Nonlinear Equations - UFRJ

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44 [CH. 4: DIFFERENTIAL FORMS Properties (4.1) and (4.2) hold in this context. We introduce the exterior derivative operator d : A k → A k+1 : dα x = ∑ 1≤i 1
[SEC. 4.2: COMPLEX DIFFERENTIAL FORMS 45 switch to another convention: if z is a complex number, x is its real part and y its imaginary part. This convention extends to vectors so z = x + √ −1 y. The sets C n and R 2n may be identified by ⎡ ⎤ x 1 y 1 z = x 2 . ⎢ ⎥ ⎣ . ⎦ y n It is possible to define alternating k-forms in C n as complex-valued alternating k-forms in R 2n . However, this approach misses some of the structure related to the linearity over C and holomorphic functions. Instead, it is usual to define A k0 as the space of complex valued alternating k-forms in C n . A basis for A k0 is given by the expressions dz i1 ∧ · · · ∧ dz ik , 1 ≤ i 1 < i 2 < · · · < i k ≤ n. They are interpreted as dz i1 ∧ · · · ∧ dz ik (u 1 , . . . , u k ) = ∑ σ∈S k (−1) |σ| u σ(1)i1 u σ(2)i2 · · · u σ(k)ik . Notice that dz i = dx i + √ −1 dy i . We may also define d¯z i = dx i − √ −1 dy i . Next we define A kl as the complex vector space spanned by all the expressions dz i1 ∧ · · · ∧ dz ik ∧ d¯z j1 ∧ · · · ∧ d¯z jl for 1 ≤ i 1 < i 2 < · · · < i k ≤ n, 1 ≤ j 1 < j 2 < · · · < j l ≤ n. Since dx i ∧ dy i = −2 √ −1 dz i ∧ d¯z i , the standard volume form in C n is (√ ) n −1 dV = dx 1 ∧ dy 1 ∧ · · · ∧ dy n = dz 1 ∧ d¯z 1 ∧ · · · ∧ d¯z n . 2 The following fact is quite useful:
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Nonlinear Equations
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Copyright © 2011 by Gregorio Malaj
Page 9 and 10: Foreword I added together the ratio
Page 11 and 12: ix another book with a systematic p
Page 13 and 14: Contents Foreword vii 1 Counting so
Page 15: CONTENTS xiii 10 Homotopy 135 10.1
Page 18 and 19: 2 [CH. 1: COUNTING SOLUTIONS if and
Page 20 and 21: 4 [CH. 1: COUNTING SOLUTIONS connec
Page 22 and 23: 6 [CH. 1: COUNTING SOLUTIONS 1.2 Sh
Page 24 and 25: 8 [CH. 1: COUNTING SOLUTIONS has ex
Page 26 and 27: 10 [CH. 1: COUNTING SOLUTIONS equat
Page 28 and 29: Chapter 2 The Nullstellensatz The s
Page 30 and 31: 14 [CH. 2: THE NULLSTELLENSATZ prin
Page 32 and 33: 16 [CH. 2: THE NULLSTELLENSATZ or a
Page 34 and 35: 18 [CH. 2: THE NULLSTELLENSATZ 3. T
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Page 38 and 39: 22 [CH. 2: THE NULLSTELLENSATZ 1. T
Page 40 and 41: 24 [CH. 2: THE NULLSTELLENSATZ Proo
Page 42 and 43: 26 [CH. 2: THE NULLSTELLENSATZ To e
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Page 46 and 47: 30 [CH. 2: THE NULLSTELLENSATZ 2.7
Page 48 and 49: 32 [CH. 2: THE NULLSTELLENSATZ Coro
Page 50 and 51: 34 [CH. 3: TOPOLOGY AND ZERO COUNTI
Page 58 and 59: Chapter 4 Differential forms Throug
Page 62 and 63: 46 [CH. 4: DIFFERENTIAL FORMS Lemma
Page 64 and 65: 48 [CH. 4: DIFFERENTIAL FORMS 2. cl
Page 66 and 67: 50 [CH. 4: DIFFERENTIAL FORMS be me
Page 68 and 69: 52 [CH. 4: DIFFERENTIAL FORMS Expan
Page 70 and 71: 54 [CH. 4: DIFFERENTIAL FORMS We co
Page 72 and 73: 56 [CH. 5: REPRODUCING KERNEL SPACE
Page 88 and 89: Chapter 6 Exponential sums and spar
Page 90 and 91: 74 [CH. 6: EXPONENTIAL SUMS AND SPA
Page 98 and 99: Chapter 7 Newton Iteration and Alph
Page 100 and 101: 84 [CH. 7: NEWTON ITERATION As long
Page 102 and 103: 86 [CH. 7: NEWTON ITERATION Proposi
Page 104 and 105: 88 [CH. 7: NEWTON ITERATION 1 y =
Page 106 and 107: 90 [CH. 7: NEWTON ITERATION t 1 t 2
Page 108 and 109: 92 [CH. 7: NEWTON ITERATION The Tay
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94 [CH. 7: NEWTON ITERATION 2 63 2
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96 [CH. 7: NEWTON ITERATION Exercis
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98 [CH. 7: NEWTON ITERATION The con
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100 [CH. 7: NEWTON ITERATION Let us
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102 [CH. 7: NEWTON ITERATION Passin
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104 [CH. 7: NEWTON ITERATION 13−3
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106 [CH. 7: NEWTON ITERATION Proof.
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108 [CH. 8: CONDITION NUMBER THEORY
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122 [CH. 9: THE PSEUDO-NEWTON OPERA
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136 [CH. 10: HOMOTOPY form for x i+
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138 [CH. 10: HOMOTOPY Again, f t is
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140 [CH. 10: HOMOTOPY We will need
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142 [CH. 10: HOMOTOPY P n+1 x i [N(
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144 [CH. 10: HOMOTOPY Let d Riem de
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146 [CH. 10: HOMOTOPY Lemma 10.13.
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148 [CH. 10: HOMOTOPY above, the se
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150 [CH. 10: HOMOTOPY Corollary 10.
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152 [CH. 10: HOMOTOPY by the geomet
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154 [CH. 10: HOMOTOPY 2. The condit
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Appendix A Open Problems, by Carlos
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[SEC. A.3: EQUIDISTRIBUTION OF ROOT
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[SEC. A.5: EXTENSION OF THE ALGORIT
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[SEC. A.7: INTEGER ZEROS OF A POLYN
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166 BIBLIOGRAPHY [12] , Smale’s 1
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168 BIBLIOGRAPHY [42] Michael R. Ga
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170 BIBLIOGRAPHY [69] , Complexity
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Glossary of notations As a general
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Index algorithm discrete, x Homotop
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INDEX 177 complexity of homotopy, 1
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Nonlinear Equations - UFRJ

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