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Number Theory in Science and Communication [electronic resource] : With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity / by Manfred R. Schroeder

Author
Schroeder, Manfred R.
Published
Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1986.
Edition
2nd ed. 1986.
Physical Description
XIX, 374 pages : online resource
Additional Creators
SpringerLink (Online service)
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Contents
A Few Fundamentals -- The Natural Numbers -- Primes -- The Prime Distribution -- Some Simple Applications -- Fractions: Continued, Egyptian and Farey -- Congruences and the Like -- Linear Congruences -- Diophantine Equations -- The Theorems of Fermat, Wilson and Euler -- Cryptography and Divisors -- Euler Trap Doors and Public-Key Encryption -- The Divisor Functions -- The Prime Divisor Functions -- Certified Signatures -- Primitive Roots -- Knapsack Encryption -- Residues and Diffraction -- Quadratic Residues -- Chinese and Other Fast Algorithms -- The Chinese Remainder Theorem and Simultaneous Congruences -- Fast Transformations and Kronecker Products -- Quadratic Congruences -- Pseudoprimes, Möbius Transform, and Partitions -- Pseudoprimes, Poker and Remote Coin Tossing -- The Möbius Function and the Möbius Transform -- Generating Functions and Partitions -- Cyclotomy and Polynomials -- Cyclotomic Polynomials -- Linear Systems and Polynomials -- Polynomial Theory -- Galois Fields and More Applications -- Galois Fields -- Spectral Properties of Galois Sequences -- Random Number Generators -- Waveforms and Radiation Patterns -- Number Theory, Randomness and "Art" -- Self Similarity, Fractals and Art -- Self-Similarity, Fractals, Deterministic Chaos and a New State of Matter.
Summary
"Beauty is the first test: there is no permanent place in the world for ugly mathematics. " - G. H. Hardy Number theory has been considered since time immemorial to be the very paradigm of pure (some would say useless) mathematics. In fact, the Chinese characters for mathematics are Number Science. "Mathematics is the queen of sciences - and number theory is the queen of mathematics," according to Carl Friedrich Gauss, the lifelong Wunderkind, who himself enjoyed the epithet "Princeps Mathematicorum. " What could be more beautiful than a deep, satisfying relation between whole numbers. (One is almost tempted to call them wholesome numbers') In fact, it is hard to come up with a more appropriate designation than their learned name: the integers - meaning the "untouched ones". How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex number- whose first names virtually exude unsavory involvement with the complex realities of everyday life! Yet, as we shall see in this book, the theory of integers can provide totally unexpected answers to real-world problems. In fact, discrete mathematics is taking on an ever more important role. If nothing else, the advent of the digital computer and digital communication has seen to that. But even earlier, in physics the emergence of quantum mechanics and discrete elementary particles put a premium on the methods and, indeed, the spirit of discrete mathematics.
Subject(s)
ISBN
9783662222461
Digital File Characteristics
PDF
text file
Part Of
Springer Nature eBook
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