Showing papers on "Bounded function published in 1992"
TL;DR: In this article, the authors define a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions, and prove that bounded sequences in $L^2 (Omega )$ are relatively compact with respect to this new type of convergence.
Abstract: Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.
2,279 citations
TL;DR: In this article, a generalization of the PAC learning model based on statistical decision theory is described, where the learner receives randomly drawn examples, each example consisting of an instance x in X and an outcome y in Y, and tries to find a hypothesis h : X < A, where h in H, that specifies the appropriate action a in A to take for each instance x, in order to minimize the expectation of a loss l(y,a).
Abstract: We describe a generalization of the PAC learning model that is based on statistical decision theory. In this model the learner receives randomly drawn examples, each example consisting of an instance x in X and an outcome y in Y , and tries to find a hypothesis h : X --< A , where h in H , that specifies the appropriate action a in A to take for each instance x , in order to minimize the expectation of a loss l(y,a). Here X, Y, and A are arbitrary sets, l is a real-valued function, and examples are generated according to an arbitrary joint distribution on X times Y . Special cases include the problem of learning a function from X into Y , the problem of learning the conditional probability distribution on Y given X (regression), and the problem of learning a distribution on X (density estimation). We give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces H , and show how this implies learnability with bounded sample size, disregarding computational complexity. As an application, we give distribution-independent upper bounds on the sample size needed for learning with feedforward neural networks. Our theorems use a generalized notion of VC dimension that applies to classes of real-valued functions, adapted from Pollard''s work, and a notion of *capacity* and *metric dimension* for classes of functions that map into a bounded metric space. (Supersedes 89-30 and 90-52.) [Also in "Information and Computation", Vol. 100, No.1, September 1992]
1,025 citations
TL;DR: In this paper, a nonlinear combination of saturation functions of linear feedbacks is proposed to stabilize a chain of integrators of arbitrary order, where the saturation function is a linear near the origin of the input.
927 citations
Book•
01 Feb 1992
TL;DR: In this paper, the authors introduce the concept of Area Preserving Maps (APM) and the notion of Stochastic Manifestations of Chaos in Quantum Scattering Processes (SOCP).
Abstract: 1. Overview / 2. Fundamental Concepts / 3. Area Preserving Maps / 4. Global Properties / 5. Random Matrix Theory / 6. Bounded Quantum Systems / 7. Manifestations of Chaos in Quantum Scattering Processes / 8. Semi-Classical Theory -- Path Integrals / 9. Time-Periodic Systems / 10. Stochastic Manifestations of Chaos / Appendices.
591 citations
TL;DR: This note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint and one of the algorithms presented here is stronglyPolynomial.
Abstract: This note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint. The main difficulty in constructing such algorithms arises since no trivial lower and upper bounds on the solution value, whose ratio is polynomially bounded, are known. In spite of this difficulty, one of the algorithms presented here is strongly polynomial. Applications to other problems are also discussed.
590 citations
Book•
30 Nov 1992
TL;DR: In this article, the authors present a classification of linear and nonlinear differential expressions with respect to their oscillation properties, including the subspace of solutions vanishing at infinity, bounded and unbounded solutions, and singular solutions.
Abstract: Series Editor's Preface. Preface. Basic Notation. I: Linear Differential Equations. 1. Equations having Properties A and GBPIiGBP. 2. Oscillatory and Nonoscillatory Equations. 3. Oscillation Properties of Solutions Vanishing at Infinity. 4. The Subspace of Solutions Vanishing at Infinity. 5. Bounded and Unbounded Solutions. 6. Asymptotic Formulas. II: Quasilinear Differential Equations. 7. Statement of the Problem Auxiliary Assertions. 8. The Family of Lh Type Solutions of the Equation. 9. L0h, LINFINITYh and Lh Type Equations. III: General Nonlinear Differential Equations. 10. Theorems on the Classification of Equations with respect to their Oscillation Properties. 11. Singular Solutions. 12. Fast Growing Solutions. 13. Kneser Solutions. 14. Proper Oscillatory Solutions. IV: Higher Order Differential Equations of Emden-Fowler Type. 15. Oscillatory Solutions. 16. Nonoscillatory Solutions. V: Second Order Differential Equations of Emden-Fowler Type. 17. Existence Theorems for Proper and Singular Solutions. 18. Oscillation and Nonoscillation Criteria for Proper Solutions. 19. Unbounded and Bounded Solutions Solutions Vanishing at Infinity. 20. Asymptotic Formulas. References. Author Index. Subject Index.
486 citations
TL;DR: In this paper, the authors apply the techniques developed in [Cl] to the problem of mappings with a convex potential between domains, and study the map vf = V W for a Lipschitz convex, such that V,/ maps Ql onto Q?2 in the a.e. sense and in some (weak) sense.
Abstract: In this work, we apply the techniques developed in [Cl] to the problem of mappings with a convex potential between domains. That is, given two bounded domains Q, Q2 of Rn and two nonnegative real functions fi defined in Qi that are bounded away from zero and infinity, we want to study the map vf = V W for a Lipschitz convex ,v, such that V ,/ maps Ql onto Q?2 in the a.e. sense and in some (weak) sense. (1) f2(VyV) det Dij V = f1 (X) . In recent work Y. Brenier showed existence and uniqueness of such a map (provided that JaQil = 0) under the obvious compatibility condition
445 citations
TL;DR: In this paper, it was proved that if Z is a bounded set of matrices such that all left infinite products converge, then 8 generates a bounded semigroup, and the equality of two differently defined joint spectral radii for a set of matrix matrices.
407 citations
TL;DR: First some well-known features of superconducting materials are reviewed and then various results concerning the model, the resultant differential equations, and their solution on bounded domains are derived.
Abstract: The authors consider the Ginzburg–Landau model for superconductivity. First some well-known features of superconducting materials are reviewed and then various results concerning the model, the resultant differential equations, and their solution on bounded domains are derived. Then, finite element approximations of the solutions of the Ginzburg–Landau equations are considered and error estimates of optimal order are derived.
398 citations
01 Jan 1992
TL;DR: In this article, the authors present two different approaches to the study of multiresolution analysis and wavelets on a bounded interval, based on the semi-orthogonal Chui-Wang spline wavelets.
Abstract: The aim of this paper is to present two different approaches to the study of multiresolution analysis and wavelets on a bounded interval. Recently, Meyer obtained orthonormal wavelets on a bounded interval by restricting Daubechies’ scaling functions and wavelets to [0, 1] and applying the Gram-Schmidt procedure to orthonormalize the restrictions. Our own approach — presented in the second part of the paper — is based on the semi-orthogonal Chui-Wang spline-wavelets. In this case we no longer have orthogonality in one scale, but there are explicit formulae for these wavelets.
342 citations
TL;DR: In this paper, the existence of positive solutions of Dirichlet problems in a bounded convex domain with smooth boundary was investigated and L ∞ a priori bounds were obtained in the same spirit as in De Figueiredo-Lions-Nussbaum.
Abstract: We investigate the existence of positive solutions of a Dirich let problem for the system \( - \Updelta u = f(v),\,\Updelta v = g(u) \) in a bounded convex domain Ω of \( {\mathbb{R}}^{N} \) with smooth boundary. In particular L ∞ a priori bounds are obtained in the same spirit as in De Figueiredo—Lions—Nussbaum [7].
TL;DR: It is proved that any functional term of appropriate type actually encodes a polynomial-time algorithm and that conversely any polynometric-time function can be obtained in this way.
TL;DR: In this article, the authors consider variational integrals defined for (sufficiently regular) functions u: Ω→Rm, where u is a bounded open subset of Rn, Du(x) denotes the gradient matrix of u at x and f is a continuous function on the space of all real m × n matrices Mm × n.
Abstract: We consider variational integralsdefined for (sufficiently regular) functions u: Ω→Rm. Here Ω is a bounded open subset of Rn, Du(x) denotes the gradient matrix of u at x and f is a continuous function on the space of all real m × n matrices Mm × n. One of the important problems in the calculus of variations is to characterise the functions f for which the integral I is lower semicontinuous. In this connection, the following notions were introduced (see [3], [9], [10]).
13 Jul 1992
TL;DR: It is shown that the general problem of computation of phytogenies for species sets is NP-Complete, and that the various finite-state approaches for bounded treewidth cannot be applied to the fixed-parameter forms of the problem.
Abstract: One of the major efforts in molecular biology is the computation of phytogenies for species sets. A longstanding open problem in this area is called the Perfect Phylogeny problem. For almost two decades the complexity of this problem remained open, with progress limited to polynomial time algorithms for a few special cases, and many relaxations of the problem shown to be NP-Complete. From an applications point of view, the problem is of interest both in its general form, where the number of characters may vary, and in its fixed-parameter form. The Perfect Phylogeny problem has been shown to be equivalent to the problem of triangulating colored graphs[30]. It has also been shown recently that for a given fixed number of characters the yes-instances have bounded treewidth[45], opening the possibility of applying methodologies for bounded treewidth to the fixed-parameter form of the problem. We show that the Perfect Phylogeny problem is difficult in two different ways. We show that the general problem is NP-Complete, and we show that the various finite-state approaches for bounded treewidth cannot be applied to the fixed-parameter forms of the problem.
TL;DR: A modified Hopfield neural network model for regularized image restoration is presented, which allows negative autoconnections for each neuron and allows a neuron to have a bounded time delay to communicate with other neurons.
Abstract: A modified Hopfield neural network model for regularized image restoration is presented. The proposed network allows negative autoconnections for each neuron. A set of algorithms using the proposed neural network model is presented, with various updating modes: sequential updates; n-simultaneous updates; and partially asynchronous updates. The sequential algorithm is shown to converge to a local minimum of the energy function after a finite number of iterations. Since an algorithm which updates all n neurons simultaneously is not guaranteed to converge, a modified algorithm is presented, which is called a greedy algorithm. Although the greedy algorithm is not guaranteed to converge to a local minimum, the l/sub 1/ norm of the residual at a fixed point is bounded. A partially asynchronous algorithm is presented, which allows a neuron to have a bounded time delay to communicate with other neurons. Such an algorithm can eliminate the synchronization overhead of synchronous algorithms. >
Posted Content•
TL;DR: In this article, it was shown that a standard multilayer feedforward network with a locally bounded piecewise activation function can approximate any continuous function to any degree of accuracy if and only if the network's activation function is not a polynomial.
Abstract: Several researchers characterized the activation function under which multilayer feedforwardnetworks can act as universal approximators. We show that most of all the characterizationsthat were reported thus far in the literature are special cases of the followinggeneral result: a standard multilayer feedforward network with a locally bounded piecewisecontinuous activation function can approximate any continuous function to any degree ofaccuracy if and only if the network's activation function is not a polynomial. We alsoemphasize the important role of the threshold, asserting that without it the last theoremdoes not hold.
TL;DR: A decomposition algorithm for solving convex programming problems with separable structure that reduces to the ordinary method of multipliers when the problem is regarded as nonseparable.
Abstract: This paper presents a decomposition algorithm for solving convex programming problems with separable structure. The algorithm is obtained through application of the alternating direction method of multipliers to the dual of the convex programming problem to be solved. In particular, the algorithm reduces to the ordinary method of multipliers when the problem is regarded as nonseparable. Under the assumption that both primal and dual problems have at least one solution and the solution set of the primal problem is bounded, global convergence of the algorithm is established.
TL;DR: In this article, the authors investigated the behavior of the solution u(x, t) of (...) where Δ = Σ 1=1 n ∂ 2 /∂ xi 2 is the Laplace operator, p > 1 is a constant, T > 0, and φ is a nonnegative bounded continuous function in R n.
Abstract: We investigate the behavior of the solution u(x, t) of (...) where Δ = Σ 1=1 n ∂ 2 /∂ xi 2 is the Laplace operator, p > 1 is a constant, T > 0, and φ is a nonnegative bounded continuous function in R n . The main results are for the case when the initial value φ has polynomial decay near x = ∞. Assuming φ ∼ λ(1+|x|) #75a with λ, a > 0, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution u(x, t) are answered in terms of simple conditions on λ, a, p and the space dimension n
TL;DR: It is established that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels, which is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm.
Abstract: We consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm. Some multiplicative variants of the multilevel methods are also considered. We establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one, which is independent of the number of levels. For a proper ordering, the iteration operators correspond to the error propagation operators of certain V-cycle multigrid methods, using Gauss-Seidel and damped Jacobi methods as smoothers, respectively.
TL;DR: In this paper, it was shown that global asymptotically flat singularity-free solutions of the spherically symmetric Vlasov-Einstein system exist for all initial data which are sufficiently small in an appropriate sense.
Abstract: We show that global asymptotically flat singularity-free solutions of the spherically symmetric Vlasov-Einstein system exist for all initial data which are sufficiently small in an appropriate sense. At the same time detailed information is obtained concerning the asymptotic behaviour of these solutions. A key element of the proof which is also of intrinsic interest is a local existence theorem with a continuation criterion which says that a solution cannot cease to exist as long as the maximum momentum in the support of the distribution function remains bounded. These results are contrasted with known theorems on spherically symmetric dust solutions.
TL;DR: In this article, the problem of deriving so-called hard error bounds for estimated transfer functions is addressed, i.e., the true system Nyquist plot will be confined with certainty to a given region, provided that the underlying assumptions are satisfied.
Abstract: The problem of deriving so-called hard-error bounds for estimated transfer functions is addressed. A hard bound is one that is sure to be satisfied, i.e. the true system Nyquist plot will be confined with certainty to a given region, provided that the underlying assumptions are satisfied. By blending a priori knowledge and information obtained from measured data, it is shown how the uncertainty of transfer function estimates can be quantified. The emphasis is on errors due to model mismatch. The effects of unmodeled dynamics can be considered as bounded disturbances. Hence, techniques from set membership identification can be applied to this problem. The approach taken corresponds to weighted least-squares estimation, and provides hard frequency-domain transfer function error bounds. The main assumptions used in the current contribution are: that the measurement errors are bounded, that the true system is indeed linear with a certain degree of stability, and that there is some knowledge about the shape of the true frequency response. >
01 Aug 1992
TL;DR: It is shown that the distance to a feasible point near the solution set can be bounded by the norm of a natural residual at that point, and this bound is used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the affine variational inequality problem.
Abstract: Consider the affine variational inequality problem. It is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point. This bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem. This latter result improves upon a recent result of Luo and Tseng that further assumes the problem to be monotone.
TL;DR: In this paper, the authors introduce a coarse homology theory using chains of bounded complexity and study some of its first properties on non-compact spaces, and show that for any nonamenable group F one can find a spin manifold with fundamental group F, with nonzero A-genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi-isometry class.
Abstract: The object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity. Manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers. We shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties. The most interesting result characterizes when H uf (X) vanishes as an analogue and strengthening of F0lner's amenability criterion for groups in terms of isoperimetric inequalities. (See [4].) One can view this result as producing a successful infinite Ponzi scheme on any nonamenable space. Each point, with only finite resources, gives to some of its neighbors some of these resources, yet receives more from the remaining neighbors. As one can imagine this is useful for eliminating obstructions on noncompact spaces. This has a number of applications. We present two of them. The first produces tilings that are "unbalanced" on any nonamenable polyhedron. Unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically. Unfortunately, imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of Euclidean space (Penrose tilings) are necessarily not accessible to our method. On the other hand, most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion. The second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature. We prove a converse to a theorem of Roe. We show that for any nonamenable group F one can find a spin manifold with fundamental group F, with nonzero A-genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi-isometry class.
TL;DR: In this article, the authors studied the multifractal structure of random cascades and provided a new role for the modified cumulant generating function (structure function) studied by Mandelbrot, Kahane and Peyriere.
Abstract: The multifractal structure of a measure refers to some notion of dimension of the set which supports singularities of a given order $\alpha$ as a function of the parameter $\alpha$. Measures with a nontrivial multifractal structure are commonly referred to as multifractals. Multifractal measures are being studied both empirically and theoretically within the statistical theory of turbulence and in the study of strange attractors of certain dynamical systems. Conventional wisdom suggests that various definitions of the multifractal structure of random cascades exist and coincide. While this is rigorously known to be the case for certain deterministic cascade measures, the same is not true for random cascades. The purpose of this paper is to pursue this theory for a class of random cascades. In addition to providing a new role for the modified cumulant generating function (structure function) studied by Mandelbrot, Kahane and Peyriere, the results have implications for the theoretical interpretation of empirical data on turbulence and rainfall distributions.
TL;DR: In this paper, the authors derived a semiclassical secular equation which applies to quantized (compact) billiards of any shape, based on the fact that the billiard boundary defines two dual problems: the inside problem of the bounded dynamics, and the outside problem which can be looked upon as a scattering from the boundary as an obstacle.
Abstract: The authors derive a semiclassical secular equation which applies to quantized (compact) billiards of any shape. Their approach is based on the fact that the billiard boundary defines two dual problems: the 'inside problem' of the bounded dynamics, and the 'outside problem' which can be looked upon as a scattering from the boundary as an obstacle. This duality exists both on the classical and quantum mechanical levels, and is therefore very useful in deriving a semiclassical quantization rule. They obtain a semiclassical secular equation which is based on classical input from a finite number of classical periodic orbits. They compare their result to secular equations which were derived by other means, and provide some numerical data which illustrate their method when applied to the quantization of the Sinai billiard.
TL;DR: In this paper, the use of finite elements to discretize the time dependent Maxwell equations on a bounded domain in 3D space is analyzed and energy norm error estimates are provided when general finite element methods are used to discrete the equations in space.
Abstract: The use of finite elements to discretize the time dependent Maxwell equations on a bounded domain in three-dimensional space is analyzed Energy norm error estimates are provided when general finite element methods are used to discretize the equations in space In addition, it is shown that if some curl conforming elements due to Nedelec are used, error estimates may also be proved in the $L^2 $ norm
TL;DR: To assess stability, which is also a precondition for scalability, the authors introduce and measure the load-sharing hit-ratio, the ratio of remote execution requests concluded successfully.
Abstract: A method for qualitative and quantitative analysis of load sharing algorithms is presented, using a number of well known examples as illustration Algorithm design choice are considered with respect to the main activities of information dissemination and allocation decision making It is argued that nodes must be capable of making local decisions, and for this efficient state, dissemination techniques are necessary Activities related to remote execution should be bounded and restricted to a small proportion of the activity of the system The quantitative analysis provides both performance and efficiency measures, including consideration of the load and delay characteristics of the environment To assess stability, which is also a precondition for scalability, the authors introduce and measure the load-sharing hit-ratio, the ratio of remote execution requests concluded successfully Using their analysis method, they are able to suggest improvements to some published algorithms >
TL;DR: In this paper, the adaptive control of a continuous-time plant of arbitrary relative degree, in the presence of bounded disturbances as well as unmodeled dynamics, is addressed, and the adaptation considered is the usual gradient update law with parameter projection, the latter being the only robustness enhancement modification employed.
Abstract: The problem of adaptive control of a continuous-time plant of arbitrary relative degree, in the presence of bounded disturbances as well as unmodeled dynamics, is addressed. The adaptation considered is the usual gradient update law with parameter projection, the latter being the only robustness enhancement modification employed. It is shown that if the unmodeled dynamics, which consists of multiplicative as well as additive system uncertainty, is small enough, then all the signals in the closed-loop system are bounded. This shows that extra modifications are not necessary for robustness with respect to bounded disturbances and small unmodeled dynamics. In the nominal case, where unmodeled dynamics and disturbances are absent, the asymptotic error in tracking a given reference signal is zero. Moreover, the performance of the adaptive controller is also robust. >
TL;DR: Positive and negative results indicate that the tail of the distribution of the Distribution of the cycle length τ
Abstract: Let X = {X(t)}t ≥ 0 be a stochastic process with a stationary version X*. It is investigated when it is possible to generate by simulation a version X˜ of X with lower initial bias than X itself, in the sense that either X˜ is strictly stationary (has the same distribution as X*) or the distribution of X˜ is close to the distribution of X*. Particular attention is given to regenerative processes and Markov processes with a finite, countable, or general state space. The results are both positive and negative, and indicate that the tail of the distribution of the cycle length t plays a critical role. The negative results essentially state that without some information on this tail, no a priori computable bias reduction is possible; in particular, this is the case for the class of all Markov processes with a countably infinite state space. On the contrary, the positive results give algorithms for simulating X˜ for various classes of processes with some special structure on t. In particular, one can generate X˜ as strictly stationary for finite state Markov chains, Markov chains satisfying a Doeblin-type minorization, and regenerative processes with the cycle length t bounded or having a stationary age distribution that can be generated by simulation.