The history of mathematics is, to a considerable extent, connected with the study of solutions of the equation f(x)=a=const for functions f(x) of one real or complex variable. Therefore, it is surprising that we know very little about solutions of u(x,y)=A=const for functions of two real variables. These two solutions, called level of sets, are very important with regard to applications in physics, biology and economics as they make a map of appropriate processes described by the function u(x,y) for given parameters (x,y). This text explores a concept, Gamma-lines, which generalizes the concept of levels of sets and, at the same time, the concept of a-points. The authors provide a book on Gamma-lines for the broad specialist and show the large range of their field of applications. The general methods proposed in this volume are useful for both physicists and engineers.
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.