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Recent documents in Mathematicsen-usWed, 23 Apr 2014 20:14:39 PDT3600The Stratified Structure of Spaces of Smooth Orbifold Mappings
http://digitalcommons.calpoly.edu/math_fac/112
http://digitalcommons.calpoly.edu/math_fac/112Wed, 05 Feb 2014 13:22:49 PST
We consider four notions of maps between smooth C^{∞} orbifolds , with compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^{r} maps between and with the C^{r} topology carries the structure of a smooth C^{∞} Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of C^{r} maps between and with the C^{r} topology carries the structure of a smooth C^{∞} Banach (r finite)/Fréchet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The C^{r} orbifold maps between and is locally a stratified space with strata modeled on smooth C^{∞} Banach (r finite)/Fréchet (r = ∞) manifolds while the set of C^{r} reduced orbifold maps between and locally has the structure of a stratified space with strata modeled on smooth C^{∞} Banach (r finite)/Fréchet (r = ∞) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show that they inherit the structure of C^{∞} Banach (r finite)/Fréchet (r = ∞) manifolds. In fact, for r finite they are topological groups, and for r = ∞ they are convenient Fréchet Lie groups.
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Joseph E. Borzellino et al.Riemannian Geometry of Orbifolds
http://digitalcommons.calpoly.edu/math_fac/111
http://digitalcommons.calpoly.edu/math_fac/111Mon, 08 Jul 2013 13:14:05 PDT
We investigate generalizations of many theorems of Riemannian geometry to Riemannian orbifolds. Basic definitions and many examples are given. It is shown that Riemannian orbifolds inherit a natural stratified length space structure. A version of Toponogov's triangle comparison theorem for Riemannian orbifolds is proven. A structure theorem for minimizing curves shows that such curves cannot pass through the singular set. A generalization of the Bishop relative volume comparison theorem is presented. The maximal diameter theorem of Cheng is generalized. A finiteness result and convergence result is proven for good Riemannian orbifolds, and the existence of a closed geodesic is shown for non-simply connected Riemannian orbifolds.
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Joseph Ernest BorzellinoOrbifolds of Maximal Diameter
http://digitalcommons.calpoly.edu/math_fac/110
http://digitalcommons.calpoly.edu/math_fac/110Mon, 08 Jul 2013 13:14:02 PDT
In this paper the Maximal Diameter Theorem of Riemannian geometry is proven for Riemannian orbifolds. In particular, it is shown that a complete Riemannian orbifold with Ricci curvature bounded below by (n−1) and diameter = π, must have constant sectional curvature 1, and must be a quotient of the sphere (Sn, can) of constant sectional curvature 1 by a subgroup of the orthogonal group O(n+1) acting discontinuously and isometrically on Sn. It is also shown that the singular locus of the orbifold forms a geometric barrier to the length minimization property of geodesics. We also extend the Bishop relative volume comparison theorem to Riemannian orbifolds.
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Joseph BorzellinoOn the existence of infinitely many closed geodesics on orbifolds of revolution
http://digitalcommons.calpoly.edu/math_fac/109
http://digitalcommons.calpoly.edu/math_fac/109Tue, 21 May 2013 10:17:49 PDT
Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing 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.
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Joseph Borzellino et al.A manifold structure for the group of orbifold diffeomorphisms of a smooth orbifold
http://digitalcommons.calpoly.edu/math_fac/108
http://digitalcommons.calpoly.edu/math_fac/108Tue, 21 May 2013 10:17:47 PDT
For a compact, smooth C^{r} orbifold (without boundary), we show that the topological structure of the orbifold diffeomorphism group is a Banach manifold for 1 ≤ r < ∞ and a Fréchet manifold if r = ∞. In each case, the local model is the separable Banach (Fréchet) space of C^{r} (C^{ ∞}, resp.) orbisections of the tangent orbibundle.
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Joseph E. Borzellino et al.Determination of the Topological Structure of an Orbifold by its Group of Orbifold Diffeomerphisms
http://digitalcommons.calpoly.edu/math_fac/107
http://digitalcommons.calpoly.edu/math_fac/107Tue, 21 May 2013 10:17:46 PDT
We show that the topological structure of a compact, locally smooth orbifold is determined by its orbifold diffeomorphism group. Let Diff^{r}_{Orb} (O) denote the C^{r} orbifold diffeomorphisms of an orbifold O. Suppose that Φ: Diff^{r}_{Orb}(O_{1}) → Diff^{r}_{Orb} (O_{2}) is a group isomorphism between the the orbifold diffeomorphism groups of two orbifolds O_{1} and O_{2}. We show that Φ is induced by a homeomorphism h: X_{O1} → X_{O}_{2} , whereX_{O}denotes the underlyingtopological space ofO. That is, Φ (ƒ) = h ƒh^{-1}for all ƒ ∈ Diff^{r}_{Orb}(O_{1}). Furthermore, ifr > 0, thenh is aC^{r}manifold diffeomorphism when restricted to the complement of the singular set of each stratum.
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Joseph E. Borzellino et al.Whose limit is it anyway?
http://digitalcommons.calpoly.edu/math_fac/106
http://digitalcommons.calpoly.edu/math_fac/106Tue, 21 May 2013 10:17:44 PDT
In a tongue-in-cheek manner, we investigate the notion of limit. We illustrate some of its shortcomings and show that addressing these shortcomings can often lead to unexpected consequences.
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Joseph E. BorzellinoClosed Geodesics on Orbifolds of Revolution
http://digitalcommons.calpoly.edu/math_fac/105
http://digitalcommons.calpoly.edu/math_fac/105Fri, 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 S^{2} 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.
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Joseph E. Borzellino et al.Elementary orbifold differential topology
http://digitalcommons.calpoly.edu/math_fac/104
http://digitalcommons.calpoly.edu/math_fac/104Tue, 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.
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Joseph E. Borzellino et al.When is a Trigonometric Polynomial Not a Trigonometric Polynomial?
http://digitalcommons.calpoly.edu/math_fac/103
http://digitalcommons.calpoly.edu/math_fac/103Tue, 26 Mar 2013 15:43:21 PDTJoseph E. Borzellino et al.Orbifold Homeomorphism and Diffeomorphism Groups
http://digitalcommons.calpoly.edu/math_fac/102
http://digitalcommons.calpoly.edu/math_fac/102Tue, 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
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Joseph E. Borzellino et al.Orbifolds with Lower Ricci Curvature Bounds
http://digitalcommons.calpoly.edu/math_fac/101
http://digitalcommons.calpoly.edu/math_fac/101Tue, 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 .
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Joseph E. BorzellinoThe Closed Geodesic Problem for Compact Riemannian 2-Orbifolds
http://digitalcommons.calpoly.edu/math_fac/100
http://digitalcommons.calpoly.edu/math_fac/100Tue, 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.
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Joseph E. Borzellino et al.The Splitting Theorem for Orbifolds
http://digitalcommons.calpoly.edu/math_fac/99
http://digitalcommons.calpoly.edu/math_fac/99Tue, 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.
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Joseph Borzellino et al.Pinching Theorems for Teardrops and Footballs of Revolution
http://digitalcommons.calpoly.edu/math_fac/98
http://digitalcommons.calpoly.edu/math_fac/98Tue, 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.
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Joseph E. BorzellinoTemporal Topos and U-Singularities
http://digitalcommons.calpoly.edu/math_fac/97
http://digitalcommons.calpoly.edu/math_fac/97Mon, 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.
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Goro C. KatoAsymptotic Functions as Kernels of the Schwartz Distributions
http://digitalcommons.calpoly.edu/math_fac/96
http://digitalcommons.calpoly.edu/math_fac/96Thu, 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.
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Todor D. TodorovAnother Proof of the Existence a Dedekind Complete Totally Ordered Field
http://digitalcommons.calpoly.edu/math_fac/95
http://digitalcommons.calpoly.edu/math_fac/95Mon, 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.
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James F. Hall et al.Completeness of the Leibniz Field and Rigorousness of Infinitesimal Calculus
http://digitalcommons.calpoly.edu/math_fac/94
http://digitalcommons.calpoly.edu/math_fac/94Mon, 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.
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James F. Hall et al.Lecture Notes: Non-Standard Approach to J.F. Colombeau’s Theory of Generalized Functions
http://digitalcommons.calpoly.edu/math_fac/93
http://digitalcommons.calpoly.edu/math_fac/93Mon, 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.
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Todor D. Todorov