Showing papers on "Bounded function published in 1969"

Observables at infinity and states with short range correlations in statistical mechanics

Abstract: We say that a representation of an algebra of local observables has short-range correlations if any observable which can be measured outside all bounded sets is a multiple of the identity, and that a state has finite range correlations if the corresponding cyclic representation does. We characterize states with short-range correlations by a cluster property. For classical lattice systems and continuous systems with hard cores, we give a definition of equilibrium state for a specific interaction, based on a local version of the grand canonical prescription; an equilibrium state need not be translation invariant. We show that every equilibrium state has a unique decomposition into equilibrium states with short-range correlations. We use the properties of equilibrium states to prove some negative results about the existence of metastable states. We show that the correlation functions for an equilibrium state satisfy the Kirkwood-Salsburg equations; thus, at low activity, there is only one equilibrium state for a given interaction, temperature, and chemical potential. Finally, we argue heuristically that equilibrium states are invariant under time-evolution.

On some problems of similarity flow of fluid with a free surface

Abstract: The paper presents the method of solving a class of two-dimensional problems of the similarity flow of an incompressible fluid with a free surface. The fluid is assumed to be non-viscous and weightless. We consider two-dimensional irrotational similarity flows with dimensionless hydrodynamic characteristics depending only on the ratios x/v0t, y/v0t, where x, y are Cartesian co-ordinates, t is time and v0 is a constant of the velocity dimension.The proposed method is based upon using the function introduced by Wagner (1932) and can be applied to the problems where the flow region is bounded by free surfaces and uniformly moving (or fixed) rectilinear impermeable boundaries. Introduction of Wagner's function makes it possible to reduce each of the problems under consideration to a non-linear singular integral equation for the real function.The method is illustrated by solving the classical problem of the uniform symmetrical entry of a wedge into a half-plane of a fluid.

Bounded perturbations of forced harmonic oscillators at resonance

Abstract: Let e be continuous and 2π-periodic, h continuous and bounded, and n>0 an integer. Sufficient conditions for the existence of 2π-periodic solutions of x″+n2x+h(x)= =e(t) are given. The proofs are based on a modification of Cesari's method and the Schauder fixed point theorem.

The nonlinear complementarity problem with applications, part 1

Abstract: The main result in this paper is an existence and uniqueness theorem for the following nonlinear complementarity problem: Given a mapping from then-dimensional Euclidean spaceE n into itself, find a nonnegative vector inE n whose image, under the given mapping, is also nonnegative, the two vectors being orthogonal to each other It is shown that the above problem has a unique solution if the given mapping is continuous and strongly monotone on the nonnegative orthantE + ofE n It is also shown that a sufficient condition for a differentiable mapping to be strongly monotone on an open set is that all the eigenvalues of the symmetric part of its Jacobian be bounded below by a positive constant on the given set

New results in linear system stability

Abstract: This paper considers connections between bounded-input, bounded-output stability and asymptotic stability in the sense of Lyapunov for linear time-varying systems. By modifying slightly the definition of bounded-input, bounded-output stability, an equivalence between the two types of stability is found for systems which are uniformly completely controllable and observable. The various matrices describing the system need not be bounded. Other results relate to the characterization of uniform complete controllability and the derivation of Lyapunov functions for linear time-varying systems.

On the Fourier transform of the indicator function of a planar set.

Abstract: Suppose C is a compact subset of the plane having a piecewise smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates, of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. In ?1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -x- 00. In ?2, we shall give applications of the results of ?1 to some questions in the geometry of numbers. 1. If AC is sufficiently smooth, and has everywhere positive Gaussian curvature, it is known that the function 'D(6) = sup, r312IF(r, 0)1 is bounded on S' (cf. [1]). If AC has points of zero curvature, this need no longer be true (cf. [3]). The following, however, remains true: THEOREM 1. If AC is of class Cn + 3, for some integer n 1, and if the Gaussian curvature of AC is nonzero at all points of AC, with the possible exception of a finite set, at each point of which the tangent line has contact of order 1. Moreover, 'D(6) is always bounded, except in neighborhoods of those points of S' which, regarded as vectors, correspond to exterior or interior normals to AC at points of zero curvature. In a neighborhood of such a point 006 'D(6) is bounded by a multiple of [dist (6, 00)] -n - 1)/2n , where dist (6, 00) is the length of the smaller arc of S' connecting 0 and 6o, and nj is the largest order of contact which can occur between AC and its tangent line, at those points of AC at which the exterior normal is either 00 or- o REMARK. Theorem 1 has analogues in higher dimensions. I shall show in another paper, by different methods, that if C is a compact convex subset of Rn, whose boundary is analytic, and if F(r, 0) is the Fourier transform, in polar coordinates, of the indicator function of the set C, then supr r(n + 1)/21 F(r, 0)1 is of class LP on Sn1, for some p>2. If C is a polygon, the estimates are of a quite different character. THEOREM 2. Suppose C is a polygon. Then

On the asymptotic behavior of the Fourier transform of the indicator function of a convex set

Abstract: Suppose C is a compact, convex subset of Rn, having a smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates (r= (x2 + + x2)12; =(x1/r, . . ., xn/r)) of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. Then it is known (cf. [1], [2]) that the function 'D(6) = supr rn + 1)/2 F(r, 0) is bounded on Sn1, provided AC is sufficiently smooth, and has everywhere positive Gaussian curvature. If AC has points of zero curvature, this need no longer be true (cf. [4]). The following, however, remains true.

Higher order averaging and related methods for perturbed periodic and quasi-periodic systems

Abstract: for 0 ? t 0. Following the basic work of Krylov and Bogoliubov [9] in 1934, the first systematic and comprehensive account of the method of averaging was given by Bogoliubov and Mitropolski [1] in 1958. In this account, the authors were primarily concerned with first and second order approximations. They laid the mathematical foundation for the method which established the validity of the first order approximation over a time interval of 0(1/E). Several authors have subsequently considered higher order averaging, e.g., Volosov [20]-[22], but have only explicitly treated first and second order averaging. In this way they establish an algorithm for the general Nth order method; but they do not establish it as an Nth order asymptotic method on 0 0. In 1964, Mitropolski [12] indicated how to establish Nth order averaging as an Nth order asymptotic method on 0 ? t 2 simply by conjecturing the general structure of the fundamental differential inequality [12, p. 339] from which the Nth order error estimate follows. More recently, Zabreiko and Ledovskaja [23] gave a set of equations defining the general Nth order method of averaging and a set of conditions which they claim are sufficient to establish Nth order averaging as an Nth order asymptotic method on 0 ? t < Llr as E -* 0. In particular, they state [23, p. 1455] that these conditions are satisfied if the right-hand side f(t, x, E) is an almost periodic function of t (and has sufficiently many derivatives in x and ? which are bounded). This statement is in general not true, the reason being that the almost periodicity off(t, x, e) in t does not, in general, imply the boundedness of the function p1(t, x) or of its first partial derivative with respect to x, where

Operators similar to their adjoints

Abstract: Here, and in what follows, all operators will be bounded linear transformations from a fixed Hilbert space into itself. W(S)is the closure of the numerical range W(S) = { (Sx, x): ||x|| = 1 } of S. We will make use of the fact that W(S)is convex and contains the spectrum a(S) of S. The purpose of this note is to present a simple theorem which explains the above result. Before doing this, however, a few remarks are pertinent. To begin with, the technique of [4] actually proves a much better result:

The range of a vector-valued measure

Abstract: Liapounoff, in 1940, proved that the range of a countably additive bounded measure with values in a finite dimensional vector space is compact and, in the nonatomic case, is convex. Later, in 1945, Liapounoff showed, by counterexample, that neither the convexity nor compactness need hold in the infinite dimensional case. The next step was taken by Halmos who in 1948 gave simplified proofs of Liapounoff's results for the finite dimensional case. In 1951, Blackwell [I] considered the case of a measure represented by a finite dimensional vector integral and obtained results similar to those of Liapounoff for these measures. Various versions of Liapounoff's theorem appeared in the 1950's and 1960's, and in 1966, Lindenstrauss [8] gave a very elegant short proof of Liapounoff's earlier result. Finally, in 1968, Olech [9] considered the case of an unbounded measure with range in a finite dimensional vector space. The purpose of this note is to demonstrate that the closure of the range of a measure of bounded variation with values in a Banach space, which is either a reflexive space or a separable dual space, is compact and, in the nonatomic case, is convex. To this end, let Q be a point set and z be a a-field of subsets of U. If X is a Banach space, then an p-valued measure is a countably additive function F defined on 2 with values in X. F is of bounded variation if

THE p-TORSION OF ELLIPTIC CURVES IS UNIFORMLY BOUNDED

Abstract: It is shown that the p-component of the torsion of elliptic curves over a numerical field is bounded.

Cebysev sets in hilbert space

Abstract: 11x-y > 11x-p(x) | if yeTandy =#p(x). In this case, we call the function x -- p(x) the metric projection onto T. If a set Tc H has the property that to each point in H there is a (not necessarily unique) nearest point in H, then we will call T an existence set. Clearly, each Cebysev set is an existence set and each existence set is norm closed. Conversely, if Tis boundedly compact, i.e. intersects each bounded and closed subset of H in a compact set, hence in particular if T is any closed subset of a finite-dimensional Hilbert space, then T is an existence set, and so is every closed subset of T. One of the most basic and elementary statements in Hilbert space theory is the theorem that every closed convex set is a Cebysev set. In 1935, Motzkin showed that if H is finite dimensional then there are no other Cebysev sets. For infinitedimensional Hilbert spaces the question of the existence of nonconvex Cebysev sets is still open. Various authors have proved the convexity of Cebysev sets under additional assumptions [6], [7] (also in more general Banach space settings) and Klee [8] has adduced plausible evidence to support the conjecture that there exist nonconvex Cebysev sets. Here we will exhibit two new additional assumptions insuring that a Cebysev set will be convex. We believe that the second one is the only known such assumption that does not explicitly mention topological properties in some form, only existence properties. 2. A convex function which is the "indefinite integral" of the metric projection. In the following, K will denote some nonconvex Cebysev set. We thus make explicitly the hypothesis that such exist, and our convexity proofs will be by contradiction. Consider the function

On the principle of exchange of stabilities

Abstract: A perturbation theorem is proved: a class of real, bounded (non-self-adjoint) perturbations of norm ϵ to real self-adjoint operators preserve the reality of the simple eigenvalues for ϵ sufficiently small. A bound is obtained on ϵ. Application is made to Benard convection with constant heat sources, radiation, particular time-dependent profiles and nonlinear equations of state and to instability of circular Couette flow for a range of gap widths. In each case the growth rate is the eigenvalue and hence if ϵ c , travelling waves (either growing or decaying) are forbidden.

Remarks on sequential point estimation

Abstract: We consider utilizing a sequential experiment for estimating the mean of a normal distribution when the variance of the distribution is unknown. For a suitable choice of the design constants of the experiment and for loss structures of practical interest, it is shown that the difference in the expected loss with optional stopping and the expected loss which would be incurred if σ were known is a bounded function of σ.

Approximation by inner functions

Abstract: Let L°°(T) denote the complex Banach algebra of (equivalence classes of) bounded measurable functions on the unit circle T, relative to Lebesgue measure m. The norm 11 /11» of an / in L^iT) is the essential supremum of |/| on T. The collection of all bounded holomorphic functions in the open unit disc U forms a Banach algebra which can be identified (via radial limits) with the norm-closed subalgebra H°° of L~(T). A function / in L°°(T) is unimodular if |/| = 1 a.e., on Tm The inner functions are the unimodular members of H°°. It is well known that they play an important role in the study of H°°. The main result (Theorem 1) is that the set of quotients of inner functions is norm-dense in the set of unimodular functions in L°°(T). One consequence of this (Theorem 7) is that the set of radial limits of holomorphic functions of bounded characteristic in U is norm-dense in L°°(T). It is also shown (Theorem 3, 4) that the Gelfand transforms of the inner functions separate points on the Silov boundary of H°°, and this is used to obtain a new proof (and generalization) of a theorem of D. J. Newman (Theorem 4).

Continuous data dependence for the equations of classical elastodynamics

Abstract: Consider a linear elastic solid occupying a bounded regular three-dimensional region B with smooth surface ∂B. The components of displacement ui referred to cartesian axes xi are then well known to satisfy the system of governing equationsin which t denotes the time variable, x = (x1, x2, x3) denotes the position vector, ρ(x) is the non-homogeneous density, assumed positive,. i(x, t) are the Cartesian components of body force per unit mass, and cijkl(x) are the non-homogeneous elasticities, which apart from certain smoothness conditions stated later, are assumed to possess the symmetryThroughout this paper, all suffixes range over the values 1, 2, 3 and the usual converition of summing over repeated indices is adopted. Except where it is in the interest of clarity we avoid explicit mention of the dependence of functions on their arguments.

Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation

Abstract: depends on the dimension d and power /, where G is the infinitesimal generator of a linear nonnegative contraction semigroup on the space B(Rd) of bounded measurable functions on Rd and c is a bounded nonnegative measurable function on Rd. This fact was recently proved by Fujita [2] when G is the Laplacian operator. In this paper we will give upper and lower bounds for the solution of (1.1) constructed by a probabilistic method (cf. (3.4) and (4.7)). As a corollary we shall obtain Fujita's result when G is a fractional power -(-A)", 0< a <2, of the Laplacian operator. Our method is based on probabilistic arguments relating to the branching Markov processes (cf. Ikeda-Nagasawa-Watanabe [3], Sirao [8] and Nagasawa [7]). The necessary facts of probabilistic arguments in this context will be summarized in ?2, while in ?3 and ?4 we shall give upper and lower bounds of the probabilistic solution of (1.1) and some applications.

On quasiconformal mappings which keep the boundary points fixed

Abstract: Introduction. From a variational point of view it is natural to consider quasiconformal selfmappings of a domain which are equal to the identity on the boundary. We say that a function / in the unit disk U= {z I Izl O such that k(+/11I) E E The last section of the present work is devoted to this question. A measurable and bounded complex valued function v in U is said to belong to the class N(Ahlfors [1]) if fu v(z)g(z) dx dy=O for every holomorphic function g in U with 11 g 1 0(2).

Production Smoothing with Stochastic Demand II: Infinite Horizon Case

Abstract: In an earlier paper [5], we generalized and extended Beckmann'a results [1] for a production and inventory problem with proportional smoothing costs and demands being random variables. Our previous results concerned the finite horizon nonstationary case. Here we consider the infinite horizon stationary case. Two curves in the plane determine an optimal policy. They are shown to have slopes between minus one and zero, to be differentiable, and to be bounded by two straight lines with a slope of minus one. These results are used (a) to accelerate each iteration of a successive approximations algorithm and (b) to formulate a linear programming problem from whose solution an optimal policy can be determined.

Periodic pulling and turbulence in a bounded plasma.

Abstract: Experimental results obtained with a $Q$ machine demonstrate the effect of a nonlinear mechanism called periodic pulling that has been proposed to explain the transition to turbulence in a bounded plasma characterized by weakly unstable modes. The effect is characteristic of any distributed-parameter system that can be described by the van der Pol equation and is related to phenomena observed in solid-state oscillators and the gas laser.

On the Thomas-fermi equation.

Abstract: A study has been made of some mathematical aspects of the Thomas-Fermi equation. This is a preliminary report on the results obtained, including (1) convergence of relevant series, (2) existence of unbounded solutions, (3) existence of solutions having an arbitrary branch point, (4) determination of a class of solutions bounded for large values of the variable, and (5) determination of a class of solutions unbounded for small values.