Showing papers on "Bounded function published in 2002"

Scalar Curvature and Stability of Toric Varieties

Abstract: We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces. We prove that, under the stability hypothesis, the Mabuchi functional is bounded below on invariant metrics, and that minimising sequences have a certain convergence property. In the reverse direction, we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature. The proofs use a general framework, developed by Guillemin and Abreu, in which invariant Kahler metrics correspond to convex functions on the moment polytope of a toric variety. This paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a Kahler metric of constant scalar curvature. The pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of “stability” in the sense of Geometric Invariant Theory. This expectation is probably now an item of folklore: going back to suggestions put forward by Yau in the case of KahlerEinstein metrics, and the many results of Tian and others in this case; reinforced by a detailed formal picture which makes clear the analogy with the well-established relation between the stability of vector bundles and Yang-Mills connections [5]. Here, we begin the investigation of

Epidemic dynamics in finite size scale-free networks

Abstract: Many real networks present a bounded scale-free behavior with a connectivity cutoff due to physical constraints or a finite network size. We study epidemic dynamics in bounded scale-free networks with soft and hard connectivity cutoffs. The finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes. The induced epidemic threshold is very small even at a relatively small cutoff, showing that the neglection of connectivity fluctuations in bounded scale-free networks leads to a strong overestimation of the epidemic threshold. We provide the expression for the infection prevalence and discuss its finite size corrections. The present paper shows that the highly heterogeneous nature of scale-free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes.

Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations

Abstract: Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

Adaptive observer for multiple-input-multiple-output (MIMO) linear time-varying systems

Abstract: For joint state-parameter estimation in linear time-varying (LTV) multiple-input-multiple-output (MIMO) systems, an approach to the design of adaptive observers is proposed. It is conceptually simple and computationally efficient. Its global exponential convergence is established for noise-free systems. In the presence of noises, it is proved that the estimation errors are bounded and converge in the mean to zero if the noises are bounded and have zero means. Potential applications are fault detection and isolation, and adaptive control.

Method for providing an attribute bounded network of computers

Abstract: An attribute bounded network of computers is created. Computers in the distributed computer network are identified by specific attributes (e.g., a geographically bounded region) and a server is used to distribute processing assignments (e.g., addresses of electronic documents to be indexed) based upon the identified attributes. A peer-to-peer computer network is also provided based upon geographically bounded regions, the peer-to-peer computer network can be used to share electronic documents. A virtual community can be created within a peer-to-peer computer network by identifying peer computer connections based upon associated attributes (e.g., a geographically bounded region). The attribute bounded network of computers provides indexes having fresher data by allowing spidering of electronic documents more often than can be done by a central server indexing site.

Centennial History of Hilbert’s 16th Problem

Abstract: The second part of Hilbert’s 16th problem deals with polynomial differential equations in the plane. It remains unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the problem inspired significant progress in the geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry. The dramatic history of the problem, as well as related developments, are presented below. §1. The problem and its counterparts What may be said about the number and location of limit cycles of a planar polynomial vector field of degree n? (The limit cycle is an isolated closed orbit of a vector field.) This second part of Hilbert’s 16th problem appears to be one of the most persistent in the famous Hilbert list [H], second only to the Riemann ζ-function conjecture. Traditionally, Hilbert’s question is split into three, each one requiring a stronger answer. Problem 1. Is it true that a planar polynomial vector field has but a finite number of limit cycles? Problem 2. Is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only? The bound on the number of limit cycles in Problem 2 is denoted by H(n) and known as the Hilbert number. Linear vector fields have no limit cycles; hence H(1) = 0. It is still unknown whether or not H(2) exists. Problem 3. Give an upper bound for H(n). A solution to any of these problems implies a solution for the previous ones. Only the first problem is solved now. The positive answer was established in [E92], [I91]. There are analytic counterparts of Problems 1 and 2. Received by the editors December 2001. 2000 Mathematics Subject Classification. Primary 34Cxx, 34Mxx, 37F75.

The solution of the Kato square root problem for second order elliptic operators on Rn

Abstract: We prove the Kato conjecture for elliptic operators on Jfin. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L =-div (AV) with bounded measurable coefficients in IEtn iS the

Exponential hedging and entropic penalties

Abstract: We solve the problem of hedging a contingent claim B by maximizing the expected exponential utility of terminal net wealth for a locally bounded semimartingale X. We prove a duality relation between this problem and a dual problem for local martingale measures Q for X where we either minimize relative entropy minus a correction term involving B or maximize the Q-price of B subject to an entropic penalty term. Our result is robust in the sense that it holds for several choices of the space of hedging strategies. Applications include a new characterization of the minimal martingale measure and risk-averse asymptotics.

Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties

Abstract: In this paper a robust model predictive control (MPC) for constrained discrete-time nonlinear system with additive uncertainties is presented. This controller uses a terminal cost, terminal constraint and nominal predictions. The terminal region and constraints on the states are computed to get robust feasibility of the closed loop system for a given bound on the admissible uncertainties. Furthermore, it is proved that the closed-loop system is input-to-state stable with relation to the uncertainties. Therefore, the closed-loop system evolves towards a compact set where it is ultimately bounded. In case of decaying uncertainties, the closed-loop system is asymptotically stable. The convergence of the closed loop system is guaranteed despite the suboptimality of the solution.

Bounded cohomology of subgroups of mapping class groups

Abstract: We show that every subgroup of the mapping class group MCG(S )o f ac ompact surface S is either virtually abelian or it has innite dimensional second bounded cohomology. As an application, we give another proof of the Farb{ Kaimanovich{Masur rigidity theorem that states that MCG(S )d oes not contain a higher rank lattice as a subgroup.

Rates of strong uniform consistency for multivariate kernel density estimators

Abstract: Let fn denote the usual kernel density estimator in several dimensions. It is shown that if {an} is a regular band sequence, K is a bounded square integrable kernel of several variables, satisfying some additional mild conditions ((K1) below), and if the data consist of an i.i.d. sample from a distribution possessing a bounded density f with respect to Lebesgue measure on R d , then lim sup n→∞ na n d log a n −1 sup t∈ R d |f n (t)−Ef n (t)|⩽C ‖f‖ ∞ ∫K 2 (x) d x a.s. for some absolute constant C that depends only on d. With some additional but still weak conditions, it is proved that the above sequence of normalized suprema converges a.s. to 2d‖f‖ ∞ ∫K 2 (x) d x . Convergence of the moment generating functions is also proved. Neither of these results require f to be strictly positive. These results improve upon, and extend to several dimensions, results by Silverman [13] for univariate densities.

Query evaluation via tree-decompositions

Abstract: A number of efficient methods for evaluating first-order and monadic-second order queries on finite relational structures are based on tree-decompositions of structures or queries. We systematically study these methods.In the first part of the article, we consider arbitrary formulas on tree-like structures. We generalize a theorem of Courcelle [1990] by showing that on structures of bounded tree-width a monadic second-order formula (with free first- and second-order variables) can be evaluated in time linear in the structure size plus the size of the output.In the second part, we study tree-like formulas on arbitrary structures. We generalize the notions of acyclicity and bounded tree-width from conjunctive queries to arbitrary first-order formulas in a straightforward way and analyze the complexity of evaluating formulas of these fragments. Moreover, we show that the acyclic and bounded tree-width fragments have the same expressive power as the well-known guarded fragment and the finite-variable fragments of first-order logic, respectively.

Continuous bounded cohomology and applications to rigidity theory

Abstract: The central theme of this paper is a product formula for (continuous) bounded cohomology, and more specifically its applications to rigidity theory for lattices — both in Lie/algebraic groups and more general topological groups. A more condensed exposition of some of the material published in the Lecture Note of the second author is followed by finiteness results for lattices. An appendix by Burger–Iozzi pins down a powerful use of boundary maps in this context.

Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular

Abstract: We establish borderline regularity for solutions of the Beltrami equation $f\sb z-\mu f\sb {\overline {z}}=0$ on the plane, where $\mu$ is a bounded measurable function, $\parallel\mu\parallel\sb \infty=k 0$. On the other hand, O. Lehto and T. Iwaniec showed that $q<1+k$ is not sufficient. In [2], the following question was asked: What happens for the borderline case $q=1+k$? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia's extrapolation technique and two-weight estimates for the [26].

Monitors: an operating system structuring concept

Abstract: This paper develops Brinch Hansen's concept of a monitor as a method of structuring an operating system. It introduces a form of synchronization, describes a possible method of implementation in terms of semaphores and gives a suitable proof rule. Illustrative examples include a single resource scheduler, a bounded buffer, an alarm clock, a buffer pool, a disk head optimizer, and a version of the problem of readers and writers.

Subexponential decay of correlations

Abstract: We describe a method for proving subexponential lower bounds for correlations functions, and apply it to study decay of correlations for maps with countable Markov partitions. One result is that LS Young’s upper estimates [Y2] are optimal in many situations. Our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on Banach spaces, which we apply to the iterates of the transfer operator.

Tracking with prescribed transient behaviour

Abstract: Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary -valued reference signal r of class W 1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that for all t ≥ 0, where φ a prescribed real-valued function of class W 1,∞ with the property that φ(s) > 0 for all s > 0 and . A simple (neither adaptive nor dynamic) error feedback control of the form $u(t)=- \alpha ({\varphi}(t)\|e(t)\|)e(t)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain .

Painleve's problem and the semiadditivity of analytic capacity

Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

Graph Nonisomorphism Has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses

Abstract: Traditional hardness versus randomness results focus on time-efficient randomized decision procedures. We generalize these trade-offs to a much wider class of randomized processes. We work out various applications, most notably to derandomizing Arthur-Merlin games. We show that every language with a bounded round Arthur-Merlin game has subexponential size membership proofs for infinitely many input lengths unless exponential time coincides with the third level of the polynomial-time hierarchy (and hence the polynomial-time hierarchy collapses). Since the graph nonisomorphism problem has a bounded round Arthur-Merlin game, this provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We also establish hardness versus randomness trade-offs for space bounded computation.

D-Equivalence and K-Equivalence

Abstract: Let X and Y be smooth projective varieties over ℂ. They are called D-equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, and K-equivalent if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.

Stability of fuzzy control systems with bounded uncertain delays

Abstract: Global exponential stability of fuzzy control systems with delays is studied. These delays in the fuzzy control systems are assumed to be any uncertain bounded continuous functions. Stability of systems with uncertain delays is interesting since in practical applications it is not easy to know the delays exactly. Conditions for global exponential stability of free fuzzy systems with uncertain delays are derived. Criteria for design of nonlinear fuzzy controllers to feedback control the stability of global nonlinear fuzzy systems are given. Theorems are proved via the method of functional differential inequalities analysis.

Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes

Abstract: Consider the partial sums {S_t} of a real-valued functional F(Phi(t)) of a Markov chain {Phi(t)} with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional F is bounded, the following conclusions are obtained: 1. Spectral theory: Well-behaved solutions can be constructed for the ``multiplicative Poisson equation''. 2. A ``multiplicative'' mean ergodic theorem: For all complex \alpha in a neighborhood of the origin, the normalized mean of \exp(\alpha S_t) converges exponentially fast to a solution of the multiplicative Poisson equation. 3. Edgeworth Expansions: Rates are obtained for the convergence of the distribution function of the normalized partial sums S_t to the standard Gaussian distribution. 4. Large Deviations: The partial sums are shown to satisfy a large deviations principle in a neighborhood of the mean. This result, proved under geometric ergodicity alone, cannot in general be extended to the whole real line. 5. Exact Large Deviations Asymptotics: Rates of convergence are obtained for the large deviations estimates above. Extensions of these results to continuous-time Markov processes are also given.

Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics

Abstract: We systematically investigate the connections between constraint satisfaction problems, structures of bounded treewidth, and definability in logics with a finite number of variables. We first show that constraint satisfaction problems on inputs of treewidth less than k are definable using Datalog programs with at most k variables; this provides a new explanation for the tractability of these classes of problems. After this, we investigate constraint satisfaction on inputs that are homomorphically equivalent to structures of bounded treewidth. We show that these problems are solvable in polynomial time by establishing that they are actually definable in Datalog; moreover, we obtain a logical characterization of the property "being homomorphically equivalent to a structure of bounded treewidth" in terms of definability in finite-variable logics. Unfortunately, this expansion of the tractability landscape comes at a price, because we also show that, for each k ? 2, determining whether a structure is homomorphically equivalent to a structure of treewidth less than k is an NP-complete problem. In contrast, it is well known that, for each k ? 2, there is a polynomial-time algorithm for testing whether a given structure is of treewidth less than k. Finally, we obtain a logical characterization of the property "having bounded treewidth" that sheds light on the complexity-theoretic difference between this property andt he property 'being homomorphically equivalent to a structure of bounded treewidth".

Time Optimal Trajectories for Bounded Velocity Differential Drive Vehicles

Abstract: This paper presents the time optimal trajectories for differential drive vehicles in the unobstructed plane. The wheel angular velocities are bounded, but may be discontinuous. The paper proves the existence of optimal controls, derives the structure of optimal trajectories, and develops an algorithm for producing a time optimal trajectory between any two configurations. Every nontrivial optimal trajectory is composed of straight segments alternating with turns about the robot's center. Optimal trajectories may have as many as five actions, but four actions are sufficient—for every optimal trajectory of five actions, there is an equally fast trajectory with four actions.