Hermann König

This monograph develops an operator viewpoint for functional equations in classical function spaces of analysis, thus filling a void in the mathematical literature. Major constructions or operations in analysis are often characterized by some elementary properties, relations or equations which they satisfy. The authors present recent results on the problem to what extent the derivative is characterized by equations such as the Leibniz rule or the Chain rule operator equation in Ck-spaces. By localization, these operator equations turn into specific functional equations which the authors then solve. The second derivative, Sturm-Liouville operators and the Laplacian motivate the study of certain "second-order" operator equations. Additionally, the authors determine the general solution of these operator equations under weak assumptions of non-degeneration. In their approach, operators are not required to be linear, and the authors also try to avoid continuity conditions. TheLeibniz rule, the Chain rule and its extensions turn out to be stable under perturbations and relaxations of assumptions on the form of the operators. The results yield an algebraic understanding of first- and second-order differential operators. Because the authors have chosen to characterize the derivative by algebraic relations, the rich operator-type structure behind the fundamental notion of the derivative and its relatives in analysis is discovered and explored. The book does not require any specific knowledge of functional equations. All needed results are presented and proven and the book is addressed to a general mathematical audience.

This monograph develops an operator viewpoint for functional equations in classical function spaces of analysis, thus filling a void in the mathematical literature. Major constructions or operations in analysis are often characterized by some elementary properties, relations or equations which they satisfy. The authors present recent results on the problem to what extent the derivative is characterized by equations such as the Leibniz rule or the Chain rule operator equation in Ck-spaces. By localization, these operator equations turn into specific functional equations which the authors then solve. The second derivative, Sturm-Liouville operators and the Laplacian motivate the study of certain "second-order" operator equations. Additionally, the authors determine the general solution of these operator equations under weak assumptions of non-degeneration. In their approach, operators are not required to be linear, and the authors also try to avoid continuity conditions. TheLeibniz rule, the Chain rule and its extensions turn out to be stable under perturbations and relaxations of assumptions on the form of the operators. The results yield an algebraic understanding of first- and second-order differential operators. Because the authors have chosen to characterize the derivative by algebraic relations, the rich operator-type structure behind the fundamental notion of the derivative and its relatives in analysis is discovered and explored. The book does not require any specific knowledge of functional equations. All needed results are presented and proven and the book is addressed to a general mathematical audience.

Der interdisziplinäre Atlas beschreibt die wesentlichen Erkrankungen und Verletzungen des Kniegelenkes - mit einer umfassenden Bildauswahl jedes einzelnen Gelenkabschnitts - von Normalbefund bis Erkrankungsvollbild. - mit ausführlichen Hinweisen auf diagnostische Fallstricke, Stadieneinteilungen bzw. therapeutische Konsequenzen. Effizient lernen, rasch nachschlagen: - Differenzierung der Bildinformation - Sichere Abklärung strittiger Befunde

With the technical advances made in MRI technology and the wider availability of MRI units, this diagnostic modality has by now - doubtedly gained a crucial role in joint imaging. The excellent detail recognition of MRI provides views of the various joint structures once only available through direct arthroscopic and surgical pro- dures. The acceptance,usefulness,and role of any diagnostic modality, however, critically relies on the experience, clinical expertise, and dedication of those who use it. With this in mind,a renowned int- disciplinary team of authors have brought together expert kno- edge from their respective fields in compiling this MRI atlas. Peter Teller and Hermann Konig are two highly experienced MRI radiologists with backgrounds in both clinical work and research. Ulrich Weber and Peter Hertel are two leading orthopedic surgeons and traumatologists in the fields of joint surgery/microsurgery and sports injuries. It is the vast radiologic experience in the interpretation of c- plex image information - an experience that takes into account the clinical requirements from the perspective of orthopedic surgeons and traumatologists - as well as the authors'didactic approach that make for the singular character of this book. Berlin,November 2001 Bernd Hamm,MD Professor and Chairman Department of Radiology Charite Medical School Humboldt-Universitat zu Berlin Preface MRI of diseases and injuries of the head, neck, and spinal column has become firmly established as a diagnostic tool since examiners could easily apply their previous experience gained in CT to MRI in these areas.