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'Et moi, ... , si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ... '; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
Sommario
1. Functions and Measures in Linear Spaces.- 1. Spaces, Mappings, Differential Operations.- 2. Measures and Integrals.- 3. Measure Differentiation.- 2. Functions and Measures on Smooth Manifolds.- 1. Smooth Manifolds and Vector Bundles.- 2. Bundle Section, Connections, Differential Operations.- 3. Hilbert and Hilbert-Schmidt Bundles.- 4. Measures on Smooth Manifolds.- 3. Stochastic Equations in Banach Spaces.- 1. Basic Notions.- 2. Stochastic Integrals for Vector and Operator Functions.- 3. Stochastic Equations.- 4. Multiplicative Functionals of Stochastic Processes.- 5. Stochastic Flow.- 4. Stochastic Equations on Smooth Manifolds.- 1. Stochastic Differentials.- 2. Stochastic Differential Equations on Manifolds.- 3. Stochastic Equations in Vector Bundles.- 5. Kolmogorov Equations.- 1. Backward Kolmogorov Equations.- 2. Quasilinear Parabolic Equations.- 3. Forward Kolmogorov Equations.- 6. Diffusion Processes on Lie Groups and Principal Fibre Bundles.- 1. Lie Groups and Smooth Bundles.- 2. Invariant Stochastic Equations.- 3. Stochastic Equations on Manifolds and their Solution Distribution Properties.- Historical Comments.- References.
