The object of these 2 volumes of collected papers is to provide insight and perspective on various research problems and theories in modern topics of Calculus of Variations, Complex Analysis, Real Analysis, Differential Equations, Geometry and their Applications, related to the work of Constantin Carathéodory. This work will be of interest both to researchers following the development of new results, and to people seeking an introduction in these fields.

The goal of this book is to investigate further the interdisciplinary interaction between Mathematical Analysis and Topology. It provides an attempt to study various approaches in the topological applications and influence to Function Theory, Calculus of Variations, Functional Analysis and Approximation Theory. The volume is dedicated to the memory of S Stoilow.

This book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry in which analogies with classical results form the right tool. The premise of analogy as a study strategy is to make the unfamiliar familiar. Accordingly, this book introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors. Gyrovectors turn out to be equivalence classes that add according to the gyroparallelogram law just as vectors are equivalence classes that add according to the parallelogram law. In the gyrolanguage of this book, accordingly, one prefixes a gyro to a classical term to mean the analogous term in hyperbolic geometry. As an example, the relativistic gyrotrigonometry of Einstein's special relativity is developed and employed to the study of the stellar aberration phenomenon in astronomy.Furthermore, the book presents, for the first time, the relativistic center of mass of an isolated system of noninteracting particles that coincided at some initial time t = 0. It turns out that the invariant mass of the relativistic center of mass of an expanding system (like galaxies) exceeds the sum of the masses of its constituent particles. This excess of mass suggests a viable mechanism for the formation of dark matter in the universe, which has not been detected but is needed to gravitationally 'glue' each galaxy in the universe. The discovery of the relativistic center of mass in this book thus demonstrates once again the usefulness of the study of Einstein's special theory of relativity in terms of its underlying hyperbolic geometry.

Constantin Carathiodory (Berlin 1873-1950 Munich) - Mathematics and Politics in Turbulent Times is a biography of a mathematician who became famous during his life, but has hitherto been ignored by historians for half a century after his death. In a thought-provoking approach, Maria Georgiadou devotes to Constantin Carathiodory all the attention such a personality deserves. With breathtaking detail and the appropriate scrutiny she elucidates his oeuvre, life and turbulent political/historical surroundings: descending from the Greek ilite of Constantinople, Carathiodory graduated from the military school of Brussels, became engineer at the Assiout dam in Egypt and finally dedicated a life of effort to mathematics and education. He studied and embarked on an international academic career, haunted by wars, catastrophes and personal tragedies. Over the last years of his life, he stayed in Munich despite World War II, an ambiguous decision upon which the author sheds unprecedented light. Carathiodory's most significant mathematical contributions were to the calculus of variations, the theory of point set measure, and the theory of functions of a real variable, pdes, also to complex function theory. The interdisciplinarity of the text allows easy access for both scholars and readers with a general interest in mathematics, politics and history. The thoroughness of the author's research and evaluations is certain to leave everyone impressed and more knowledgeable.

This book develops methods which explore some new interconnections and interrelations between Analysis and Topology and their applications. Emphasis is given to several recent results which have been obtained mainly during the last years and which cannot be found in other books in Nonlinear Analysis. Interest in this subject area has rapidly increased over the last decade, yet the presentation of research has been confined mainly to journal articles.

The volume will consist of about 40 articles written by some very influential mathematicians of our time and will expose the latest achievements in the broad area of nonlinear analysis and its various interdisciplinary applications.