Preface
List of Contributors
Part I Value Distribution of Complex and P-adic Functions
Chapter 1 The Second Main Theorem on Generalized Parabolic Manifolds
1.1 Monge-Ampere equations and generalized parabolic manifolds
1.2 Projectivized bundles over Stein manifolds
1.3 Meromorphic global forms
1.4 Analytic and algebraic Plucker Formulas: The classical case
1.5 Plucker’s formulas for generalized parabolic manifolds
1.6 An analogue of the Ahlfors-Stoll estimate
1.7 The second main theorem
Chapter 2 P-adic Value Distribution
2.1 Ultrametric analytic functions
2.2 Lazard’s problem and p-adic Nevanlinna theory
2.3 Applications of the Nevanlinna theory
Chapter 3 Survey on Meromorphic Functions of Uniqueness
3.1 Introduction and basic results
3.2 Main results and examples
Chapter 4 A Survey on Uniqueness Polynomials and Unique Range Sets
4.1 Meromorphic functions sharing points
4.2 Unique range set for meromorphic functions
4.3 Uniqueness polynomials for meromorphic functions
Chapter 5 On Petrenko’s Theory of Growth of Meromorphic Functions
5.1 The growth of functions meromorphic in the plane
5.2 The growth of functions meromorphic in the disc
5.3 Separated maximum modulus points of entire and meromorphic functions
5.4 Strong asymptotic values and strong asymptotic spots of entire and meromorphic functions
Chapter 6 Linear Operators，Fourier Transforms and the Riemann ξ-function
6.1 The Laguerre-Polya class and the Riemann ξ-function
6.2 Complex zero decreasing sequences and λ-sequences
6.3 The distribution of zeros of Fourier transforms
6.4 Infinite order differential operators and the Riemann ξ-function
Chapter 7 Hyperbolic Hypersurfaces of Lower Degrees
7.1 Introduction and main techniques
7.2 Hyperbolic curves
7.3 Hyperbolic surfaces
7.4 Hyperbolic hypersurfaces in P4(C)
Chapter 8 Admissible Solutions of Functional Equations of Diophantine Type
8.1 Introduction
8.2 Background and results
8.3 Preliminary lemmas
8.4 Proofs of the results
Part II New Applications of the Concept of Differentiability
Chapter 9 Recent Developments in Applied Pseudoanalytic Function Theory
9.1 Introduction
9.2 Some definitions and results from pseudoanalytic function theory
9.3 Solutions of second order elliptic equations as real components of complex pseudoanalytic functions
9.4 Complete systems of solutions for second-order equations
9.5 A remark on orthogonal coordinate systems in a plane
9.6 Explicit construction of a generating sequence
9.7 Explicit construction of complete systems of solutions of second-order elliptic equations
Chapter 10 Biquaternions for Analytic and Numerical Solution of Equations of Electrodynamics
10.1 Introduction
10.2 Biquaternionic fundamental solutions
10.3 Biquaternionic reformulation of Maxwell’s equations in chiral media
10.4 Completeness of a system of biquaternionic fundamental solutions
10.5 Numerical realization
10.6 Time-dependent Maxwell’s equations for chiral media
10.7 Field equations in a biquaternionic form
10.8 Green function for the operator M
10.9 Inhomogeneous media
Chapter 11 Asymptotic Behavior of Subfunctions of Time-Independent Schrodinger Operators
11.1 Introduction
11.2 Subfunctions of the operator Lc
11.3 Phragm′en-Lindel¨of theorem for subfunctions in an n-dimensional cone
11.4 Bilinear series and estimates of the Green’s function of the operator Lq with a radial potential in a cone
11.5 The Blaschke theorem
11.6 Generalization of the Hayman-Azarin theorem
11.7 Subfunctions in tube domains
11.8 Green’s function of the operator Lc
11.9 Eigenfunctions of the Laplace-Beltrami operator on the unit sphere
11.10 Asymptotic behavior of solutions of the equation y"(r)+(n-1)r-1y'(r)-(λr-2+q(r)y(r)=0
Chapter 12 The Bank-Laine Conjecture-a Survey
12.1 Introduction
12.2 Elementary computations and examples
12.3 First results towards the Bank-Laine conjecture
12.4 Solutions with prescribed zero-sequences
12.5 Bank-Laine sequences
12.6 Bank-Laine functions
12.7 The case of a meromorphic A(z)
12.8 Bank-Laine functions in the unit disc
Chapter 13 On Complex Solutions to Functional and Non-linear Partial Differential Equations
13.1 Introduction
13.2 The equation f2+g2=1 and related eikonal type equations
13.3 Factorization of partial derivatives and its relation to partial differential equations
13.4 Exact entire solutions of generalized Eikonal and Fermat equations
13.5 Eikonal and Fermat equations with arbitrarily many terms
Chapter 14 Clifford Algebras Depending on Parameters and Their Applications to Partial Differential Equations
14.1 Introduction
14.2 Clifford algebras depending on parameters
14.3 Generalized Clifford analysis
14.4 Concluding remarks
Chapter 15 Application of Complex Analytic Method to Equations of Mixed (Elliptic-Hyperbolic) Type
15.1 Oblique derivative problem for second order equations of mixed type with parabolic degeneracy
15.2 Oblique derivative problem for second order elliptic equations with parabolic degeneracy
15.3 Oblique derivative problem for degenerate hyperbolic equations of second order
15.4 Oblique derivative problem for second order equations of mixed type with parabolic degeneracy
15.5 Complex analytic method for mixed equations with parabolic degeneracy and some open problems
Chapter 16 The Tricomi Problem for Second Order Equations of Mixed Type
16.1 Formulation of the Tricomi problem for mixed equations with degenerate rank 0
16.2 Representation of solutions of Tricomi problem for the degenerate mixed equations
16.3 Existence of solutions of the Tricomi problem for degenerate mixed equations
16.4 Existence of solutions of the Tricomi problem for elliptic equations with parabolic degeneracy
Part III Boundary Value Problems
Preface of G.Akhalaia and N.Manjavidze
Introduction to the Chapters 17-19
Chapter 17 The Problem of Linear Conjugation and Systems of Singular Integral Equations
17.1 Formulation of the problem
17.2 Boundary value problem of linear conjugation with continuous coefficients
17.3 Boundary value problems with piecewise continuous coefficients
17.4 Systems of singular integral equations
17.5 Differentiability of solutions and singular integral equations
Chapter 18 Linear Conjugation with Displacement for Analytic Functions
18.1 Introduction and auxiliary propositions
18.2 Linear conjugation with displacement in case of continuous coefficients
18.3 Linear conjugation with displacement in case of piecewise continuous coefficients
18.4 Boundary value problems with displacement containing complex conjugate values of the desired functions
Chapter 19 Linear Conjugation with Displacement for Generalized Analytic Functions and Vectors
19.1 Definitions and notations
19.2 Relation between linear conjugation with displacement and generalized analytic functions
19.3 Boundary value problem of linear conjugation with displacement for generalized analytic vectors
19.4 The problem of linear conjugation with displacement for an elliptic system of differential equations
19.5 Differential boundary value problems for generalized analytic vectors
Chapter 20 On Boundary Value Problems for Non-Linear Systems of Partial Differential Equations in the Plane
20.1 Introduction
20.2 Dirichlet problem in simply connected domains
20.3 Dirichlet problem in multiply connected domains
20.4 Riemann-Hilbert problem for simply connected domains
20.5 Application of the Schauder principle |
