Superanalysis

defined as elements of Grassmann algebra (an algebra with anticom­ muting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used. Later, durin...

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Main Author: Khrennikov, Andrei Y.
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Dordrecht Springer Netherlands 1999, 1999
Edition:1st ed. 1999
Series:Mathematics and Its Applications
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • I Analysis on a Superspace over Banach Superalge-bras
  • 1. Differential Calculus
  • 2. Cauchy-Riemann Conditions and the Condition of A-Linearity of Derivatives
  • 3. Integral Calculus
  • 4. Integration of Differential Forms of Commuting Variables
  • 5. Review of the Development of Superanalysis
  • 6. Unsolved Problems and Possible Generalizations
  • II Generalized Functions on a Superspace
  • 1. Locally Convex Superalgebras and Supermodules
  • 2. Analytic Generalized Functions on the Vladimirov-Volovich Superspace
  • 3. Fourier Transformation of Superanalytic Generalized Functions
  • 4. Superanalog of the Theory of Schwartz Distributions
  • 5. Theorem of Existence of a Fundamental Solution
  • 6. Unsolved Problems and Possible Generalizations
  • III Distribution Theory on an Infinite-Dimensional Superspace
  • 1. Polylinear Algebra over Commutative Supermodules
  • 2. Banach Supermodules
  • 3. Hilbert Supermodules
  • 4. Duality of Topological Supermodules
  • 6. Cauchy Problem for Partial Differential Equations with Variable Coefficients
  • 7. Non-Archimedean Supersymmetrical Quantum Mechanics
  • 8. Trotter Formula for non-Archimedean Banach Algebras
  • 9. Volkenborn Distribution on a non-Archimedean Super-space
  • 10. Infinite-Dimensional non-Archimedean Superanalysis
  • 11. Unsolved Problems and Possible Generalizations
  • VII Noncommutative Analysis
  • 1. Differential Calculus on a Superspace over a Noncommutative Banach Algebra
  • 2. Differential Calculus on Noncommutative Banach Algebras and Modules
  • 3. Generalized Functions of Noncommuting Variables
  • VIII Applications in Physics
  • 1. Quantization in Hilbert Supermodules
  • 2. Transition Amplitudes and Distributions on the Space of Schwinger Sources
  • References
  • 5. Differential Calculus on a Superspace over Topological Supermodules
  • 6. Analytic Distributions on a Superspace over Topological Supermodules
  • 7. Gaussian and Feynman Distributions
  • 8. Unsolved Problems and Possible Generalizations
  • IV Pseudodifferential Operators in Superanalysis
  • 1. Pseudodifferential Operators Calculus
  • 2. The Correspondence Principle
  • 3. The Feynman-Kac Formula for the Symbol of the Evolution Operator
  • 4. Unsolved Problems and Possible Generalizations
  • V Fundamentals of the Probability Theory on a Superspace
  • 1. Limit Theorems on a Superspace
  • 2. Random Processes on a Superspace
  • 3. Axiomatics of the Probability Theory over Superalgebras
  • 4. Unsolved Problems and Possible Generalizations
  • VI Non-Archimedean Superanalysis
  • 1. Differentiale and Analytic Functions
  • 2. Generalized Functions
  • 3. Laplace Transformation
  • 4. Gaussian Distributions
  • 5. Duhamel non-Archimedean Integral. Chronological Exponent