Showing papers on "Space (mathematics) published in 1971"

Introduction to Fourier Analysis on Euclidean Spaces.

Abstract: The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.

On Weak Convergence of Stochastic Processes with Multidimensional Time Parameter

Abstract: The well-known space $D\lbrack 0, 1\rbrack$ is generalized to $k$ time dimensions and some properties of this space $D_k$ are derived. Then, following the "classical" lines as presented in Billingsley [1], a Skorohod-metric, tightness criteria and some other results concerning weak convergence are given. The theory is applied to prove weak convergence of two generalizations of the one-dimensional empirical process and of the Kolmogorov-Smirnov test statistic of independence. Stochastic processes with multidimensional time parameter and their weak convergence have been investigated by several authors. Dudley [4] established a theory of convergence of stochastic processes with sample functions in nonseparable metric spaces. Later on, Wichura [11] (see also Wichura [12]) modified the concepts of Dudley and developed them systematically. He applied his theory to a space which is with minor changes our space $D_k$. Weak convergence in the sense of Wichura [12] and ours differ usually, but both concepts coincide if the limit process has--with probability one--continuous sample functions only. From here it follows that the results of Dudley and Wichura concerning weak convergence of multivariate empirical processes are equivalent to ours. At least two further authors proved the convergence of multivariate empirical processes, namely LeCam [8] and Bickel [1]. Our proof follows the classical approach of Parthasarathy [9] using an argument of Kuelbs [7] to carry over the proof from 1 to $k$ dimensions. Kuelbs however deals properly with the "interpolated sum" process for two-dimensional time parameter. The space $D_k$ seems to be defined for the first time in connection with multivariate processes by Winkler [13], yet his investigations are not concerned with weak convergence. Another generalization of the space $D\lbrack 0, 1\rbrack$ and the Skorohod metric to functions on more general spaces than $E_k$ is given in the paper [10] of Straf, in which there are applications to genuinely discontinuous limit processes.

Theory of symmetry in nuclear magnetic relaxation including applications to high resolution N.M.R. line shapes

Abstract: It is shown in discussing problems involving magnetic relaxation in liquids that, whilst the usual Hilbert space spanned by all the eigenkets of the spin hamiltonian does not reflect any symmetry inherent in the spin system, the vector space (Liouville space) comprising all operators of the spin system does so. The transformation properties of the Liouville operator, as reflected by those of the high resolution spin hamiltonian and relaxation operators whose effects are introduced by means of Redfield relaxation theory, with respect to arbitrary rotations of the coordinate system are investigated. The use of irreducible tensor operators as a set of basis operators spanning Liouville space is stressed, since it is shown that their super-matrix elements of the Liouville operator are given by the Wigner-Eckart theorem provided that relaxation by anisotropy of the chemical shift or anisotropic random fields is absent. These arguments are independent of the fine details of molecular reorientation in the extrem...

Dissipative periodic processes

Abstract: : The objective of this paper is to develop a general and meaningful theory of dissipative periodic systems. For ordinary periodic differential equations one studies the iterates of a map T of a state space into itself where the map T is topological and the space is locally compact (n-dimensional Euclidean space). However, for the applications the authors have in mind, the solutions will be unique only in the forward direction of time and the state spaces are not locally compact. Because of this generalization of the results for ordinary differential equations is by no means trivial. The basic theory of dissipative periodic processes on Banach spaces are developed in Sections 2 and 3 of the paper. How this applies to retarded functional differential equations of retarded type is discussed in the fourth section. Two sufficient conditions for dissipativeness are given in terms of Liapunov functions. They formalize the intuitive notion that many systems for large displacements dissipate energy. Application of these results is illustrated. (Author)

Linear Problems of Complex Analysis

Abstract: This article attempts to give a linearized form of the basic theorems of complex analysis (the Oka-Cartan theory). With this aim we study simultaneously: a) the isomorphism problem for spaces of holomorphic functions and , ; b) the existence of a linear separation of singularities for the space , where , and () are holomorphically convex domains in a complex manifold , and, in a more general setting, the splitting of the Cech complex of a coherent sheaf over a holomorphically convex domain ; c) the existence of a linear extension for holomorphic functions on a submanifold , and more generally, the splitting of a global resolution of a coherent sheaf. In several cases (for strictly pseudoconvex domains) these questions can be answered affirmatively. The proofs are based on the theory of Hilbert scales and bounds for solutions of the -problem in weighted -spaces. Counterexamples show that the same questions may also have negative answers.

Regular representations of Dirichlet spaces

Abstract: We construct a regular and a strongly regular Dirichlet space which are equivalent to a given Dirichlet space in the sense that their associated function algebras are isomorphic and isometric. There is an appropriate strong Markov process called a Ray process on the underlying space of each strongly regular Dirichlet space.

Supports of continuous functions

Abstract: Gillman and Jerison have shown that when A" is a realcompact space, every function in C(X) that belongs to all the free maximal ideals has compact support. A space with the latter property will be called fi-compact. In this paper we give several characterizations of /?-compact spaces and also introduce and study a related class of spaces, the ^-compact spaces ; these are spaces X with the property that every function in C(X) with pseudocompact support has compact support. It is shown that every realcompact space is ^-compact and every i/i-compact space is /?-compact. A family & of subsets of a space X is said to be stable if every function in C(X) is bounded on some member of #". We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.

Application of Regge Calculus to the Schwarzschild and Reissner‐Nordstro/m Geometries at the Moment of Time Symmetry

Abstract: Applied to 3‐dimensional space, Regge calculus approximates a curved space by a collection of tetrahedrons or other simple solid blocks. Within each block the geometry is Euclidean. Curvature is idealized as concentrated at the edge common to two or more of these solids. We specialize to a static geometry endowed with spherical symmetry and to a radial electric field produced by the flux of electric lines of force trapped in a throat connecting two quasi‐Euclidean regions of space. The one relevant Einstein field equation takes the form ∑ all edges which meet at a given vertex(length of edge of prism)(deficit between (1) sum of dihedral angles which meet at that edge and (2) normal value of 2π)=(factor proportional to square of electric field). In method (1) the space is decomposed into shells separated from one another by icosahedral surfaces, all having a common center. Method (2) is even simpler: Space is decomposed into successive spherical shells of area 4π ρi2 separated by a proper distance d. Regge...

Anomalies of the Axial-Vector Current in Two Dimensions

Abstract: In a world with one space and one time dimension, it is shown that the axial-vector current in a vectorgluon model exhibits anomalies in perturbation theory analogous to those found by Adler in fermion electrodynamics. The analysis is extended to include a four-fermion (Thirring) interaction; this model has been solved exactly by Sommerfield, permitting an explicit verification of the perturbation-theory calculations. In analogy to results in four dimensions, a model is presented in which the anomalous properties of the axial-vector current, both in its divergence and in its commutation relations, follow immediately from the canonical structure of the theory.

On the motion of space charge in a dielectric medium

Abstract: A general relation is derived which describes the spatio-temporal behaviour of space charge in an ideal dielectric medium. (Diffusion effects are neglected.) This result is used to determine the behaviour of a parallel plate capacitor which contains space charges near its electrodes. Expressions are derived for the variation with time of the potential difference between the plates following the removal of an applied voltage and for the external current which flows when the plates are connected together. Symmetrical and asymmetrical charge distributions are considered.

Boundary Value Control Theory of the Higher-Dimensional Wave Equation, Part II

Abstract: The present work extends controllability results obtained for the wave equation in two or more space variables in an earlier article [13] In the earlier paper approximate controllability was established for time $T > 2T_0 $, where $T_0 $ is a constant related to the wave propagation speeds in the medium, but only for three or fewer space dimensions In the present article we establish this result for an arbitrary space dimension We also examine the controllability problem for $T = 2T_0 $, the “critical time,” and show that here controllability depends upon certain relationships between the coefficients of the partial differential equation and the shape of the spatial domain under consideration

Invariant Approach to a Space-Time Symmetry

Abstract: A study is made of Killing vector fields in vacuum Einstein spaces with a restriction primarily to those fields whose associated bivector is nonnull. However, a well‐known theorem of Robinson is modified slightly to show that if such a space admits a null bivector associated with a Killing vector, the space must be algebraically special. Consequently, all algebraically general spaces admit a nonnull Killing bivector (KBV) if they admit a symmetry at all. Furthermore, it is shown that if a vacuum Einstein space admits a spacelike or timelike Killing vector field whose associated KBV is nonnull, the Killing trajectories are not geodesics. Computation of invariants from the curvature tensor and the KBV allows an approach which gives a general classification to such spaces which admit at least one hypersurface orthogonal Killing vector field. A few geometrical properties involving the principal null directions of the KBV are also derived for the hypersurface orthogonal cases. In addition, a topological result...

Spaces of network functions

Abstract: In this paper, difference analogs of Sobolev–Slobodeckiĭ spaces are studied: , where is either the whole space or a halfspace. One obtains difference analogs of imbedding theorems, on traces and on extensions of network functions from a halfspace to the whole space with preservation of class. Bibliography: 19 items.