Number Theory and Fourier Analysis Applications in Physics, Acoustics and Computer Science

摘要: Number theory is traditionally considered a rather abstract field, far removed from practical applications. In the recent past, however, “higher arithmetic” has provided highly useful answers to numerous real—world problems. Many of these uses depend on special correlation and Fourier Transform properties certain real complex sequences derived different branches number theory, particularly finite fields quadratic residues. The applications include design new musical scales, powerful cryptographic systems, diffraction gratings for acoustic electromagnetic waves with unusually broad scatter, in radar camouflage, laser speckle removal, noise abatement, concert hall acoustics. Another prime domain construction very effective error—correction codes, such as those used picture transmission space vehicles compact discs (CDs). Other schemes spread—spectrum communication, “error—free” computing, fast computational algorithms, precision measurements (of interplanetary distances, example) at extremely low signal—to—noise ratios. this manner “fourth prediction” General Relativity (the slowing radiation gravitation fields, predicted by Einstein early 1907) been fully confirmed. contemporary physics quasiperiodic route chaos nonlinear dynamical systems double—pendulum three—body problem, mention two simple examples) are being analyzed terms theoretic concepts continued fractions, Fibonacci numbers, golden mean Farey trees. Even recently discovered state matter, christened quasicrystals, most effectively described arithmetic principles. And last not least, whose distribution combines predictable regularity surprising randomness, rich source pleasing artistic - either directly or through Transform.