Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation, Vol. I

Cover......Page 1
Asymptotics in Dynamics,\rGeometry and PDEs;\rGeneralized Borel\rSummation,\rvol. I......Page 4
ISBN: 9788876423741......Page 5
Table of Contents\r......Page 6
Introduction......Page 12
Authors’ affiliations......Page 14
1 Introduction......Page 17
2 Previous results on complex classical mechanics......Page 18
3 Classical particle in a complex elliptic potential......Page 28
4 Summary and discussion......Page 31
References......Page 32
1 Introduction......Page 35
2 Parabolic dynamics in one variable......Page 38
3 Germs tangent to the identity......Page 39
4.1 Semiattractive germs......Page 42
4.2 Quasi-parabolic germs......Page 43
5 One-resonant germs......Page 45
References......Page 47
Power series with sum-product Taylor coefficients and their resurgence algebra......Page 51
The notion of SP series......Page 53
Overview......Page 54
A gratifying surprise: the mir-transform......Page 55
1.3 The four gates to the inner algebra......Page 56
2.1 Resurgent functions and their three models......Page 59
Resurgent algebras: the multiplicative structure......Page 61
The upper/lower Borel-Laplace transforms......Page 62
The pros and cons of the upper/lower choices......Page 63
2.2 Alien derivations as a tool for Riemann surface description......Page 64
Definition of the operators ω and ω......Page 65
Lateral and median singularities......Page 66
Strong versus weak resurgence......Page 67
The pros and cons of the 2πi factor......Page 68
Retrieving closest singularities......Page 69
Retrieving distant singularities......Page 70
The Bernoulli numbers and polynomials......Page 71
3.2 Resurgence of the Gamma function......Page 73
3.3 Monomial/binomial/exponential factors......Page 75
Monomial factors......Page 76
3.4 Resummability of the total ingress factor......Page 77
4.1 Some heuristics......Page 78
4.2 The long chain behind nir//mir......Page 83
Details of the nine steps:......Page 84
Integral expression of nir......Page 85
4.4 The reciprocation transform......Page 86
4.5 The mir transform......Page 89
4.6 Translocation of the nir transform......Page 92
4.7 Alternative factorisations of nir. The lir transform......Page 96
4.8 Application: kernel of the nir transform......Page 98
4.9 Comparing/extending/inverting nir and mir......Page 99
5.1 Some heuristics......Page 101
The short, four-link chain:......Page 102
Details of the four steps:......Page 103
Details of the nine steps:......Page 104
5.3 The nur transform......Page 105
5.4 Expressing nur in terms of nir......Page 107
5.5 The mur transform......Page 108
5.7 Removal of the ingress factor......Page 109
6.1 “Variable” and “covariant” differential equations......Page 110
Existence and calculation of the covariant ODEs for f (0) = 0......Page 113
Existence and calculation of the covariant ODEs for f (0)  = 0......Page 114
Dimensions of spaces of variable ODEs......Page 115
Tables of dimensions for low degrees r = deg( f )......Page 116
Differential polynomial P in the noncommuting variables (n, ν)......Page 117
Basic symmetric functions x∗s , x∗∗ s , ν∗s , ν∗∗......Page 118
Centered β-coefficients......Page 119
6.3 Explicit ODEs for low-degree polynomial inputs f......Page 120
Input f of degree 1......Page 121
Input f of degree 3......Page 122
Input f of degree 4......Page 126
6.4 The global resurgence picture for polynomial inputs f......Page 128
6.5 The antipodal exchange for polynomial inputs f......Page 130
6.6 ODEs for monomial inputs F......Page 134
6.7 Monomial inputs F: global resurgence......Page 136
The simple-crossing matricesMk ,Mk......Page 138
The auxiliary polynomials Hδd (x, y)......Page 140
The left normalising matrix L......Page 141
Normalisation identities for the full-turn matricesMp+k,k......Page 142
Defining identities for the normalising matrices L,R......Page 143
Complement: partial rotations with   =  1......Page 144
6.9 Ramied monomial inputs F: innite order ODEs......Page 146
Special values of the T2 k-polynomials......Page 148
6.10 Ramified monomial inputs F: arithmetical aspects......Page 149
6.11 From flexible to rigid resurgence......Page 152
7 The general resurgence algebra for SP series......Page 157
7.1.1 From original to outer......Page 158
7.1.4 From exceptional to inner......Page 160
7.1.5 From inner to inner. Ping-pong resurgence......Page 161
7.1.6 Recapitulation. One-way arrows, two-way arrows......Page 162
7.2 Meromorphic input F: the general picture......Page 163
Converting ν-singularities into ζ -singularities......Page 164
Resurgence properties of the 0-based singularities......Page 165
7.4 Rational inputs F: the inner algebra......Page 166
8.1 Polynomial inputs f . Examples......Page 167
8.3 Rational inputs F. Examples......Page 168
8.4 Holomorphic/meromorphic inputs F. Examples .......Page 174
9.1 The knot 41 and the attached power series GNP, GP......Page 176
9.2 Two contingent ingress factors......Page 178
9.4 Two outer generators Lu and Luu......Page 179
9.6 One exceptional generator Le......Page 180
9.7 A complete system of resurgence equations......Page 181
9.8.1 From Li to Lii and back (inner to inner)......Page 184
9.8.2 From Lo to Li (original to close-inner)......Page 185
9.8.3 From Lo to Lii (original to distant-inner)......Page 186
9.8.4 From Lo to Lu (original to outer)......Page 188
9.8.5 From Lu to Li and Lii (outer to inner)......Page 190
9.8.7 From Loo to Lii (original to distant-inner)......Page 191
9.8.9 From Luu to Li and Lii (outer to inner)......Page 192
10.1.1 Functional transforms......Page 194
10.1.3 Parity relations......Page 195
10.2.2 Compact form......Page 196
10.3.1 Tangency 0, ramification 1......Page 198
10.3.2 Tangency 1, ramification 2......Page 199
10.3.3 Tangency 2, rami cation 3......Page 200
10.3.4 Tangency 3, ramification 4......Page 201
10.4.1 Tangency 0, ramification 1......Page 202
10.4.2 Tangency 1, ramification 2......Page 203
10.4.3 Tangency 2, ramification 3......Page 204
10.4.4 Tangency 3, rami cation 4......Page 205
10.5.2 Tangency > 0......Page 206
10.6.1 Standard case......Page 207
10.6.2 Free-β case......Page 208
11.1 The original generators Lo and Loo......Page 209
11.2 The outer generators Lu and Luu......Page 213
11.3 The inner generators Li and Lii......Page 214
11.4 The exceptional generator Le......Page 215
References......Page 216
1 Introduction......Page 217
2 Formally linearizable germs......Page 221
3 Convergence under the reduced Brjuno condition......Page 225
4 Russmann condition vs. reduced Brjuno condition......Page 232
References......Page 233
1 Introduction......Page 235
2 Symmetric functions......Page 236
3.1 Basic definitions......Page 240
3.2 Generators and linear bases......Page 242
3.3 Solomon’s descent algebras......Page 243
3.5 The Hausdorff series......Page 245
3.7 Lie idempotents as noncommutative symmetric functions......Page 249
3.8 The (1 − q)-transform......Page 250
3.9 Hopf algebras enter the scene......Page 251
3.10 A one parameter family of Lie idempotents......Page 252
3.11 The iterated q-bracketing and its diagonalization......Page 253
3.12 Noncommutative symmetric functions and mould calculus......Page 255
3.12.2 S and  : symmetral/alternel......Page 256
3.12.3 An alternal mould: the Magnus expansion......Page 258
3.12.5 Moulds related to the Fer-Zassenhaus expansion......Page 259
3.13 Noncommutative symmetric functions and alien calculus......Page 260
4.1 Free quasi-symmetric functions......Page 262
4.2 From permutations to binary trees......Page 264
4.3 Trees from functional equations......Page 265
5.2 A Hopf algebra on parking functions......Page 267
5.3 A Catalan algebra related to quasi-symmetric functions......Page 268
References......Page 269
CRM Series Publications by the Ennio De Giorgi Mathematical Research Center Pisa......Page 275