This book is a self-contained account of the method based on Carleman estimates for inverse problems of determining spatially varying functions of differential equations of the hyperbolic type by non-overdetermining data of solutions. The formulation is different from that of Dirichlet-to-Neumann maps and can often prove the global uniqueness and Lipschitz stability even with a single measurement. These types of inverse problems include coefficient inverse problems of determining physical parameters in inhomogeneous media that appear in many applications related to electromagnetism, elasticity, and related phenomena. Although the methodology was created in 1981 by Bukhgeim and Klibanov, its comprehensive development has been accomplished only recently. In spite of the wide applicability of the method, there are few monographs focusing on combined accounts of Carleman estimates and applications to inverse problems. The aim in this book is to fill that gap. The basic tool is Carleman estimates, the theory of which has been established within a very general framework, so that the method using Carleman estimates for inverse problems is misunderstood as being very difficult. The main purpose of the book is to provide an accessible approach to the methodology. To accomplish that goal, the authors include a direct derivation of Carleman estimates, the derivation being based essentially on elementary calculus working flexibly for various equations. Because the inverse problem depends heavily on respective equations, too general and abstract an approach may not be balanced. Thus a direct and concrete means was chosen not only because it is friendly to readers but also is much more relevant. By practical necessity, there is surely a wide range of inverse problems and the method delineated here can solve them. The intention is for readers to learn that method and then apply it to solving new inverse problems.

These proceedings of the international Conference "Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis," held at the Samarkand State University, Uzbekistan in September 2000 bring together fundamental research articles in the major areas of the numerated fields of analysis and mathematical physics. The book covers the following topics: theory of ill-posed problems inverse problems for differential equations boundary value problems for equations of mixed type integral geometry mathematical modelling and numerical methods in natural sciences

This book aims to establish a foundation for fractional derivatives and fractional differential equations. The theory of fractional derivatives enables considering any positive order of differentiation. The history of research in this field is very long, with its origins dating back to Leibniz. Since then, many great mathematicians, such as Abel, have made contributions that cover not only theoretical aspects but also physical applications of fractional calculus. The fractional partial differential equations govern phenomena depending both on spatial and time variables and require more subtle treatments. Moreover, fractional partial differential equations are highly demanded model equations for solving real-world problems such as the anomalous diffusion in heterogeneous media. The studies of fractional partial differential equations have continued to expand explosively. However we observe that available mathematical theory for fractional partial differential equations is not still complete. In particular, operator-theoretical approaches are indispensable for some generalized categories of solutions such as weak solutions, but feasible operator-theoretic foundations for wide applications are not available in monographs. To make this monograph more readable, we are restricting it to a few fundamental types of time-fractional partial differential equations, forgoing many other important and exciting topics such as stability for nonlinear problems. However, we believe that this book works well as an introduction to mathematical research in such vast fields.

The December 1937 incident that has come to be known as the Rape of Nanking is, without doubt, a tragedy that will not soon be forgotten. While acknowledging that a tremendous loss of life occurred, this study challenges the current prevailing notion that the incident was a deliberate, planned effort on the part of the Japanese military and analyzes events to produce an accurate estimate of the scale of the atrocities. Drawing on Chinese, Japanese, and English sources, Yamamoto determines that what happened at Nanking were unfortunate atrocities of conventional war with precedents in both Eastern and Western military history. He concludes that post-war events such as the war crimes trials and the impact of the Holocaust in Europe affected public opinion regarding Nanking and led to a dramatic reinterpretation of events.

The Rape of Nanking consisted of two distinct phases: the mass execution of prisoners of war (as well as conscription age men who appeared to be combatants) and the delinquent acts of individual soldiers. The first phase, which occurred immediately after Nanking's fall and which claimed most of the atrocity victims, was the result of the Japanese military's attempt to clear the city of Chinese soldiers thought to be in plain clothes. The second phase, which lasted approximately six weeks, was horrible, but resulted in a much smaller number of fatalities. It was characterized by numerous criminal acts, ranging from rape and murder to arson and theft, committed by unrestrained Japanese soldiers. The root cause for both phases was the Japanese military's bureaucratic inefficiency and command irresponsibility. While both Chinese and American contemporary sources initially attributed the incident to these causes, subsequent Japanese atrocities against both military and civilian Allied personnel during World War II and evidence presented at war crimes trials would come to reshape perceptions of the Nanking events as an Asian counterpart to the Nazi Holocaust.