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Joseph Fourier 250th Birthday. Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century
Frédéric Barbaresco and Jean-Pierre Gazeau
2019
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For the 250th birthday of Joseph Fourier, born in 1768 in Auxerre, France, this MDPI Special Issue will explore modern topics related to Fourier Analysis and Heat Equation. Modern developments of Fourier analysis during the 20th century have explored generalizations of Fourier and Fourier–Plancherel formula for non-commutative harmonic analysis, applied to locally-compact, non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups. One should add the developments, over the last 30 years, of the applications of harmonic analysis to the description of the fascinating world of aperiodic structures in condensed matter physics. The notions of model sets, introduced by Y. Meyer, and of almost periodic functions, have revealed themselves to be extremely fruitful in this domain of natural sciences. The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties. One last comment concerns the fundamental contributions of Fourier analysis to quantum physics: Quantum mechanics and quantum field theory. The content of this Special Issue will highlight papers exploring non-commutative Fourier harmonic analysis, spectral properties of aperiodic order, the hypoelliptic heat equation, and the relativistic heat equation in the context of Information Theory and Geometric Science of Information.
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Keywords
- affine group
- Born–Jordan quantization
- continuum thermodynamic systems
- covariant integral quantization
- cubature formulas
- discrete multivariate sine transforms
- discrete thermodynamic systems
- dynamical systems
- Fourier analysis
- fourier transform
- Guyer-Krumhansl equation
- harmonic analysis on abstract space
- heat equation on manifolds and Lie Groups
- heat pulse experiments
- higher order thermodynamics
- homogeneous manifold
- homogeneous spaces
- interconnection
- Irreversible processes
- Lévy processes
- Lie group machine learning
- Lie groups
- Lie groups thermodynamics
- metrics
- non-equilibrium processes
- non-equivariant cohomology
- non-Fourier heat conduction
- Nonequilibrium thermodynamics
- nonholonomic constraints
- Orthogonal polynomials
- Partial Differential equations
- poly-symplectic manifold
- pseudo-temperature
- quantum mechanics
- Reference, information & interdisciplinary subjects
- Research & information: general
- rigged Hilbert spaces
- rigid body motions
- short-time propagators
- Signal processing
- Souriau-Fisher metric
- special functions
- stochastic differential equations
- symplectization
- thermal expansion
- Thermodynamics
- time-slicing
- Van Vleck determinant
- variational formulation
- Weyl quantization
- Weyl-Heisenberg group
- Wigner function
Links
DOI: 10.3390/books978-3-03897-747-6Editions