Number Theory in Science and Communication

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Number Theory in Science and Communication / Manfred Schroeder

資料種別:: 電子ブック
出版情報:: SpringerLink Books - AutoHoldings
Springer Berlin Heidelberg, 2009
著者名:: Schroeder, Manfred
ISBN:: 9783540852971 [3540852972]
9783540852988 [3540852980]
9783642099014 [3642099017]
書誌ID:: OB00500974

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言語:

英語

目次情報:

A Few Fundamentals / Part I：

Introduction / 1：

The Family of Numbers

Fibonacci, Continued Fractions and the Golden Ratio / 1.1：

Fermat, Primes and Cyclotomy / 1.2：

Euler, Totients and Cryptography / 1.3：

Gauss, Congruences and Diffraction / 1.4：

Galois, Fields and Codes / 1.5：

The Natural Numbers / 2：

The Fundamental Theorem / 2.1：

The Least Common Multiple / 2.2：

Planetary "Gears" / 2.3：

The Greatest Common Divisor / 2.4：

Human Pitch Perception / 2.5：

Octaves, Temperament, Kilos and Decibels / 2.6：

Coprimes / 2.7：

Euclid's Algorithm / 2.8：

The Decimal System Decimated / 2.9：

Primes / 3：

How Many Primes are There? / 3.1：

The Sieve of Eratosthenes / 3.2：

A Chinese Theorem in Error / 3.3：

A Formula for Primes / 3.4：

Mersenne Primes / 3.5：

Repunits / 3.6：

Perfect Numbers / 3.7：

Fermat Primes / 3.8：

Gauss and the Impossible Heptagon / 3.9：

The Prime Distribution / 4：

A Probabilistic Argument / 4.1：

The Prime-Counting Function [pi](x) / 4.2：

David Hilbert and Large Nuclei / 4.3：

Coprime Probabilities / 4.4：

Primes in Progressions / 4.5：

Primeless Expanses / 4.6：

Squarefree and Coprime Integers / 4.7：

Twin Primes / 4.8：

Prime Triplets / 4.9：

Prime Quadruplets and Quintuplets / 4.10：

Primes at Any Distance / 4.11：

Spacing Distribution Between Adjacent Primes / 4.12：

Goldbach's Conjecture / 4.13：

Sum of Three Primes / 4.14：

Some Simple Applications / Part II：

Fractions: Continued, Egyptian and Farey / 5：

A Neglected Subject / 5.1：

Relations with Measure Theory / 5.2：

Periodic Continued Fractions / 5.3：

Electrical Networks and Squared Squares / 5.4：

Fibonacci Numbers and the Golden Ratio / 5.5：

Fibonacci, Rabbits and Computers / 5.6：

Fibonacci and Divisibility / 5.7：

Generalized Fibonacci and Lucas Numbers / 5.8：

Egyptian Fractions, Inheritance and Some Unsolved Problems / 5.9：

Farey Fractions / 5.10：

Farey Trees / 5.10.1：

Locked Pallas / 5.10.2：

Fibonacci and the Problem of Bank Deposits / 5.11：

Error-Free Computing / 5.12：

Congruences and the Like / Part III：

Linear Congruences / 6：

Residues / 6.1：

Some Simple Fields / 6.2：

Powers and Congruences / 6.3：

Diophantine Equations / 7：

Relation with Congruences / 7.1：

A Gaussian Trick / 7.2：

A Stamp Problem / 7.3：

Nonlinear Diophantine Equations / 7.4：

Triangular Numbers / 7.5：

Phthagorean Numbers / 7.6：

Exponential Diophantine Equations / 7.7：

Fermat's Last "Theorem" / 7.8：

The Demise of a Conjecture by Euler / 7.9：

A Nonlinear Diophantine Equation in Physics and the Geometry of Numbers / 7.10：

Normal-Mode Degeneracy in Room Acoustics (A Number-Theoretic Application) / 7.11：

Waring's Problem / 7.12：

The Theorems of Fermat, Wilson and Euler / 8：

Fermat's Theorem / 8.1：

Wilson's Theorem / 8.2：

Euler's Theorem / 8.3：

The Impossible Star of David / 8.4：

Dirichlet and Linear Progression / 8.5：

Permutations, Cycles and Derangements / 9：

Permutations / 9.1：

Binomial Coefficients / 9.2：

The Binomial and Related Distributions / 9.3：

Permutation Cycles / 9.4：

Derangements / 9.5：

Ascents and Descents / 9.6：

Quantum Decrypting / 9.7：

Decrypting without Factoring / 9.8：

Quantum Cryptography / 9.9：

One-Time Pads / 9.10：

The Bennet-Brossard Key Distribution Scheme (BB84) / 9.11：

Cryptography and Divisors / Part IV：

Euler Trap Doors and Public-Key Encryption / 10：

A Numerical Trap Door / 10.1：

Digital Encryption / 10.2：

Public-Key Encryption / 10.3：

A Simple Example / 10.4：

Repeated Encryption / 10.5：

Summary and Encryption Requirements / 10.6：

The Divisor Functions / 11：

The Number of Divisors / 11.1：

The Average of the Divisor Function / 11.2：

The Geometric Mean of the Divisors / 11.3：

The Summatory Function of the Divisor Function / 11.4：

The Generalized Divisor Functions / 11.5：

The Average Value of Euler's Function / 11.6：

The Prime Divisor Functions / 12：

The Number of Different Prime Divisors / 12.1：

The Distribution of [omega](n) / 12.2：

The Number of Prime Divisors / 12.3：

The Harmonic Mean of [Omega](n) / 12.4：

Medians and Percentiles of [Omega](n) / 12.5：

Implications for Public-Key Encryption / 12.6：

Certified Signatures / 13：

A Story of Creative Financing / 13.1：

Certified Signature for Public-Key Encryption / 13.2：

Primitive Roots / 14：

Orders / 14.1：

Periods of Decimal and Binary Fractions / 14.2：

A Primitive Proof of Wilson's Theorem / 14.3：

The Index-A Number-Theoretic Logarithm / 14.4：

Solution of Exponential Congruences / 14.5：

What is the Order T[subscript m] of an Integer m Modulo a Prime p? / 14.6：

Index "Encryption" / 14.7：

A Fourier Property of Primitive Roots and Concert Hall Acoustics / 14.8：

More Spacious-Sounding Sound / 14.9：

Galois Arrays for X-Ray Astronomy / 14.10：

A Negative Property of the Fermat Primes / 14.11：

Knapsack Encryption / 15：

An Easy Knapsack / 15.1：

A Hard Knapsack / 15.2：

Residues and Diffraction / Part V：

Quadratic Residues / 16：

Quadratic Congruences / 16.1：

Euler's Criterion / 16.2：

The Legendre Symbol / 16.3：

A Fourier Property of Legendre Sequences / 16.4：

Gauss Sums / 16.5：

Pretty Diffraction / 16.6：

Quadratic Reciprocity / 16.7：

A Fourier Property of Quadratic-Residue Sequences / 16.8：

Spread Spectrum Communication / 16.9：

Generalized Legendre Sequences Obtained Through Complexification of the Euler Criterion / 16.10：

Chinese and Other Fast Algorithms / Part VI：

The Chinese Remainder Theorem and Simultaneous Congruences / 17：

Simultaneous Congruences / 17.1：

The Sino-Representation: A Chinese Number System / 17.2：

Aplications of the Sino-Representation / 17.3：

Discrete Fourier Transformation in Sino / 17.4：

A Sino-Optical Fourier Transformer / 17.5：

Generalized Sino-Representation / 17.6：

Fast Prime-Length Fourier Transform / 17.7：

Fast Transformation and Kronecker Products / 18：

A Fast Hadamard Transform / 18.1：

The Basic Principle of the Fast Fourier Transforms / 18.2：

Application of the Chinese Remainder Theorem (CRT) / 19：

Pseudoprimes, Mobius Transform, and Partitions / Part VII：

Pseudoprimes, Poker and Remote Coin Tossing / 20：

Pulling Roots to Ferret Out Composites / 20.1：

Factors from a Square Root / 20.2：

Coin Tossing by Telephone / 20.3：

Absolute and Strong Pseudoprimes / 20.4：

Fermat and Strong Pseudoprimes / 20.5：

Deterministic Primality Testing / 20.6：

A Very Simple Factoring Algorithm / 20.7：

Factoring with Elliptic Curves / 20.8：

Quantum Factoring / 20.9：

The Mobius Function and the Mobius Transform / 21：

The Mobius Transform and Its Inverse / 21.1：

Proof of the Inversion Formula / 21.2：

Second Inversion Formula / 21.3：

Third Inversion Formula / 21.4：

Fourth Inversion Formula / 21.5：

Riemann's Hypothesis and the Disproof of the Mertens Conjecture / 21.6：

Dirichlet Series and the Mobius Function / 21.7：

Generating Functions and Partitions / 22：

Generating Functions / 22.1：

Partitions of Integers / 22.2：

Generating Functions of Partitions / 22.3：

Restricted Partitions / 22.4：

From Error Correcting Codes to Covering Sets / 23：

Covering Sets in Coding Theory / 23.1：

Discrete Covering Sets / 23.2：

Cyclotomy and Polynomials / Part VIII：

Cyclotomic Polynomials / 24：

How to Divide a Circle into Equal Parts / 24.1：

Gauss's Great Insight / 24.2：

Factoring in Different Fields / 24.3：

Cyclotomy in the Complex Plane / 24.4：

How to Divide a Circle with Compass and Straightedge / 24.5：

Rational Factors of z[superscript N]-1 / 24.5.1：

An Alternative Rational Factorization / 24.6：

Relation Between Rational Factors and Complex Roots / 24.7：

How to Calculate with Cyclotomic Polynomials / 24.8：

Linear Systems and Polynomials / 25：

Impulse Responses / 25.1：

Time-Discrete Systems and the z Transform / 25.2：

Discrete Convolution / 25.3：

Cyclotomic Polynomials and z Transform / 25.4：

Polynomial Theory / 26：

Some Basic Facts of Polynomial Life / 26.1：

Polynomial Residues / 26.2：

Chinese Remainders for Polynomials / 26.3：

Euclid's Algorithm for Polynomials / 26.4：

Galois Fields and More Applications / Part IX：

Galois Fields / 27：

Prime Order / 27.1：

Prime Power Order / 27.2：

Generation of GF(2[superscript 4]) / 27.3：

How Many Primitive Elements? / 27.4：

Recursive Relations / 27.5：

How to Calculate in GF(p[superscript m]) / 27.6：

Zech Logarithm, Doppler Radar and Optimum Ambiguity Functions / 27.7：

A Unique Phase-Array Based on the Zech Logarithm / 27.8：

Spread-Spectrum Communication and Zech Logarithms / 27.9：

Spectral Properties of Galois Sequences / 28：

Circular Correlation / 28.1：

Application to Error-Correcting Codes and Speech Recognition / 28.2：

Application to Precision Measurements / 28.3：

Concert Hall Measurements / 28.4：

The Fourth Effect of General Relativity / 28.5：

Toward Better Concert Hall Acoustics / 28.6：

Higher-Dimensional Diffusors / 28.7：

Active Array Applications / 28.8：

Random Number Generators / 29：

Pseudorandom Galois Sequences / 29.1：

Randomness from Congruences / 29.2：

"Continuous" Distributions / 29.3：

Four Ways to Generate a Gaussian Variable / 29.4：

Pseudorandom Sequences in Cryptography / 29.5：

Waveforms and Radiation Patterns / 30：

Special Phases / 30.1：

The Rudin-Shapiro Polynomials / 30.2：

Gauss Sums and Peak Factors / 30.3：

Galois Sequences and the Smallest Peak Factors / 30.4：

Minimum Redundancy Antennas / 30.5：

Golomb Rulers / 30.6：

Number Theory, Randomness and "Art" / 31：

Number Theory and Graphic Design / 31.1：

The Primes of Gauss and Eisenstein / 31.2：

Galois Fields and Impossible Necklaces / 31.3：

"Baroque" Integers / 31.4：

Self-Similarity, Fractals and Art / Part X：

Self-Similarity, Fractals, Deterministic Chaos and a New State of Matter / 32：

Fibonacci, Noble Numbers and a New State of Matter / 32.1：

Cantor Sets, Fractals and a Musical Paradox / 32.2：

The Twin Dragon: A Fractal from a Complex Number System / 32.3：

Statistical Fractals / 32.4：

Some Crazy Mappings / 32.5：

The Logistic Parabola and Strange Attractors / 32.6：

Conclusion / 32.7：

Glossary of Symbols

References

Name Index

Subject Index

A Few Fundamentals / Part I：

Introduction / 1：

The Family of Numbers

Fibonacci, Continued Fractions and the Golden Ratio / 1.1：

Fermat, Primes and Cyclotomy / 1.2：

Euler, Totients and Cryptography / 1.3：

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