Showing papers on "Space (mathematics) published in 1980"

Entire spacelike hypersurfaces of constant mean curvature in Minkowski space

Abstract: for spacelike hypersurfaces in Minkowski Space possess a remarkable Bernstein property in the maximal case, H = 0 . Calabi [4] has shown for n__<4 and later Cheng and Yau 1-5] for all n that entire, maximal, spacelike solutions of (1) are linear. We investigate to what extent uniqueness holds when the mean curvature H is a positive constant. In that case Eq. (1) has many solutions which we are able to classify by their projective boundary values at infinity in Theorem 1. We can also construct solutions to Eq.(1) asymptotic to arbitrary C 2 perturbations of the light cone in Theorem 2. The Lorentz-Minkowski Space L "+L is IR "+1 endowed with the metric ds 2

Ordered topological spaces

Abstract: Publisher Summary This chapter presents a few topics from the theory of GO spaces. It highlights areas in which there has been recent progress and describes the way in which researchers in ordered spaces view the subject. The chapter also focuses on a LOTS or a GO space, which is a topological space already equipped with a compatible ordering. Over the years, some effort has been devoted to giving a characterization of those topological spaces for which some compatible ordering can be constructed. Results of that type are called orderability theorems. Characterizations of the arc, Cantor set, and space of irrationals might be viewed as orderability theorems. Every GO space is collectionwise normal so that, in the light of general theory, many well-known covering properties, for example sub-para-compactness, θ-refinability, and meta-compactness, are equivalent to para-compactness in a GO space.

On geometry of Orlicz spaces

Abstract: In terms of the function Φ it is established when Orlicz space LΦ does not contain l ∞ n uniformly and when it has some type or cotype.

On conjectures of Mahowald, Segal and Sullivan

Abstract: In this paper we prove some results about the stable homotopy and cohomotopy of spaces related to the infinite real protective space RP∞. These include M. E. Mahowald's conjecture on the limit of stable homotopy of stunted real projective spaces RP2N+m/RP2N−m as N, m → ∞, G. Segal's Burnside ring conjecture forand the stable analogue of a conjecture of D. Sullivan on RP∞.

Propagation of singularities for semilinear hyperbolic equations in one space variable

Abstract: In this paper we investigate the propagation of singularities for general semi-linear hyperbolic equations in two variables. In order to make clear the motivation for this work, we give a little history. In [91, Reed showed that in one space dimension solutions ju = f (u, Du) are C~o except at points Kx, t> where the backward characteristics through the point intersect singular points of the initial data at t = 0; thus, as in the linear case, the singularities lie on rays issuing from singularities at t = 0. The counterexample of Lascar [61 and the theorems of Rauch [71 show that when the number of space dimensions is greater than one, the solutions u of Elu = f (u) may have other singularities too. Roughly what is true is the following. Suppose that the solution u is in Hk and that the backward characteristics through Kx, t> do not intersect the singular support of the initial data. Then u will be in HP at Kx, t> where p > k depends on k and the number of space dimensions. Such results are valid for higher order equations and more strongly nonlinear equations. In this paper we reexamine the result of Reed to understand what is special about one space dimension. First, we prove that the C~o result holds in one space dimension for all semilinear hyperbolic equations of second order. Second, we construct a third order semi-linear counterexample in which the propagation of singularities is not that predicted by the highest order linear part. A new singularity is created when two singularities cross and the new singularity propagates in the direction of the third characteristic. Our example makes it clear that the same phenomenon will also occur for higher order equations. Thus, we have the following general picture: the only semi-linear hyperbolic equations of order >2 for which the propagation of singularities is like the linear case are second order

The significance of Weinhold’s length

Abstract: An interpretation is given for the proposed metric of weinhold1 for the space of thermodynamic states. The proposed metric is nij=∂2U/∂Xi∂Xi where X’di and Xj are extensive variables. The physical significance of Nij is discussed. (AIP)

Large spaces of matrices of bounded rank

Abstract: IN THIS paper we consider subspaces X of M^*, the space of all m x n matrices with entries in some given field, with the property that each matrix of X has rank at most r. In [2] Flanders showed that such spaces necessarily have dimension at most max (mr, nr) and he determined the spaces of precisely this dimension. We shall extend this work by classifying the spaces of dimension slightly lower than this upper bound. Our results depend on the (often unstated) assumption that the ground field has at least r +1 elements but, unlike Flanders, we do not need to exclude the characteristic 2 case. If every matrix in the space X has rank at most r the same is clearly true of the space PXQ = {PXQ: X e X} where P, Q are non-singular mxtn, nxn matrices respectively. This equivalent space PXQ can also be derived from X by performing row and column operations to all matrices of X simultaneously. A wide class of examples is provided by spaces equivalent to subspaces of the space 9i(p, q) of all matrices of the

Equivariant classifying spaces and fibrations

Abstract: Explicit classifying spaces for equivariant fibrations are constructed using the geometric two-sided bar construction. The constructions are then extended to classify stable equivariant spherical fibrations and equivariant K-theory. The ambient groups is assumed compact Lie. In order to be able to prove an equivariant version of the Adams Conjecture [Wal], it is certainly helpful to have a classifying space for equivariant stable spherical fibrations, and to prove that they lead to a generalized equivariant cohomology theory [MHW]. Stasheff first constructed classifying spaces for various categories of fibrations in [Stl], and these have proved to be an indispensable tool for homotopy theorists. The purpose of this paper is to construct explicit classifying spaces for various categories of stable and unstable equivariant fibrations over suitable base spaces. This will be done using a generalized "classifying space machine" based largely on that of Peter May in Classifying spaces and fibrations [Mal]. As a by-product, we shall also obtain explicit classifying spaces for the various categories of (stable and unstable) equivariant bundles, thereby providing alternate models of spaces constructed by R. Lashof and M. Rothenberg in [Lal], as well as new versions of the classifying spaces for equivariant K-theory. (See, for example, [Mol].) In order to be able to construct universal G-fibrations and to prove a classification theorem, the foundations of G-homotopy theory and G-cellular theory must be put in order. This is done in [Wa2] for G-compact Lie, and will enable us to prove our classification theorem with the full generality of G-compact Lie. The foundational theory of G-fibrations is discussed in [Wa3] and will be referred to here as needed. The technique of our approach to the classification will be to restrain the fibers to lie in an appropriate "category of fibers" which (usually) contains a prototype space F with varying actions of closed subgroups of G. F serves as a homotopy model for the fibers of a given fibraton. The concept of a G-fibration with fiber F [Wa3] may serve as a motivating example. (Without such a prototype, the family of classes of equivariant fibrations over a point are too large to be a set.) When F is compact, we can classify up to strict fiberwise G-homotopy, and when F is not compact, we are still able to classify up to weak fiberwise G-homotopy equivalence. Received by the editors September 7, 1978 and, in revised form, January 15, 1979. AMS (MOS) subject classifications (1970). Primary 54H15.

Comparison of two leading uniform theories of edge diffraction with the exact uniform asymptotic solution

Abstract: The diffraction of an arbitrary cylindrical wave by a half-plane has been treated by Rahmat-Samii and Mittra who used a spectral domain approach. In this paper, their exact solution for the total field is expressed in terms of a new integral representation. For large wave number k, two rigorous procedures are described for the exact uniform asymptotic expansion of the total field solution. The uniform expansions obtained are valid in the entire space, including transition regions around the shadow boundaries. The final results are compared with the formulations of two leading uniform theories of edge diffraction, namely, the uniform asymptotic theory and the uniform theory of diffraction. Some unique observations and conclusions are made in relating the two theories.

Gauge field configurations in curved space-times. IV. Self-dual SU(2) fields in multicenter spaces

Abstract: Finite-action, self-dual SU(2) gauge fields are constructed for the multi-Taub-NUT (Newman-Unti-Tamburino) and the asymptotically, locally Euclidean multicenter metrics.

Theory of Nonlinear Spin Relaxation

Abstract: A rigorous quantum theory is developed on the nonlinear spin relaxation process. With the use of the phase space method, the quantum mechanical equation for a density matrix is mapped onto a c -number space. A quasi-probability function is then expanded in terms of spherical harmonics. It is clarified that the nonlinear relaxation process is characterized by 2 J ( J being the magnitude of a spin) different relaxation times. For J =1, time-evolutions of the first moment, fluctuation and probability function are exactly solved. An enhancement of the fluctuation is observed in the course of time development: This is similar to the phenomenon called “anomalous fluctuation” found in certain nonequilibrium systems.

Topological entropy of block maps

Abstract: We show that h(fm.) = log 2 where f.O is the map on the space of sequences of zeros and ones induced by the block map f(xo, . . ., xk) = xo + lk. -(xi + b,) where k > 2 and the k-block bI . . . bk is aperiodic.

Solution modules and system equivalence

Abstract: In this paper we consider a system as a relation between input, internal variables and output. This relation is given by the solution space of the system's equations. For time invariant linear systems in differential operator representation the solution space carries a K[s]-module structure defined by the ordinary differential operator. This algebraic structure is exploited systematically to develop a self-contained theory of strict system equivalence in time domain. The module of free motions is considered as space of initial conditions. An algebraic characterization of systems having the same solution space is presented. System homomorphisms are defined as special K[s] homomorphisms between the solution modules. Two systems are called system-equivalent, if there exists a system-isomorphism between their solution spaces. It turns out that, this concept coincides with Rosenbrock's concept of strict. system equivalence. It. is shown that further concepts and results of linear system theory (constr...

A systematic method of estimating the performance of X‐ray optical systems for synchrotron radiation. II. Treatment in position‐angle–wavelength space

Abstract: Position–angle–wavelength space is introduced to describe X-ray optical properties of various optical elements and their combined systems. This three-dimensional space is a combination of the position-angle space used in the phase-space method and the angle-wavelength space used in the DuMond diagram. Representations in the position–angle–wavelength space are shown for a synchrotron radiation source, a flat perfect crystal, a curved crystal and a double-crystal arrangement. A method of estimating the spatial width, angular divergence, wavelength spread and intensity of the X-ray beam emitted by a given optical system is described by making use of descriptions in the position–angle–wavelength space.

Partial inner product spaces. III. Compatibility relations revisited

Abstract: This paper continues our systematic study of partial inner product spaces. We show here that a linear compatibility relation on a vector space V is characterized by special families of vector subspaces of V, called involutive coverings, and vice versa. This result provides the link between partial inner product spaces, defined in an intrinsic way, and various concrete structures, such as rigged or nested Hilbert spaces. Given a linear compatibility, generating sets (’’rich subsets’’) are discussed, and several examples are worked out. Finally, we introduce an order relation among all linear compatibilities on the same vector space.

Superluminal transformations in complex Minkowski spaces

Abstract: We calculate the mixing of real and imaginary components of space and time under the influence of superluminal boosts in thex direction. A unique mixing is determined for this superluminal Lorentz transformation when we consider the symmetry properties afforded by the inclusion of three temporal directions. Superluminal transformations in complex six-dimensional space exhibit unique tachyonic connections which have both remote and local space-time event connections.