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The Theory of Finslerian Laplacians and Applications

  • Book
  • © 1998

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Editors:
  1. Peter L. Antonelli

    1. Department of Mathematical Sciences, University of Alberta, Edmonton, Canada

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    You can also search for this editor in PubMed   Google Scholar

  2. Bradley C. Lackey

    1. Department of Mathematical Sciences, University of Alberta, Edmonton, Canada

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    You can also search for this editor in PubMed   Google Scholar

Part of the book series: Mathematics and Its Applications (MAIA, volume 459)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xxx
    Download chapter PDF

  2. Finsler Laplacians in Application

    1. Introduction to Diffusion on Finsler Manifolds

      • P. L. Antonelli, T. J. Zastawniak
      Pages 1-12

    2. Density Dependent Host/Parasite Systems of Rothschild Type and Finslerian Diffusion

      • P. L. Antonelli, T. J. Zastawniak
      Pages 13-31

    3. Stochastic Finsler Geometry in the Theory of Evolution by Symbiosis

      • P. L. Antonelli, T. J. Zastawniak
      Pages 33-46

  3. Stochastic Analysis and Brownian Motion

    1. Diffusions on Finsler Manifolds

      • P. L. Antonelli, T. J. Zastawniak
      Pages 47-62

    2. Stochastic Calculus on Finsler Manifolds and an Application in Biology

      • P. L. Antonelli, T. J. Zastawniak
      Pages 63-88

    3. Diffusion on the Tangent and Indicatrix Bundles of a Finsler Manifold

      • P. L. Antonelli, T. J. Zastawniak
      Pages 89-110

  4. Stochastic Lagrange Geometry

    1. Diffusion on the Total Space of a Vector Bundle

      • DragoÅŸ Hrimiuc
      Pages 111-121

    2. Diffusions and Laplacians on Lagrange Manifolds

      • P. L. Antonelli, D. Hrimiuc
      Pages 123-131

    3. Ï•-Lagrange Laplacians

      • P. L. Antonelli, D. Hrimiuc
      Pages 133-139

  5. Mean-Value Properties of Harmonic Functions

    1. Diffusion, Laplacian and Hodge Decomposition on Finsler Spaces

      • P. L. Antonelli, T. J. Zastawniak
      Pages 141-149

    2. A Mean-Value Laplacian For Finsler Spaces

      • Paul Centore
      Pages 151-186

  6. Analytical Constructions

    1. The Non-Linear Laplacian for Finsler Manifolds

      • Zhongmin Shen
      Pages 187-198

    2. A Bochner Vanishing Theorem for Elliptic Complices

      • Brad Lackey
      Pages 199-226

    3. A Lichnerowicz Vanishing Theorem for Finsler Spaces

      • Brad Lackey
      Pages 227-243

    4. A Geometric Inequality and a Weitzenböck Formula for Finsler Surfaces

      • David Bao, Brad Lackey
      Pages 245-275

    5. Spinors on Finsler Spaces

      • Francis J. Flaherty
      Pages 277-282
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Keywords

About this book

Finslerian Laplacians have arisen from the demands of modelling the modern world. However, the roots of the Laplacian concept can be traced back to the sixteenth century. Its phylogeny and history are presented in the Prologue of this volume.
The text proper begins with a brief introduction to stochastically derived Finslerian Laplacians, facilitated by applications in ecology, epidemiology and evolutionary biology. The mathematical ideas are then fully presented in section II, with generalizations to Lagrange geometry following in section III. With section IV, the focus abruptly shifts to the local mean-value approach to Finslerian Laplacians and a Hodge-de Rham theory is developed for the representation on real cohomology classes by harmonic forms on the base manifold. Similar results are proved in sections II and IV, each from different perspectives.
Modern topics treated include nonlinear Laplacians, Bochner and Lichnerowicz vanishing theorems, Weitzenböck formulas, and Finslerian spinors and Dirac operators. The tools developed in this book will find uses in several areas of physics and engineering, but especially in the mechanics of inhomogeneous media, e.g. Cofferat continua.
Audience: This text will be of use to workers in stochastic processes, differential geometry, nonlinear analysis, epidemiology, ecology and evolution, as well as physics of the solid state and continua.

Editors and Affiliations

  • Department of Mathematical Sciences, University of Alberta, Edmonton, Canada

    Peter L. Antonelli, Bradley C. Lackey

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