Showing papers on "Space (mathematics) published in 1970"

On some topological invariants of algebraic functions

Abstract: There are some interesting connections between the theory of algebraic functions and Artin’s theory of braids. For instance, the space G n of polynomials of degree n not having multiple roots is the space K(π ,1) for the group B(n) of braids with n strings:

The Theory of Superspace

Abstract: In this work a theory of superspace is introduced. The superspace, or space of all geometries, of a fixed closed (compact without boundary) 3-dimensional manifold M is defined as the orbit space \(S(M) = \frac{{Riem (M)}}{{Diff (M)}}\) of the group of diffeomorphisms, Diff (M), acting by “coordinate-transformation” on the space of Riemannian metrics, Riem (M). A geometry is then a point in S(M), i.e. an equivalence class of isometric Riemannian metrics. Superspace, as the space of physically distinguishable states, is the proper configuration space for a dynamical theory of relativity. It is the space in which the momentary geometries of space itself evolve. Our first result states that this space is in fact a metric space.

On metric multidimensional unfolding

Abstract: The problem of locating two sets of points in a joint space, given the Euclidean distances between elements from distinct sets, is solved algebraically. For error free data the solution is exact, for fallible data it has least squares properties.

Indefinite Finsler Spaces and Timelike Spaces

Abstract: In this paper we investigate indefinite Finsler spaces in which the metric tensor has signature n — 2. These spaces are a generalization of Lorentz manifolds. Locally a partial ordering may be defined such that the reverse triangle inequality holds for this partial ordering. Consequently, the spaces we study may be made into what Busemann [3] terms locally timelike spaces. Furthermore, sufficient conditions are obtained for an indefinite Finsler space to be a doubly timelike surface (see [2; 4]). In particular, all two-dimensional pseudo-Riemannian spaces are shown to be doubly timelike surfaces.

On Wilton's ripples: a special case of resonant interactions

Abstract: The phenomenon of second harmonic resonance for capillary-gravity waves is reconsidered here by the asymptotic method of multiple time and space scales. The periodic finite amplitude waves of permanent form found by Wilton in 1915 which correspond to this configuration are shown to be no more than a special case of the more general resonant interaction theory, and owe their existence to a critical choice of initial conditions. It is further suggested that the influence of viscous dissipation will render this solution virtually undetectable in a real liquid.

On the Global Isometric Embedding of Pseudo-Riemannian Manifolds

Abstract: It is shown that any pseudo-Riemannian manifold has (in Nash's sense) a proper isometric embedding into a pseudo-Euclidean space, which can be made to be of arbitrarily high differentiability. The application of this to the positive definite case treated by Nash gives a new proof using a Euclidean space of substantially lower dimension. The general result is applied to the space-time of relativity, and the dimensions and signatures of the spaces needed to embed various cases are evaluated.

Alternating direction methods for parabolic equations in two space dimensions with a mixed derivative

Abstract: Alternating direction implicit (A.D.I.) methods are proposed for solving the parabolic equation with variable coefficients in two and three space dimensions with mixed derivatives. The methods require the solution of two tridiagonal sets of equations at each time step in the two space dimensional case, and three tridiagonal sets of equations in the three space dimensional case. Several theorems are stated showing the methods to be unconditionally stable for certain ranges of an auxiliary parameter. Reference is made to other authors and numerical results are mentioned.

Nonlinear Dynamic Stability: A Formal Theory

Abstract: where Lz and Nz are linear and nonlinear operators, respectively, defined on a space o( functions satisfying appropriate boundary and smoothness conditions. Both Lz and Nz are independent of the time variable t, with one or both depending on a parameter A. To study the stability of a stationary (time-independent) solution uo of (1), we must determine whether it can sustain itself against perturbations (to which all physical systems are subjected). That is, we must see whether all perturbations decay, and the system reverts to uo, or whether some perturbations grow (perhaps into new stationary solutions). Since the problem depends on a parameter A, uo may be stable for some values of A, e.g., below a critical value 'c and unstable for others. We assume L(O) = N(O) = 0 and that the boundary conditions are homogeneous, and investigate the stability of the solution uo = 0. The stability of solutionis uo : 0 and satisfying inhomogeneous boundary conditions is treated in a similar fashion, by considering the problem for

Spinor Treatment of Stationary Space‐Times

Abstract: A generalized SU(2) spinor calculus is established on the ``background space'' V3 of the stationary space‐time. The method of spin coefficients is developed in three dimensions. The stationary field equations can be put to a form which in V3 is analogous to the Newman‐Penrose equations. A V3 filling family of curves is determined by the gravitational field and is called the eigenray congruence. Stationary space‐times may be characterized by the geometric properties of eigenrays. The relation of this classification to the algebraic ones is discussed. The method of solving the equations obtainable for various classes is illustrated on the case of nonshearing geodetic eigenrays. Assuming asymptotic flatness, we obtain the Kerr metric.

Einstein Spaces with Symmetry Groups

Abstract: In this paper, we construct all possible groups of motion (symmetry groups) for empty Einstein spaces admitting a diverging, geodesic, and shear‐free ray congruence. (Minkowski space is excluded throughout the discussion.) It is proved that any such Einstein space cannot admit a symmetry group with dimension greater than four. Although the field equations are not solved completely for spaces with groups of dimension one or two, a generalization of the Kerr spinning‐mass solution is obtained from the 2‐dimensional class. It is shown that all such spaces with 4‐dimensional symmetry groups are well known: Schwarzschild, NUT (Newman, Unti, and Tamborino), and a particular hypersurface orthogonal Kerr‐Schild metric. The only member of these spaces admitting a 3‐dimensional symmetry group is a Petrov Type III hypersurface orthogonal metric.

Space in Time

Abstract: Concepts of Space The History of Theories of Space in Physics. By Max Jammer. Second edition. Pp. xv + 221. (Harvard University Press: Cambridge, Massachusetts; Oxford University Press: London, September 1969.) 52s.

Jet streams in space

Abstract: Ifviscosity is taken into account, Keplerian motion of a large number of grains in a gravitational field has a tendency to lead to the formation ofjet streams.

Metrics on test function spaces for canonical field operators

Abstract: In a canonical field theory, the field Φ(f) and momentum π(g) are assumed defined for test functionsf andg which are elements of linear vector spaces and , respectively. Generally, the continuity of the map onto the unitary Weyl operatorsU(f),V(g) is taken as ray continuity, the barest minimum to recover the field operators as their generators, i.e.,U(f)=e iΦ(f) ,V(g)=e iπ(g) . This leaves open the question of whether any wider continuity properties follow and what form they would take. We show that much richer continuity properties do follow in a natural fashion for every cyclic representation of the canonical commutation relations. In particular, we show that the test function space may be taken as a metric space, that the space may be uniquely completed in this topology, and that the map into the unitary Weyl operators is strongly continuous in this topology. The topology induced by this metric is minimal in the sense that it is the weakest vector topology for which the mapsf→U(f),g→V(g) are strongly continuous. An expression for a suitable metric can easily be given in terms of a simple integral over a state on the Weyl operators.

The Theory of Best Approximation in Normed Linear Spaces

Abstract: It is well known that the problem of best approximation of a function consists in the determination of a function belonging to a fixed family such that its deviation from the given function is a minimum, This problem was first formulated by P. L. Chebyshev, who investigated the approximation of continuous functions by algebraic polynomials of given degree and by rational fractions with numerators and denominators of fixed degree. As a measure of the deviation between two functions, Chebyshev used the maximum of the absolute value of their difference. Subsequently, a number of mathematicians have studied other specialized problems of best approximation whose content is defined by some choice of the measure of deviation and the function set used for the approximation. Among these, we should in the first place note A.A. Markov, Jackson, Bernshtein, de la Vallee-Poussin, Haar, and Kolmogorov. With the development of the theory of normed spaces it became clear that a wide range of problems of best approximation can be put into a general formulation in terms of normed spaces, if the norm of the space is taken as the measure of deviation.

On the singularities of cosmological solutions of the gravitational equations.

Abstract: Publisher Summary This chapter examines the general properties of the cosmological solutions of the gravitational equations near a time singularity. The customarily used (Friedmann) cosmological solution of Einstein's gravitational equations is based on the assumption that matter is distributed in space homogeneously and isotropically. This assumption is very far-fetched mathematically, apart from the fact that its fulfillment in a real world can at best be only approximate. The solution obtained for the centrally symmetrical problem is actually a particular case of a more general class of solutions. In the absence of matter, there is naturally no centrally-symmetrical solution as the free gravitational field cannot have such symmetry.

An Abstract Concept of the Sum of a Numerical Series

Abstract: Our aim in this paper, generally stated, is to formulate an abstract concept of the notion of the sum of a numerical series. More particularly, it is a study of the type of sequence space called “sum space”. The idea of sum space arose in connection with two distinct problems. 1.1 The Kothe-Toeplitz dual of a sequence space T consists of all sequences t such that st ∈ l 1 (absolutely summable sequences) for each s ∈ T. It is known that if cs or bs is used in place of l 1, an analogous theory of duality for sequence spaces can be developed (cf. [2]). What other spaces of sequences can play a role analogous to l 1? This problem is treated in [6]. 1.2. Let {xn , fn } be a complete biorthogonal sequence in (X, X*), where X is a locally convex linear topological space and X* is its topological dual space.