A non-trivial example of an L-topological space, the fuzzy real line is examined. Various L-topological properties and their relationships are developed. Extension theorems on the L-fuzzy real line as well as extension theorems on more general L-topological spaces follow. Finally, a theory of L-fuzzy vector spaces leads up to a fuzzy version of the Hahn-Banach theorem.

Let U be an n-dimensional vector space over an algebraically closed field. Let [formula omitted] denote the [formula omitted] space spanned by all Grassmann products [formula omitted]. Subsets of vectors of [formula omitted] denoted by [formula omitted] and [formula omitted] are defined as follows [formula omitted]. A vector which is in [formula omitted] or is zero is called pure or decomposable. Each vector in [formula omitted] is said to have rank one. Similarly each vector in [formula omitted] has rank two. A subspace of H of [formula omitted] is called a rank two subspace If [formula omitted] is contained in [formula omitted]. In this thesis we are concerned with investigating rank two subspaces. The main results are as follows: If dim [formula omitted] such that every nonzero vector [formula omitted] is independent in U. The rank two subspaces of dimension less than four are also characterized. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

Thesis (Dr. rer. nat.)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2001. / Includes bibliographical references (p. 87-90).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

For the case where L is an ecl-premonoid, we explore various characterizations of SL-topological spaces, in particular characterization in terms of a convergence function lim: FS L(X) ! LX. We find we have to introduce a new axiom , L on the lim function in order to completely describe SL-topological spaces, which is not required in the case where L is a frame. We generalize the classical Kowalski and Fischer axioms to the lattice context and examine their relationship to the convergence axioms. We define the category of stratified L-generalized convergence spaces, as a generalization of the classical convergence spaces and investigate conditions under which it contains the category of stratified L-topological spaces as a reflective subcategory. We investigate some subcategories of the category of stratified L-generalized convergence spaces obtained by generalizing various classical convergence axioms.

Suppose q₁,q₂,…,qn are the generalised coordinates of a mechanical system moving with constraints expressed by r non-integrable equations (r〈n) (1) [equation omitted] where the dots denote differentiation with respect to the time t, and fα are nonlinear in the q’s. The equations (1) are said to represent nonlinear nonholonomic constraints and the system moving with such constraints is called nonlinear nonholonomic. From a purely analytical point of view, the author has obtained the equations of motion for a nonlinear nonholonomic mechanical system in many a different form. The importance of these forms lies in their simplicity and novelty. Some of these forms are deduced from the principle of d'Alembert-Lagrange using the definition of virtual (possible) displacements due to Četaev [ll] . The others are obtained as a result of certain transformations. Moreover, these different forms of equations of motion are written either in terms of the generalised coordinates or in terms of nonlinear nonholonomic coordinates introduced by V.S. Novoselov [23]. These forms involve the energy of acceleration of the system or the kinetic energy or some new functions depending upon the kinetic energy of the system. Two of these new functions, denoted by R (Sec. 2.3) and K (Sec. 2.4), can be identified, to a certain approximation, with the energy of acceleration of the system and the Gaussian constraint, respectively. An alternative proof (Sec.2.5) is given to the fact that, if virtual displacements are defined in the sense of N.G. Četaev [ll], the two fundamental principles of analytical dynamics - the principle of d'Alembert-Lagrange and the principle of least constraint of Gauss -are consistent. If the1 constraints are rheonomic but linear, a generalisation of the classical theorem of Poisson is obtained in terms of quasi-coordinates and the generalised Poisson's brackets introduced by V.V. Dobronravov [17]. The advantage of the various novel forms for the equations of motion is illustrated by solving a few problems. / Science, Faculty of / Mathematics, Department of / Graduate

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: leaves 106-109. / by Rae Michael Andrew Shortt. / Ph.D.

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Introduction: The discovery of new solutions to Einstein's field equations has long been a problem in General Relativity. However due to new techniques of Newman and Penrose [1], Carter [2] and others there has been a considerable proliferation of new solutions in recent times. Consequently a new problem has arisen. How are we to interpret the new solutions physically? The tools available, despite a spate of papers in the past fifteen years, remain inadequate although often sophisticated. Any attempts at physical interpretations of metrics are beset with difficulties. There is always the possibility that two entirely different physical pictures will emerge. For example a direct approach would be to attempt an "infilling" of the metric, that is, an extension of the metric into the region occupied by the gravitating matter. However even for the Kerr [1] metric the infilling is by no means unique, in fact a most natural "infilling" turns out to be unphysical (Israel [1]). Yet few people would doubt the physical significance of the Kerr metric. Viewed in this light our attempt to discuss, among other things, the physical interpretation of type D metrics is slightly ambitious. However the problems with regard to this type of metric are not as formidable as for most of the other metrics, since we have been able to integrate the geodesic equations. Nevertheless it is still not possible to produce complete answers to all the questions posed. After a chapter on Mathematical preliminaries the study divides naturally into four sections. We start with an outline of the Hamilton-Jacobi theory of Rund [1] and then go on to show how this theory can be applied to the Carter [2] metrics. In the process we lay a foundation in the calculus of variations for Carter's work. This leads us to the construction of Killing tensors for all but one of the Kinnersley [1] type D vacuum metrics and the Cartei [2] metrics which are not necessarily vacuum metrics. The geodesic equations, for these metrics, are integrated using the Hamilton-Jacobi procedure. The remaining chapters are devoted to the Kinnersley [1] type D vacuum metrics. We omit his class I metrics since these are the Schwarzschild metrics, and have been studied in detail before. Chapter three is devoted to a general study of his class II a metric, a generalisation of the Kerr [1] and NUT (Newman, Tamburino and Unti [1]) metrics. We integrate the geodesic equations and discuss certain general properties: the question of geodesic completeness, the asymptotic properties, and the existence of Killing horizons. Chapter four is concerned with the interpretation of the new parameter 'l', that arises in the class II a and NUT metrics. This parameter was interpreted by Demianski and Newman [1] as a magnetic monopole of mass. Our work centers on the possibility of obtaining observable effects from the presence of 'l'. We have been able to show that its presence is observable, at least in principle, from a study of the motion of particles in the field. In the first place, if l is comparable to the mass of the gravitating system, a comparatively large perihelion shift is to be expected. The possibility of anomalous behaviour in the orbits of test particles, quite unlike anything that occurs in a Newtonian or Schwarzschild field, also arises. In the fifth chapter the Kinnersley class IV metrics are considered. These metrics, which in their simplest form have been known for some time, present serious problems and no interpretations have been suggested. Our discussion is essentially exploratory and the information that does emerge takes the form of suggestions rather than conclusions. Intrinsically the metrics give the impression that interesting results should be obtainable since they are asymptotically flat in certain directions. However the case that we have dealt with does not appear to represent a radiation metric.