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Number theory in science and communication : with applications in cryptography, physics, digital information, computing, and self-similarity 2nd enl. ed

Number theory in science and communication : with applications in cryptography, physics, digital information, computing, and self-similarity 2nd enl. ed

Material type
단행본
Personal Author
Schroeder, M. R. (Manfred Robert), 1926-
Title Statement
Number theory in science and communication : with applications in cryptography, physics, digital information, computing, and self-similarity / M.R. Schroeder.
판사항
2nd enl. ed.
Publication, Distribution, etc
Berlin ;   New York :   Springer-Verlag,   c1986.  
Physical Medium
xix, 374 p. : ill. ; 24 cm.
Series Statement
Springer series in information sciences ;v. 7.
ISBN
0387158006 (U.S. : pbk.)
General Note
Includes indexes.  
Bibliography, Etc. Note
Bibliography: p. [353]-362.
Subject Added Entry-Topical Term
Number theory. Nombres, theorie des.
000 00971camuuu200277 a 4500
001 000000110700
005 19980706103212.0
008 850718s1986 gw a b 001 0 eng
010 ▼a 85017260 //r90
020 ▼a 0387158006 (U.S. : pbk.)
040 ▼a DLC ▼c DLC ▼d FTV
049 1 ▼l 421041362 ▼f 과학
050 0 0 ▼a QA241 ▼b .S318 1986
082 0 0 ▼a 512/.7 ▼2 19
090 ▼a 512.7 ▼b S381n
100 1 ▼a Schroeder, M. R. ▼q (Manfred Robert), ▼d 1926-
245 1 0 ▼a Number theory in science and communication : ▼b with applications in cryptography, physics, digital information, computing, and self-similarity / ▼c M.R. Schroeder.
250 ▼a 2nd enl. ed.
260 ▼a Berlin ; ▼a New York : ▼b Springer-Verlag, ▼c c1986.
300 ▼a xix, 374 p. : ▼b ill. ; ▼c 24 cm.
440 0 ▼a Springer series in information sciences ; ▼v v. 7.
500 ▼a Includes indexes.
504 ▼a Bibliography: p. [353]-362.
650 0 ▼a Number theory.
650 7 ▼a Nombres, theorie des. ▼2 ram.

Holdings Information

No. Location Call Number Accession No. Availability Due Date Make a Reservation Service
No. 1 Location Science & Engineering Library/Sci-Info(Stacks2)/ Call Number 512.7 S381n Accession No. 421041362 Availability Available Due Date Make a Reservation Service B M

Contents information

Table of Contents

권호명 : 1

CONTENTS
Part Ⅰ A Few Fundamentals
  1. Introduction = 1
    1.1 Fibonacci, Continued Fractions and the Golden Ratio = 4
    1.2 Fermat, Primes and Cyclotomy = 7
    1.3 Euler, Totients and Cryptography = 9
    1.4 Gauss, Congruences and Diffraction = 10
    1.5 Galois, Fields and Codes = 12
  2. The Natural Numbers = 17
    2.1 The Fundamental Theorem = 17
    2.2 The Least Common Multiple = 18
    2.3 Planetary "Gears" = 19
    2.4 The Greatest Common Divisor = 20
    2.5 Human Pitch Perception = 22
    2.6 Octaves, Temperament, Kilos and Decibels = 22
    2.7 Coprimes = 25
    2.8 Euclid's Algorithm = 25
  3. Primes = 26
    3.1 How Many Primes are There? = 26
    3.2 The Sieve of Eratosthenes = 27
    3.3 A Chinese Theorem in Error = 29
    3.4 A Formula for Primes = 29
    3.5 Mersenne Primes = 30
    3.6 Repunits = 34
    3.7 Perfect Numbers = 35
    3.8 Fermat Primes = 37
    3.9 Gauss and the Impossible Heptagon = 38
  4. The Prime Distribution = 40
    4.1 A Probabilistic Argument = 40
    4.2 The Prime-Counting Function π(x) = 43
    4.3 David Hilbert and Large Nuclei = 47
    4.4 Coprime Probabilities = 48
    4.5 Twin Primes = 51
    4.6 Primeless Expanses = 53
    4.7 Square-Free and Coprime Integers = 54
Part Ⅱ Some Simple Applications
  5. Fractions : Continued, Egyptian and Farey = 55
    5.1 A Neglected Subject = 55
    5.2 Relations with Measure Theory = 60
    5.3 Periodic Continued Fractions = 60
    5.4 Electrical Networks and Squared Squares = 64
    5.5 Fibonacci Numbers and the Golden Ratio = 65
    5.6 Fibonacci, Rabbits and Computers = 70
    5.7 Fibonacci and Divisibility = 72
    5.8 Generalized Fibonacci and Lucas Numbers = 73
    5.9 Egyptian Fractions, Inheritance and Some Unsolved Problems = 76
    5.10 Farey Fractions = 77
    5.11 Fibonacci and the Problem of Bank Deposits = 80
    5.12 Error-Free Computing = 81
Part Ⅲ Congruences and the Like
  6. Linear Congruences = 87
    6.1 Residues = 87
    6.2 Some Simple Fields = 90
    6.3 Powers and Congruences = 92
  7. Diophantine Equations = 95
    7.1 Relation with Congruences = 95
    7.2 A Gaussian Trick = 96
    7.3 Nonlinear Diophantine Equations = 98
    7.4 Triangular Numbers = 100
    7.5 Pythagorean Numbers = 102
    7.6 Exponential Diophantine Equations = 103
    7.7 Format's Last "Theorem" = 104
    7.8 The Demise of a Conjecture by Euler = 105
    7.9 A Nonlinear Diophantine Equation in Physics and the Geometry of Numbers = 106
    7.10 Normal-Mode Degeneracy in Room Acoustics (A Number-Theoretic Application) = 108
    7.11 Waring's Problem = 109
  8. The Theorems of Format, Wilson and Euler = 111
    8.1 Format's Theorem = 111
    8.2 Wilson's Theorem = 112
    8.3 Euler's Theorem = 113
    8.4 The Impossible Star of David = 115
    8.5 Dirichlet and Linear Progression = 116
Part Ⅳ Cryptography and Divisors
  9. Euler Trap Doors and Public-Key Encryption = 118
    9.1 A Numerical Trap Door = 118
    9.2 Digital Encryption = 119
    9.3 Public-Key Encryption = 121
    9.4 A Simple Example = 123
    9.5 Repeated Encryption = 123
    9.6 Summary and Encryption Requirements = 125
  10. The Divisor Functions = 127
    10.1 The Number of Divisors = 127
    10.2 The Average of the Divisor Function = 130
    10.3 The Geometric Mean of the Divisors = 131
    10.4 The Summatory Function of the Divisor Function = 131
    10.5 The Generalized Divisor Functions = 132
    10.6 The Average Value of Euler's Function = 133
  11. The Prime Divisor Functions = 135
    11.1 The Number of Different Prime Divisors = 135
    11.2 The Distribution of w(n) = 138
    11.3 The Number of Prime Divisors = 141
    11.4 The Harmonic Mean of Ω(n) = 144
    11.5 Medians and Percentiles of Ω(n) = 146
    11.6 Implications for Public-Key Encryption = 147
  12. Certified Signatures = 149
    12.1 A Story of Creative Financing = 149
    12.2 Certified Signature for Public-Key Encryption = 149
  13. Primitive Roots = 151
    13.1 Orders = 151
    13.2 Periods of Decimal and Binary Fractions = 154
    13.3 A Primitive Proof of Wilson's Theorem = 157
    13.4 The Index - A Number-Theoretic Logarithm = 158
    13.5 Solution of Exponential Congruences = 159
    13.6 What is the Order Tm of an Integer m Modulo a Prime p? = 161
    13.7 Index "Encryption" = 162
    13.8 A Fourier Property of Primitive Roots and Concert Hall Acoustics = 163
    13.9 More Spacious-Sounding Sound = 164
    13.10 A Negative Property of the Fermat Primes = 167
  14. Knapsack Encryption = 168
    14.1 An Easy Knapsack = 168
    14.2 A Hard Knapsack = 169
Part Ⅴ Residues and Diffraction
  15. Quadratic Residues = 172
    15.1 Quadratic Congruences = 172
    15.2 Euler's Criterion = 173
    15.3 The Legendre Symbol = 175
    15.4 A Fourier Property of Legendre Sequences = 176
    15.5 Gauss Sums = 177
    15.6 Pretty Diffraction = 179
    15.7 Quadratic Reciprocity = 179
    15.8 A Fourier Property of Quadratic-Residue Sequences = 180
    15.9 Spread Spectrum Communication = 183
    15.10 Generalized Legendre Sequences Obtained Through Complexification of the Euler Criterion = 183
Part Ⅵ Chinese and Other Fast Algorithms
  16. The Chinese Remainder Theorem and Simultaneous Congruences = 186
    16.1 Simultaneous Congruences = 186
    16.2 The Sino-Representation : A Chinese Number System = 187
    16.3 Applications of the Sino-Representation = 189
    16.4 Discrete Fourier Transformation in Sino = 190
    16.5 A Sino-Optical Fourier Transformer = 191
    16.6 Generalized Sino-Representation = 192
    16.7 Fast Prime-Length Fourier Transform = 194
  17. Fast Transformations and Kronecker Products = 196
    17.1 A Fast Hadamard Transform = 196
    17.2 The Basic Principle of the Fast Fourier Transforms = 199
  18. Quadratic Congruences = 201
    18.1 Application of the Chinese Remainder Theorem(CRT) = 201
Part Ⅶ Pseudoprimes, M o ·· bius Transform, and Partitions
  19. Pseudoprimes, Poker and Remote Coin Tossing = 203
    19.1 Pulling Roots to Ferret Out Composites = 203
    19.2 Factors from a Square Root = 205
    19.3 Coin Tossing by Telephone = 206
    19.4 Absolute and Strong Pseudoprimes = 209
    19.5 Fermat and Strong Pseudoprimes = 211
    19.6 Deterministic Primality Testing = 212
    19.7 A Very Simple Factoring Algorithm = 213
  20. The M o ·· bius Function and the M o ·· bius Transform = 215
    20.1 The M o ·· bius Transform and Its Inverse = 215
    20.2 Proof of the Inversion Formula = 217
    20.3 Second Inversion Formula = 218
    20.4 Third Inversion Formula = 219
    20.5 Fourth Inversion Formula = 219
    20.6 Riemann's Hypothesis and the Disproof of the Mertens Conjecture = 219
    20.7 Dirichlet Series and the M o ·· bius Function = 220
  21. Generating Functions and Partitions = 223
    21.1 Generating Functions = 223
    21.2 Partitions of Integers = 225
    21.3 Generating Functions of Partitions = 226
    21.4 Restricted Partitions = 227
Part Ⅷ Cyclotomy and Polynomials
  22. Cyclotomic Polynomials = 232
    22.1 How to Divide a Circle into Equal Parts = 232
    22.2 Gauss's Great Insight = 235
    22.3 Factoring in Different Fields = 240
    22.4 Cyclotomy in the Complex Plane = 240
    22.5 How to Divide a Circle with Compass and Straightedge = 242
      22.5.1 Rational Factors of zN - 1 = 243
    22.6 An Alternative Rational Factorization = 244
    22.7 Relation Between Rational Factors and Complex Roots = 245
    22.8 How to Calculate with Cyclotomic Polynomials = 247
  23. Linear Systems and Polynomials = 249
    23.1 Impulse Responses = 249
    23.2 Time-Discrete Systems and the z Transform = 250
    23.3 Discrete Convolution = 251
    23.4 Cyclotomic Polynomials and z Transform = 251
  24. Polynomial Theory = 253
    24.1 Some Basic Facts of Polynomial Life = 253
    24.2 Polynomial Residues = 254
    24.3 Chinese Remainders for Polynomials = 256
    24.4 Euclid's Algorithm for Polynomials = 257
Part Ⅸ Galois Fields and More Applications
  25. Galois Fields = 259
    25.1 Prime Order = 259
    25.2 Prime Power Order = 259
    25.3 Generation of GF( 24 ) = 262
    25.4 How Many Primitive Elements? = 263
    25.5 Recursive Relations = 264
    25.6 How to Calculate in GF( pm ) = 266
    25.7 Zech Logarithm, Doppler Radar and Optimum Ambiguity Functions = 267
    25.8 A Unique Phase-Array Based on the Zech Logarithm = 271
    25.9 Spread-Spectrum Communication and Zech Logarithms = 272
  26. Spectral Properties of Galois Sequences = 274
    26.1 Circular Correlation = 274
    26.2 Application to Error-Correcting Codes and Speech Recognition = 277
    26.3 Application to Precision Measurements = 278
    26.4 Concert Hall Measurements = 279
    26.5 The Fourth Effect of General Relativity = 280
    26.6 Toward Better Concert Hall Acoustics = 281
    26.7 Higher-Dimensional Diffusors = 287
    26.8 Active Array Applications = 287
  27. Random Number Generators = 289
    27.1 Pseudorandom Galois Sequences = 290
    27.2 Randomness from Congruences = 291
    27.3 "Continuous" Distributions = 292
    27.4 Four Ways to Generate a Gaussian Variable = 293
    27.5 Pseudorandom Sequences in Cryptography = 295
  28. Waveforms and Radiation Patterns = 296
    28.1 Special Phases = 297
    28.2 The Rudin-Shapiro Polynomials = 299
    28.3 Gauss Sums and Peak Factors = 300
    28.4 Galois Sequences and the Smallest Peak Factors = 302
    28.5 Minimum Redundancy Antennas = 305
  29. Number Theory, Randomness and "Art" = 307
    29.1 Number Theory and Graphic Design = 307
    29.2 The Primes of Gauss and Eisenstein = 309
    29.3 Galois Fields and Impossible Necklaces = 310
Part Ⅹ Self-Similarity, Fractals and Art
  30. Self-Similarity, Fractals, Deterministic Chaos and a New State of Matter = 315
    30.1 Fibonacci, Noble Numbers and a New State of Matter = 319
    30.2 Cantor Sets, Fractals and a Musical Paradox = 324
    30.3 The Twin Dragon : a Fractal from a Complex Number System = 330
    30.4 Statistical Fractals = 331
    30.5 Some Crazy Mappings = 333
    30.6 The Logistic Parabola and Strange Attractors = 337
    30.7 Conclusion = 340
Appendix = 341
  A. A Calculator Program for Exponentiation and Residue Reduction = 341
  B. A Calculator Program for Calculating Fibonacci and Lucas Numbers = 345
  C. A Calculator Program for Decomposing an Integer According to the Fibonacci Number System = 346
Glossary of Symbols = 349
References = 353
Name Index = 363
Subject Index = 367

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