Showing papers on "Bounded function published in 1979"

Symmetry and related properties via the maximum principle

Abstract: We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.

Discrete-time multivariable adaptive control

Abstract: This paper establishes global convergence for a class of adaptive control algorithms applied to discrete time multi-input multi-output deterministic linear systems. It is shown that the algorithms will ensure that the system inputs and outputs remain bounded for all time and that the output tracking error converges to zero.

Function theory on manifolds which possess a pole

Abstract: Preliminaries.- The Hessian comparison theorem.- Sub-mean-value theorem for subharmonic functions.- Quasi-isometry of the exponential map and the absence of positive harmonic functions.- New criterion of hyperbolicity.- Bounded exhaustion functions and a new class of hyperbolic Kahler manifolds.- A digression.- The Bergman metric.- Manifolds biholomorphic to ?n.

Direct Velocity Feedback Control of Large Space Structures

Abstract: which relates the displacements u(x,t) of the equilibrium position of a flexible structure ft (a bounded open connected set with smooth boundary dfi in /7-dimensiona l space R") to the applied force distribution F(x,t). The mass density m(x) is a positive function of the location x on the structure. The change of variables u(x,t) ^u(x,t)/m(x) l/2 eliminates m(x) without changing the properties of Eq. (1) and, henceforth, assume m(x) = \. The non-negative real number £ is the damping coefficient of the structure; it is quite small for LSS. The operator A is a symmetric, non-negative, time-invariant differential operator with compact resolvent and a square root A'/2. The domain D(A) of A contains all sufficiently differentiable functions which satisfy the appropriate boundary conditions for the LSS. D(A) is dense in the Hilbert space //=L 2 (Q) with the usual inner product (.,.)

A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian systems

Abstract: Regular and irregular motions of bounded conservative Hamiltonian systems of N degrees of freedom can be distinguished by the structure of the frequency spectrum of a single trajectory. The spectral entropy S is introduced which provides a measure of the distribution of the frequency components. Numerical calculations on the model Henon and Heiles system and a realistic molecular model are performed. Power spectra are obtained from numerical solutions to Hamilton's equations using fast Fourier transforms and the Hanning method. For regular trajectories S is found to stabilise after a finite time of integration, while for irregular cases S increases erratically. Estimates of the relative volume of regular regions of phase space as a function of energy are given for the two systems.

Stability of dynamical systems: A constructive approach

Abstract: A set A of n \times n complex matrices is stable if for every neighborhood of the origin U \subset C^{n} , there exists another neighborhood of the origin V , such that for each M \in A' (the set of finite products of matrices in A), MV\subset U . Matrix and Liapunov stability are related. Theorem: A set of matrices A is stable if and only if there exists a norm, |\cdot | , such that for all M \in A , and all z \in C^{n} , |Mz| . The norm is a Liapunov function for the set A . It need not be smooth; using smooth norms to prove stability can be inadequate. A novel central result is a constructive algorithm which can determine the stability of A based on the following. Theorem: A,={M0,Mj,. .,Mmi) is a set of m distinct complex matrices. Let Wo be a bounded neighborhood of the origin. Define for k > 0 , Wk =convexhbull ~ Mk'Wk - I where k'=(k- 1) mod m . Then A isstableifand only if V-U Q,_ is bounded. W* is the norm of the first theorem. The constructive algorithm represents a convex set by its extreme points and uses linear programming to construct the successive W_k . Sufficient conditions for the finiteness of constructing W_k from W_{k-1} , and for stopping the algorithm when either the set is proved stable or unstable are presented. A is generalized to be any convex set of matrices. A dynamical system of differential equations is stable if a corresponding set of matrices --associated with the logarithmic norms of the matrices of the linearized equations--is stable. The concept of the stability of a set of matrices is related to the existence of a matrix norm. Such a norm is an induced matrix norm if and only if the set of matrices is maximally stable (ie., it cannot be enlarged and remain stable).

Removable singularities in Yang-Mills fields

Abstract: We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every Yang-Mills field overR 4 with bounded functional (L 2 norm) may be obtained from a field onS 4=R 4∪{∞}. Hodge (or Coulomb) gauges are constructed for general small fields in arbitrary dimensions including 4.

On rational approximations of semigroups

Abstract: We show that if $r^n (hA),nh = t$, is an A-acceptable rational approximation of a strongly continuous semigroup $e^{tA} $ on a Banach space, then for t bounded, $\| {r^n (hA)} \| \leqq Cn^{{1/2}} $, with certain improvements under additional hypotheses on r. We also discuss the convergence of $r^n (hA)v$ to $e^{tA} v$ as $h \to 0$ under various assumptions on r and v.

Ergodic theorems for spatial processes

Abstract: We investigate the ergodic properties of spatial processes, i.e. stochastic processes with an index set of bounded Borel subsets in ℝv, and prove mean and individual ergodic theorems for them. As important consequences we get a generalization of McMillan's theorem due to Fritz [4]; the existence of specific energy for a large class of interactions in the case of marked point processes in ℝv and the existence of the specific Minkowski Quermasintegrals for Boolean models in ℝv with convex, compact grains.

Some properties of Γ-limits of integral functionals

Abstract: We prove some integral representation theorems for the Γ limit in L1 of sequences of the form\(\int\limits_\Omega {f_h (x,Du)}\) dx in coercivity and bounded growth hypothesis which are optimal as it is checked with examples. These results are utilized to describe the L1 lower semicontinuous envelope of a given functional. We consider also the stability of Γ limits with respect to obstacle type perturbations and prove the homogenization formulas in conditions more generals of those already considered by several authors.

Contributions to the Theory of Nonparametric Regression, with Application to System Identification

Abstract: The objective in nonparametric regression is to infer a function $m(x)$ on the basis of a finite collection of noisy pairs $\{(X_i, m(X_i) + N_i)\}^n_{i=1}$, where the noise components $N_i$ satisfy certain lenient assumptions and the domain points $X_i$ are selected at random. It is known a priori only that $m$ is a member of a nonparametric class of functions (that is, a class of functions like $C\lbrack 0, 1\rbrack$ which, under customary topologies, does not admit a homeomorphic indexing by a subset of a Euclidean space). The main theoretical contribution of this study is to derive uniform convergence bounds and uniform consistency on bounded intervals for the Nadaraya-Watson kernel estimator and its derivatives. Also, we obtain the corresponding convergence results for the Priestly-Chao estimator in the case that the domain points are nonrandom. With these developments we are able to apply nonparametric regression methodology to the problem of identifying noisy time-varying linear systems.

Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements

Abstract: The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Local error estimates are given for the case when the finite element partitions are refined in a systematic fashion near corners. 0. Introduction. We assume that the reader is familiar with Part 1, [21, of this paper; some notation is briefly recollected in Section 1. General references to the literature were given in the Bibliography of Part 1. Of these references, the following are particularly relevant to our present situation: Babuska [1], Babuska and Aziz [21, Babuska and Rheinboldt [4], Babuska and Rosenzweig [5], Eisenstat and Schultz [1 1], Thatcher [36]. Let Q be a bounded simply connected plane polygonal domain with interior angles 0 2 denote the optimal order of the parameter h to which the spaces S" can approximate smooth functions in Lq norms. Furthermore, let Q2j, j = 1, . .. , M, be the intersection of Q with a disc of radius Rj centered at the jth vertex and such that Q2j contains no other vertex, and set Qo = 2\(UL, , 1). Also, put f3 = 7r/aj. In Part 1 we showed that with e > 0 arbitrarily small (see Part 1, Theorem 4.1 Received March 1, 1978. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. * This work was supported in part by the National Science Foundation. ? 1979 American Mathematical Society 0025-571 8/79/0000-0051 /$08.00 465 This content downloaded from 157.55.39.161 on Mon, 23 May 2016 06:01:22 UTC All use subject to http://about.jstor.org/terms 466 A. H. SCHATZ AND L. B. WAHLBIN for the precise hypotheses), IIU UhIIL (Q2i) r, we may take hM,k h (i.e., no refinement is necessary); whereas if OM r/2, no refinement is necessary at that vertex. If g3 2) the refinement process can be taken to start fairly close to the corner according to (0.10), and is less stringent than at the Mth vertex (even if f3 OM)The conditions (0.10)-(0.12) can also be motivated from simple approximation considerations, see Section 4. Let us remark that if an hr-E rate of convergence is desired only on the interior domain QO, then the weaker kind of refinement described in (0.10)-(0.12) suffices at each corner. To elucidate the above, let us give three examples. Example 0.1. A procedure for placing the nodes in the radial direction near VM. Consider the problem of how to place N + 1 nodes over [0, 1] so as to obtain an efficient approximation of the function xg (= gM) with piecewise polynomials of degree r 1. This problem was solved by Rice [1], who explicitly prescribed the location of the nodes so as to obtain a good approximation, asymptotically as N oo. Essentially, the N + 1 nodes xi, i =0, . . ., N, were taken as xi = (jIN)rl. In the two dimensional situation, one can, e.g., construct a triangular mesh near VM in the following fashion, Figure 1. Draw N + 1 radial lines (including the boundaries) from vM; along each of these mark down the N + 1 points xi. Then connect the ith points on the successive radial lines, thus obtaining a cobweb-like set of quadrilaterals. Now triangulate those by drawing one diagonal in each. The family of triangulations obtained in this simple way will, as N X oo, satisfy a maximum angle condition, but not a minimum angle one. In order to satisfy the latter, a more complicated construction would be necessary.

Global Optimization Using Interval Analysis: The One-Dimensional Case

Abstract: We show how interval analysis can be used to compute the minimum value of a twice continuously differentiable function of one variable over a closed interval. When both the first and second deriva- tives of the function have a finite number of isolated zeros, our method never fails to find the global minimum. Consider a function f(x) in C 2. We shall describe a method for computing the minimum value of f(x) on a closed interval (a, b). We shall see that, if f'(x) and f"(x) have only a finite number of isolated zeros, our method always converges. In a subsequent paper, we shall show how the method can be extended to the case in which x is a vector of more than one variable. Moreover, it will be extended to the constrained case, and a modified method will remove the differentiability condition. The present paper serves to introduce the necessary ideas. In practice, we can only compute minima in a bounded interval. Hence, it is no (additional) restriction to confine our attention to a closed interval. The term global minimum used herein refers to the fact that we find the smallest value of f(x) throughout (a, b). We shall not mistake a local minimum for the global one. Indeed, our method will usually not find local minima, unless forced to do so. Its efficiency would then be degraded if it did. In our method, we iteratively delete subintervals of (a, b) until the remaining set is sufficiently small. These subintervals consist of points at which either f(x) is proved to exceed the minimum in value or else the derivative is proved to be nonzero.

Aspect and the bounded/unbounded (telic/atelic) distinction

Abstract: This paper deals with the bounded/unbounded (telicjatelic) distinction which is relevant to the study of aspect (more specifically, 'Aktionsart') but which has been defined in different ways and has been applied to different objects in the linguistic literature. In this paper the author aims (1) to clear up some of the confusion by formulating an accurate definition; (2) to show that the distinction has been wrongly defined in terms of two values ('-Abounded' and '—bounded'), as some sentences clearly require a neutral value ('&bounded'); (3) to make clear that this triple distinction also provides a basis for distinguishing between relevant classes ofdurational adverbials; and (4) to throw some light on the factor s that determine the +bounded, —bounded, or to-bounded character of linguistic objects. In most recent articles on aspect and related semantic problems a distinction is made between verb phrases like drink beer and drink three glasses of beer. The former are said to be 'unbounded' (atelic, durative, imperfective, nonconclusive, activities), the latter have been referred to as 'bounded' (telic, nondurative, perfective, conclusive, terminative, resultative, performances, accomplishments). The distinction (which appears to go back to Aristotle) has proved very useful for explaining the meaning and use of particular verb forms in many languages. Thus, Garey (1957) relies on it to explain the difference between the French 'imparfait' and the 'passe compose'; Bauer (1970) and Zydatiss (1976) use the distinction to account for the different meanings and uses of the English perfect; and many linguists have applied it in making clear the different uses and implications of simple and progressive tense forms in English. In spite of the fact that it has been widely made use of, the distinction has not been applied in a uniform way. In the linguistic literature we find it applied to various objects, such as verbs (e.g. Garey, 1957; Kenny, 1963; Bauer, 1970), verb phrases (e.g. Dowty, 1977), sentences (e.g. Zydatiss, 0024-3949/79/0017-0761 $2.00 Linguistics 17 (1979), 761-794. © Mouton Publishers, The Hague

The Sharing Problem

Abstract: Many important problems are concerned with the equitable distribution of resources. A new approach to resource distribution problems is presented as a network flow problem with a maximum objective function. The value of the smallest linear tradeoff function at the terminal points in a capacitated network is maximized. We develop a polynomially bounded algorithm and give computational experience. We illustrate the importance and usefulness of the sharing-problem model by considering the equitable distribution of coal during a prolonged coal strike.

A note on singular integrals

Abstract: In this article we discuss what happens when we consider a convolution operator whose kernel is a Calder6n-Zygmund kernel multiplied by a bounded radial function. Some generalizations are obtained. The purpose of this note is to obtain the following results. THEOREM. Let U(x)/lxl = K(x) be a Calderon-Zygmund kernel in R', n > 2 (i.e., let Q(x) be homogeneous of degree 0, f s'-' Q(x)do(x) = 0, and suppose Q satisfies a Lipschitz condition of positive order on Sni). Let h(x) be any bounded radial function, and put H(x) = h(x)* K(x). Then the convolution operator T(f) = f * H is bounded on LP(Rn), if 1 21y1 because H could be so rough. Nevertheless, we shall prove the somewhat stronger theorem below, whose theme is that smoothness in the radial direction for a convolution kernel is unnecessary in order to have the boundedness of the corresponding operator. THEOREM. (a) Suppose that for each r > 0, we are given a function Qr defined on SnI in such a way that the family {Qr} is uniformly in the Dini class (i.e., if 0*(8)= sup{Ir(x)Qr(y): y x, y E 5n-1, Ix -yI 0), then K*(8)d8/ < x) and also f Qr(x)da(x) = 0. Let H(x) = 0 jxi(x/IXI)/IxIn. Then IIH *f112 < CIIfII2 (b) Suppose in part (a) we replace the Dini class by a Lipschitz class of some positive order. Then IIH *fIIp < CpIIf I I 1 < p < 0o. At this point, we would like to remark that in this work, we were very much motivated by the work of E. M. Stein on maximal spherical averages (see [3]). PROOF OF (a). We estimate the Fourier transform of H: = A s2Qxj(x//xI) '" dx ] A dr Received by the editors May 15, 1978 and, in revised form, July 17, 1978. AMS (MOS) subject classifications (1970). Primary 42A40.

Fixed point iterations using infinite matrices

Abstract: This chapter discusses fixed point iterations using infinite matrices. It presents an assumption as per which there is a normed linear space X ; C is a nonempty, closed, bounded and convex subset of X ; T : C → C is a mapping with at least one fixed point; and A is an infinite matrix. Given the iteration scheme x 0 = x 0 in C ; x n +1 = Tx n , n = 0, 1, 2 …, the chapter discusses the restriction on the matrix A that is necessary and/or sufficient to guarantee that the iteration scheme converges to a fixed point of T . Using these iteration schemes, results have been obtained for certain class of infinite matrices. This chapter presents the generalizations of several of these results. It presents a proof of a theorem for the solution of operator equations in a Banach space involving generalized contraction mappings and also presents a few results as corollaries.

Abstract Linear and Nonlinear Volterra Equations Preserving Positivity

Abstract: Let X be a real or complex Banach space. We study the Volterra equation \[({\text{v}})\qquad u(t) + \int_0^t {a(t - s)Au(s)\,ds} = f(t)\quad (0 \leqq t \leqq T,T > 0),\] where a is a given kernel, A is a bounded or unbounded linear operator from X to X, and f is a given function with values in X. (Of particular importance is the case $f = u_0 + a * g$, $u_0 \in X$, $g \in L^1 (0,T;X)$, where $ * $ denotes the convolution). We establish existence, uniqueness, continuity results and sufficient conditions involving a, A, f which insure that solutions of (v) are positive by using certain representation formulas for solutions of (v). We also discuss the positivity of solutions of (v) when A is a nonlinear (m-accretive) operator and we discuss several examples when A is a partial differential operator.

Geometric Modeling Using the Euler Operators

Abstract: A recent advance In the modeling of three-dimensional shapes is the Joint development of bounded shape models, capable of representing complete and well-formed arbitrary polyhedra, and operators for manipulating them. Two approaches have been developed thus far in forming bounded shape models: to combine a given fixed set of primitive shapes into other possibly more complex ones using the spatial set operators, and/or to apply lower level operators that define and combine faces, edges, loops and vertices to directly construct a shape. The name that has come to be applied to these latter operators is the Eider operators. This paper offers a description of the Euler operators, In a form expected to be useful for prospective implementers and others wishing to better understand their function and behavior. It includes considerations regarding their specification in terms of being able to completely describe different classes of shapes, how to properly specify them and the extent of their well-formedness, especially in terms of their Interaction with geometric operations. Example specifications are provided as well as some useful applications. The Euler operators provide different capabilities from the spatial set operators. An extensible CAD/CAM facility needs them both.

Globally hypoelliptic and globally solvable first-order evolution equations

Abstract: We consider global hypoellipticity and global solvability of abstract first order evolution equations defined either on an interval or in the unit circle, and prove that it is equivalent to certain conditions bearing on the total symbol We relate this to known results about hypoelliptic vector fields on the 2-torus 0 Introduction Let A denote a linear selfadjoint operator, densely defined in a complex Hilbert space H, which is unbounded, positive, and has a bounded inverse A Such an operator defines a scale of Sobolev spaces HS (s E R) Their intersection fHl is denoted by H ' and their union U sH byH Let Q be either an open interval of R or the unit circle S' We will consider first-order evolution operators of the form L = at + b(t, A)A, t EC 0Q, (0 1) where a, means a/at and the coefficient b(t, A) belongs to the ring 2A of series in the nonnegative powers of A -1 with complex coefficients in C o(Q), which converge in E(H, H) as well as each one of their t-derivatives, uniformly with respect to t in compact subsets of U (For more details on these definitions see [1], [3]) We denote by C'(Q; H ) (C'(Q; H -)) the space of smooth functions defined in Q and valued in H (H -), and by C,'(Q; H ?) the compactly supported functions of C '(Q; H + O) REMARK 01 The requirement that A be strictly positive is inessential, since one can always work with the scale of spaces defined by (I + A2)1/2 That is, for instance, the case when A = (1/ i)(a/ax) in R or Sl DEFINITION 01 Let to be any point of U We say that L is locally solvable at to if there is an open neighborhood V c Q of to such that, to every f E Cc ( V; H ) there is u E C??(Q; H ) satisfying Lu = f in V (02) We say that L is locally solvable in Q if L is locally solvable for all to E U Received by the editors October 28, 1977 AMS (MOS) subject classifications (1970) Primary 35H05, 43A75

Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces

Abstract: The following theorem is proved: Let S ( t ), t ≧0 be a dynamical system in an infinite dimensional Banach space X such that S ( t ) = S 1 ( t )+ S 2 ( t ) for t ≧0, where (1) uniformly in bounded sets of x in X , and (2) S 2 ( t ) is compact for t sufficiently large. Then, if the orbit { S ( t ) x : t ≧0} of x ∈ X is bounded in X , it is precompact in X . Applications are made to an age dependent population model, a non-linear functional differential equation on an infinite interval, and a non-linear Volterra integrodifferential equation.

The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension

Abstract: where u0(x) is a bounded continuous nonnegative function whose support is a bounded interval. The functions 9p and A are smooth nonnegative functions satisfying 9p(0) = 99'(0) = A(O) = 0, and T9'(u), T9"(u), 4'(u) > 0 if u > 0. Thus equation (1.1) is nonlinear and degenerates when u = 0. According to [8], equation (1.1) occurs in the theory of nonlinear heat conduction with absorption. Existence and uniqueness of continuous bounded nonnegative solutions possessing bounded weak derivatives a3T(u)/ ax was established in [8]. If 4 _ 0 then (1.1) is called the porous medium equation and this equation has been studied by many authors [1]-{5], [7]{-15], [17]{-21]. The special case u, = (uI)xxCu' (1.3) where C > 0, p > 1, and i > 0 has been studied in [9]. ?2 is devoted to stating the assumptions about the functions 9p and 4 in (1.1) and defining a useful transformation of the solution u. Some of our main results for problem (1.1), (1.2) are given in ?3. There we establish the connectedness of the open set P[u] = {(x, t) E R2 x (0, oo)Iu(x, t) > 0). We also prove that, under certain conditions on 9p and 4, the open set P[u] n { t = T-) is an interval (t,(T), T2(T)) for each T > 0, where the functions Di belong to C0[ 0, oo) n C 01(0, oo), t1, and t2I* In the special case

Generalized disks of contractivity for explicit and implicit Runge-Kutta methods

Abstract: The A-contractivity of Runge-Kutta methods with respect to an inner product norm was investigated thoroughly by Butcher and Burrage (who used the term B-stability). Their theory is extended to contractivity in a region bounded by a circle through the origin. The largest possible circle is calculated for many known explicit Runge-Kutta methods. As a rule it is considerably smaller than the stability region, and in several cases it degenerates to a point. It is shown that an explicit Runge-Kutta method cannot be contractive in any circle of this class if it is more than fourth order accurate.

Green's functions, bi-linear forms, and completeness of the eigenfunctions for the elastostatic strip and wedge

Abstract: The bi-harmonic Green's functionG(r′,r) for the infinite strip region -1≤y≤1, -∞