Logicoalgebraic structures I

In this survey article, some basic properties of quantum logics are described, such as physical motivation, basic examples and constructions, and state spaces and observables. Some recent results concerning joint distributions and commutators and their relations to uncertainty relations, Bell inequalities, independence of center, automorphisms and state space, are included.     

Effect test spaces

An effect test space, or E-test space, for short, is a generalization of a test space that is able to describe unsharp measurements. Effects in an E-test space correspond to yes-no measurements, and observables correspond to general measurements that may have more than two values. Sharpness, compatibility, and orthogonality of effects are considered. It is shown that every observable is determined by its eigenvalues and eigeneffects. The spectrum of an observable is studied and special types of observables are investigated. Orthocomplements and a natural local sum on an E-test space are introduced. Relationships between the resulting structures and previously studied frameworks are presented.     

Symmetric spaces and star representations: II. Causal symmetric spaces

We construct and identify star representations canonically associated with holonomy-reducible simple symplectic symmetric spaces. This leads a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley-type and Hermitian symmetric spaces of tube-type.     

Trilocal structures. II. Expansion

A procedure is described by which a set of expansion functions is generated for the trilocal wave function. The first 49 functions in the set are listed, along with matrix elements of the Hamiltonian that are generated through this same procedure.     

Hamiltonian structures for homogeneous spaces

The definition and classification of classical relativistic particles requires the classification of certain invariant tensor fields on the inhomogeneous Lorentz group. The entire 10-parameter set is exhibited. At the same time, a much larger class of Lie groups is treated. The connection with particles will be presented in the succeeding article.     

Black holes in the Einstein-Cartan theory. II. Riemann-Cartan spaces

The Lagrangian of the fermionic matter is considered. The minimum mass of the black hole for it is estimated as ～10¹⁹ gram.     

Real structures in asymptotically flat ℋ spaces

A real structure is defined in asymptotically flat ? spaces and investigated in connection with the equations of motion in ? space.     

A note on asymptotically flat spaces. II

After brief reviews of the Geroch and spin-coefficient formalism approaches to null infinity, we present a dictionary which translates between the two formalisms.     

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     

Twistor spaces of generalized complex structures

The twistor construction is applied for obtaining examples of generalized complex structures (in the sense of Hitchin) that are not induced by a complex or a symplectic structure.     

Partial isometries and quotient structures in Hilbert spaces

Partial isometries are studied as the natural framework both for the representation of semi-groups on Hilbert spaces and for the mapping of operators with different spectra. The general theory is illustrated by examining several pertinent problems from conventional quantum mechanics. Families of partial isometries are found to induce quotient structures on Hilbert space. Embedding in appropriate tensor product spaces allows the representation of such families by a single isometry.     

Conformally flat spaces of locally constant connection. II

This paper continues the study of conformally flat spaces of locally constant connection to the case in which the global vector field defined by the connection in such spaces is null.Supported by NSERC A-7667 to P. L. Antonelli.     

Recursion operators and bi-Hamiltonian structures in multidimensions. II

We analyze further the algebraic properties of bi-Hamiltonian systems in two spatial and one temporal dimensions. By utilizing the Lie algebra of certain basic (starting) symmetry operators we show that these equations possess infinitely many time dependent symmetries and constants of motion. The master symmetries for these equations are simply derived within our formalism. Furthermore, certain new functionsT ₁₂ are introduced, which algorithmically imply recursion operators ₁₂. Finally the theory presented here and in a previous paper is both motivated and verified by regarding multidimensional equations as certain singular limits of equations in one spatial dimension.     

Effect test spaces and effect algebras

The concept of an effect test space, which is equivalent to a D-test space of Dvurečenskij and Pulmannová, is introduced. Connections between effect test space. (E-test space, for short) morphisms, and event-morphisms as well as between algebraic E-test spaces and effect algebras, are studied. Bimorphisms and E-test space tensor products are considered. It is shown that any E-test space admits a unique (up to an isomorphism) universal group and that this group, considered as a test group, determines the E-test space uniquely (up to an isomorphism).     

Dislocation structures. Part II. Slip system dependence

Part I established, via extensive transmission electron microscopy investigations, that the type of dislocation structure formed in metals of medium-to-high stacking fault energy upon deformation in tension or rolling to moderate strain levels (≤0.8) depends strongly on crystallographic grain orientation. This paper analyzes the grain orientation-dependent structures in terms of the active slip systems, focusing on the crystallographic plane of extended planar boundaries (geometrically necessary boundaries). The analysis establishes slip systems as the factor controlling the dislocation structure. Five fundamental slip classes, consisting of one to three active slip systems, have been identified. Multiple activation of these slip classes is also considered. The slip classes give rise to different types of dislocation structure, of which all except one contains geometrically necessary planar boundaries aligning with unique crystallographic planes (not necessarily slip planes). A slip class leads to the same type of structure, irrespective of the macroscopic deformation mode, as also demonstrated by successful predictions for shear deformation.     

Diffraction from modulated structures II. Quasiperiodic model of the modulated structures

The present model of modulated structure assumes the variation of the number of atomic planes in the inner part of the individual zones in their array. This variation is described by the interzone correlation function. The analytical expression for the diffracted intensity from this aperiodic array of zones is also given.     

Atomic ordering in ternary alloys with hexagonal-close-packed structures. II

This paper constitutes a theoretical study of the influence of interstitial impurity atoms on the type of order-disorder transition in an alloy of the AB₃-C_in type with an hcp structure. Allowance is made for a variety of types of interstices over the range 0 1. The free energy is calculated by the Gorskii-Bragg-Williams method. In the presence of an impurity there may be a change in the magnitude of the jump taking place in the degree of long-range order at the transition point, and also a change in the type of order-disorder transition occurring.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 20–26, March, 1974.     

New approach to Einstein-Petrov type I spaces. II. A classification scheme

A classification scheme is presented which enables one to have an overall look at Einstein-Petrov type I spaces and in particular to pick out those spaces whose Petrov scalars are functionally independent. It is shown that some subclasses defined by quite severe simplifying assumptions (which lead to considerable simplification of the field equations) still retain functionally independent Petrov scalars. For these subclasses three coupled differential equations (two of which are wave equations) are calculated for the Petrov scalars.     

Conformal Einstein spaces in N-dimensions: Part II

This paper generalizes the main result of “Conformal Einstein spaces in N-dimensions” published in Ann. Global Anal. Geom. 20(2) (2001). We present necessary and sufficient tensorial conditions for a certain class of semi-Riemannian manifolds to be conformally related to Einstein spaces.