Showing papers on "Space (mathematics) published in 1973"

Nonuniqueness of canonical field quantization in Riemannian space-time.

Abstract: We point out and discuss an ambiguity which arises in the quantum theory of fields when the background metric is not explicitly Minkowskian-in other words, when an external gravitational field, real or apparent, is present. A general theory of a canonical neutral scalar field in a static universe, including the construction of a Fock space, is presented. It is applied to a portion of two-dimensional flat space-time equipped with a non-Cartesian space-time coordinate system with respect to which the metric is nonetheless static. The resulting particle interpretation of the field is shown to be different from the standard one in special-relativistic free-field theory. The ambiguity frustrates an attempt to define uniquely the energy-momentum tensor by the usual method of normal ordering. We discuss various suggestions for (1) distinguishing a unique correct quantization in a given physical situation, or (2) reinterpreting seemingly inequivalent theories as physically equivalent. In passing it is shown that the vacuum state and the energy density of a free field in a box with periodic boundary conditions differ from those associated with a region of the same size in infinite space; this result should be of interest outside the gravitational context.

Variational properties of functions of the mean curvatures for hypersurfaces in space forms

Abstract: M port function, and 2Q is the square of the length of the position vector. Many of our results could be derived from the theory in [13] but it appears that because we study a less general case here our methods are more elementary than those of [13]. We begin by deriving the formula for the first variation of our integral as well as the formula for the second variation in those cases (see above) studied

Global analysis of the Kerr-Taub-NUT metric

Abstract: The Kerr‐Taub‐NUT metric is a local analytic solution of the vacuum Einstein‐Maxwell equations. When the metric is expressed in Schwarzschild‐like coordinates, two types of coordinate singularity are present. One occurs at certain values of the ``radial'' coordinate where grr becomes infinite and corresponds to bifurcate Killing horizons; the other occurs at θ=0,π, where the determinant of the components of the metric vanishes. It is shown that for nonzero NUT parameter the fixed points of the bifurcate Killing horizons and the degeneracies at θ=0,π cannot all be covered in one manifold. A maximal analytic manifold is constructed which covers the degeneracies at θ=0,π. It is non‐Hausdorff but contains maximal Hausdorff subspaces, topologically S3×R, which reduce to Taub‐NUT space for vanishing Kerr parameter. Kerr‐Taub space can be interpreted as a closed, inhomogeneous electromagnetic‐gravitational wave undergoing gravitational collapse. Another maximal analytic manifold is constructed which covers the f...

Instability of closed spaces in general relativity

Abstract: There are strong restrictions on the solutions of the initial value constraints of General Relativity when the spatial hypersurface is closed. In particular, closed flat space is unstable: not all solutions of the linearized constraints correspond to nearby solutions of the constraints themselves. For example, no nearby solutions whatever exist which are time symmetric. Other restrictions, which limit perturbations of non-flat closed initial solutions, are also exhibited.

Photon and Particle Interactions with Surfaces in Space

Abstract: The 6th ESLAB Symposium, organised by the Space Science Department (formerly ESLAB) of the European Space Research and Technology Center, was held in Noord wijk from 26-29 September 1972. This year the theme was "Photon and Particle Interactions with Surfaces in Space." More than 60 scientists attended mainly from ESRO Member States and from America. The first part of the Symposium was devoted to introductory lectures and to papers on interactions with spacecraft. The second half dealt with the photon and particle interactions with celestial objects, and ended with a general discussion and presenta tions of areas where new developments are required. The purpose of this Symposium was to throw light on the importance of the prob lems which are evoked by E. A. Trendelenburg in his introductory remarks, and to sum up our present understanding of these phenomena. It is hoped that this book will prove useful to physicists and engineers who are actually involved in space ex periments and are concerned with interactions of these types. R. J. L. GRARD OPENING ADDRESS Gentlemen, I should like to welcome you to the 6th ESLAB Symposium. In the past we have always organised this Symposium jointly with our sister in stitute, ESRIN, in Frascati, but unfortunately reductions in the scientific budget have forced ESRO to terminate the activities of that laboratory. Nevertheless, we have decided to carryon the tradition, and we shall continue on our own organising this series of symposia on specialised subjects."

Linear Integral Equations of the Third Kind

Abstract: Linear integral equations of the third kind are studied as equations in two different spaces of generalized functions. In the first space $D_\tau $, which consists of linear combinations of delta functions and continuous functions, the equation of the third kind has properties similar to those of the Fredholm equation of the second kind. The second space $P_\tau $ is comprised of linear combinations of delta functions and functions continuous except for poles, integration over the poles being defined by Cauchy’s principal value. $\ln P_\tau $ the behavior of the third-kind equation is essentially different from that of second-kind Fredholm equations. Solutions in both $D_\tau $ and $P_\tau $ may be constructed explicitly via Fredholm theory. Examples showing the suitability of these spaces in physical problems are cited, and earlier literature on third-kind equations is surveyed briefly.

The approximation property does not imply the bounded approximation property

Abstract: There is a Banach space which has the approxi- mation property but fails the bounded approximation property. The space can be chosen to have separable conjugate, hence there is a nonnuclear operator on the space which has nuclear adjoint. This latter result solves a problem of Grothendieck (2).

A Simple Approach to the Perron-Frobenius Theory for Positive Operators on General Partially-Ordered Finite-Dimensional Linear Spaces*

Abstract: This paper presents simple proofs of the principal results of the Perron-Frobe- nius theory for linear mappings on finite-dimensional spaces which are nonnegative relative to a general partial ordering on the space. The principal tool for these proofs is an applica- tion of the theory of norms in finite dimensions to the study of order inequalities of the form Ax 0 where A _ 0. This approach also permits the derivation of various inclusion and comparison theorems. 1. Introduction. The results of Perron (1907) and Frobenius (1908)-(1912) concerning spectral properties of matrices with nonnegative elements have become an important tool in the study of iterative methods for linear equations in Rn. These results have been generalized in various ways; see, for example, Krein and Rutman (1950) and Schaefer (1966) for general extensions to infinite-dimensional spaces and further references. Simple proofs of the Perron-Frobenius results for matrices can be found in Varga (1962) and Householder (1964). These proofs, however, do not appear to carry over to the case of linear mappings on a finite-dimensional space which are nonnegative under a general partial ordering on the space. For this case, it is necessary either to emulate the infinite-dimensional proofs by using the Brouwer fixed point theorem (see, e.g., Fan (1958)) or to depend heavily on the spectral theory of finite-dimensional linear maps and the Jordan form of a matrix (see Birkhoff (1967) and Vandergraft (1968)). In this paper, elementary proofs are presented of the principal results of the Perron-Frobenius theory for general partially-ordered finite-dimensional spaces. Our basic tools are some results about norms and a consistent use of simple order- bound concepts. No use is made of the spectral theory of linear mappings. These proofs are similar in spirit to the cited proofs of Varga and Householder for the case of the componentwise ordering. They also emulate some techniques of Bohl (1966) and Schneider and Turner (1972) which were employed by these authors in connection with discussions of the infinite-dimensional case.

Introduction to measure and integration

Abstract: Preface 1. Theory of sets 2. Point set topology 3. Set functions 4. Construction and properties of measure 5. Definitions and properties of the integral 6. Related Spaces and measures 7. The space of measurable functions 8. Linear functionals 9. Structure of measures in special spaces Index of notation General index.

Positive definite functions on spheres

Abstract: Positive definite functions on metric spaces were considered by Schoenberg (26). We write σk for the unit hypersphere in (k + 1)-space; then σk is a metric space under geodesic distance. The functions which are positive definite (p.d.) on σk were characterized by Schoenberg (27), who also obtained a necessary condition for a function to be p.d. on the it sphere σ∞ in Hilbert space. We extend this result by showing that Schoenberg's necessary condition for a function to be p.d. on σ∞ is also sufficient.

Maximal foliations of extended Schwarzschild space

Abstract: Smooth families of spherically symmetric maximal surfaces which are spacelike except at r = 2m are explicitly constructed in Schwarzschild space. Such surfaces should be useful in the study of initial value problems.

Resolution space; operators and systems

Abstract: 1. Causality.- A. Resolution Space.- B. Causal Operators.- C. Closure Theorems.- D. The Integrals of Triangular Truncation.- E. Strictly Causal Operators.- F. Operator Decomposition.- G. Problems and Discussion.- 2. Feedback Systems.- A. Well-Posedness.- B. Stability.- C. Sensitivity.- D. Optimal Controllers.- E. Problems and Discussion.- 3. Dynamical Systems.- A. State Decomposition.- B. Controllability, Observability and Stability.- C. The Regulator Problem.- D. Problems and Discussion.- 4. Time-Invariance.- A. Uniform Resolution Space.- B. Spaces of Time-Invariant Operators.- C. The Fourier Transform.- D. The Laplace Transform.- E. Problems and Discussion.- Appendices.- A. Topological Groups.- A. Elementary Group Concepts.- B. Character Groups.- C. Ordered Groups.- D. Integration on (LCA) Groups.- E. Differentiation on (LCA) Groups.- B. Operator Valued Integration.- A. Operator Valued Measures.- B. The Lebesgue Integral.- C. The Cauchy Integrals.- D. Integration over Spectral Measures.- C. Spectral Theory.- A. Spectral Theory for Unitary Groups.- B. Spectral Multiplicity Theory.- C. Spectral Theory for Contractive Semigroups.- D. Representation Theory.- A. Resolution Space Representation Theory.- B. Uniform Resolution Space Representation Theory.- References.

Generalized Scaling Hypothesis in Multicomponent Systems. I. Classification of Critical Points by Order and Scaling at Tricritical Points

Abstract: The goal of this work is to provide an analysis of spaces of critical points for multicomponent systems. First, we propose the geometric concept of order $\mathcal{O}$ for critical points; we distinguish it from a previous definition of a "multicritical" point. Specifically, we may define the intersection of spaces of critical points of order $\mathcal{O}$ to be a space of critical points of order ($\mathcal{O}+1$). Ordinary critical points are defined to be of order $\mathcal{O}=2$, so that the tricritical points introduced by Griffiths are of order $\mathcal{O}=3$. We discuss more general examples of critical spaces of order $\mathcal{O}=3$ which are known for a wide variety of systems; we also propose several examples of models of magnetic systems showing critical points of order $\mathcal{O}=4$---i.e., systems having intersecting lines of tricritical points. The analysis of critical and coexistence spaces also provides a new form of the Gibbs phase rule suitable for complex magnetic models. Next we define---for the critical points of order $\mathcal{O}$ of which examples have been given---special directions in terms of which to make a scaling hypothesis. We give the hypothesis for simple systems and then for tricritical points, and then, in a subsequent paper, part II, the special directions are used to make a scaling hypothesis at spaces of critical points of any order. Certain predictions (e.g., scaling laws and "single-power" scaling functions) follow in a simple and straightforward fashion. We consider the scaling hypothesis at a critical space of order $\mathcal{O}$ in terms of a group of transformations. We can define a set of invariants of the group. It is possible, for $\mathcal{O}\ensuremath{\ge}3$, to make a second scaling hypothesis for the space of order $\mathcal{O}\ensuremath{-}1$ using certain of these invariants as independent variables. This is advantageous because certain "double-power" scaling functions then follow directly; these predict that for $\mathcal{O}=3$, experimental data collapse from a volume onto a line. This prediction is to be contrasted with ordinary scaling functions, which predict that data collapse by only a single dimension (e.g., from a volume onto a surface or from a surface onto a line).

Inclusion theorems for $K$-spaces

Abstract: A sequence space is a vector subspace of the space ω of all real (or complex) sequences. A sequence space E with a locally convex topology τ is called a K- space if the inclusion map E → ω is continuous, when ω is endowed with the product topology . A K-space E with a Frechet (i.e., complete, metrizable and locally convex) topology is called an FK-space; if the topology is a Banach topology, then E is called a BK-space.

Solution of the Space-Dependent Reactor Kinetics Equations in Three Dimensions

Abstract: A general class of two-step alternating-direction semi-implicit methods is proposed for the approximate solution of the semi-discrete form of the space-dependent reactor kinetics equations. An expo...

On the Variational Method for the Existence of Solutions of Nonlinear Equations of Hammerstein Type

Abstract: Let Y be a real Banach space and X* its conjugate Banach space. Let A be an unbounded monotone linear mapping from A"to X* and Na potential mapping from X* to X. In this paper we establish the existence of a solution of the equation u + ANu = v for a given v in X* using variational method. Our method consists in using a splitting of A via an auxiliary Hubert space and solving an equivalent equation in this auxiliary Hubert space. In §2, we prove the same result in the case when Y is a Hubert space using the natural splitting of A in terms of its square root. We do this to compare and contrast the proofs in the two cases. Introduction. Let Y be a real Banach space and Y* denote its conjugate Banach space. Let A be an unbounded monotone linear mapping from .Yto X* and A/a nonlinear mapping from X* to Y satisfying no monotone hypothesis. In this paper we study the solvability of the equation

H-closed extensions I

Abstract: In this chapter we begin a detailed investigation of the set H(X) of all H-closed extensions of a space X. We begin by considering strict and simple extensions of a space. We then construct and study the Fomin extension σX of an arbitrary space X, the Banaschewski-Fomin-Sanin extension μX of a semiregular space X, and one-point H-closed extensions of locally H-closed spaces. Next we consider the interrelationships among certain partitions of σX\X and the poset structure of H(X). We characterize and study those f ∈ C(X,Y) that can be extended to a function κf ∈ C(κX,κY). The chapter concludes with the study of Θ-equivalent H-closed extensions.

Regional stationary principles and regional virial theorems

Abstract: If the Born‐Oppenheimer Hamiltonian operator for a molecular system is averaged over all space for electrons 2,3, ··· , N, but for electron 1 is averaged only over the volume A, with surface S, it is shown that the regional expectation value so defined is stationary with respect to change of a parameter ζ in the exact wavefunction, provided that ∫S[(∂/∂ n1) Pζ(1,1′)]1=1′ dS1=0, where, for nodeless ψ, Pζ(1,1′)=∫ ··· ∫ψ*(1′,2,··· , N) ψ (1,2, ··· , N) × ∂ ln[ψ(1, 2, ··· , N)/ ψ*(1′, 2, ··· , N)]∂ ζ |ζ=1 dτ2 ··· dτN. Under this condition with ζ a scale parameter, a regional virial theorem is shown to hold for the volume A, in the sense previously described by Bader and co‐workers [J. Am. Chem. Soc. 93, 3095 (1971): Chem. Phys. Lett. 8, 29 (1971), J. Chem. Phys. 56, 3320 (1972); 58, 557 (1973)]. This condition differs from the corresponding condition suggested by Bader (∂ / ∂ n1)ρ (r1)=0 on S, but in most cases the surfaces determined by it should be similar to those determined by Bader's condition. Correspon...

Temperature and speed responsive fluid coupling

Abstract: A fluid friction coupling, especially for the fan drive of internal combustion engines in which at least one working space provided with a working gap is arranged between the driving and driven part and in which additionally a reservoir space is provided which is in communication with the working space by way of an aperture or apertures, whereby the cross section of this aperture or apertures is controllable both by temperature as well as rotational speed; mutually independent control means are provided to control the aperture or apertures as a function of temperature and independently thereof as a function of rotational speed.

The structure of groups of motions admitted by Einstein-Maxwell space-times

Abstract: The known symmetry of the non-null electromagnetic field, which acts as the source of a four-dimensional space-time satisfying the Einstein-Maxwell equations, is used to show that when such a space-time admits a group of motions, generated by a Killing vector, the structure constants for the group must satisfy an additional relation to the known relations of group theory.

Sets of points of discontinuity

Abstract: In order that a subset F of a topological space coincide with the set of points of discontinuity of a real-valued function on the space, it is necessary that F be an Fff-set devoid of isolated points. It is shown that this condition is also sufficient if the space is "almost-resolvable", and in particular if the space is either separable, first countable, locally compact Hausdorff, or topological linear.

The local b-completeness of space-times

Abstract: It is shown that any point of space-time has a neighbourhoodU such that theb-boundaryŪ ofU coincides withŪ/U.

A study of monotonically normal spaces

Abstract: We obtain various characterizations of monoton- ically normal spaces which not only answer various questions of Zenor but also allow an elementary proof of a result of Heath and Lutzer. We also prove that elastic spaces are monotonically normal. Recently P. Zenor introduced the class of monotonically normal spaces (his results will appear in (3)). Soon afterward, R. Heath and D. Lutzer proved that each linearly ordered topological space is monotonically normal (see (3)). Since Zenor did not know if monotonically normal spaces were hereditarily monotonically normal and the result of Heath and Lutzer had a very long proof, my interest in the study of these spaces was aroused. The characterizations of monotonically normal spaces which follow not only answer Zenor's questions but also permit us to prove, quite easily, that linearly ordered topological spaces and elastic spaces are monotonically normal. (Our proof of the first result appears in (2). For the second result, see Theorem 2.3.) 1. Characterizations of monotonically normal spaces. For the sake of completeness, we will first define this class of spaces. Definition 1.1. For any space X, let 3)X={(A, B)\A and B are disjoint closed subsets of A"}. The Trspace X is said to be monotonically normal provided that to each (A, B) e @)x one can assign an open subset G(A, B) of X such that (a) A<=G(A,B)<=G(A,B)-^X-B, (b) G(A, B)^G(A', B'), whenever A <=A' and B'<=B. The function G is called a monotone normality operator.