Uniqueness

The articles in this volume reflect a subsequent development after a scientific meeting entitled Carleman Estimates and Control Theory, held in Cartona in September 1999. The 14 research-level articles, written by experts, focus on new results on Carleman estimates and their applications to uniqueness and controlla bility of partial differential equations and systems. The main topics are unique continuation for elliptic PDEs and systems, con trol theory and inverse problems. New results on strong uniqueness for second or higher order operators are explored in detail in several papers. In the area of control theory. the reader will find applications of Carleman estimates to stabiliza tion, observability and exact control for the wave and the SchrOdinger equations. A final paper presents a challenging list of open problems on the topic of control lability of linear and sernilinear heat equations. The papers contain exhaustive and essentially self-contained proofs directly ac cessible to mathematicians, physicists, and graduate students with an elementary background in PDEs. Contributors are L. Aloui, M. Bellassoued, N. Burq, F. Colombini, B. Dehman, C. Grammatico, M. Khenissi, H. Koch, P. Le Borgne, N. Lerner, T. Nishitani. T. Okaji, K.D. Phung, R. Regbaoui, X. Saint Raymond, D. Tataru, and E. Zuazua.

This monograph aims to fill a void by making available a source book which first systematically describes all the available uniqueness and nonuniqueness criteria for ordinary differential equations, and compares and contrasts the merits of these criteria, and second, discusses open problems and offers some directions towards possible solutions.

My Red Shirt and Me The red shirt incident begins with a rather ordinary red shirt. Not a brightly colored red shirt, not a dramatic cherry or firehouse red, more like a faded burgundy. But, for several days, my very iden tity was bound up in its redness. It was me, and I wore it with the pride a matador takes in his splendid cape, a hero in his medals of bravery, or a nun in her religious habit. I'll never forget the bound less joy I felt wearing that simple, pullover, short-sleeved red shirt in the hospital--or the rush of relief that I experienced when, at last, I decided to surrender it. However, we are getting ahead of our story, which starts a short time earlier with a most unfortunate accident. A light flurry of wet snow had begun to fall as the university limousine turned the corner on its way from the Bronx campus of New York University to the downtown campus. Although eight of us were packed into the car and had resigned ourselves to the usual boring faculty meeting awaiting us, somehow a spontaneous air of joviality was created.

To make this work accessible to logicians as well as set theorists and analysts, classical and modern theory of sets of uniqueness are covered as well as the relevant parts of descriptive set theory.

The concept of the uniqueness of Christianity often blocks attempts at dialogue with other religions. Traditionally, the argument goes: if Christianity is unique, then to dialogue with others somehow diminishes the weight of the claim that Jesus and the Gospel are unique. But what if uniqueness, properly defined, actually constitutes the key for understanding both Jewish and Christian traditions? Author Gabriel Moran frames his analysis of uniqueness by discussing the implications of that question. In this fluent and conversational work, Moran examines the paradox surrounding the concept of uniqueness in Christian and Jewish religious traditions. He uncovers the layers of meaning that accrue in a word that is in some sense both illogical and yet indispensable for human religious conversation. Tracing a logic of uniqueness embodied in revelation, faith, chosenness, covenant, and mediator, Moran opens a conversation between Jews and Christians that will lead readers to greater wisdom and religious depth.

This book addresses both probabilists working on diffusion processes and analysts interested in linear parabolic partial differential equations with singular coefficients. The central question discussed is whether a given diffusion operator, i.e., a second order linear differential operator without zeroth order term, which is a priori defined on test functions over some (finite or infinite dimensional) state space only, uniquely determines a strongly continuous semigroup on a corresponding weighted Lp space. Particular emphasis is placed on phenomena causing non-uniqueness, as well as on the relation between different notions of uniqueness appearing in analytic and probabilistic contexts.

The classical result for uniqueness in elasticity theory is due to Kirchhoff. It states that the standard mixed boundary value problem for a homogeneous isotropic linear elastic material in equilibrium and occupying a bounded three-dimensional region of space possesses at most one solution in the classical sense, provided the Lame and shear moduli, A and J1 respectively, obey the inequalities (3 A + 2 J1) > 0 and J1>O. In linear elastodynamics the analogous result, due to Neumann, is that the initial-mixed boundary value problem possesses at most one solution provided the elastic moduli satisfy the same set of inequalities as in Kirchhoffs theorem. Most standard textbooks on the linear theory of elasticity mention only these two classical criteria for uniqueness and neglect altogether the abundant literature which has appeared since the original publications of Kirchhoff. To remedy this deficiency it seems appropriate to attempt a coherent description ofthe various contributions made to the study of uniqueness in elasticity theory in the hope that such an exposition will provide a convenient access to the literature while at the same time indicating what progress has been made and what problems still await solution. Naturally, the continuing announcement of new results thwarts any attempt to provide a complete assessment. Apart from linear elasticity theory itself, there are several other areas where elastic uniqueness is significant.

YIS has been thought as an annual series of volumes collecting contributes aimed at developing the integration of idiographic and nomothetic approaches in psychological and more in general social science. At the beginning, 3 years ago, we got an agreement with an Italian publisher (FGP - Firera Publishing Group) interested in the scientific project and therefore willing to help the start up of this scientific enterprise. After publishing the first volume (YIS 2008- yet published in 2009 - the Volume is freely available on the FPG's website) we have had many positive feedbacks and signals of interests, as well as several submissions, from many parts of the world . This has provided an acceleration of the following issues - Above all, this led us to realize that it was time to give an editorial collocation to YIS that can be more consistent with the interest it has raised and that can ulteriorly raise. FPG does not put constraint on this perspective, being aware and agreed of the necessity of a worldwide context for the YIS's development. Moreover, there are no constraints in the possibility of going on in using the label "YIS," starting from Volume 4 The Series addresses a quite large potential public - students and researchers interested to theoretical and methodological development of psychology and, more in general, social science. Persons engaged with qualitative, dynamic informed models of analysis will find YIS a precious tool as well as a context enabling to develop a worlwide network of practices and cultures of research. The first three volumes' TOC witness how large and constantly increasing is the interest around the scientific project.