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Showing papers on "Space (mathematics) published in 1982"



Journal ArticleDOI
TL;DR: In this paper, a brief review of the classical gravitational instabilities, as well as the stability of flat space, are given, and the effect due to the Schwarzschild instanton is analyzed and a negative mode is discovered.
Abstract: The instabilities of quantum gravity are investigated using the path-integral formulation of Einstein's theory. A brief review is given of the classical gravitational instabilities, as well as the stability of flat space. The Euclidean path-integral representation of the partition function is employed to discuss the instability of flat space at finite temperature. Semiclassical, or saddle-point, approximations are utilized. We show how the Jeans instability arises as a tachyon in the graviton propagator when small perturbations about hot flat space are considered. The effect due to the Schwarzschild instanton is studied. The small fluctuations about this instanton are analyzed and a negative mode is discovered. This produces, in the semiclassical approximation, an imaginary part of the free energy. This is interpreted as being due to the metastability of hot flat space to nucleate black holes. These then evolve by evaporation or by accretion of thermal gravitons, leading to the instability of hot flat space. The nucleation rate of black holes is calculated as a function of temperature.

428 citations


Journal ArticleDOI
TL;DR: A computational theory of the interpolation of surfaces from visual information is presented, and it is shown that there is a vector space of possible functionals that measure surface consistency, this vector space being spanned by thefunctional of quadratic variation and the functional of square Laplacian.
Abstract: Computational theories of structure-from-motion and stereo vision only specify the computation of three-dimensional surface information at special points in the image. Yet the visual perception is clearly of complete surfaces. To account for this a computational theory of the interpolation of surfaces from visual information is presented. The problem is constrained by the fact that the surface must agree with the information from stereo or motion correspondence, and not vary radically between these points. Using the image irradiance equation, an explicit form of this surface consistency constraint can be derived. To determine which of two possible surfaces is more consistent with the surface consistency constraint, one must be able to compare the two surfaces. To do this, a functional from the space of possible functions to the real numbers is required. In this way, the surface most consistent with the visual information will be that which minimizes the functional. To ensure that the functional has a unique minimal surface, conditions on the form of the functional are derived. In particular, if the functional is a complete semi-norm that satisfies the parallelogram law, or the space of functions is a semi-Hilbert space and the functional is a semi-inner product, then there is a unique (to within possibly an element of the null space of the functional) surface that is most consistent with the visual information. It can be shown, based on the above conditions plus a condition of rotational symmetry, that there is a vector space of possible functionals that measure surface consistency, this vector space being spanned by the functional of quadratic variation and the functional of square Laplacian. Arguments based on the null spaces of the respective functionals are used to justify the choice of the quadratic variation as the optimal functional. Possible refinements to the theory, concerning the role of discontinuities in depth and the effects of applying the interpolation process to scenes containing more than one object, are discussed.

147 citations



Journal ArticleDOI
TL;DR: It is shown that the problem is log space complete for deterministic polynomial time, so the maximum flow problem probably has no algorithm which needs only O(logk n) storage space for any constant k.

131 citations


Book ChapterDOI
TL;DR: The Kothe sets and Kothe sequence spaces were studied by G. Kothe and O. Toeplitz as mentioned in this paper prior to the development of general tools available through the present day theory of topological vector spaces; Kothe's early work with sequence spaces has helped point the way in establishing a general theory.
Abstract: Publisher Summary This chapter discusses the Kothe sets and Kothe sequence spaces. Echelon and co-echelon spaces had been studied by G. Kothe (and O. Toeplitz) prior to the development of general tools available through the present day theory of topological vector spaces; Kothe's early work with sequence spaces has helped point the way in establishing a general theory. In its turn, however, this general theory has been successfully utilized in the study of sequence spaces, while echelon and co-echelon spaces have continued to serve as a ready source for examples and counter examples. The chapter presents the fundamental definitions and establish the notation and treats the role of the space K p in the duality of echelon and co-echelon spaces. A condition on the Kothe matrix A is considered and is preferred to phrase in terms of the corresponding decreasing sequence—namely, the sequence space analog of the property that is called “regularly decreasing.”

124 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that any static metric on a space-time manifold with nonvanishing first Betti number (R}_{1}$ is a family of solutions to the electromagnetic Aharonov-Bohm effect.
Abstract: It is well known that gravitational fields may be locally the same but globally distinct due to differences in the topology of their underlying manifolds. Globally stationary but locally static gravitational fields provide an example of gravitational fields which are locally the same but globally distinct in spite of the homeomorphism of their underlying manifolds. Any static metric on a space-time manifold with nonvanishing first Betti number ${R}_{1}$ is shown to generate an ${R}_{1}$-parameter family of such solutions. These fields are seen to provide a gravitational analog of the electromagnetic Aharonov-Bohm effect. The exterior field of a rotating infinite cylinder of matter is discussed as an exactly soluble example.

120 citations


Journal ArticleDOI
TL;DR: The space of solutions of Einstein's vacuum equations has conical singularities at each spacetime possessing a compact Cauchy surface of constant mean curvature and a set of Killing fields as mentioned in this paper.

116 citations


Journal ArticleDOI
TL;DR: In this paper, the threshold behavior of the stochastic spatial general epidemic model on a discrete location space is investigated by making use of the general percolation theory of McDiarmid.
Abstract: The threshold behaviour of the stochastic spatial general epidemic model on a discrete location space is investigated by making use of the general percolation theory of McDiarmid.

116 citations


Journal ArticleDOI
TL;DR: In this article, an orthogonalization process for spaces which are realizations of abstract Hilbert spaces is proposed, which is simpler than the Gram-Schmidt process and it is shown that the generalized Langevin equation is contained therein.
Abstract: An orthogonalization process is proposed, applicable to spaces which are realizations of abstract Hilbert space. It is simpler than the Gram-Schmidt process. A recurrence relation which orthogonalizes a physical space is proposed and it is shown that the generalized Langevin equation is contained therein. This process serves as a basis for understanding the nature of the dynamic many-body formalism.

114 citations


Journal ArticleDOI
TL;DR: In this article, a sounding rocket experiment involving the injection of a barium gas jet in the upper ionosphere provided an opportunity of investigating quantitatively several aspects of the beam-plasma interaction that is the substance of Alfven's critical velocity effect.
Abstract: A sounding rocket experiment involving the injection of a barium gas jet in the upper ionosphere provided an opportunity of investigating quantitatively several aspects of the beam-plasma interaction that is the substance of Alfven’s critical velocity effect. Whereas the experimental data are presented elsewhere, this paper contains some theoretical considerations of (1) the limiting neutral density for which the ionization process can operate, (2) the interaction of the freshly injected ions with the background plasma, and (3) the microprocess which leads to collisionless electron tail formation. The observed distribution of injected ions is consistent with the Townsend condition on the beam-plasma discharge. The mass loading on the ambient plasma, although locally high, has a weak effect on the dynamics of the involved flux-tube because of the limited extent of the beam. The most likely process by which energy is transferred from the freshly generated ions to the electrons is an ion beam instability leading to the excitation of lower hybrid waves

Journal ArticleDOI
TL;DR: In this article, a theory of non-commutative stochastic integration analogous to the Ito-theory is presented, and a Doob-Meyer decomposition for supermartingales is obtained.

Journal ArticleDOI
TL;DR: In this article, Noller highlighted the value of joint microwave and photoelectron experiments as general readily applicable techniques for detecting moderately unstable species, such as small thiocarbonyls and selenocarbone.
Abstract: 3 Studies of Semistable Molecules A. Thiocarbonyls and Selenocarbony1s.-It was work on some small thiocarbonyls which originally highlighted the value of joint microwave and photoelectron experiments as general readily applicable techniques for detecting moderately unstable species. There must have been many attempts in the past to make sulphur analogues of formaldehyde, acetaldehyde, and acetone. Indeed Noller


Journal ArticleDOI
TL;DR: In this paper, a posteriori estimation of the space discretization error in the finite element method of lines solution of parabolic equations is analyzed for time-independent space meshes, and the effectiveness of the estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size.
Abstract: In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.

Journal ArticleDOI
TL;DR: In this paper, the real line is replaced by either a non-Abelian group or a symmetric space, and exact analogues of these commutation results with the real lines replaced by non-abelian groups or symmetric spaces are shown.
Abstract: Slepian, Landau and Pollak found that a certain finite convolution integral operator on the real line commutes with a much simpler second order differential operator. This opens the way to a detailed analysis of the space of “time and band limited functions” which has found applications in several fields. Here we exhibit some exact analogues of these commutation results with the real line replaced by either a non-Abelian group or a symmetric space. The sphere may be the most natural example for the applications.

Journal ArticleDOI
TL;DR: In this paper, various Hartree-Fock iteration techniques are discussed, all derived from the simple gradient iteration step where one proceeds along the slope of steepest descent of the energy functional.


Journal ArticleDOI
TL;DR: In this paper, it was shown that there do not exist discrete arithmetic groups generated by reflections in Lobachevsky spaces if the dimension of the space is greater than 15 and the degree of the ground field is sufficiently large.
Abstract: It is proved that there do not exist discrete arithmetic groups generated by reflections in Lobachevsky spaces if the dimension of the Lobachevsky space is greater than 15 and the degree of the ground field is sufficiently large. Bibliography: 24 titles.

Journal ArticleDOI
TL;DR: In this paper, the construction of the reduced space and the reduced Hamiltonian for the semisimple 1:1 resonance case was described, and the results were related to problems in physics on "degeneracies" due to symmetries of classical two-dimensional harmonic oscillators and their quantum analogues for the hydrogen atom.

Journal ArticleDOI
TL;DR: In this article, the irreducible band representations of a space group are induced from a set of inequivalent relevant symmetry centers in the Wigner-Seitz cell, and a connection is established between representations and band representations by using the Born-von K\'arm\'an boundary conditions.
Abstract: All the irreducible band representations of a space group are shown to be induced from a set of inequivalent relevant symmetry centers in the Wigner-Seitz cell. A connection is established between representations and band representations of space groups by using the Born---von K\'arm\'an boundary conditions. Continuity chords are used for proving the equivalency theorem which enables one to distinguish between equivalent and inequivalent band representations. As examples we consider a one-dimensional crystal and the ${D}_{6h}^{4}$ space group for a hexagonal close-packed structure.

Book
01 Jan 1982
TL;DR: In this article, the authors define full bundles and bundles with completely regular base space, including bundles with locally paracompact base spaces, and bundles of operators with continuous lattices.
Abstract: Notational remarks.- Basic definitions.- Full bundles and bundles with completely regular base space.- Bundles with locally paracompact base spaces.- Stone - Weierstrass theorems for bundles.- An alternative description of spaces of sections: Function modules.- Some algebraic aspects of ?-spaces.- A third description of spaces of sections: C(X)-convex modules.- C(X)-submodules of ?(p).- Quotients of bundles and C(X)-modules.- Morphisms between bundles.- Bundles of operators.- Excursion: Continuous lattices and bundles.- M-structure and bundles.- An adequate M-theory for ?-spaces.- Duality.- The closure of the "unit ball" of a bundle and separation axioms.- Locally trivial bundles: A definition.- Local linear independence.- The space Mod(?(p),C(X)).- Internal duality of C(X)-modules.- The dual space ?(p)' of a space of sections.

Journal ArticleDOI
TL;DR: In this article, the authors consider Lagrangians s−equivalent to T−V, where T is flat space kinetic energy and V is a spherically symmetric potential.
Abstract: Two Lagrangians are s‐equivalent (s for ‘‘solution’’) if they yield equations of motion having the same set of solutions. We consider Lagrangians s‐equivalent to T−V, where T is flat space kinetic energy and V is a spherically symmetric potential. We show that for n=dimension of space ≥3, there are many s‐equivalent Lagrangians which cannot be formed from T−V by multiplication by a constant or addition of a total time derivative. In general these s‐equivalent Lagrangians lead to inequivalent quantum theories in the sense that the energy spectra are different.


Journal ArticleDOI
TL;DR: In this article, the Douady space of compact complex subspaces of a complex space is defined and the corresponding universal family of subspace families of complex spaces of the same complex space are defined.
Abstract: Let X be a complex space. Let Dx be the Douady space of compact complex subspaces of X [6] and px : Zx→ Dx the corresponding universal family of subspaces of X.

Journal ArticleDOI
TL;DR: In this paper, the authors use the same general argument to characterize the onto linear isometries of the weighted Bergman spaces over balls and polydiscs, and apply it to describe the equimeasurability of the Hp spaces over bounded Runge domains.
Abstract: In [2], [8] and [10], Forelli, Rudin and Schneider described the isometries of the Hp spaces over balls and polydiscs. Koranyi and Vagi [6] noted that their methods could be used to describe the isometries of the Hp spaces over bounded symmetric domains. Recently Kolaski [4] observed that the algebraic techniques used above and Rudin's theorem on equimeasurability extended to the Bergman spaces over bounded Runge domains. In this paper we use the same general argument to characterize the onto linear isometries of the weighted Bergman spaces over balls and polydiscs, (all isometries referred to are assumed to be linear). 2. Preliminaries. Horowitz [3] first defined the weighted Bergman space Ap,α (0 < p < ∞, 0 < α < ∞) to be the space of holomorphic functions f in the disc which satisfy (1)

Journal ArticleDOI
TL;DR: In this article, it was shown that the dynamical group of collective states is then the group Spc(2,R), which is the restriction to the collective subspace of the group of linear canonical transformations in n dimensions conserving the O(n) symmetry.
Abstract: In the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an A‐nucleon system, defined as those A‐particle states invariant under the orthogonal group O(n) associated with the n = A−1 Jacobi vectors. The present paper discusses the case of a hypothetical one‐dimensional space. Simple invariance considerations show that the dynamical group of collective states is then the group Spc(2,R), which is the restriction to the collective subspace of the group Sp(2,R) of linear canonical transformations in n dimensions conserving the O(n) symmetry. In addition to the well‐known realization of the dynamical group in the Schrodinger representation based upon the Dzublik–Zickendraht transformation, two new realizations are proposed. The first acts in a Barut Hilbert space, which is the subspace of a Bargmann Hilbert space of analytic functions left invariant by O(n). A unitary mapping is established between the ordi...


Journal ArticleDOI
TL;DR: In this article, the energy level set topologies in the abstract nuclear charge space Z of molecular systems are defined and analyzed, and two theorems, one on the general convexity of level sets in Z, another on homotopies of boundaries of level set, induced by nuclear geometry variations in the nuclear configuration space R, are proven.
Abstract: Electronic energy level set topologies in the abstract nuclear charge space Z of molecular systems are defined and analyzed. Two theorems, one on the general convexity of level sets in Z, another on homotopies of boundaries of level sets, induced by nuclear geometry variations in the nuclear configuration space R, are proven. The applications of the two theorems are illustrated by examples of various molecules and ions.