Bøger udgivet af Birkhauser Boston

This monograph is based on the author's results on the Riemannian ge- ometry of foliations with nonnegative mixed curvature and on the geometry of sub manifolds with generators (rulings) in a Riemannian space of nonnegative curvature. The main idea is that such foliated (sub) manifolds can be decom- posed when the dimension of the leaves (generators) is large. The methods of investigation are mostly synthetic. The work is divided into two parts, consisting of seven chapters and three appendices. Appendix A was written jointly with V. Toponogov. Part 1 is devoted to the Riemannian geometry of foliations. In the first few sections of Chapter I we give a survey of the basic results on foliated smooth manifolds (Sections 1.1-1.3), and finish in Section 1.4 with a discussion of the key problem of this work: the role of Riemannian curvature in the study of foliations on manifolds and submanifolds.

This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera- tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "e;noisy"e;. We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre- sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in reservoir theory and related areas. 3) The theory should be strong and efficient enough to allow us to solve th,~se SPDEs explicitly, or at least provide algorithms or approximations for the solutions.

Overview This book is intended as a textbook in probability for graduate students in math­ ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a 'difficult' but productive marriage of mathemat­ ical abstraction and everyday intuition, and we have attempted to exhibit this fact. Thus we may appear at times to be obsessively careful in our presentation of the material, but our experience has shown that many students find them­ selves quite handicapped because they have never properly come to grips with the subtleties of the definitions and mathematical structures that form the foun­ dation of the field. Also, students may find many of the examples and problems to be computationally challenging, but it is our belief that one of the fascinat­ ing aspects of prob ability theory is its ability to say something concrete about the world around us, and we have done our best to coax the student into doing explicit calculations, often in the context of apparently elementary models. The practical applications of probability theory to various scientific fields are far-reaching, and a specialized treatment would be required to do justice to the interrelations between prob ability and any one of these areas. However, to give the reader a taste of the possibilities, we have included some examples, particularly from the field of statistics, such as order statistics, Dirichlet distri­ butions, and minimum variance unbiased estimation.

Infinite dimensional systems can be used to describe many phenomena in the real world. As is well known, heat conduction, properties of elastic- plastic material, fluid dynamics, diffusion-reaction processes, etc., all lie within this area. The object that we are studying (temperature, displace- ment, concentration, velocity, etc.) is usually referred to as the state. We are interested in the case where the state satisfies proper differential equa- tions that are derived from certain physical laws, such as Newton's law, Fourier's law etc. The space in which the state exists is called the state space, and the equation that the state satisfies is called the state equation. By an infinite dimensional system we mean one whose corresponding state space is infinite dimensional. In particular, we are interested in the case where the state equation is one of the following types: partial differential equation, functional differential equation, integro-differential equation, or abstract evolution equation. The case in which the state equation is being a stochastic differential equation is also an infinite dimensional problem, but we will not discuss such a case in this book.

as anywhere today, it is becoming more d- ficult to tell the truth. To be sure, our store of accurate facts is more plentiful now than it has ever been, and the minutest details of history are being thoroughly recorded. Scientists, - men and scholars vie with each other in publishing excruciatingly definitive accounts of all that happens on the natural, political and historical scenes. Unfortunately, telling the truth is not quite the same thing as reciting a rosary of facts. Jos6 Ortega y Gasset, in an adm- able lesson summarized by Antonio Machado's three-line poem, prophetically warned us that the reason people so often lie is that they lack imagination: they don't realize that the truth, too, is a matter of invention. Sometime, in a future that is knocking at our door, we shall have to retrain ourselves or our children to properly tell the truth. The exercise will be particularly painful in mathematics. The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics. Shocking as it may be to a conservative logician, the day will come when currently MATHEMATICS, IN vague concepts such as motivation and purpose will be made formal and accepted as constituents of a revamped logic, where they will at last be allotted the equal status they deserve, si- by-side with axioms and theorems.

These two volumes on Estrogens, Progestins, and Their Antagonists repre­ sent a thematic extension of the series, Hormones in Health and Disease. The first publication in the series, Steroid Hormone Receptors: Basic and Clinical Aspects, focused on recent advances in the anatomy of steroid receptors and members of the steroid receptor superfamily. Consistent with the spirit of the series, the authors addressed issues of clinical significance of steroid receptor detection in hormone-related disorders. The second volume in the series, Hormones and Cancer, attempted a more direct examination of ac­ tions of hormones in cancerous tissues and cells. In these two volumes, which together form the third in the series, the editor, Dr. Edward Pavlik, has introduced a team of leading investigators engaged in research on various aspects of the steroids that regulate female reproductive physiology. Estrogens and progestins, the main components of the most widely used contraceptive pills, have found a variety of uses in clinical endocrinology. These volumes contain discussions that range from the introduction of novel hormonal ligands to hormonal antagonism by steroid analogs. A balanced treatment is provided of applications of the steroids in treatment and management of hormone-dependent conditions and malignancies. The remarkable synthesis of literature contained in this volume will provide a reader with both the fundamental concepts underly­ ing steroid hormone physiology and the clinical applications of observations made on basic aspects of hormone action. I congratulate the editorial leadership of Dr.

In 1991, a small annual meeting named "International Winter Conference on Neurodegeneration (lWCN)" was established; the aim of this meeting is to review the neurodegenerative disorders and to attempt to explore how progress might be made in this field, as the neurodegenerative disorders have been emerging to be one of the major causes of morbidity and mortality in modern societies. The first meeting took place in Seefeld, Austria, in February 1992; the topics for the first IWCN were chosen to provide a broad foundation of clinical science, which included the problem of aging, classification of neurodegenerative disorders and of Alzheimer's dis­ natural history, pathology, and clinical neurology ease, Parkinson's disease, and amyotrophic lateral sclerosis. The fundamental pathology underlying these neurodegenerative disorders is neuronal cell death. For the understanding of pathophysiol­ ogy and the development of neuroprotective treatment for these dis­ orders, elucidation of the mechanism of neuronal cell death at the cellular and molecular level is essential. With this concept in mind, the second IWCN was held in Whistler Village in Canada in January 1993. Funding was generously provided by Schering AG, Berlin, and for the excellent organization we have to thank Ms. Ingeborg Runge.

Modeling and Applied Mathematics Modeling the behavior of real physical systems by suitable evolution equa­ tions is a relevant, maybe the fundamental, aspect of the interactions be­ tween mathematics and applied sciences. Modeling is, however, only the first step toward the mathematical description and simulation of systems belonging to real world. Indeed, once the evolution equation is proposed, one has to deal with mathematical problems and develop suitable simula­ tions to provide the description of the real system according to the model. Within this framework, one has an evolution equation and the re­ lated mathematical problems obtained by adding all necessary conditions for their solution. Then, a qualitative analysis should be developed: this means proof of existence of solutions and analysis of their qualitative be­ havior. Asymptotic analysis may include a detailed description of stability properties. Quantitative analysis, based upon the application ofsuitable methods and algorithms for the solution of problems, ends up with the simulation that is the representation of the dependent variable versus the independent one. The information obtained by the model has to be compared with those deriving from the experimental observation of the real system. This comparison may finally lead to the validation of the model followed by its application and, maybe, further generalization.

Oxygen free radicals and other reactive oxygen species are being postulated as causal agents in an increasing number of pathological conditions. Indeed, some investigators are suggesting that highly destructive reactive oxygen species are the final common path lead­ ing to tissue damage following a wide variety of insults including trauma, hypoxia, ischemia, hyperoxia, radiation, some toxins, and even strenuous athletic pursuits. But, as Robert Floyd points out, "Proof of the importance of oxygen free radicals and the oxidative damage they initiate depend on unequivocal evidence for the pres­ ence of free radicals and a clear association of their formation with the induction of the dysfunction of pathological conditions. " Since such proof does not come easily, there have been and will continue to be many controversies regarding the role played by reactive oxygen species in tissue damage. There have been many recent reviews of the chemistry and pos­ sible role of reactive oxygen species in many types of organ dys­ functions, tissue damage, degenerative diseases, and aging. This book is not such a review. Rather it presents for a diverse audience of physical-organic chemists, biochemists, medical researchers, and other investigators of pathophysiology, discussions of a variety of is­ sues important for understanding reactive oxygen species and their role in tissue damage.

The impact of molecular genetics on plant breeding and, consequently, agri­ culture, is potentially enonnous. Understanding and directing this potential im­ pact is crucial because of the urgent issues that we face concerning sustainable agriculture for a growing world population as well as conservation of the world's rapidly dwindling plant genetic resources. This book is largely devoted to the applications of genetic markers that have been developed by the application of molecular genetics to practical problems. These are known as DNA markers. They have gained a certain notoriety in foren­ sics, but can be used in a variety of practical situations. We are going through a period of accelerated breakthroughs in molecular ge­ netics. Therefore, the authors of each chapter were encouraged to speculate about both current bottlenecks and the future of their subfields of research. We can cer­ tainly apply molecular genetic tools and approaches to help resolve crucial ge­ netic resource problems that face humanity. However, little has been discussed with respect to when or how we should use such tools, nor to who specifically should use them; therefore, social and economic analyses are important in the planning stages of projects that are aimed at practical results.

Differential equations is a subject of wide applicability, and knowledge of dif- Differential equations is a subject of wide applicability, and knowledge of dif- ferential ferential equations equations topics topics permeates permeates all all areas areas of of study study in in engineering engineering and and applied applied mathematics. mathematics. Some Some differential differential equations equations are are susceptible susceptible to to analytic analytic means means of of so- so- lution, lution, while while others others require require the the generation generation of of numerical numerical solution solution trajectories trajectories to to see see the the behavior behavior of of the the system system under under study. study. For For both both situations, situations, the the software software package package Maple Maple can can be be used used to to advantage. advantage. To To the the student student Making Making effective effective use use of of differential differential equations equations requires requires facility facility in in recognizing recognizing and and solving solving standard standard "e;tractable"e; "e;tractable"e; problems, problems, as as well well as as having having the the background background in in the the subject subject to to make make use use of of tools tools for for dealing dealing with with situations situations that that are are not not amenable amenable to to simple simple analytic analytic approaches. approaches.