The structure of the b-completion of space-time

Structured space, as a natural generalization of the manifold concept, is defined to be a topological space with a sheaf of real function algebras which are suitably localized and closed with respect to composition with smooth Euclidean functions. Vector fields, differential forms, linear connection and curvature are introduced on structured spaces. It is shown that structured spaces correctly model space-times with singularities. Schmidt's b-boundary of space-time is constructed in the category of structured spaces, and well known difficulties with the b-boundaries of the closed Friedman and Schwarzschild space-times are disentangled. It is argued that the b-boundary of space-time, when considered in the category of structured spaces, can serve as a good definition of classical singularities.     

Directional radiation pattern in structural-acoustic coupled system

In this paper we demonstrate the possibility of designing a radiator using structural-acoustic interaction by predicting the pressure distribution and radiation pattern of a structural-acoustic coupling system that is composed by a wall and two spaces. If a wall separates spaces, then the wall's role in transporting the acoustic characteristics of the spaces is important. The spaces can be categorized as bounded finite space and unbounded infinite space. The wall considered in this study composes two plates and an opening, and the wall separates one space that is highly reverberant and the other that is unbounded without any reflection. This rather hypothetical circumstance is selected to study the general coupling problem between the finite and infinite acoustic domains. We developed an equation that predicts the energy distribution and energy flow in the two spaces separated by a wall, and its computational examples are presented. Three typical radiation patterns that include steered, focused, and omnidirected are presented. A designed radiation pattern is also presented by using the optimal design algorithm.     

Universal Groups of Effect Spaces

Various axiomatic models for unsharp quantum measurements are investigated. These include effect spaces (E-spaces), effect test spaces (E-test spaces), effect algebras, and test groups. It is shown that a test group G is the universal group of an E-test space if and only if G is strongly atomistic. It follows that if G is strongly atomistic, then G is an interpolation group. We then demonstrate that if G is an interpolation group, then G is the universal group of an E-space. Finally, it is shown that an E-space is isomorphic to an E-test space if and only if it is strongly atomistic.     

The topology of homophase misorientation spaces

Homophase misorientation spaces are investigated with a focus on the effect of symmetry operations on their topology and their minimum embedding dimensions in Euclidean space. Whereas the topology of rotation space is well established and requires a minimum of five variables for a one-to-one and continuous mapping, the spaces of orientations and misorientations are quotient spaces of the rotation space and are obtained by applying various equivalence relations. The equivalence relations for orientation spaces only involve the rotational symmetries of the underlying crystals. These spaces are classified under the three-dimensional manifolds called the spherical 3-manifolds, which have a non-trivial fundamental group, are not simply connected spaces, and do not embed in three-dimensional Euclidean space. In the case of homophase misorientation spaces, however, in addition to rotational symmetry operations there is a further ‘grain exchange symmetry’, which is shown to simplify the topology considerably. In some important cases this symmetry also reduces the number of Euclidean dimensions required to embed these misorientation spaces. The homophase misorientation spaces for the dihedral point groups D ₂(222), D ₄(422) and D ₆(622), the tetrahedral point group T(23), and the octahedral group O(432) are all found to be embeddable in only three dimensions, two dimensions less than required for rotations. Hence, these misorientation systems can be represented using three variables in a one-to-one and continuous manner.     

A BOSON-FERMION DESCIPTION OF THE NUCLEAR MANY-BODY PROBLEM

A new formalism of the nuclear many-body problem is established in which one ean work with various physical state vector spaces consisting of both Fermions and Bosons, all being equivalent to the original one consisting only of Fermions.With the help of the usual commutation relations and anticommutation relations between the annihilation and creation operators of Boson and Fermion, a generalized state vector space is established, which contains all the physical spaces each being equivalent to the original physical space in terms of pure Fermion operators. Transformation between the state vectors in various equivalent physical spaces are constructed. Basic operators in the original state space are transformed into effective operators in the new physical spaces.     

On weakly symmetric Finsler spaces

In this paper, we study weakly symmetric Finsler spaces. We first study an existence theorem of weakly symmetric Finsler spaces. Then we study some geometric properties of these spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each weakly symmetric Finsler space is of Berwald type.     

The geometrical structure of the parameter space of the two-dimensional normal distribution

The differential geometrical consideration of the parameter space, especially as a Riemannian geometry, was initiated by C.R. Rao in 1945. This approach appears to be important for the problem of estimation and test of hypotheses as well as for applications to physical problems. It has been shown that the parameter spaces of univariate normal distribution, univariate exponential distribution and multinomial distribution are Riemannian spaces of constant curvature. In the present paper the discussion is confined to the parameter space of the two-dimensional normal distribution. It has been shown that in general the parameter space is not necessarily of a constant curvature and that, if the correlation coefficient vanishes, the parameter space becomes an Einstein space. In addition, some invariant quantities, as sectional curvature, mean curvature and scalar curvature, have been calculated.     

Fourier analysis on space-like surfaces I. the case of mass ≠ 0

Some spaces of functions or functionals on any space-like surface and those on the momentum space are considered. Fourier transformations are defined appropriately on these spaces, and it follows that these transformations are continuous and have their respective continuous inverses. Invariant singular functions are defined, and many physically important relations hold properly with respect to these singular functions.     

Determination of kerr congruences by algebraic methods in the theory of mapping of pseudo-Riemann into Riemann spaces

The spaces of general relativity theory are known to be pseudo-Riemannian. Sometimes, however, it is useful to transform to Riemann spaces of a definite signature (++++). The mapping of pseudo-Riemann spaces into Riemann spaces by means of a timelike congruence has been investigated by Mitskevich [1]. In the present article we elaborate certain aspects of the theory of such mappings; in particular, we introduce the concept of double mapping of a pseudo-Riemann space into Minkowski space and investigate methods for the determination of specific congruences in the example of the Kerr field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 50–53, January, 1982.     

Conformal Einstein spaces

We study conformal transformations in four-dimensional manifolds. In particular, we present a new set of two necessary and sufficient conditions for a space to be conformal to an Einstein space. The first condition defines the class of spaces conformal to C spaces, whereas the last one (the vanishing of the Bach tensor) gives the particular subclass ofC spaces which are conformally related to Einstein spaces.This work has been partly supported bym a grand from the National Science Foundation.     

Separation of variables in Einstein spaces. II. No ignorable coordinates

We prove that the only Einstein spaces which admit a coordinate system with no ignorable coordinates which separates the Hamilton-Jacobi equation are certain symmetric spaces of Petrov typeD due to Kasner and the constant-curvature de Sitter spaces. We also show that a space admitting a coordinate system with no ignorable coordinates which separates the Helmholtz (Schrödinger) equation must be of Petrov type     

When Can Statistical Theories Be Causally Closed?

The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in particular that no probability space with a finite number of random events can contain common causes of all the correlations it predicts; however, it is demonstrated that probability spaces even with a finite number of random events can be common cause closed with respect to a causal independence relation that is stronger than logical independence. Furthermore it is shown that infinite, atomless probability spaces are always common cause closed in the strongest possible sense. Open problems concerning common cause closedness are formulated and the results are interpreted from the perspective of Reichenbach's Common Cause Principle (RCCP).     

Integrability and Huygens' principle on symmetric spaces

The explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found. These symmetric spaces have the following characteristic property: all multiplicities of their restricted roots are even. As a corollary in the odd-dimensional case one has that the Huygens' principle in Hadamard's sense for these equations is fulfilled. We consider also the heat and Laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions-heat kernel and Green's function. This continues our previous investigations [16] of the spherical functions on the same symmetric spaces based on the fact that the radial part of the Laplace-Beltrami operator on such a space is related to the algebraically integrable case of the generalised Calogero-Sutherland-Moser quantum system. In the last section of this paper we apply the methods of Heckman and Opdam to extend our results to some other symmetric spaces, in particular to complex and quaternian grassmannians.     

The spaces of trial and generalized functions of infinite number of variables

In this article the spaces of trial and generalized functions of infinite number of variables closely connected with the equipment of Fock space are introduced. The spaces introduced are described in new terminology. The properties of continuity and differentiability of trial functions are studied.     

Tempered distributions in infinitely many dimensions

The space of testing functions for tempered distributions is characterized in an abstract way as the maximal space in a certain class of locally convex topological vector-spaces. The main characteristic of this class is stability under the differentiation and multiplication operators.The ensuing characterization of tempered distributions may readily be generalized to the case of infinitely many dimensions, and a certain class of such generalizations is studied. The spaces of testing elements are required to be stable under the action of the canonical field operators of the quantum theory of free fields, and it is shown that extreme spaces of testing elements exist and have simple properties. In fact, the maximal space is a Montel space, and the minimal complete space is a direct sum of such spaces.The formalism is applied to the problem of extending the canonical field operators, and a number of extension theorems are derived. In a forthcoming paper the theory of tempered distributions in infinitely many variables will be applied to a structurally simple linear operator equation.     

Twistor integral representations of fundamental solutions of massless field equations

We consider the general dimensional (complex) Minkowski spaces and the extended twistor spaces. We show that the fundamental solutions of the complex wave or Laplace equations are explicitly represented by the integrals of some closed forms on the twistor spaces. The closed form is defined from labeled trees explained in graphs theory, and is written, as the cohomology class, by the linear combination of the logrithmic forms on some hyperplane configuration complement in some complex affine space.     

Letter: Global Hyperbolicity of Sliced Spaces

We show that for generic sliced spacetimes global hyperbolicity is equivalent to space completeness under the assumption that the lapse, shift and spatial metric are uniformly bounded. This leads us to the conclusion that simple sliced spaces are timelike and null geodesically complete if and only if space is a complete Riemannian manifold.     

Around Polygons in ℝ3 and S3

We survey certain moduli spaces in low dimensions and some of the geometric structures that they carry, and then construct identifications among all of these spaces. In particular, we identify the moduli spaces of polygons in ℝ³ and S ³, the moduli space of restricted representations of the fundamental group of a punctured 2-sphere, the moduli space of flat connections on a punctured sphere, the moduli space of parabolic bundles on a sphere, the moduli space of weighted points on ℂℙ¹ and the symplectic quotient of SO(3) acting diagonally on (S ²) ⁿ . All of these spaces depend on parameters and some the above identifications require the parameters to be small. One consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given. Received: 20 September 1999 / Accepted: 28 November 2000