Showing papers on "Uniform boundedness published in 1987"

Oscillations and concentrations in weak solutions of the incompressible fluid equations

Abstract: The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

Global existence and boundedness in reaction-diffusion systems

Abstract: In many applications, systems of reaction-diffusion equations arise in which the nature of the nonlinearity in the reaction terms renders ineffective the standard techniques (such as invariant sets and differential inequalities) for establishing global existence, boundedness, and asymptotic behavior of solutions. In this paper we prove global existence and uniform boundedness for a class of reaction-diffusion systems involving two unknowns in which an a priori bound is available for one component as long as solutions exist. Among this class of systems is the so-called Brusselator, a model from the study of instabilities in chemical processes.

On the convergence of a finite element method for a nonlinear hyperbolic conservation law

Abstract: We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers' equation). The finite element method is higher-order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. We first prove that if a sequence of finite element solutions converges boundedly almost everywhere (as the mesh size tends to zero) to a function u, then u is an entropy solution of the conservation law. This result may be extended to systems of conservation laws with convex entropy in several dimensions. We then prove, using a compensated compactness result of Murat-Tartar, that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of Burgers' equation. We also consider a further modification of the test functions giving a method with improved shock capturing. Finally, we present the results of some numerical experiments.

Approximate recovery of periodic functions of several variables

Abstract: A linear method is constructed for recovery of functions with bounded mixed difference from values at a fixed number of points. The method gives a recovery error close to the corresponding Kolmogorov width of the class of functions with bounded mixed difference.Bibliography: 13 titles.

High velocity particles in a collisionless plasma

Abstract: We prove that smooth solutions of the relativistic Vlasov-Maxwell equations exist for all time provided the kinetic energy density is uniformly bounded.

Deterministic Control of Large-scale Uncertain Dynamical Systems☆

Abstract: We consider the design of control schemes for large-scale systems composed of interconnected subsystems with time-varying uncertainties. The local control, utilizing solely the state of the subsystem, and the global control, utilizing the states of the neighboring subsystems as additional information, are proposed to guarantee certain deterministic performance, including uniform boundedness and uniform ultimate boundedness. It can be shown that the local control is applicable provided a test matrix T satisfies certain requirements. The global control, on the other hand, is applicable if the interconnections between subsystems satisfy certain structural constraints. The salient features of these control schemes is that the only information required is the bounds of uncertainties.

One-way functions and circuit complexity

Abstract: A finite function f is a mapping of {0, 1}n into {0, 1}m⌣{#}, where “#” is a symbol to be thought of as “undefined.” A family of finite functions is said to be one-way (in a circuit complexity sense) if it can be computed with polynomial-size circuits, but every family of inverses of these functions cannot. In this paper we show that, provided functions that are not one-to-one are allowed, one-way functions exist if and only if the satisfiability problem SAT does not have polynomial-size circuits. A family of functions fi(x) can be checked if some family of polynomial-size circuits with inputs x and y can determine if fi(x) = y. A family of functions fi(x) can be evaluated if some family of polynomial-size circuits with input x can compute fi(x). Can all families of total functions that can be checked also be evaluated? We show that this is true if and only if the nonuniform versions of the complexity classes P and UP ⋔ co -UP are equal. A family of functions fi is one-way for constant depth circuits if fi can be computed with unbounded famin circuits of polynomial size and constant depth, but every family of inverses fi−1 cannot. We give two provably one-way functions (in fact permutaions) for constant-depth circuits. The second example has the stronger property that no bit of its inverse can be computed in polynomial size and constant depth.

A Decomposition Theorem for Binary Markov Random Fields

Abstract: Consider a binary Markov random field whose neighbor structure is specified by a countable graph with nodes of uniformly bounded degree. Under a minimal assumption we prove a decomposition theorem to the effect that such a Markov random field can be represented as the nodewise modulo 2 sum of two independent binary random fields, one of which is white binary noise of positive weight. Said decomposition provides the information theorist with an exact expression for the per-site rate-distortion function of the random field over an interval of distortions not exceeding this weight. We mention possible implications for communication theory, probability theory and statistical physics.

An extension of billingsley's uniform boundedness theorem to higher dimensional m-processes

Abstract: For random functions belonging to the C[0,1] or D[0,1] space, Theorem 12.2 of Billingsley's monograph [2] relates to a probability inequality for the supremum, and it plays a vital role in the proof of the tightness of these processes. In the context of tightness of multi-parameter processes (i.e., for random functions belonging to the D[0,1]* space, for some q g 1), various extensions of the Billingsley inequality have been considered by a host of workers (viz., Bickel and Wichura [1] and references cited therein). For robust estimation in general linear models (viz., Jureckova and Sen [5] and the references cited therein), it may be convenient to consider some general multi-parameter M-processes and to exploit their asymptotic linearity results in the study of the properties of the derived estimators. In this context, a basic requirement is the uniform boundedness in probability of such M-processes. Such a result can, of course, be derived through the weak convergence of such processes (viz., Jureckova and Sen [3], [4] and others). However, this may demand comparatively more stringent regularity conditions. For this reason, for a general class of multi-dimensional M-processes, an extension of Billingsley's uniform boundedness theorem is considered under less stringent regularity conditions, and applications of this result in statistical inference are stressed. Along with the preliminary notions, the main theorem is presented in Section 2. Applications are considered in the last section.

Prophet Compared to Gambler: An Inequality for Transforms of Processes

Abstract: A prophet is a player with complete foresight; a gambler knows only the past and the present, but not the future. If each of them bets on differences of consecutive nonnegative random variables $X_i$ such that $E(X_i|X_{i - 1}) = EX_i$, the players multiplying their stakes by uniformly bounded variables, then the expected gain of the prophet is at most three times that of the gambler. The constant 3 is optimal.

Sharp order estimates for best rational approximations in classes of functions representable as convolutions

Abstract: Let be a function of bounded variation, , and the Weyl kernel of order , i.e. , . Denote by and the classes of functions represented by the corresponding formulas The conjugate classes of functions and are also considered; they are convolutions of conjugate Weyl kernels with functions of bounded variation.The following main result is proved: where is the best uniform approximation by trigonometric rational functions of order at most , and is one of the classes Bibliography: 13 titles.

Discrete velocity models with small data

Abstract: It is shown that all discrete velocity models conserving mass, momentum and energy and without self-interactions have global, uniformly bounded solutions for small enough initial data with compact support. This is interesting because of a recent example with small data inL1 ∩L∞ whose solution is not bounded, and shows the unboundedness is due to mass being fed in from infinity.

The space of totally bounded analytic functions

Abstract: This paper is a continuation of our project on “inverse interpolation”, begun in [6]. In brief, the task of inverse interpolation is to deduce some property of a function f from some given property of the set L of its Lagrange interpolants. In the present work, the property of L is that it be a uniformly bounded set of functions when restricted to the domain of f . In particular (see Section 3), when the domain is a disc, we deduce sharp bounds on the successive derivatives of f . As a result, f must extend to be an analytic function (of restricted growth) in the concentric disc of thrice the original radius.

Some topological properties of the solution sets of parametrized minimax problems

Abstract: Sufficient conditions are given to guarantee the closedness and uniform boundedness of the solution sets corresponding to a pair of dual parametrized minimax problems. The parameter of the minimax problems belongs to a metrizable space and affects not only the objective function, but also the feasible sets.

Linear Feedback Control of Uncertain Robotic Manipulators with Guaranteed Boundedness

Abstract: Guaranteed uniform boundedness of states and the possibility of transient performance adjustment are demonstrated for robotic manipulators with uncertain parameters.

SPACE C_h, BOUNDEDNESS AND PERIODIC SOLUTIONS OF FDE WITH INFINITE DELAY

Abstract: This paper will introduce a new space C_k with norm |*|_h which is effective for FDE with infinite delay. We combine C_h and Liapunov's second method and obtain the conditions ensuringh-uniform boundedness and h-ultimate uniform boundedness of the solutions to(2). A theorem due to T. Yoshizawa regarding the existence of periodic solutions is generalized.

Robust state feedback stabilization of discrete-time uncertain dynamical systems

Abstract: The problem of stabilization of a class of discrete-time dynamic systems containing uncertain elements using the direct method of Lyapunov is considered in this paper. State feedback controllers that guarantee uniform boundedness and uniform ultimate boundedness of the closed loop system are proposed.

Some Theorems Concerning Holomorphic

Abstract: II~~~C-~-~+~~l~Ialf~~~ll,~~~ and ll~~~C-~~+~~I~l~lf~~+~~~Il,,~ is uniformly bounded in y. (b) f(z) is an entire function and for every E >0 but for no E > 0,