robustness properties of the M estimator from which they came. Attempting to robustify the generalized k means method, the generalized trimmed k means method arises from combining k means idea with a so-called impartial trimming procedure. In this article study generalized k means and generalized trimmed k means performance from the viewpoint of Hampel's robustness criteria; that is, we investigate the influence function, breakdown point, and qualitative robustness, confirming the superiority provided by the trimming. We inc...

Boundary observability for the space semi-discretizations of the 1 - D wave equation

Abstract: We consider space semi-discretizations of the 1 − d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability,i.e., the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h ! 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a uniform bound in a subspace of solutions generated by the low frequencies of the discrete system. When h ! 0 this nite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We consider both nite-dierence and nite-element semi-discretizations.

Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains: 1. Theory and computational approach

Abstract: We consider the effect of measuring randomly varying hydraulic conductivitiesK(x) on one's ability to predict numerically, without resorting to either Monte Carlo simulation or upscaling, steady state flow in bounded domains driven by random source and boundary terms. Our aim is to allow optimum unbiased prediction of hydraulic heads h(x) and fluxes q(x) by means of their ensemble moments, 〈h(x)〉c and 〈q(x)〉c, respectively, conditioned on measurements of K(x). These predictors have been shown by Neuman and Orr [1993a] to satisfy exactly an integrodifferential conditional mean flow equation in which 〈q(x)〉c is nonlocal and non-Darcian. Here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty; obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter σY which represents the standard estimation error of ln K(x); and show how to solve these equations to first order in σY2 by finite elements on a rectangular grid in two dimensions. In the special case where one treats K(x) as if it was locally homogeneous and mean flow as if it was locally uniform, one obtains a localized Darcian approximation 〈q(x)〉c ≈ −Kc(x)∇〈h(x)〉c in which Kc(x) is a space-dependent conditional hydraulic conductivity tensor. This leads to the traditional deterministic, Darcian steady state flow equation which, however, acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique. It further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database. Localized equations yield no information about predictive uncertainty. Our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux, obtained by traditional inverse methods, are generally smaller (often considerably so) than measures of corresponding predictive uncertainty, which can be assessed only by means of stochastic models such as ours. We present a detailed comparison between finite element solutions of nonlocal and localized moment equations and Monte Carlo simulations under superimposed mean-uniform and convergent flow regimes in two dimensions. Paper 1 presents the theory and computational approach, and paper 2 [Guadagnini and Neuman, this issue] describes unconditional and conditional computational results.

Isoperimetric and universal inequalities for eigenvalues

Abstract: This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H.C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S n or H n . Introduction The Eigenvalue Problems The first eigenvalue problem we shall introduce is that of the fixed membrane, or Dirichlet Laplacian . We consider the eigenvalues and eigenfunctions of –Δ on a bounded domain (=connected open set) Ω in Euclidean space R n , i.e., the problem It is well-known that this problem has a real and purely discrete spectrum where Here each eigenvalue is repeated according to its multiplicity. An associated orthonormal basis of real eigenfunctions will be denoted u 1 , u 2 , u 3 , …. In fact, throughout this paper we will assume that all functions we consider are real. This entails no loss of generality in the present context. The next problem we introduce is that of the free membrane, or Neumann Laplacian .

Construction of symbolic dynamics from experimental time series

Abstract: Symbolic dynamics play a central role in the description of the evolution of nonlinear systems. Yet there are few methods for determining symbolic dynamics of chaotic data. One difficulty is that the data contains random fluctuations associated with the experimental process. Using data obtained from a magnetoelastic ribbon experiment we show how a topological approach that allows for experimental error and bounded noise can be used to obtain a description of the dynamics in terms of subshift dynamics on a finite set of symbols.

Bounds for the Entries of Matrix Functions with Applications to Preconditioning

Abstract: Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A)are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A)in terms of Riemann-Stieltjes integrals and by approximating such integrals by Gaussian quadrature rules are also considered. Applications of these bounds to preconditioning are suggested and illustrated by a few numerical examples.