Invariant differential equations and the Adler-Gel'fand-Dikii bracket

In this paper we find an explicit formula for the most general vector evolution of curves on RP^{n-1} invariant under the projective action of SL(n,R). When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that this evolution is identical to the second KdV Hamiltonian evolution under appropriate conditions. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of SL(n,R), namely, the SL(n,R) invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudo-differential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary n

Differential invariant signatures and flows in computer vision : a symmetry group approach

Includes bibliographical references (p. 40-44).Supported by the National Science Foundation. DMS-9204192 DMS-8811084 ECS-9122106 Supported by the Air Force Office of Scientific Research. AFOSR-90-0024 Supported by the Rothschild Foundation-Yad Hanadiv and by Image Evolutions, Ltd.Peter J. Olver, Guillermo Sapiro, Allen Tannenbaum

Field theoretic formulation and empirical tracking of spatial processes

Spatial processes are attacked on two fronts. On the one hand, tools from theoretical and statistical physics can be used to understand behaviour in complex, spatially-extended multi-body systems. On the other hand, computer vision and statistical analysis can be used to study 4D microscopy data to observe and understand real spatial processes in vivo. On the rst of these fronts, analytical models are developed for abstract processes, which can be simulated on graphs and lattices before considering real-world applications in elds such as biology, epidemiology or ecology. In the eld theoretic formulation of spatial processes, techniques originating in quantum eld theory such as canonical quantisation and the renormalization group are applied to reaction-di usion processes by analogy. These techniques are combined in the study of critical phenomena or critical dynamics. At this level, one is often interested in the scaling behaviour; how the correlation functions scale for di erent dimensions in geometric space. This can lead to a better understanding of how macroscopic patterns relate to microscopic interactions. In this vein, the trace of a branching random walk on various graphs is studied. In the thesis, a distinctly abstract approach is emphasised in order to support an algorithmic approach to parts of the formalism. A model of self-organised criticality, the Abelian sandpile model, is also considered. By exploiting a bijection between recurrent con gurations and spanning trees, an e cient Monte Carlo algorithm is developed to simulate sandpile processes on large lattices. On the second front, two case studies are considered; migratory patterns of leukaemia cells and mitotic events in Arabidopsis roots. In the rst case, tools from statistical physics are used to study the spatial dynamics of di erent leukaemia cell lineages before and after a treatment. One key result is that we can discriminate between migratory patterns in response to treatment, classifying cell motility in terms of sup/super/di usive regimes. For the second case study, a novel algorithm is developed to processes a 4D light-sheet microscopy dataset. The combination of transient uorescent markers and a poorly localised specimen in the eld of view leads to a challenging tracking problem. A fuzzy registration-tracking algorithm is developed to track mitotic events so as to understand their spatiotemporal dynamics under normal conditions and after tissue damage.Open Acces

Traditional logic and the early history of sets, 1854–1908

Ministerio de Educación y Ciencia (España

Poincaré and the Three Body Problem

The purpose of the thesis is to present an account of Henri Poincare's famous memoir on the three body problem, the final version of which was published in Acta Mathematica in 1890 as the prize-winning entry in King Oscar II's 60th birthday competition. The memoir is reknowned both for its role in providing the foundations for Poincare's celebrated three volume Méthodes Nouvelles de la Mécanique Céleste, and for containing the first mathematical description of chaotic behaviour in a dynamical system. A historical context is provided both through consideration of the problem itself and through a discussion of Poincaré's earlier work which relates to the mathematics developed in the memoir. The organisation of the Oscar competition, which was undertaken by Gösta Mittag-Leffler, is also described. This not only provides an insight into the late 19th century European mathematical community but also reveals that after the prize had been awarded Poincare found an important error in his work and substantially revised the memoir prior to its publication in Acta. The discovery of a printed version of the original memoir personally annotated by Poincaré has allowed for a detailed comparative study of the mathematics contained in both versions of the memoir. The error is explained and it is shown that it was only as a result of its correction that Poincaré discovered the chaotic behaviour now associated with the memoir. The contemporary reception of the memoir is discussed and Poincaré's subsequent work in celestial mechanics and related topics is examined. Through the consideration of sources up to 1920 the influence and impact of the memoir on the progress of the three body problem and on dynamics in general is assessed

Spaces of Appearance: Writings on Contemporary Theatre and Performance

This thesis, a collection of previously published materials, reflects a plural and evolving engagement with theatre and performance over the past fifteen years or so: as researcher, writer, editor, teacher, practitioner, spectator. These have rarely been discreet categories for me, but rather different modalities of exploration and enquiry, interrelated spaces encouraging dynamic connectivities, flows and further questions. Section 1 offers critical accounts of the practices of four contemporary theatre directors: Jerzy Grotowski, Robert Wilson, Peter Brook and Ariane Mnouchkine. Section 2 draws on elements of contemporary philosophy and critical thinking to explore the mutable parameters of performance. lt proposes performative mappings of certain unpredictable, energetic events 'in proximity of performance', to borrow Matthew Goulish's phrase: contact, fire, animals, alterity, place. Section 3 contains examples of documentation of performance practices, including a thick description of a mise en scene of a major international theatre production, reflections on process, training and dramaturgy, a performance text with a framing dramaturgical statement, and personal perspectives on particular collaborations. The external Appendix comprises a recently published collection of edited and translated materials concerning five core collaborative projects realised by Ariane Mnouchkine and the Theatre du Soleil at their base in the Cartoucherie de Vincennes, Paris. The core concerns of this thesis include attempts to think through: • the working regimes, poetics and pedagogies of certain directors, usually in collaborative devising contexts within which the creative agency of performers is privileged; • the processes and micro-politics of collaboration, devising, and dramaturgical composition; the dramaturgical implications of trainings, narrative structures, spaces, mise en scene, and of images as multi-layered, dynamic 'fields'; • the predicament and agency of spectators in diverse performance contexts, and the ways in which spectatorial roles are posited or constructed by dramaturgies; • the imbrication of embodiment, movement and perception in performance, and the plurality of modes of perception; • the critical and political functions of theatre and theatre criticism as cultural/social practice and 'art of memory' (de Certeau), of dramaturgies as critical historiographies, and of theatre cultures (and identities) as plural, dynamic, and contested; • performance as concentrated space for inter-subjectivity and the flaring into appearance of the 'face-to-face' (Levinas); the possibility of ethical, 'response-able' encounter and exchange with another; identity as relational and in-process, alterity as productive event, the inter-personal as political; • the poetics and politics of what seems an unthinkable surplus (and constitutive 'outside') to the cognitive reach of many conventional frames and maps in theatre criticism and historiography; an exploration of acts of writing as performative propositions and provocations ('critical fictions') to think the event of meanings at/of the limits of knowledge and subjectivity. This partial listing of recurrent and evolving concerns within the thesis traces a trajectory in my evolution as a writer and thinker, a gradual displacement from the relatively 'solid ground' of theatre studies and theatre history towards more fluid and tentative articulations of the shifting 'lie of the land' in contemporary performance and philosophy. This trajectory reflects a growing fascination with present process, conditions, practices, perceptions 'in the middle', and ways of writing (about) performance as interactive and ephemeral event

Lie Algebras In Particle Physics

In this book, the author convinces that Sir Arthur Stanley Eddington had things a little bit wrong, as least as far as physics is concerned. He explores the theory of groups and Lie algebras and their representations to use group representations as labor-saving tools

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