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Showing papers on "Bounded function published in 2009"


01 Jan 2009
TL;DR: This article surveys a technique called Bounded Model Checking (BMC), which uses a propositional SAT solver rather than BDD manipulation techniques, and is widely perceived as a complementary technique to BDD-based model checking.
Abstract: Besides Equivalence Checking [KK97, KPKG02] the most important industrial application of SAT is currently Bounded Model Checking (BMC) [BCCZ99]. Both techniques are used for formal hardware verification in the context of electronic design automation (EDA), but have successfully been applied to many other domains as well. In this chapter, we focus on BMC. In practice, BMC is mainly used for falsification resp. testing, which is concerned with violations of temporal properties. However, the original paper on BMC [BCCZ99] already discussed extensions that can prove properties. A considerable part of this chapter discusses these complete extensions, which are often called “unbounded” model checking techniques, even though they are build upon the same principles as plain BMC. Two further related applications, in which BMC becomes more and more important, are automatic test case generation for closing coverage holes, and disproving redundancy in designs. Most of the techniques discussed in this chapter transfer to this more general setting as well, even though our focus is on property verification resp. falsification. The basic idea of BMC is to represent a counterexample-trace of bounded length symbolically and check the resulting propositional formula with a SAT solver. If the formula is satisfiable and thus the path feasible, a satisfying assignment returned by the SAT solver can be translated into a concrete counterexample trace that shows that the property is violated. Otherwise, the bound is increased and the process repeated. Complete extensions to BMC allow to stop this process at one point, with the conclusion that the property cannot be violated, hopefully before the available resources are exhausted.

689 citations


Journal ArticleDOI
TL;DR: For Riemannian manifolds with a measure (M, g, edvolg) as mentioned in this paper showed that the Ricci curvature and volume comparison can be improved when the Bakry-Emery Ricci tensor is bounded from below.
Abstract: For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.

572 citations


Posted Content
TL;DR: In this article, the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions was considered and a priori estimates of Gidas-Spruck type were obtained.
Abstract: We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

456 citations


Journal ArticleDOI
TL;DR: In this article, Li et al. developed an approach to find ground state solutions, i.e., nontrivial solutions with least possible energy, based on a direct reduction of the indefinite variational problem to a definite one.

431 citations


Journal ArticleDOI
TL;DR: A fully sequential sampling policy is proposed called the knowledge-gradient policy, which is provably optimal in some special cases and has bounded suboptimality in all others and it is demonstrated how this policy may be applied to efficiently maximize a continuous function on a continuous domain while constrained to a fixed number of noisy measurements.
Abstract: We consider a Bayesian ranking and selection problem with independent normal rewards and a correlated multivariate normal belief on the mean values of these rewards. Because this formulation of the ranking and selection problem models dependence between alternatives' mean values, algorithms may use this dependence to perform efficiently even when the number of alternatives is very large. We propose a fully sequential sampling policy called the knowledge-gradient policy, which is provably optimal in some special cases and has bounded suboptimality in all others. We then demonstrate how this policy may be applied to efficiently maximize a continuous function on a continuous domain while constrained to a fixed number of noisy measurements.

421 citations


Posted Content
TL;DR: In this article, the polynomial decay of orbits of Hilbert space semigroups is characterized in resolvent terms, and results of the same type for general Banach space semiigroups and functions obtained recently in the paper by C.J.Batty and T.K.Duyckaerts are sharp.
Abstract: We characterize the polynomial decay of orbits of Hilbert space $C_0$-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in the paper by C.J.K.Batty and T.Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces (J. Evol. Eq. 2008), are sharp. This settles a conjecture posed in the paper by C.J.K.Batty and T.Duyckaerts.

336 citations


Journal ArticleDOI
TL;DR: In this article, Chen, Strain, Tsai and Yau showed that for axi-symmetric Navier-Stokes equations with bounded velocity, the solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions.
Abstract: We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in Rn × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].

314 citations


Journal ArticleDOI
TL;DR: In this paper, the authors develop a theory of Hardy and BMO spaces associated to L, which includes a molecular decomposition, maximal and square function characterizations, duality of Hardy spaces, and a John-Nirenberg inequality.
Abstract: Consider a second order divergence form elliptic operator L with complex bounded measurable coefficients. In general, operators based on L, such as the Riesz transform or square function, may lie beyond the scope of the Calderon–Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some L p spaces. In this work we develop a theory of Hardy and BMO spaces associated to L, which includes, in particular, a molecular decomposition, maximal and square function characterizations, duality of Hardy and BMO spaces, and a John–Nirenberg inequality.

292 citations


Book ChapterDOI
16 Apr 2009
TL;DR: It is shown that unlike "normal" PRFs, wPRFs are seed-incompressible, in the sense that the output of a wPRF is pseudorandom even if a bounded amount of information about the key is leaked.
Abstract: A weak pseudorandom function (wPRF) is a cryptographic primitive similar to --- but weaker than --- a pseudorandom function: for wPRFs one only requires that the output is pseudorandom when queried on random inputs. We show that unlike "normal" PRFs, wPRFs are seed-incompressible, in the sense that the output of a wPRF is pseudorandom even if a bounded amount of information about the key is leaked. As an application of this result we construct a simple mode of operation which --- when instantiated with any wPRF --- gives a leakage-resilient stream-cipher. The implementation of such a cipher is secure against every side-channel attack, as long as the amount of information leaked per round is bounded, but overall can be arbitrary large. The construction is simpler than the previous one (Dziembowski-Pietrzak FOCS'08) as it only uses a single primitive (a wPRF) in a straight forward manner.

292 citations


Journal ArticleDOI
TL;DR: In this paper, the authors study the transport and mixing properties of flows in a variety of settings, connecting the classical geometrical approach via invariant manifolds with a probabilistic approach via transfer operators.

285 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the multidimensional aggregation equation with a mild singularity at the origin (Lipschitz or better) and showed that the Osgood condition for well-posedness of the ODE characteristics determines global in time wellposedness in the PDE with compactly supported bounded nonnegative initial data.
Abstract: We consider the multidimensional aggregation equation ut − ∇ (u∇K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717–35), whereas for C2 kernels there is no finite-time blow-up. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blow-up that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally, we conclude with a discussion of similarity solutions for the case K(x) = |x| and some open problems.

Journal ArticleDOI
TL;DR: In this article, Pejsachowicz et al. formulated general second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains into nonlinear mappings between Sobolev spaces and showed that the linearized mapping is a Fredholm operator of index zero.

Journal ArticleDOI
TL;DR: In this article, it was shown that there are infinitely often primes differing by 16 or less in the Elliott-Halberstam conjecture and that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing.
Abstract: We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, lim inf n→∞ Pn+1-Pn/log Pn/log = 0. We will quantify this result further in a later paper.

Journal ArticleDOI
TL;DR: The basic theory of toric dynamical systems is developed in the context of computational algebraic geometry and it is shown that the associated moduli space is a toric variety, which has a unique point within each invariant polyhedron.

Journal ArticleDOI
TL;DR: In this paper, Meerschaert et al. extended the approach of Meershaert and Scheffler (23) to fractional Cauchy problems on bounded domains and constructed stochastic solutions via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative.
Abstract: Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain DR d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brow- nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. 1. Introduction. In this paper, we extend the approach of Meerschaert and Scheffler ( 23) and Meerschaert et al. (24) to fractional Cauchy problems on bounded domains. Our methods involve eigenfunction expansions, killed Markov processes and inverse stable subordinators. In a recent related paper (7), we establish a connection between fractional Cauchy problems with index β = 1/2 on an unbounded domain, and iterated Brownian motion (IBM), defined as Zt = B(|Yt|), where B is a Brownian motion with values in R d and Y is an independent one-dimensional Brownian motion. Since IBM is also the stochastic solution to a Cauchy problem involving a fourth-order derivative in space (2, 14), that paper also establishes a connection between certain higher-order Cauchy problems and their time-fractional analogues. More generally, Baeumer, Meerschaert and Nane (7) shows a connection between fractional Cauchy problems with β = 1/2 and higher-order Cauchy problems that involve the square of the generator. In the present paper, we

Journal ArticleDOI
TL;DR: The Lyapunov-Krasovskii method is extended to linear time-delay systems in a Hilbert space to provide effective tools for stability analysis and control synthesis of distributed parameter systems.

Posted Content
TL;DR: In this paper, the authors give two meta-theorems on kernelzation for planar graph problems and show that all problems expressible in counting monadic second order logic and satisfying a coverability property admit a kernel on graphs of bounded genus.
Abstract: In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two meta-theorems on kernelzation. The first theorem says that all problems expressible in Counting Monadic Second Order Logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every coherent or convex monetary risk measure on an Orlicz heart which is real-valued on a set with non-empty algebraic interior admits a robust representation as maximal penalized expectation with respect to different probability measures.
Abstract: Coherent, convex, and monetary risk measures were introduced in a setup where uncertain outcomes are modeled by bounded random variables. In this paper, we study such risk measures on Orlicz hearts. This includes coherent, convex, and monetary risk measures on Lp-spaces for 1 ≤p < ∞ and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. We prove that every coherent or convex monetary risk measure on an Orlicz heart which is real-valued on a set with non-empty algebraic interior is real-valued on the whole space and admits a robust representation as maximal penalized expectation with respect to different probability measures. We also show that penalty functions of such risk measures have to satisfy a certain growth condition and that our risk measures are Luxemburg-norm Lipschitz-continuous in the coherent case and locally Luxemburg-norm Lipschitz-continuous in the convex monetary case. In the second part of the paper we investigate cash-additive hulls of transformed Luxemburg-norms and expected transformed losses. They provide two general classes of coherent and convex monetary risk measures that include many of the currently known examples as special cases. Explicit formulas for their robust representations and the maximizing probability measures are given.

01 Jan 2009
TL;DR: For any integer d ≥ 3 and positive real e, it was shown in this article that satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd √ log n−1/e) where n is the number of bits of communication from the first player to the second player.
Abstract: Consider the following two-player communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to decide cooperatively whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player.For any integer d ≥ 3 and positive real e, we show that, if satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd − e), then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for e = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs.By reduction, similar results hold for other NP-complete problems. For the vertex cover problem on n-vertex d-uniform hypergraphs, this statement holds for any integer d ≥ 2. The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k2 − e) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.

Journal ArticleDOI
TL;DR: In this paper, the completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs, and an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them.
Abstract: The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them.

Journal ArticleDOI
TL;DR: This paper studies quantized and delayed state-feedback control of linear systems with given constant bounds on the quantization error and on the time-varying delay and proposes decomposition of the quantizations into a sum of a saturation and of a uniformly bounded disturbance.

Journal ArticleDOI
TL;DR: In this paper, the authors prove approximate controllability of the bilinear Schrodinger equation in the case of the uncontrolled Hamiltonian having a discrete non-resonant spectrum.
Abstract: We prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials.

Journal ArticleDOI
TL;DR: In this article, the adaptive observers for a class of uniformly observable MIMO nonlinear systems with general nonlinear parameterizations were proposed and the state and unknown parameters of the considered systems are supposed to lie in bounded domains which size can be arbitrarily large and the exponential convergence of the observers is shown to result under a well defined persistent excitation condition.

Journal ArticleDOI
01 Oct 2009
TL;DR: A decentralized adaptive methodology is presented for large-scale nonlinear systems with model uncertainties and time-delayed interconnections unmatched in control inputs and it is proved that all the signals in the closed-loop system are semiglobally uniformly bounded.
Abstract: A decentralized adaptive methodology is presented for large-scale nonlinear systems with model uncertainties and time-delayed interconnections unmatched in control inputs. The interaction terms with unknown time-varying delays are bounded by unknown nonlinear bounding functions related to all states and are compensated by choosing appropriate Lyapunov-Krasovskii functionals and using the function approximation technique based on neural networks. The proposed memoryless local controller for each subsystem can simply be designed by extending the dynamic surface design technique to nonlinear systems with time-varying delayed interconnections. In addition, we prove that all the signals in the closed-loop system are semiglobally uniformly bounded, and the control errors converge to an adjustable neighborhood of the origin. Finally, an example is provided to illustrate the effectiveness of the proposed control system.

Journal ArticleDOI
TL;DR: In this article, the authors studied embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A p -condition and obtained sufficient conditions for boundedness and compactness of the embedding operator for smooth domains and domains with boundary singularities.
Abstract: In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A p -condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion). The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Holder singularities demonstrate the sharpness of our machinery comparatively with known results.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the global existence of the inviscid Boussinesq system with bounded vorticity and showed that there exists a global unique solution under a natural additional assumption on the initial temperature.
Abstract: The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. No smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the Benard system are also discussed.

Journal ArticleDOI
TL;DR: In this paper, the uniqueness of a DG enhancement for triangulated categories of coherent sheaves and perfect complexes on quasi-projective schemes has been shown for the case of perfect complexes.
Abstract: The paper contains general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded derived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded derived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.

Journal ArticleDOI
TL;DR: In this article, an existence and uniqueness theorem for isometric embeddings in the Minkowski space has been proved and the boundary value problem for Jang's [13] equation has been solved.
Abstract: The definition of quasi-local mass for a bounded space-like region Ω in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface $${\Sigma=\partial \Omega}$$ and should be independent of whichever space-like region $${\Sigma}$$ bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been taken by Brown-York [4,5] and Liu-Yau [16,17] (see also related works [3,6,9,12,14,15,28,32]) to define such notions using the isometric embedding theorem into the Euclidean three space. However, there exist surfaces in the Minkowski space whose quasilocal mass is strictly positive [19]. It appears that the momentum information needs to be accounted for to reconcile the difference. In order to fully capture this information, we use isometric embeddings into the Minkowski space as references. In this article, we first prove an existence and uniqueness theorem for such isometric embeddings. We then solve the boundary value problem for Jang’s [13] equation as a procedure to recognize such a surface in the Minkowski space. In doing so, we discover a new expression of quasi-local mass for a large class of “admissible” surfaces (see Theorem A and Remark 1.1). The new mass is positive when the ambient space-time satisfies the dominant energy condition and vanishes on surfaces in the Minkowski space. It also has the nice asymptotic behavior at spatial infinity and null infinity. Some of these results were announced in [29].

Journal ArticleDOI
TL;DR: It turns out that, for any bounded time-varying delays, the magnitude of the delays does not affect the stability of these systems and system stability is completely determined by the system matrices.
Abstract: This brief addresses stability of the discrete-time positive systems with bounded time-varying delays and establishes some necessary and sufficient conditions for asymptotic stability of such systems. It turns out that, for any bounded time-varying delays, the magnitude of the delays does not affect the stability of these systems. In other words, system stability is completely determined by the system matrices.

Journal ArticleDOI
TL;DR: It is shown that, when the system observation matrix restricted to the observable subspace is invertible, the known lower bound on pc is, in fact, the exact critical probability.
Abstract: We consider the problem of Kalman filtering when observations are available according to a Bernoulli process. It is known that there exists a critical probability pc such that, if measurements are available with probability greater than pc, then the expected prediction covariance is bounded for all initial conditions; otherwise, it is unbounded for some initial conditions. We show that, when the system observation matrix restricted to the observable subspace is invertible, the known lower bound on pc is, in fact, the exact critical probability. This result is based on a novel decomposition of positive semidefinite matrices.