In this paper, we present how to use an interval arithmetics framework based on free algebra construction, in order to build better defined inclusion function for interval semi-group and for its associated vector space. One introduces the ψ-algorithm, which performs set inversion of functions and exhibits some numerical examples developped with the python programming langage.
ABSTRACT. One develops a new mathematical tool, the complex (min, +)-analysis which permits to define a new variational calculus analogous to the classical one (Euler-Lagrange and Hamilton Jacobi equations), but which is well-suited for functions defined from C n to C. We apply this complex variational calculus to Born-Infeld theory of electromagnetism and show why it does not exhibit nonlinear effects.
One shows that the Feynman's Path Integral designed for quantum mechanics has an analogous in classical mechanics, the so-called (min, +) Path Integral. This former is build on (min, +)-algebra and (min, +)-analysis which permit to handle in a linear way non-linear problems occurring in mathematical physics. The Hamilton-Jacobi equations and their solutions within this mathematical framework, are introduced and yield to a new interpretation expressed in a duality between action field and particle.
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