Gamma-Lines

Preface. Tangent Variation Principle. Satellite Principles. Modification of Length-area Principle. Tangent Variation Principle. Estimates for collections of Gamma-Lines. Estimates of lengths of Gamma-Lines for angular-quasiconformal mappings. Remarks on applying of estimates of L (D, Gamma). Nevanlinna and Ahilfors Theories. Additions. Basic concepts and outcomes of Nevanlinna Value Distribution theory and Ahlfors theory of covering surfaces. Geometric deficient values. On some additions to L. Ahlfor's theory of covering surfaces. Bounds of some integrals. Gamma-Lines Approach in the Theory of Meromorphic Functions. Principle of closeness of sufficiently large sets of Alpha-points of meromorphic functions. Integrated Version of the Principle. Connections with known classes of functions. Distribution of Gamma-Lines for Functions Meromorphic in C. Applications. The main results on distribution of Gamma-Lines. "Wingdings" of Gamma-Lines. Average lengths of Gamma-Lines along concentric circles and the deficient values. Distribution of Gamma-Lines and value distribution of subclasses of modules and real parts of mermorphic functions. The number of Gamma-Lines crossing rings. Distribution of Gelfond points. Nevalinna's dream-description of transcendental ramification of Riemann surfaces. The proximity property of Alpha-points of meromorphic functions. A proof of the proximity property of Alpha-points based only on investigation of Gamma-Lines. Some Applied Problems. Gamma-Lines in Physics. On the cross road of value distribution, Gamma-Lines, free boundary theories and applied mathematics. "Pointmaps" of physical processes and Alpha-points of general classes of functions Principles. Nevanlinna and Ahilfors Theories. Additions. Gamma-Lines Approach in the Theory of Meromorphic Functions. Distribution of Gamma-Lines for Functions Mermorphic in C. Applications. Some Applied Problems.