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Herman J., Simsa J., Kucera R. — Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory
Herman J., Simsa J., Kucera R. — Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory

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Название: Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory

Авторы: Herman J., Simsa J., Kucera R.

Аннотация:

This book presents methods of solving problems in three areas of classical elementary mathematics: Equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are immediately followed by carefully worked out examples of increasing degrees of difficulty, and by exercises which range from routine to rather challenging problems. While this book emphasizes some methods that are not usually covered in beginning university courses, it nevertheless teaches techniques and skills which are useful not only in the specific topics covered here. There are approximately 330 examples and 760 exercises.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 344

Добавлена в каталог: 22.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel, Niels Henrik      25
Algebra      23
Algorithm, division      28
Algorithm, Euclidean      178 181
Alternating sums      15
AM-GM inequality      109 144 151 155 160 162 163 167 232
AM-GM inequality for weighted means      162
Arithmetic mean      33 109 150
Arithmetic progression      5
Base of the natural logarithm      113
Base representations      258
Bernoulli numbers      13
Bernoulli’s formula      12
Bernoulli’s inequality      109 141 166 168 233
Bernoulli’s inequality, general      165
Bezout’s equality      179
Bezout’s Theorem      28
Binomial coefficients      2 75
Binomial coefficients, generalized      276
Binomial theorem      2 8 175 263
Cauchy’s inequality      127 131 133 149 166
Chebyshev’s inequality      145 148 149 150 159
Chebyshev’s theorem      187
Chinese remainder theorem      211
Combinatorial identities      6
Combinatorial numbers      1
Common divisor      178 180
Common multiple      178 180
Completing the square      113
Complex number      75
Composite number      183
Congruences      189
Congruences in one variable      201
Congruences, linear      202
Congruences, of higher degree      211
Congruences, quadratic      216
Conjecture, Goldbach      173
Conjecture, Shimura — Taniyama      173
Conjecture, twin prime      173
conjugate      65
Coprime      180
Cramer’s Rule      46
Cubic polynomial      24
De Moivre’s Theorem      75 77 78 84
Decimal expansion      261
Degree of a polynomial      24 38
Degree of a power mean      167
Digit patterns      265
Digit sum      261 267
DIGITS      258
Digits, final      261
Diophantine equations      217
Diophantine equations, cardinality of solution set      245
Diophantine equations, linear      217
Diophantine equations, linear in some variable      220
Diophantus of Alexandria      217
Dirichlet’s principle      268
Dirichlet’s Theorem      187
Discriminant      124
Divisibility of numbers      174
Division algorithm      28
Division theorem      176
Eisenstein’s irreducibility criterion      280
Elementary symmetric polynomials      32 39
Elementary symmetric polynomials in three variables      43
Elementary symmetric polynomials in two variables      39
Elimination method      54
Equivalent transformations      94
Estimation method      101
Euclidean algorithm      178 181
Euler’s $\varphi$-function      194
Euler’s criterion      216
Euler’s theorem      198 262 264 266
Factorial      1
Factorial, prime decomposition of      255
False roots      61
Fermat’s Last Theorem      173
Fermat’s theorem      197
Finite induction      136
Finite sums      5
Fractional part      251
Fundamental mean property      165
Fundamental theorem of algebra      29
Gauss, Carl Friedrich      189
Gaussian elimination      46
Generating functions      9
Geometric mean      151 167
Geometric progression      6
Geometric series      110
Geometric-power mean inequalities      167
Goldbach conjecture      173
Greatest common divisor      178 180
Harmonic mean      156
Hoelder’s inequality      166
Homogeneous inequality      105
Homogeneous polynomial      38
Implication method      61
Incomplete quotient      176
Induction principle      135 136
Induction principle, finite      134
Inequality      89
Inequality, AM-GM      109 144 151 155 160 162 163 167 232
Inequality, AM-GM, for weighted means      162
Inequality, Bernoulli’s      109 141 166 168 233
Inequality, Bernoulli’s, general      165
Inequality, between power means      167
Inequality, Cauchy’s      127 131 133 149 166
Inequality, Chebyshev’s      145 148 149 150 159
Inequality, geometric-power mean      167
Inequality, Hoelder’s      166
Inequality, homogeneous      105
Inequality, Jensen’s      105 169
Inequality, Minkowski’s      130 166
Inequality, symmetric      104
Inequality, triangle      129
Inequality, weak      91
Inequality, Young’s      163
Infinite product      140
Infinitude of primes in arithmetic progressions      187
Infinitude of primes of the form 3k+2      187
Infinitude of primes of the form 4k+3      187
Integer part      251 267
Integer-valued polynomials      276
Interpolation      23
Irrational equations      60
Irrational equations, with a parameter      71
Irrational numbers      169
Irreversible transformations      96
Jensen’s Inequality      105 169
Least common multiple      178 180
Length of a vector      130
Linear congruences      202
Linear diophantine equations      217
Linear polynomial      24
Lower bound      101
Mathematical induction      5 135
Mean of degree zero      167
Mean value      150
Method of squares      113
Method of Substitution      66
Method of symmetric polynomials      55
Method of undetermined coefficients      19
Minkowski’s inequality      130 166
Monomial      38
Multiple      174
Multiple root      28
Multiple zero      28
Multiplicity of a zero      28
Nth roots      60
Number theory      173
Ordering of an n-tuple      146
Pairwise relatively prime      180
Partial fraction decomposition      21
Partial summation      16
Pigeonhole Principle      268
Polynomial division      27
Polynomials      23 84 274
Polynomials in several variables      38
Polynomials, cubic      24
Polynomials, degree of      24 38
Polynomials, elementary symmetric      32 39 43
Polynomials, homogeneous      38
Polynomials, integer-valued      276
Polynomials, irreducible      280
Polynomials, linear      24
Polynomials, Taylor      23
Polynomials, with integer coefficients      274
Power mean      151 166 167
Prime number      183
Pythagorean equation      237
Quadratic congruences      216
Quadratic polynomial      24
Quotient      28 176
Rational zeros      35
Regular n-gon      80
Relatively prime      180
Remainder      28 176
Repunit      265
Root of a polynomial equation      25
Scalar product      128
Shimura — Taniyama conjecture      173
Sierpinski, W.      187
Simple zero      28
Strict inequalities      91
Sum of digits      261 267
Sums of powers      11 39
Symmetric inequality      104
Symmetric polynomial      38
Systems of equations      46
Systems of equations of higher degree      54
Systems of irrational equations      73
Systems of linear congruences in one variable      206
Systems of linear equations      46
Systems of linear equations with parameters      48
Taylor polynomials      23
Theorem, Bezout’s      28
Theorem, binomial      2 8 175 263
Theorem, Chebyshev’s      187
Theorem, Chinese remainder      211
Theorem, de Moivre’s      75 77 78 84
Theorem, Dirichlet’s      187
Theorem, division      176
Theorem, Euler’s      198 262 264 266
Theorem, Fermat’s      197
Theorem, Fermat’s last      173
Theorem, Wilson’s      204
Transitivity of inequalities      91
Triangle inequality      129
Trichotomy Law      91
Trigonometric functions      78
Twin Prime Conjecture      173
Unique factorization      185
Upper bound      101
Vieta’s relations      32 38 41 56 78 80 82 83 125 132
Vinogradov, I. M.      216
Weak inequalities      91
Weighted arithmetic mean      162
Weighted geometric mean      162
Weighted means      151 162
Wiles, Andrew      173
Wilson’s Theorem      204
Young’s inequality      163
Zero of a polynomial      23 25 275
Zero polynomial      24 38
Zero, multiple      28
Zero, simple      28
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