11,154 research outputs found
Discrete Harmonic Analysis. Representations, Number Theory, Expanders and the Fourier Transform
This self-contained book introduces readers to discrete harmonic analysis with an emphasis on the Discrete Fourier Transform and the Fast Fourier Transform on finite groups and finite fields, as well as their noncommutative versions. It also features applications to number theory, graph theory, and representation theory of finite groups. Beginning with elementary material on algebra and number theory, the book then delves into advanced topics from the frontiers of current research, including spectral analysis of the DFT, spectral graph theory and expanders, representation theory of finite groups and multiplicity-free triples, Tao's uncertainty principle for cyclic groups, harmonic analysis on GL(2,Fq), and applications of the Heisenberg group to DFT and FFT. With numerous examples, figures, and over 160 exercises to aid understanding, this book will be a valuable reference for graduate students and researchers in mathematics, engineering, and computer science
QUANTUM DISSIPATION AND QUANTUM GROUPS
We discuss the r\^ole of quantum deformation of Weyl-Heisenberg algebra in
dissipative systems and finite temperature systems. We express the time
evolution generator of the damped harmonic oscillator and the generator of
thermal Bogolubov transformations in terms of operators of the quantum
Weyl-Heisenberg algebra. The quantum parameter acts as a label for the
unitarily inequivalent representations of the canonical commutation relations
in which the space of the states splits in the infinite volume limit.Comment: to appear in Annals of Physics (N.Y.
- …