Aldrovandi R. Mathematical Physics and Matrix Representations...2025

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Aldrovandi R. Mathematical Physics and Matrix Representations...2025

Textbook in PDF format This book expounds three kinds of matrices that are of physical interest, centering on physical examples. Stochastic matrices describe dynamical systems of many different types, involving (or not) phenomena like transience, dissipation, ergodicity, nonequilibrium and hypersensitivity to initial conditions. The main characteristic is growth by agglomeration, as in glass formation. Circulants are the building blocks of elementary Fourier analysis and provide a natural gateway quantum mechanics, noncommutative geometry and random walks, including some phenomenology like diffusion-advection equation and prey-predator chains. We also present two applications: chemical reaction and genetics. The last subject may seem specially out of place in "Mathematical Physics". Our excuse is that Mendelism and blood types are here presented by using just the same methods of the other chapters. Bell polynomials offer closed expressions for many formulas concerning Lie algebra invariants, differential geometry and real gases, and their matrices are instrumental in the study of chaotic mappings. Ruben Aldrovandi’s book on matrices is a unique blend of mathematical rigour, pedagogical skill and a plethora of novel and often unexpected applications in Physics, Chemistry, Biology and Genetics. It should be read and studied accordingly, by the graduates, doctoral students and the high-level researchers alike