February 26, 2017

Numerical and Computational Challenges in Science and Engineering Program

June 21, 10:00 am

Gregory Litvinov, International Sophus Lie Centre

Dequantization of Mathematics: An Introduction to Idempotent Mathematics
G.L. Litvinov and V.P. Maslov

Idempotent Mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking pure imaginary values. There exists a correspondence between important, interesting, and useful constructions and results over the fields of real and complex numbers and similar constructions and results over semirings with idempotent addition (this means that x+x = x) in the spirit of N. Bohr's correspondence principle in Quantum Mechanics. A systematic and consistent application of the "idempotent" correspondence principle leads to a variety of results, often quite unexpected. For instance, the well-known Legendre transform is nothing more that an idempotent version of the traditional Fourier transform. The Hamilton- Jacobi equation (which is the basic equation in Classical Mechanics) is an idempotent version of the Schroedinger equation. The least action principle in Classical Mechanics can be considered as an idempotent version of the R. Feynman's approach to Quantum Mechanics via path integrals. Some problems that are nonlinear in the traditional sense turn out to be linear over suitable idempotent semirings (the idempotent superposition principle). For example, the Hamilton-Jacobi equation and different versions of the Bellman equation (which is the basic equation in the optimization theory) are linear over suitable idempotent semirings.

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