Complex Analysis and Potential Theory: Proceedings of the Conference Satellite to ICM 2006

Tahir Aliyev Azeroglu, Promarz M. Tamrazov9789812705983, 981-270-598-8

This volume gathers the contributions from outstanding mathematicians, such as Samuel Krushkal, Reiner Kühnau, Chung Chun Yang, Vladimir Miklyukov and others.It will help researchers to solve problems on complex analysis and potential theory and discuss various applications in engineering. The contributions also update the reader on recent developments in the field. Moreover, a special part of the volume is completely devoted to the formulation of some important open problems and interesting conjectures.

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Table of contents :
CONTENTS……Page 10
Preface……Page 6
Participants……Page 8
Part A TALKS……Page 14
1.1…….Page 16
2. Sketch of the proof of Theorem 1……Page 18
2.2…….Page 19
2.4…….Page 21
2.5…….Page 22
3.3…….Page 25
3.5…….Page 27
References……Page 28
1. Introduction and Results……Page 30
2. Some Lemmas……Page 33
3. Proof of Theorem 1……Page 42
6. Proof of Corollary 2……Page 43
7. Proof of Corollary 3……Page 44
References……Page 45
1. Isothermal coordinates……Page 46
2. Canonical homeomorphisms……Page 49
3. Main Theorem……Page 51
4. Remarks on W1,2 majorized functions……Page 53
References……Page 56
1. Introduction……Page 59
2. Proof of Theorem 2……Page 62
3. Examples of mappings (9)……Page 63
4. Analogue: Golusin’s inequalities……Page 65
References……Page 66
1. Introduction……Page 67
2.1. Fixed membrane……Page 68
2.2. Free membrane……Page 71
3. Stekloff eigenvalue problems……Page 74
3.1. Stekloff problem……Page 75
3.2. Mixed Stelcloff problem……Page 76
References……Page 77
Geometry of the General Beltrami Equations B. Bojarski……Page 79
1. Generating the Beltrami equations……Page 81
2. Principal homeomorphisms’ of the Beltrami equations……Page 82
3. Structure theorem for general Beltrami equations……Page 86
4. Primary solutions of the general Beltrami equations……Page 90
5. Lavrentiev fields and quasiconformal mappings……Page 92
6. Uniqueness in the general measurable Riemann mapping theorem……Page 94
References……Page 95
1. Biharmonic boundary value problems……Page 97
2. A representation formula……Page 98
3. A polyharmonic Dirichlet problem……Page 107
4. Appendix……Page 122
References……Page 127
1. Introduction……Page 129
2. Main Results……Page 132
3. Proof of Theorems……Page 134
References……Page 137
A Generalized Schwartz Lemma at the Boundary T. Aliyev Azeroilu and B. N . Ornek……Page 138
References……Page 142
1. Introduction…….Page 144
2. A concrete case: the case of the nonlinear Robin boundary conditions……Page 149
References……Page 150
1. Klein surface of a real function field……Page 153
2. Abelian Differentials……Page 155
References……Page 157
Combinatorial Theorems of Complex Analysis Yu. B. Zelinskii……Page 158
References……Page 160
1. The Cauchy theorems for univalent functions…….Page 161
2. Main result for locally univalent functions…….Page 163
3. Generalizations of the Bohr and Menshoff theorems for continuous functions…….Page 164
4. Generalized quasiconformal mappings in Rn…….Page 165
5. Equivalence of analytic and geometric descriptions…….Page 167
References……Page 168
1.1. Separately subharmonic functions…….Page 169
1.2. Functions subharmonic in one variable and harmonic in the other…….Page 170
2.2…….Page 171
3.1…….Page 172
4. The result of Cegrell and Sadullaev……Page 173
5. The result of Kolodziej and Thornbiornson……Page 174
References……Page 177
1. Introduction……Page 179
2. A problem about extracting harmonic triad of vectors……Page 180
3. Monogenic functions in an infinite-dimensional harmonic algebra……Page 181
4. Monogenic functions associated with axial-symmetric potential fields……Page 184
5. Integral expressions for axial-symmetric potential and the the Stokes flow function……Page 185
References……Page 186
2. Results……Page 187
3. Proofs……Page 188
References……Page 190
1. Introduction……Page 191
2. Notations and preliminaries……Page 192
3. Martin boundary associated with (S)……Page 193
4. Restricted mean value property……Page 195
References……Page 199
An Implicit Function Theorem for Sobolev Mappings I. V. Zhuravlev……Page 200
References……Page 203
2. Ramanujan’s Integral formula……Page 204
4-1 Fourier Bore1 transform……Page 205
4-2 Avanissian-Gay transform……Page 206
6. A relation between Ramanujan’s Integral formula and Shannon’s sampling theorem……Page 207
8. The meaning of Plana’s summation formula in the theory of analytic functionals……Page 208
References……Page 209
1. Introduction……Page 211
2. Preliminaries……Page 212
3. Asymptotic expansions of the solutions to the heat equations with the tempered distributions initial data……Page 213
4. Asymptotic expansions of the solutions to the heat equations with the distributions of exponential growth initial data……Page 216
References……Page 219
1. Introduction……Page 220
2. Separation of singularities for harmonic differential forms……Page 221
3. Mittag – Leffler theorem for harmonic differential forms……Page 224
4. Weierstrass’s theorem for harmonic differential forms……Page 225
References……Page 226
1. Introduction and main results……Page 227
2. Auxiliary results……Page 230
References……Page 233
1. Introduction……Page 235
2.2. Harmonic polynomials……Page 236
3.1. Harmonic transfinite diameter……Page 237
3.2. Harmonic Chebyshev constants……Page 238
3.3. Relation between the characteristics……Page 240
4. Prolate spheroids……Page 241
References……Page 243
1. Introduction……Page 244
2. Estimates for uniform moduli of smoothness of arbitrary order……Page 245
3. Estimates for local moduli of smoothness of arbitrary order……Page 246
4. Estimates for integral moduli of smoothness of arbitrary order……Page 249
References……Page 250
1. Introduction……Page 252
2. Definition and Notation……Page 253
3. Main Results……Page 254
References……Page 261
1. Introduction……Page 262
2. Homogeneous problem……Page 263
3. Nonhomogeneous problem……Page 264
References……Page 268
1. Introduction. Modified Crank-Nicholson Difference Schemes……Page 269
2. Theorem on Stability……Page 270
3. Applications……Page 276
4. Numerical Analysis……Page 279
References……Page 284
Part B OPEN PROBLEMS……Page 286
Some Old (Unsolved) and New Problems and Conjectures on Functional Equations of Entire and Meromorphic Functions C.-C. Yang……Page 288
References……Page 291
References……Page 292
Open Problems on Hausdorff Operators E. Laflyand……Page 293
References……Page 297
Author Index……Page 300