A stability theorem for minimal foliations on a torus

This paper is concerned with minimal foliations; these are foliations whose leaves are extremals of a prescribed variational problem, as for example foliations consisting of minimal surfaces. Such a minimal foliation is called stable if for any small perturbation of the variational problem there exists a minimal foliation conjugate under a smooth diffeomorphism to the original foliation. In this paper the stability of special foliations of codimension 1 on a higher-dimensional torus is established. This result requires small divisor assumptions similar to those encountered in dynamical systems. This theorem can be viewed as a generalization of the perturbation theory of invariant tori for Hamiltonian systems to elliptic partial differential equations for which one obtains quasi-periodic solution

Computer-assisted proofs in geometry and physics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references.In this dissertation we apply computer-assisted proof techniques to two problems, one in discrete geometry and one in celestial mechanics. Our main tool is an effective inverse function theorem which shows that, in favorable conditions, the existence of an approximate solution to a system of equations implies the existence of an exact solution nearby. This allows us to leverage approximate computational techniques for finding solutions into rigorous computational techniques for proving the existence of solutions. Our first application is to tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence of many hitherto unknown tight regular simplices in quaternionic projective spaces and in the octonionic projective plane. We also consider regular simplices in real Grassmannians. The second application is to gravitational choreographies, i.e., periodic trajectories of point particles under Newtonian gravity such that all of the particles follow the same curve. Many numerical examples of choreographies, but few existence proofs, were previously known. We present a method for computer-assisted proof of existence and demonstrate its effectiveness by applying it to a wide-ranging set of choreographies.by Gregory T. Minton.Ph.D

Geometric Flows of Diffeomorphisms

The idea of this thesis is to apply the methodology of geometric heat flows to the study of spaces of diffeomorphisms. We start by describing the general form that a geometrically natural flow must take and the implications this has for the evolution equations of associated geometric quantities. We discuss the difficulties involved in finding appropriate flows for the general case, and quickly restrict ourselves to the case of surfaces. In particular the main result is a global existence, regularity and convergence result for a geometrically defined quasilinear flow of maps u between flat surfaces, producing a strong deformation retract of the space of diffeomorphisms onto a finite-dimensional submanifold. Partial extensions of this result are then presented in several directions. For general Riemannian surfaces we obtain a full local regularity estimate under the hypothesis of bounds above and below on the singular values of the first derivative. We achieve these gradient bounds in the flat case using a tensor maximum principle, but in general the terms contributed by curvature are not easy to control. We also study an initial-boundary-value problem for which we can attain the necessary gradient bounds using barriers, but the delicate nature of the higher regularity estimate is not well-adapted for obtaining uniform estimates up to the boundary. To conclude, we show how appropriate use of the maximum principle can provide a proof of well-posedness in the smooth category under the assumption of estimates for all derivatives

Foundations of Mechanics, Second Edition

Preface to the Second Edition. Since the first edition of this book appeared in 1967, there has been a great deal of activity in the field of symplectic geometry and Hamiltonian systems. In addition to the recent textbooks of Arnold, Arnold-Avez, Godbillon, Guillemin-Sternberg, Siegel-Moser, and Souriau, there have been many research articles published. Two good collections are "Symposia Mathematica," vol. XIV, and "Géométrie Symplectique el Physique Mathématique," CNRS, Colloque Internationaux, no. 237. There are also important survey articles, such as Weinstein [1977b]. The text and bibliography contain many of the important new references we are aware of. We have continued to find the classic works, especially Whittaker [1959], invaluable. The basic audience for the book remains the same: mathematicians, physicists, and engineers interested in geometrical methods in mechanics, assuming a background in calculus, linear algebra, some classical analysis, and point set topology. We include most of the basic results in manifold theory, as well as some key facts from point set topology and Lie group theory. Other things used without proof are clearly noted. We have updated the material on symmetry groups and qualitative theory, added new sections on the rigid body, topology and mechanics, and quantization, and other topics, and have made numerous corrections and additions. In fact, some of the results in this edition are new. We have made two major changes in notation: we now use f^* for pull-back (the first edition used f[sub]*), in accordance with standard usage, and have adopted the "Bourbaki" convention for wedge product. The latter eliminates many annoying factors of 2. A. N. Kolmogorov's address at the 1954 International Congress of Mathematicians marked an important historical point in the development of the theory, and is reproduced as an appendix. The work of Kolmogorov, Arnold, and Moser and its application to Laplace's question of stability of the solar system remains one of the goals of the exposition. For complete details of all tbe theorems needed in this direction, outside references will have to be consulted, such as Siegel-Moser [1971] and Moser [1973a]. We are pleased to acknowledge valuable assistance from Paul Chernoff, Wlodek Tulczyjew, Morris Hirsh, Alan Weinstein, and our invaluable assistant authors, Richard Cushman and Tudor Ratiu, who all contributed some of their original material for incorporation into the text. Also, we are grateful to Ethan Akin, Kentaro Mikami, Judy Arms, Harold Naparst, Michael Buchner, Ed Nelson, Robert Cahn, Sheldon Newhouse, Emil Chorosoff, George Oster, André Deprit, Jean-Paul Penot, Bob Devaney, Joel Robbin, Hans Duistermaat, Clark Robinson, John Guckenheimer, David Rod, Martin Gutzwiller, William Satzer, Richard Hansen, Dieter Schmidt, Morris Kirsch, Mike Shub, Michael Hoffman, Steve Smale, Andrei Iacob, Rich Spencer, Robert Jantzen, Mike Spivak, Therese Langer, Dan Sunday, Ken Meyer, Floris Takens, [and] Randy Wohl for contributions, remarks, and corrections which we have included in this edition. Further, we express our gratitude to Chris Shaw, who made exceptional efforts to transfom our sketches into the graphics which illustrate the text, to Peter Coha for his assistance in organizing the Museum and Bibliography, and to Ruthie Cephas, Jody Hilbun, Marnie McElhiney, Ruth (Bionic Fingers) Suzuki, and Ikuko Workman for their superb typing job. Theoretical mechanics is an ever-expanding subject. We will appreciate comments from readers regarding new results and shortcomings in this edition. RALPH ABRAHAM, JERROLD E. MARSDEN</p

Preliminary Study on Deterministic Chaotic System From Physical Approach

Chaotic phenomenon widely exists in the nonlinear dynamic systems. Such phenomenon is described to be highly related the initial conditions and intrinsic system properties. Traditionally, large effort has been put on evaluation of the existence of chaos, such as Lyapunov exponent or power spectrum methods. In addition, iteration methods are usually employed to explore the states of system. In this work, we will study the chaotic system in a different point of view. The system properties of physics quantities will be considered firstly. Starting from these physics quantities, it is expected to potentially determine the final states of the system. Then a relationship between the initial conditions and final states will be established. The objective of this work is to preliminary study the feasibility of employing such method for predetermine the final state of reappear initial conditions. The main approach is building a strong relationship between each initial conditions and final states within limited computation. Once the relationship is established, we could know the system states distribution and achieve the exploitation on the system. In this work, we will focus on the deterministic chaotic systems that the final states are deterministic and non-periodic. In order to perform the proof-of-concept, a typical chaotic system, the magnetic pendulum system, will be employed for demonstration. A program written in Java will display the pendulum movement in real-time and also the basin diagram that shows the relationship between initial conditions and final states. All of code for this program are constructed from scratch. After the implementation of our program, we will compare results from it and that of the separate numerical simulation of the same system. The numerical simulation will be performed through the commercially available software Mathematica. Further investigation will also be discussed with the results from numerical simulation. Three conclusions could draw from this work. The first is a proof-of-concept has been established for the opportunity to develop a new approach to study the deterministic chaotic system based on the physical properties methods. The second is the relationship between the initial condition and final states are proved to be work in states predetermination. The third is that relationship between the intrinsic system parameter and the final system states distributions has been found, which could be a guiding line for the similar deterministic chaotic system study in the future. It is also worthy to mention that herein the magnetic pendulum is only an example to testify our approach. And this approach should also be valid for other general deterministic chaotic systems. In this work, we only perform the preliminary study on this approach. In the future, our study will be extended to more systems and even the system in real world. It is expected such approach would build a new viewpoint on understanding the chaotic system, and a potentially new method to understand data

Characterisation, emulation and by-emitter degradation analysis of high power semiconductor laser diodes

The characterisation, emulation and by-emitter degradation analysis of two types of high power semiconductor laser diodes are presented in this thesis as part of an European Union (EU) project. An attempt is made using an accurate laser simulator called Speclase to learn more about the degradation of high power semiconductor laser diodes. Speclase being a single emitter simulation tool was transformed to model a bar i.e. multiple emitters, which we have named Barlase, through an external control interface written in Labview. The concept of Barlase was based on the fact that a bar is a monolithic block of multiple emitters connected in parallel with each other with a common voltage connected across them. This tool is capable of performing simulation in different modes of operation (i.e. constant current or constant power). The tool is designed to examine and emulate the degradation processes at both the laser bar and individual emitter levels of operation. It is known that, emitter degradation is faster for emitters within a bar than for identical single emitters due to a combination of packaging-induced strain and current competition between emitters amongst others. This tool shows clear evidence of the benefits of using by-emitter degradation analysis for gaining detailed understanding of individual emitters operating in a bar and for determining bar degradation mechanisms. The tool complement to the by-emitter analysis, allowing the effects of certain factors that affect the degradation of laser bars to be investigated. Various intervention measures were taken to improve upon the results of the emulation such as modifying the trap density through local heating and the use of the global thermal solver. The modification of the trap density allowed the acceptance of a spatially variable local trap density distribution that gave a more realistic and accurate simulation of the degradation behaviour. The introduction of the global thermal solver allowed the modelling of thermal cross-talk communication between the emitters, which brings about the frown shaped current/power profiles for the unaged bars (though not as pronounced as in the experiment). An attempt was made to employ this tool in the emulation of experimentally observed degradation behaviour in a 975 nm, 16 emitter infrared tapered laser bar with each group of 4 mini-array emitters. The laser bar was first calibrated to achieve a reasonable agreement between the experimental P-I curves of unaged emitters assuming identical emitters with the simulated P-I curves. The simulated P-I curve was then used to perform simulations to emulate the degradation of the laser. The simulated output power profile did not correspond well with the experimental power profile, but a good agreement was realised between the combined output powers of the bar. Better correlation was observed between the experimental and the simulated temperature profiles. This was expected since the experimental temperature was set as input for the heatsink temperature profile. This agreement therefore must not be over-emphasised. The bar emulation model was enhanced by including a global thermal solver to model the thermal crosstalk between emitters. Emulations using this model showed a clearly defined frown shaped profile in the output current and power profiles but the change was minimal. As the emulation of laser bar degradation has not been attempted before, this work is still at a very early stage. Therefore, further work is needed to achieve better agreement in the output current/power profiles and to better the model

Characterisation, emulation and by-emitter degradation analysis of high power semiconductor laser diodes

The characterisation, emulation and by-emitter degradation analysis of two types of high power semiconductor laser diodes are presented in this thesis as part of an European Union (EU) project. An attempt is made using an accurate laser simulator called Speclase to learn more about the degradation of high power semiconductor laser diodes. Speclase being a single emitter simulation tool was transformed to model a bar i.e. multiple emitters, which we have named Barlase, through an external control interface written in Labview. The concept of Barlase was based on the fact that a bar is a monolithic block of multiple emitters connected in parallel with each other with a common voltage connected across them. This tool is capable of performing simulation in different modes of operation (i.e. constant current or constant power). The tool is designed to examine and emulate the degradation processes at both the laser bar and individual emitter levels of operation. It is known that, emitter degradation is faster for emitters within a bar than for identical single emitters due to a combination of packaging-induced strain and current competition between emitters amongst others. This tool shows clear evidence of the benefits of using by-emitter degradation analysis for gaining detailed understanding of individual emitters operating in a bar and for determining bar degradation mechanisms. The tool complement to the by-emitter analysis, allowing the effects of certain factors that affect the degradation of laser bars to be investigated. Various intervention measures were taken to improve upon the results of the emulation such as modifying the trap density through local heating and the use of the global thermal solver. The modification of the trap density allowed the acceptance of a spatially variable local trap density distribution that gave a more realistic and accurate simulation of the degradation behaviour. The introduction of the global thermal solver allowed the modelling of thermal cross-talk communication between the emitters, which brings about the frown shaped current/power profiles for the unaged bars (though not as pronounced as in the experiment). An attempt was made to employ this tool in the emulation of experimentally observed degradation behaviour in a 975 nm, 16 emitter infrared tapered laser bar with each group of 4 mini-array emitters. The laser bar was first calibrated to achieve a reasonable agreement between the experimental P-I curves of unaged emitters assuming identical emitters with the simulated P-I curves. The simulated P-I curve was then used to perform simulations to emulate the degradation of the laser. The simulated output power profile did not correspond well with the experimental power profile, but a good agreement was realised between the combined output powers of the bar. Better correlation was observed between the experimental and the simulated temperature profiles. This was expected since the experimental temperature was set as input for the heatsink temperature profile. This agreement therefore must not be over-emphasised. The bar emulation model was enhanced by including a global thermal solver to model the thermal crosstalk between emitters. Emulations using this model showed a clearly defined frown shaped profile in the output current and power profiles but the change was minimal. As the emulation of laser bar degradation has not been attempted before, this work is still at a very early stage. Therefore, further work is needed to achieve better agreement in the output current/power profiles and to better the model