Steele J.M. — Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities :: Электронная библиотека попечительского совета мехмата МГУ

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Steele J.M. — Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities

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Название: Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities

Автор: Steele J.M.

Аннотация:

Michael Steele describes the fundamental topics in mathematical inequalities and their uses. Using the Cauchy-Schwarz inequality as a guide, Steele presents a fascinating collection of problems related to inequalities and coaches readers through solutions, in a style reminiscent of George Polya, by teaching basic concepts and sharpening problem solving skills at the same time. Undergraduate and beginning graduate students in mathematics, theoretical computer science, statistics, engineering, and economics will find the book appropriate for self-study.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 316

Добавлена в каталог: 11.06.2008

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Предметный указатель

1-trick      110 144 146 215 219 227 231
1-trick, defined      226
1-trick, first used      12
1-trick, refinement      205 288
Abel inequality      208 221
Abel, Niels Henrik      208
Aczel, J.      287
Additive bound      4 9 66 106
Additive bound and Hoelder      137 264
Akerberg's re?nement      34
al-Khazin, Abu Ja'far      286
Alexanderson, G.      286
AM-GM inequality      20 21
AM-GM inequality and investments      100
AM-GM inequality and Kantorovich      247
AM-GM inequality and the exponential      91
AM-GM inequality via cyclic shifts      84
AM-GM inequality via integral analog      114
AM-GM inequality via rearrangement      84
AM-GM inequality via Schur concavity      194
AM-GM inequality, Akerberg's refinement      34
AM-GM inequality, algorithmic proof      245
AM-GM inequality, betweeness induction      83
AM-GM inequality, general weights      23
AM-GM inequality, geometric interpretation      33
AM-GM inequality, integral analog      113
AM-GM inequality, Polya's proof      23
AM-GM inequality, rational weights      22
AM-GM inequality, smoothing proof      244
AM-GM inequality, stability      35
American Mathematical Monthly      96 192 253
Andreescu, T.      251
Anti-symmetric forms      237
Arithmetic mean-geometric mean inequality      see AM-GM inequality
Arithmetica of Diophantus      40
Arrangement of spheres      52
Artin, Emil      47
Backtracking      137
Backwards induction      236
Baseball inequality      82
Belentepe, C.      vi
Bennett, C.      288 289
Bernoulli inequality      31
Bessel inequality      71 225
Betweeness, exploitation of      74
Birkhoff's theorem      207
Birthday problem      206
Bledsoe, W.W.      283
Bombieri, E.      284
Box, thinking outside      52
Brahmagupta      286
Brahmagupta identity      47
Brunn — Minkowski inequality      245 287
Bunyakovsky, Victor Yacovlevich      10 190 285
Bunyakovsky, Victor Yacovlevich and AM-GM inequality      115
Bunyakovsky, Victor Yacovlevich vis-a-vis Schwarz      11
Bunyakovsky, Victor Yacovlevich, Chebyshev contact      76
Buzano, M.L.      286
Cai, T.      vi 246
Cancellation, origins of      210
Carleman inequality      118
Carleman inequality, Carleson proof      173
Carleman inequality, Polya's proof      27
Carleman inequality, refined via Knopp      128
Carleson inequality      173
Carlson inequality      165
Cauchy inequality      1
Cauchy inequality and quadratic forms      14
Cauchy inequality by induction      2
Cauchy inequality in an array      16
Cauchy inequality via inner product identity      68
Cauchy inequality via monotone interpolation      85
Cauchy inequality, beating      15
Cauchy inequality, case of equality      37
Cauchy inequality, interpolation      48
Cauchy inequality, Polya — Szegoe converse      83
Cauchy inequality, three term      13
Cauchy inequality, vector-scalar melange      69
Cauchy — Binet identity      49
Cauchy — Schwarz inequality      8
Cauchy — Schwarz inequality as accidental corollary      57
Cauchy — Schwarz inequality via Gram — Schmidt      71
Cauchy — Schwarz inequality, cross term defects      83
Cauchy — Schwarz inequality, geometric proof      58
Cauchy — Schwarz inequality, self-generalization      16
Cauchy's induction argument, AM-GM inequality      20
Cauchy's induction argument, Cauchy inequality      31
Cauchy's induction argument, Jensen inequality      101
Cauchy, Augustin-Louis      10
Centered inequality      115
Chebyshev order inequality      76
Chebyshev tail inequality      86
Chebyshev, Pafnuty Lvovich      76 287
Chong, K.M.      244
Chung, F.R.K.      289
Clevenson, M.L.      277
Closest point problem      56
Completing the square      57
Conjugate powers      137
Consistency principles      134
Convex functions, continuity of      254
Convex minorant      98
Convexity via transformation      102
Convexity, defined      87
Convexity, differential criterion      90
Convexity, geometric characterizations      87
Convexity, strict      89
Convexity, versus J-convexity      101
Cours d'Analyse Algebrique      10
Cramer — Rao inequality      18
Cronin-Kardon, C.      vi
Crystallographic inequality      13
Cyclic shifts      84
Cyclic sum inequality      104
D'Angelo, J.P      242
Davis, P.J.      287
Debeau, F.      285
Dellacherie, C.      vi 254 290
Determinant      49
Diaconis, P.      vi
Diophantus      40 286
Diophantus identity and Brahmagupta      47 237
Diophantus identity and complex factorization      237
Diophantus identity and Pythagoras      47 237
Dissection of integrals      106
Doubly stochastic      196
Dragomir, S.S.      241 285
Dudley, R.M.      vi
Elementary symmetric function      178
Elliot, E.B.      290
Enflo inequality      225
Engel, A.      94 246 251
Equality in AM-GM inequality      22
Equality in Cauchy inequality      5 37
Equality in Cauchy — Schwarz      8
Equality in Hoelder      137
Equality in Jensen inequality      89
Erdoes, P.      261
Euclidean distance in $\mathbb{R}^d$       51
Euclidean distance, triangle inequality      53
Exponential sums      210
Extend and conquer      222
Feng, Z.      251
Fermat, Pierre de      40
fibonacci      286
Fisher information      18
Flor, P.      238
Four letter identity      49
Fuji, M.      287
Gamma function      165

Gauss, C.F.      288
General means      120
Generic improvement      94
Geometric mean as a limit      120
Geometric mean as minimum      133
Geometric mean, superadditivity      100 133
George, C.      231 288
Gram — Schmidt process      70
Gross — Erdmann, K.-G.      290
Gruess inequality      119
Hadwiger — Finsler inequality      102
Hajela, D.      289
Halmos, P.      278
Hammer, D.      231
Hardy's inequality      166
Hardy's inequality and Hilbert      176
Hardy's inequality in       176
Hardy's inequality, discrete      169
Hardy's inequality, geometric refinement      177
Hardy's inequality, special instance      177
Hardy, G.H.      197 290
Harker — Kasper inequality      14
Harmonic mean      126
Harmonic mean, minimax characterization      132
Harmonic series, divergence      99 255
Heisenberg principle      288
Hewitt, E.      243
Hilbert inequality      155
Hilbert inequality, homogeneous kernel      164
Hilbert inequality, integral version      163
Hilbert inequality, max version      163
Hilbert inequality, via Toeplitz method      165
Hilbert's 17th problem      46
Hilbert, David      46 55
HM-GM and HM-AM inequalities      126
Hoelder inequality      135 136
Hoelder inequality, case of equality      137
Hoelder inequality, converse      139
Hoelder inequality, defect estimate      94
Hoelder inequality, historical form      151
Hoelder inequality, stability      144 145
Hoelder, Otto Ludwig      94 135 263 290
Homogeneity in $\Sigma$       132
Homogenization trick      189
How to Solve It      30
Humble bound      284
Inclusion radius      148
Inner product space, Cauchy — Schwarz inequality      8
Inner product space, definition      7
Integral representations      116
Interpolation in Cauchy inequality      48
Intuition, how much      55
Intuition, refining      53
Investment inequalities      100
Isometry      60
Isoperimetric property      19
Isoperimetric property for the cube      34
Israel, R.      253
Iteration and discovery      149
J-convexity      101
Janous, Walther      177
Jensen inequality      87 263
Jensen inequality and Schur convexity      201
Jensen inequality for integrals      113
Jensen inequality via Cauchy's induction      101
Jensen inequality, case of equality      89
Jensen inequality, geometric applications      93
Jensen inequality, Hoelder's defect estimate      94
Jensen inequality, implies Minkowski      150
Jensen, J.L.W.V.      101
Joag-Dev, K.      277
Kahane, J.-P.      vi
Karayannakis, D.      286
Kedlaya, K.      vi 152 264 274
Knuth, D.      260
Koerner, T.      vi
Komlos, J.      277
Kronecker's lemma      177
Kubo, F.      287
Kufner, A.      290
Kuniyeda, M.      262
Lagrange identity      39
Lagrange identity, continuous analogue      48
Lagrange identity, simplest case      42
Lagrange, Joseph Louis de      40
Landau's notation      120
Leap-forward fall-back induction      20 31 36 101
Lee, H.      vi 247 251
Light cone defined      62
Light cone inequality      63 245
Littlewood, J.E.      197
Looking back      25 26
Loomis — Whitney inequality      16
Lorentz product      62
Lovasz, L.      278
Lozansky, E.      264
Lyusternik, L.A.      245
Maclaurin inequality      285
Magiropoulos, M.      286
Majorant principles      284
Majorization      191
Maligranda, L.      vi 264 288 289
Marriage lemma      206
Marshall, A.      277
Matousek, J.      vi 285
McConnell, T.R.      277
Meng, X.      vi
Mengoli, P.      99 248
Method of halves      122
Method of parameterized parameters      164
Metric space      54
Mignotte, M.      262
Milne inequality      50
Minimal surfaces      10
Minkowski inequality      141
Minkowski inequality via Jensen      150
Minkowski inequality, Riesz proof      141
Minkowski's conjecture      44 46
Minkowski's light cone      62
Minkowski, Hermann      44
Mitrinovic, D.S.      277 283
Moebius transformation      242
Moment sequences      149
Monotonicity and integral estimates      118
Montgomery, H.L.      284
Motzkin, T.S.      46 286
Muirhead condition and majorization      195
Nakhash, A.      253
Names of inequalities      11
Naor, E.      259
Needham, T.      242
Nesbitt inequality      84 131 246
Neyman — Pearson lemma      287
Niculescu, C.P.      290
Nievergelt, Y.      253
Niven, I.      260
Nonnegative polynomials      43
Norm, defined      55
Norm, p-norm, or -norm      140
Normalization method      5 25 26 66
Normed linear space      55
Obvious and not      56
Obvious Hilbert story      56
Obvious triangle inequality      56
Olkin, I.      277
One-trick      see 1-trick
Opic, B.      290
Optimality principles      33
Order inequality      76
Order relationship to quadratic      76

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