scholarly journals Graded manifold theory as the geometry of supersymmetry

1979 ◽  
Vol 66 (3) ◽  
pp. 197-221 ◽  
Author(s):  
John Dell ◽  
Lee Smolin
Keyword(s):  
Graded Manifold ◽  
Manifold Theory

scholarly journals Theory of Connections on Graded Principal Bundles

1998 ◽  
Vol 10 (01) ◽  
pp. 47-79 ◽  
Author(s):  
T. Stavracou
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Lie Superalgebra ◽  
Principal Bundles ◽  
Graded Manifold ◽  
Definition Of ◽  
Manifold Theory ◽  
Free Actions

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant–Berezin–Leites. We first review the basic elements of this theory establishing at the same time supplementary properties of graded Lie groups and their actions. Particular emphasis is given in introducing and studying free actions in the graded context. Next, we investigate the geometry of graded principal bundles; we prove that they have several properties analogous to those of ordinary principal bundles. In particular, we show that the sheaf of vertical derivations on a graded principal bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited. Finally, we define the curvature for the graded connection and we prove that the curvature controls the involutivity of the horizontal graded distribution corresponding to the graded connection.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

The Shape of Time

2018 ◽  
Author(s):  
Donald C. Williams
Keyword(s):  
Time Travel ◽  
Arrow Of Time ◽  
The Past ◽  
Time Traveler ◽  
Dimensional Shape ◽  
Manifold Theory ◽  
The Universe

This chapter is about the arrow or direction of time against the backdrop of the pure manifold theory. It is accepted that the fact that time has a direction ought to be explained. It is proposed that the arrow of time is grounded in deeper facts about the four-dimensional nature of each object in the manifold and in facts about the overall four-dimensional shape of the universe. Towards the end of the chapter the possibility of time travel is discussed. It is argued that time travel is metaphysically possible and that there is a reasonable and intelligible sense in which a time traveler can and cannot change the past, according to the pure manifold theory.

The Nature of Time

2018 ◽  
Author(s):  
Donald C. Williams
Keyword(s):  
Large Scale ◽  
Time Travel ◽  
Dimensional Manifold ◽  
Nature Of Time ◽  
Manifold Theory ◽  
The Universe ◽  
Theoretical Cost ◽  
Do So

This chapter provides a fuller treatment of the pure manifold theory with an expanded discussion of competing doctrines. It is argued that competing doctrines fail to account for the extensive and/or transitory aspect(s) of time, or they do so at great theoretical cost. The pure manifold theory accounts for the extensive aspect of time because it admits a four-dimensional manifold and it accounts for the transitory aspect of time because it hypothesizes that the increase of entropy is the thing that is ‘felt’ in veridical cases of felt passage. A four-dimensionalist theory of time travel is outlined, along with a sketch of large-scale cosmological traits of the universe.

The Sea Fight Tomorrow

2018 ◽  
Author(s):  
Donald C. Williams
Keyword(s):  
Three Dimensions ◽  
The Past ◽  
The World ◽  
The Future ◽  
Metaphysics Of Time ◽  
Manifold Theory ◽  
Dimensions Of Space

This chapter is the first of this book to deal specifically with the metaphysics of time. This chapter defends the pure manifold theory of time. On this view, time is just another dimension of extent like the three dimensions of space, the past, present, and future are equally real, and the world is at bottom tenseless. What is true is eternally true. For example, it is now true that there will be a sea fight tomorrow or that there will not be a sea fight tomorrow. It is argued that the pure manifold theory does not entail fatalism and that contingent statements about the future do not imply that only the past and present exist.

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

MODELING THE IMPACT OF VOLUNTARY TESTING AND TREATMENT ON TUBERCULOSIS TRANSMISSION DYNAMICS

2012 ◽  
Vol 05 (04) ◽  
pp. 1250029 ◽  
Author(s):  
S. MUSHAYABASA ◽  
C. P. BHUNU
Keyword(s):  
Deterministic Model ◽  
Reproduction Number ◽  
Positive Impact ◽  
Transmission Dynamics ◽  
Effective Control ◽  
Reproductive Number ◽  
Voluntary Testing ◽  
Manifold Theory ◽  
The Impact ◽  
Disease Free

A deterministic model for evaluating the impact of voluntary testing and treatment on the transmission dynamics of tuberculosis is formulated and analyzed. The epidemiological threshold, known as the reproduction number is derived and qualitatively used to investigate the existence and stability of the associated equilibrium of the model system. The disease-free equilibrium is shown to be locally-asymptotically stable when the reproductive number is less than unity, and unstable if this threshold parameter exceeds unity. It is shown, using the Centre Manifold theory, that the model undergoes the phenomenon of backward bifurcation where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. The analysis of the reproduction number suggests that voluntary tuberculosis testing and treatment may lead to effective control of tuberculosis. Furthermore, numerical simulations support the fact that an increase voluntary tuberculosis testing and treatment have a positive impact in controlling the spread of tuberculosis in the community.

scholarly journals Statistics on the Stiefel manifold: Theory and applications

2019 ◽  
Vol 47 (1) ◽  
pp. 415-438 ◽  
Author(s):  
Rudrasis Chakraborty ◽  
Baba C. Vemuri
Keyword(s):  
Stiefel Manifold ◽  
Manifold Theory

Application of Center Manifold Theory to Regulation of a Flexible Beam

1997 ◽  
Vol 119 (2) ◽  
pp. 158-165 ◽  
Author(s):  
Amir Khajepour ◽  
M. Farid Golnaraghi ◽  
Kirsten A. Morris
Keyword(s):  
Control Strategy ◽  
Center Manifold ◽  
Flexible Beam ◽  
Trial And Error ◽  
Center Manifold Theory ◽  
Control Techniques ◽  
Lumped Parameter ◽  
The Stability ◽  
Frequency Relationship ◽  
Manifold Theory

In this paper we consider the problem of regulation of a flexible lumped parameter beam. The controller is an active/passive mass-spring-dashpot mechanism which is free to slide along the beam. In this problem the plant/controller equations are coupled and nonlinear, and the linearized equations of the system have two uncontrollable modes associated with a pair of pure imaginary eigenvalues. As a result, linear control techniques as well as most conventional nonlinear control techniques can not be applied. In earlier studies Golnaraghi (1991) and Golnaraghi et al. (1994) a control strategy based on Internal resonance was developed to transfer the oscillatory energy from the beam to the slider, where it was dissipated through controller damping. Although these studies provided very good understanding of the control strategy, the analytical method was based on perturbation techniques and had many limitations. Most of the work was based on numerical techniques and trial and error. In this paper we use center manifold theory to address the shortcomings of the previous studies, and extend the work to a more general control law. The technique is based on reducing the dimension of system and simplifying the nonlinearities using center manifold and normal forms techniques, respectively. The simplified equations are used to investigate the stability and to develop a relation for the optimal controller/plant natural frequencies at which the maximum transfer of energy occurs. One of the main contributions of this work is the elimination of the trial and error and inclusion of damping in the optimal frequency relationship.

Travelling fronts in a food-limited population model with time delay

2002 ◽  
Vol 132 (1) ◽  
pp. 75-89 ◽  
Author(s):  
S. A. Gourley ◽  
M. A. J. Chaplain
Keyword(s):  
Population Model ◽  
Time Delays ◽  
Front Solutions ◽  
Heteroclinic Connection ◽  
Non Local ◽  
Spatial Terms ◽  
Travelling Fronts ◽  
Manifold Theory ◽  
And Diffusion ◽  
Existence Question

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.

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