Showing papers on "Space (mathematics) published in 1968"

Event horizons in static electrovac space-times

Abstract: The following theorem is established. Among all static, asymptotically flat electrovac fields with closed, simply-connected equipotential surfacesg0 0=const.. the only ones which have regular event horizonsg0 0=0 are the Reissner-Nordstrom family of spherisymmetric solutions withm≧G1/2|e|/c. In the special case where the gravitational coupling of the electromagnetic energy density is neglected (G=0) all solutions are computed explicitly, thus extending an earlier result ofGinzburg for a magnetic dipole inSchwarzschild's space-time. Possible implications for gravitational collapse are briefly discussed.

The Stefan problem in several space variables

Abstract: Introduction. The Stefan problem is a free boundary problem for parabolic equations. The solution is required to satisfy the usual initial-boundary conditions, but a part of the boundary is free. Naturally, an additional condition is imposed at the free boundary. A two-phase problem is such that on both sides of the free boundary there are given parabolic equations and initial-boundary conditions, and neither of the solutions is identically constant. In case the space-dimension is one, there are numerous results concerning existence, uniqueness, stability, and asymptotic behavior of the solution; we refer to [1] and the literature quoted there (see also [8]). In the case of several space variables the problem is much harder. The difficulty is not merely due to mathematical shortcomings but also to complications in the physical situation. Thus, even if the data are very smooth the solution need not be smooth, in general. For example, when a body of ice having the shape

Numerical advection experiments1

Abstract: First, second, and fourth order finite difference approximations to the color equation in both advection and conservation form are considered in one and two space dimensions. All schemes considered are based on forward time differences and most involve centered space differences. All are shown to be numerically stable for |uΔt/Δx| ≤ 1. Test calculations indicate that for the same order of accuracy, the conservation form produces more accurate solutions than the advection form. For either conservation or advection form, fourth order schemes are shown to be more accurate than second or first order schemes in terms of both amplitude and phase errors.

Coherent Soft‐Photon States and Infrared Divergences. I. Classical Currents

Abstract: As a first step toward a treatment of soft‐photon processes which is free of infrared divergences and avoids the necessity of introducing a fictitious photon mass, the specification of asymptotic photon states belonging to non‐Fock representations is discussed. As in the work of Chung, a basis consisting of generalized coherent states is used, but in contrast to his work, these states are rigorously defined in terms of von Neumann's infinite tensor product. It is shown that the states must be given an additional label which serves to distinguish various ``weakly equivalent'' vectors, and which corresponds formally to an infinite phase factor. A nonseparable Hilbert space HIR is defined (as a subspace of the infinite tensor‐product space) which may be regarded as the space of all possible asymptotic photon states. The interaction of the electromagnetic field with a prescribed classical current distribution is discussed, and it is shown that a unitary S operator, all of whose matrix elements are finite, may...

Space filling curves and mathematical programming

Abstract: The problem of finding {if314-1} in n dimensional Euclidean space such that {if314-2}, i = 1, 2, ···, N , is considered. The only assumption on the f i is that a solution exists in the quantized unit hypercube. Implicitly exhaustive solution procedures, which obtain solutions by implicitly considering every point in the quantized space without making computations at each point, are studied. The implicitly exhaustive feature is made possible by adapting “space filling curves≓ to discrete spaces of general dimensionality. Several space filling curves are surveyed, and Peano's continuous mapping from the unit interval onto the unit square is used as a basis for defining a mapping from the unit quantized interval onto the unit quantized hypercube, and inversely. Ternary arithmetic is the basis for the required functional relationships in the discrete mapping. The discrete mapping has attributes of quasi-continuity, and specific numerical bounds are derived in this respect. It is shown that these bounds are of optimal order dependence on the relevant variables. It is shown how to use knowledge of first and second order properties of the f i to obtain solutions on a digital computer using space filling curves as a basis for the concept of implicitly exhaustive search. The only global properties assumed are bounds on first, or possibly second, order variations. Concluding remarks bear on the ultimate practicality of the method, and present a limited amount of experimental data.

On the Stability of Nonlinear Space-Dependent Reactor Kinetics

Abstract: The spatial and temporal behavior of neutron distributions governed by the nonlinear diffusion equation approximation to neutron transport theory are considered in this paper. Stability criteria fo...

Asymptotic Fields in Some Models of Quantum Field Theory. III

Abstract: A quantum field with nonlocal interaction is considered. We prove under a proper smoothness condition on the interaction that the asymptotic limits of the annihilation‐creation operators exist. The asymptotic limits are then used to prove that the state space decomposes as a tensor product of an incoming (outgoing) Fock space and a zero‐particle space.

from Analysis of the Echo Signal Propagating in Space

Abstract: RADAR target scintillation is observed in every type of RADAR system and has generally been analyzed on the basis of the performance of specific types of RADAR systems. However, the target scintillation phenomenon, including Doppler scintillation, may be expressed as distortions of the RADAR echo signal propagating in space, independent of RADAR system parameters. In this form it is convenient for visualizing the overall effects on RADAR systems and how these effects are altered by the RADAR system parameters. Past literature has demonstrated the target angle scintillation as a distortion of the RADAR echo signal phase front. Extension of this approach by a Poynting-vector analysis of the RADAR echo signal from a complex source shows a corresponding deviation of the direction of power flow consistent with the phase-front distortion theory and target scintillation measurements as well as describing all other target scintillation characteristics. The analysis demonstrates that deviations in the direction of the echo signal power flow from a complex target can be so large that the apparent source falls many target spans away from the actual target location. This is demonstrated by both tracking RADAR and single-beam search-t ype RADAR experiments. Although the theoretical angle deviations approach infinite error in target location, RADAR parameters, such as the finite size antenna aperture which perform a space integration of the echo signal, impose practical limitations. Typically, when a complex target such as an aircraft subtends an angle approaching a few tenths of a beamwidth, the antenna aperture integration will significantly limit the rms angle scintillation. Furthermore, the ways in which intentional means, such as diversity techniques, may be employed to reduce the effects of target scintillation on a RADAR are observed in Poynting-vector analysis. The analysis of the echo signal propagating in space provides a readily visualized basis for derivation of the Doppler scintillation caused by the airframe (rigid body portion) of a complex target which spreads the Doppler over a finite bandwidth when it has random yaw, pitch, and roll motion typical of aircraft in flight. The derivation relates the Doppler scintillation to the angle scintillation and the random motions typical of aircraft targets. A typical aircraft target with Gaussian-distributed angle scintillation and Gaussian-distributed rates of random motion will have a spike-shaped Doppler spectrum described by the modified Hankel function K/sub 0/ where the parameters are determined from the values of the rms angle scintillation and the rms angular rates of random motion. These values can be closely approximated without extensive measurements on the target. Experimental results verify the theory. The expressions used to derive the Doppler spectrum may be modified to accommodate non-Gaussian distributed angle scintillation and rates of angle motion.

Related First Integral Theorem: A Method for Obtaining Conservation Laws of Dynamical Systems with Geodesic Trajectories in Riemannian Spaces Admitting Symmetries

Abstract: In this paper we develop in detail a unified method, referred to as the Related First Integral Theorem, for obtaining ``derived'' first integrals (i.e., constants of the motion) of mass‐pole test particles with geodesic trajectories in a Riemannian spece. By this method, which is based upon a process of Lie differentiation, additional conservation laws in the form of mth order first integrals can be generated from a given mth order first integral (conservation law), provided the space admits symmetries in the form of continuous groups of projective collineations (which include affine collineations and motions as special cases). We give in tensor form a reformulation of the well‐known Poisson's theorem on constants of the motion for particles with geodesic trajectories. We then show for this class of trajectories that, as a method for generating mth order first integrals from a given mth order first integral, Poisson's theorem is a special case of the Related First Integral Theorem. It is also shown that d...

Constants and Related Information for Astrodynamic Calculations, 1968

Abstract: Principal constants and related data for space navigation, trajectory, and orbit calculations

Brans--dicke cosmology.

Abstract: We discuss the implications of the Brans-Dicke scalar-tensor theory for cosmology with particular emphasis on the primordial element abundances that would obtain. Two general classes of models are found. Models of one class expand through the nuclear burning stage slightly more rapidly than the general relativistic case: models of the other class may expand at any rate whatsoever. The first class of models yeilds primordial abundances of D, He3 and He4 in agreement with their general relativistic values if the present mass density is low. High-density cosmologies, however, would produce too much He4. The second class of models yields element abundances which are far too high unless the expansion rate was quite large: in this case no He4 at all is produced. Finally, we determine the rate of change of the constant of gravitationG at the present epoch. For all but a very small class of models Ġ is negative at the present epoch. Models with positive values of Ġ at the present epoch produce no primordial He4 whatsoever, and have ages significantly lower than the corresponding general relativistic ages.

Simple computation-universal cellular spaces and self-reproduction

Abstract: Cellular spaces computationally equivalent to any given Turing machine are exhibited which are simple in the sense that each cell has only a small number of states and a small neighborhood. Neighborhood reduction theorems are derived in this interest, and several simple computationuniversal cellular spaces are presented. Conditions for computation-universality of a cellular space are investigated, and, in particular, the conjecture that unbounded but boundable propagation in a space is a sufficient condition is refuted. Finally, the computation-universal spaces derived in the study are used to introduce, via recursive function theory, examples of simple self-reproducing universal Turing machine configurations in one and two dimensions.

A note on compactifications and semi-normal spaces

Abstract: Recently Orrin Frink (see [2]) gave a neat internal characterization of Tychonoff or completely regular T spaces. This characterization was given in terms of the notion of a normal base for the closed sets of a space X . A normal base for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of . In a normal space the ring of closed sets is a normal base.