Mathematics

Closed Geodesics on Orbifolds of Revolution

Joseph E. Borzellino et al. — Fri, 05 Apr 2013 15:24:11 PDT

Using the theory of geodesics on surfaces of revolution, we show that any two-dimensional orbifold of revolution homeomorphic to S² must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be regarded as a topological two-sphere with metric singularities, we will have extended Bangert's theorem on the existence of infinitely many closed geodesics on any smooth Riemannian two-sphere. In addition, we give an example of a two-sphere cone-manifold of revolution which possesses a single closed geodesic, thus showing that Bangert's result does not hold in the wider class of closed surfaces with cone manifold structures.

Elementary orbifold differential topology

Joseph E. Borzellino et al. — Tue, 26 Mar 2013 15:43:22 PDT

Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f:O→P between smooth orbifolds O and P. We show that Sardʼs theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence of a smooth orbifold map imposes on local isotropy groups. As an application, we prove a Borsuk no retraction theorem for compact orbifolds with boundary and some obstructions to the existence of real-valued orbifold maps from local model orbifold charts.

When is a Trigonometric Polynomial Not a Trigonometric Polynomial?

Joseph E. Borzellino et al. — Tue, 26 Mar 2013 15:43:21 PDT

Orbifold Homeomorphism and Diffeomorphism Groups

Joseph E. Borzellino et al. — Tue, 26 Mar 2013 15:43:19 PDT

In this paper we outline results on orbifold diffeomorphism groups that were presented at the International Conference on Infinite Dimensional Lie Groups in Geometry and Representation Theory at Howard University, Washington DC on August 17-21, 2000. Specifically, we define the notion of reduced and unreduced orbifold diffeomorphism groups. For the reduced orbifold diffeomorphism group we state and sketch the proof of the following recognition result: Let O1 and O2 be two compact, locally smooth orbifolds. Fix r ≥ 0. Suppose that Φ : Diffr (O1) → Diffr (O2) is a group isomorphism. Then Φ is induced by redred a (topological) homeomorphism h : X→ XThat is, Φ(f)= hfh−1 for O1 O2 . r all f ∈ Diffr (O1). Furthermore, if r>0, his a Cmanifold diffeomorphism red when restricted to the complement of the singular set of each stratum. We then show that if we replace the reduced orbifold diffeomorphism group by the unreduced orbifold diffeomorphism group in the above theorem, we can strengthen the homeomorphism hto an orbifold homeomorphism (orbifold structure preserving). Lastly, we state a structure theorem for the orbifold diffeomorphism group, showing that it is a Banach manifold for 1 ≤ r

Orbifolds with Lower Ricci Curvature Bounds

Joseph E. Borzellino — Tue, 26 Mar 2013 15:43:13 PDT

We show that the first betti number of a compact Riemannian orbifold with Ricci curvature and diameter is bounded above by a constant , depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the is bounded above by the dimension , and that if, in addition, , then is a flat torus .

The Closed Geodesic Problem for Compact Riemannian 2-Orbifolds

Joseph E. Borzellino et al. — Tue, 26 Mar 2013 15:43:10 PDT

In this paper it is shown that any compact Riemannian 2-orbifold whose underlying space is a (compact) manifold without boundary has at least one closed geodesic.

The Splitting Theorem for Orbifolds

Joseph Borzellino et al. — Tue, 26 Mar 2013 15:43:07 PDT

In this paper we wish to examine a generalization of the splitting theorem of Cheeger–Gromoll [CG] to Riemannian orbifolds. Roughly speaking, a Riemannian orbifold is a metric space locally modelled on quotients of Rie- mannian manifolds by finite groups of isometries. The term orbifold was coined by W. Thurston [T] sometime around the year 1976–77. The term is meant to suggest the orbit space of a group action on a manifold. A similar concept was introduced by I. Satake in 1956, where he used the term V–manifold (See [S1]). The “V” was meant to suggest a cone–like singularity. Since then, orbifold has become the preferred terminology.

Pinching Theorems for Teardrops and Footballs of Revolution

Joseph E. Borzellino — Tue, 26 Mar 2013 15:43:04 PDT

We give explicit optimal curvature pinching constants for the Riemannian (p, q)- football orbifolds under the assumption that they are realised as surfaces of revolution in ³. We show that sufficiently pinched sectional curvature assumptions imply that a (p, q)-football must be good.

Temporal Topos and U-Singularities

Goro C. Kato — Mon, 16 Apr 2012 11:40:20 PDT

Several papers and books by C. Isham, C.Isham-A. Doering, F. Van Oystaeyen, A.Mallios-I. Raptis, C. Mulvey, and Guts and Grinkevich, have been published on the methods of categories and sheaves to study quantum gravity. Needless to say, there are well-written treatises on quantum gravity whose methods are non-categorical and non-sheaf theoretic. This paper may be one of the first papers explaining the methods of sheaves with minimally required background that retains experimental applications. Temporal topos (t-topos) is related to the topos approach to quantum gravity being developed by Prof. Chris Isham of the Oxford-Imperial research group (with its foundations inthe work of F. W. Lawvere). However, in spite of strong influence from papers by Isham, our method of t-topos is much more direct in the following sense. Our approach is much closer to the familiar applications of the original algebraic geometric topos where little logic is involved.

Asymptotic Functions as Kernels of the Schwartz Distributions

Todor D. Todorov — Thu, 01 Dec 2011 11:33:45 PST

Using a version of the sequential method we introduce a class of generalized functions called here "asymptotic functions''. This class contains kernels of all Schwartz distributions and is equipped with a correctly defined multiplication operation. So, in a sense, one solves the problem of "multiplication of Schwartz distributions" although the solution refers to the class of the asymptotic functions and not to the Schwartz distributions themselves. The paper is a continuation of a series of works [1-10] but here only part of the results of [5], [6] and [8] will be needed.

Another Proof of the Existence a Dedekind Complete Totally Ordered Field

James F. Hall et al. — Mon, 14 Nov 2011 12:15:21 PST

We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis.

Completeness of the Leibniz Field and Rigorousness of Infinitesimal Calculus

James F. Hall et al. — Mon, 14 Nov 2011 12:15:18 PST

We present a characterization of the completeness of the field of real numbers in the form of a collection of ten equivalent statements borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. As an application we exploit one of our results to argue that the Leibniz infinitesimal calculus in the 18^th century was already a rigorous branch of mathematics – at least much more rigorous than most contemporary mathematicians prefer to believe. By advocating our particular historical point of view, we hope to provoke a discussion on the importance of mathematical rigor in mathematics and science in general. We believe that our article will be of interest for those readers who teach courses on abstract algebra, real analysis, general topology, logic and the history of mathematics.

Lecture Notes: Non-Standard Approach to J.F. Colombeau’s Theory of Generalized Functions

Todor D. Todorov — Mon, 14 Nov 2011 12:14:39 PST

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau’ theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau’ theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau’s theory are rings with zero divisors. The improvement of the scalars leads to other improvements and simplifications of Colombeau’s theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau’s theory, other seems to be without counterpart. We present applications of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau’s community, the readers who are already familiar with non-standard methods might also find a short and comfortable way to learn about Colombeau’s theory: a new branch of functional analysis which naturally generalizes the Schwartz theory of distributions with numerous applications to partial differential equations, differential geometry, relativity theory and other areas of mathematics and physics.

A Lost Theorem: Definite Integrals in An Asymptotic Setting

Ray Cavalcante et al. — Mon, 14 Nov 2011 12:14:27 PST

Monads and Realcompactness

Sergio Salbany et al. — Mon, 14 Nov 2011 12:14:23 PST

We give a quantifier free characterization of realcompactness and ordered realcompactness in terms of monads. We also present simple proofs of some topological facts concerning realcompact spaces.

Nonstandard and Standard Compactifications of Ordered Topological Spaces

Sergio Salbany et al. — Mon, 14 Nov 2011 12:14:18 PST

We construct the Nachbin ordered compactification and the ordered realcompactification, a notion defined in the paper, of a given ordered topological space as nonstandard ordered hulls. The maximal ideals in the algebras of the differences of monotone continuous functions are completely described. We give also a characterization of the class of completely regular ordered spaces which are closed subspaces of products of copies of the ordered real line, answering a question of T.H. Choe and Y.H. Hong. The methods used are topological (standard) and nonstandard.

Operations with Distribution Vectors

Brian Fisher et al. — Mon, 14 Nov 2011 12:14:06 PST

Asymptotic Functions and the Problem of Multiplication of Distributions

Todor D. Todorov — Mon, 14 Nov 2011 12:14:00 PST

The asymptotic functions are a new type of generalized functions. But they are not functionals on some space of test-functions as the Schwartz distributions. They are mappings of the set of the asymptotic numbers (1, 3, 5, 6) into itself. On its part, the set of the asymptotic numbers is a totally-ordered set of generalized numbers including the systems of real and complex numbers, as well as infinitesimals and infinitely large numbers. Every two asymptotic functions can be multiplied. On the other hand, the Schwartz distributions have realizations, in a certain sense, as asymptotic functions. The motivations of this work are connected with some physical problems of quantum theory [18, 25].

The Products δn²(x), δ(x). X^-n, ϴ(x). X^-n, etc. in the Class of the Asymptotic Functions

Todor D. Todorov — Mon, 14 Nov 2011 12:13:53 PST

Several products like δⁿ(x), δ(x)ϴ(x), δ^(m)(x). X^-n, ϴ(x). X^-n, etc., where δ(x), ϴ(x), X^-n, etc., are kernels of the corresponding Schwartz distributions, are studied in the framework of the class of the asymptotic functions F₀ introduced in a previous paper [11]. In some particular cases many formulae are derived and several examples are presented. The work is of mathematical type but its motivations lie in some problems in quantum theory. It is closely connected with a series of previous works [1 - 11] and first of all with [11].

Quasi-Extended Asymptotic Functions

Todor D. Todorov — Mon, 14 Nov 2011 12:13:45 PST

The class F of "quasi-extended asymptotic functions" introduced in the present paper contains all extended asymptotic functions [8, (3.1)] (in particular, all examples constructed in [9, Sec. 1 ]). But F contains also some new asymptotic functions very similar to tht Schwartz distributions. On the other hand, every two quasi-extended asymptotic functions can be multiplied as opposed to the Schwartz distributions; in particular, the square &# 948;² of an asymptotic function &# 948; similar to Dirac's delta-function is constructed as an example. The connection with the asymptotic functions introduced in [2] and [4] is established.