Showing papers on "Space (mathematics) published in 1977"

On the theory of identical particles

Abstract: The classical configuration space of a system of identical particles is examined. Due to the identification of points which are related by permutations of particle indices, it is essentially different, globally, from the Cartesian product of the one-particle spaces. This fact is explicity taken into account in a quantization of the theory. As a consequence, no symmetry constraints on the wave functions and the observables need to be postulated. The two possibilities, corresponding to symmetric and antisymmetric wave functions, appear in a natural way in the formalism. But this is only the case in which the particles move in three- or higher-dimensional space. In one and two dimensions a continuum of possible intermediate cases connects the boson and fermion cases. The effect of particle spin in the present formalism is discussed.

The large-scale structure of space-time.

Abstract: Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving behind a 'black hole' in space; and secondly, that there will exist singularities in space-time itself. These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large. Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. A discussion of the Cauchy problem for General Relativity is also included in this 1973 book.

Integral functionals, normal integrands and measurable selections

Abstract: : A fundamental notion in many areas of mathematics, including optimal control, stochastic programming, and the study of partial differential equations, is that of an integral functional. By this is meant an expression of the form If(x) = integral over S of f(s,x(s))mu(DS), x is a member of X where X is a linear space of measurable functions defined on a measure space (S, A, mu) and having values in a linear space E. This paper provides a thorough treatment of the properties of such functionals in the case of E = R to the n-th power, including properties of continuity convexity and duality.

Axiomatic basis for spaces with noninteger dimension

Abstract: Five structural axioms are proposed which generate a space SD with ’’dimension’’ D that is not restricted to the positive integers. Four of the axioms are topological; the fifth specifies an integration measure. When D is a positive integer, SD behaves like a conventional Euclidean vector space, but nonvector character otherwise occurs. These SD conform to informal usage of continuously variable D in several recent physical contexts, but surprisingly the number of mutually perpendicular lines in SD can exceed D. Integration rules for some classes of functions on SD are derived, and a generalized Laplacian operator is introduced. Rudiments are outlined for extension of Schrodinger wave mechanics and classical statistical mechanics to noninteger D. Finally, experimental measurement of D for the real world is discussed.

The monotone and planar circuit value problems are log space complete for P

Abstract: Ladner [3] ]has shown that the circuit value problem (CV) is log space complete for P, adding to the list of such problems found by Cook [iJ and Jones and Laaser [2]. Ladner's simulation of Turing mac]hines by boolean circuits seems to require an \"adequate\" set of gates, such as AND and NOT. We show that the same simulation is possible with monotone circuits using AND and OR gates only. Theorem The monotone circuit value problem (MCV) is log space complete for P. Proof Clearly MCV • P, and the construction below shows that CV -

Degenerating Families of Riemann Surfaces

Abstract: For several years it has been conjectured that the moduli of degenerating families of conformally finite Riemann surfaces with signature may be studied using the Bers embedding T(G) of the Teichmuiller space. In a previous paper ([2], see also Marden [17]) we examined a distinguished classthe regular b-groups-of Kleinian groups lying in AT(G). The augmented Teichmiiller space T(G) is the union of T(G) with the regular b-groups on its boundary.

A special-relativistic theory of the locally anisotropic space-time

Abstract: The relativistic mechanics and electrodynamics have been considered in a flat locally anisotropic space. Relativistic expressions for energy and momentum, as well as the geometry of the 4-momentum space, have been found. The laws governing the transformations of the time intervals and space volunes, 4-momenta and electrodynamic quantities have been established. Special attention has been paid to the Doppler effect, since by making use of it one can attempt to discover experimentally the local anisotropy of space-time. This work has resulted in formulating a principle which generalizes Einstein's special principle of relativity within the framework of the postulate of local anisotropy of space and time.

Subspaces and Quotient Spaces of (s)

Abstract: The main intention of this article is to give a survey of results which have been obtained by the author on subspaces and by the author together with M.J. Wagner on quotient spaces of s, the space of rapidly decreasing sequences of real or complex numbers (§ § 1,…,6). The results in these paragraphs are contained in Vogt [20] and Vogt-Wagner [21]. They are presented with only minor changes in arrangement and details of proofs. There is given a complete characterization of subspaces and quotient spaces of s, which in the case of nuclear (F)-spaces with basis (=sequence spaces) has been obtained independently and using other methods by Dubinsky [3] and Dubinsky-Robinson [4]. For the development of the concept of classes (DN) and (Ω) see [18], [19], [22]. In the proof of our results splitting theorems for exact sequences of (F)-spaces (s. §2) play an important role. These are also important in other fields of analysis and functional analysis. The last two paragraphs are concerned with related results and applications of these theorems.

Orlicz sequence spaces without local convexity

Abstract: In this paper we continue the study of Orlicz sequence spaces initiated by Lindberg (5) and Lindenstrauss and Tzafriri (7), (8) and (9). Our main concern is to investigate features of the theory which occur when the restriction of local convexity is lifted. It is clear that some results will hold with identical proofs, at least when the space is locally bounded. However, we are chiefly interested in the differences which arise. We always assume that the Orlicz function F satisfies the Δ 2 -condition.

On the Mixtures of Probability Distributions

Abstract: The problem of mixtures of probability distri- butions in a general situation, where the parameter set and the observational space are measurable spaces, is considered here. A general theory is developed for identifiability of a class of mixing distributions. A characterization theorem of identifia- bility is given and a relationship between the identifiability of the mixing distributions relative to multivariate distributions and to that of corresponding marginals is established. Simpler proofs with lesser restrictions of some well known results are given and are supported by examples. Finally, a general method of estimating the mixing distribution in a metric space is considered. A consistent estimator of the mixing distribution, when the class of mixing distribution is compact in the topology of weak convergence, is constructed.

Uniform quality Gaussian basis sets

Abstract: A direct optimization technique was applied to determine uniformly balanced, optimum (4s2p), (6s3p), (8s4p), and (10s5p) Gaussian basis sets for the first row atoms. These bases formally correspond to 2G, 3G, 4G, and 5G function representations per symmetry type per shell. The basis sets are throroughly balanced and all satisfy a rather rigorous quality criterion in terms of the local properties of the energy hypersurface over the space of the orbital exponents.

Dilations of Positive Contractions on Lp Spaces

Abstract: Throughout this article p denotes a fixed number such that 1 ≤ p < ∞. The definition of a real Lp space associated with a measure space is well known. These spaces are Banach Spaces and, with the usual partial ordering of (equivalence classes of) functions, also Banach Lattices.

Lack of interpolation of linear operators in spaces of smooth functions

Abstract: We prove that , the space of times continuously differentiable functions on the closure of a region in a finite-dimensional manifold, is not an interpolation space between and for . We find analogous results for the Sobolev-Stein spaces. In the class of spaces , defined by the modulus of continuity, we describe all interpolation spaces between and .Bibliography: 34 titles.

Scattering from infinite sheets

Abstract: We prove the existence and completeness of the wave operators for quantum-mechanical scattering by a potential which does not decrease to zero at infinity in two of the three space directions. We also obtain a new abstract result concerning the continuous dependence of the wave operators on the potential.

Asymptotic behaviour of solutions of a nonlinear diffusion equation

Abstract: where t and x denote, respectively, a time and a space coordinate, and the subscripts t and x denote partial differentiation with respect to these variables. The diffusion coefficient D(s) is defined on IR; it will be assumed that De C 2 + ' (R) (0 A > 0 for all s eX. We shall discuss two problems. I. The Cauchy problem in the strip S T = ( ~ , oo) • (0, T], where T is some fixed positive number, which eventually will tend to infinity. At t = 0 we prescribe

A fourth note on recent research in geometrical probability

Abstract: Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, line-segments and flats in Euclidean spaces, the random division of space, coverage, packing, random sets, stereology and probabilistic aspects of integral geometry.