The aim of this monograph is to present a self-contained introduction to some geometric and analytic aspects of the Yamabe problem. The book also describes a wide range of methods and techniques that can be successfully applied to nonlinear differential equations in particularly challenging situations. Such situations occur where the lack of compactness, symmetry and homogeneity prevents the use of more standard tools typically used in compact situations or for the Euclidean setting. The work is written in an easy style that makes it accessible even to non-specialists. After a self-contained treatment of the geometric tools used in the book, readers are introduced to the main subject by means of a concise but clear study of some aspects of the Yamabe problem on compact manifolds. This study provides the motivation and geometrical feeling for the subsequent part of the work. In the main body of the book, it is shown how the geometry and the analysis of nonlinear partial differential equations blend together to give up-to-date results on existence, nonexistence, uniqueness and a priori estimates for solutions of general Yamabe-type equations and inequalities on complete, non-compact Riemannian manifolds.

This book presents very recent results involving an extensive use of analytical tools in the study of geometrical and topological properties of complete Riemannian manifolds. It analyzes in detail an extension of the Bochner technique to the non compact setting, yielding conditions which ensure that solutions of geometrically significant differential equations either are trivial (vanishing results) or give rise to finite dimensional vector spaces (finiteness results). The book develops a range of methods from spectral theory and qualitative properties of solutions of PDEs to comparison theorems in Riemannian geometry and potential theory.

All needed tools are described in detail, often with an original approach. Some of the applications presented concern the topology at infinity of submanifolds, Lp cohomology, metric rigidity of manifolds with positive spectrum, and structure theorems for KAhler manifolds.

The book is essentially self-contained and supplies in an original presentation the necessary background material not easily available in book form.

I.A.U. symposium No. 110 on VLBI and Compact Radio Sources was held in Bologna, Italy from June 27 to July 1, 1983. 166 participants from 19 countries were registered and 106 invited and contributed papers were registered. The scientific presentations and discussion concentrated on VLBI observation and interpretation of galactic and extragalactic radio sour ces, including topics as diverse as quasars and galactic nuclei, inter stellar masers, pulsars, and astrometry. Geodetic applications and tech nical development were treated only briefly, as these topics have been the subject of other recent international symposia. Since the first VLBI observations in 1967, sensitivity, resolution, and image quality have improved dramatically. Radio maps shown at the symposium were of comparable quality to conventional synthesis maps be ing made at the time of the first VLBI experiments 15 years ago, but with a resolution more than a factor of 1000 better. We wanted to accommodate the large number of contributed papers in this rapidly developing field, but there was inadequate time for normal oral presentations and discussion. We therefore asked that all contrib uted papers be put on display for at least 24 hours prior to a brief oral summary. A question and discussion period followed groups of oral pres entations on the same or similar topic. In this way the opportunity for interactive discussion, not available in conventional poster displays, was preserved."

The aim of this monograph is to present a self-contained introduction to some geometric and analytic aspects of the Yamabe problem. The book also describes a wide range of methods and techniques that can be successfully applied to nonlinear differential equations in particularly challenging situations. Such situations occur where the lack of compactness, symmetry and homogeneity prevents the use of more standard tools typically used in compact situations or for the Euclidean setting. The work is written in an easy style that makes it accessible even to non-specialists. After a self-contained treatment of the geometric tools used in the book, readers are introduced to the main subject by means of a concise but clear study of some aspects of the Yamabe problem on compact manifolds. This study provides the motivation and geometrical feeling for the subsequent part of the work. In the main body of the book, it is shown how the geometry and the analysis of nonlinear partial differential equations blend together to give up-to-date results on existence, nonexistence, uniqueness and a priori estimates for solutions of general Yamabe-type equations and inequalities on complete, non-compact Riemannian manifolds.

The proceedings of the international conference held in honor of Lodewijk Woltjer, focus on the physics, origin and space distribution of AGNs and quasars and their relationship to the environment, a very wide subject which has attracted much of Lodewijk Wolter's research interest. A number of lectures were also dedicated to reviewing the recent observational advances and those that may be attained by the introduction of new and powerful astronomical instrumentation both from the ground and from space, in recognition of the central role played by L. Woltjer in the promotion of the ESO VLT and of his involvement in shaping ESA's space programme Horizon 2000+.

This volume contains a series of lectures presented at the 4th Course of the International School of Astrophysics, held in Erice (Sicily) from July 9 - July 20, 1977 at the "E. Majorana" Centre for Scientific Culture. The course was fully supported by a grant from the NATO Advanced Study Institute Programme. It was attended by 82 participants from 15 countries. Even though the infrared portion of the electromagnetic spectrum covers an extensive interval from the red region of the optical spectrum (10,000 A) to the microwave radio region (1 mm), its role in astronomy has been minimal until the last two decades. Until very recently, the only objects observed were the sun, the moon and the planets. A primary reason for this late development was the lack of sensitive detectors and the necessary cryogenic technology that must accompany their use. Recent progress in this technology has been paralleled by an ever increasing interest of astronomers in infrared observations, leading to a number of ex tremely important results in different branches of astronomy. This becomes evident when one realizes that in many astrophysical conditions most of the energy is found to be channeled into the infrared portion of the electromagnetic spectrum. Stars were detected that yield most of their radiation in the infrared; these objects present a new view of stellar evolution, both in the birth and death stages."

by the completion of large aperture synthesis radio telescopes, which have permitted a detailed mapping of radio sources, and by the development of very long base-line interferometry (VLBI), with 4 3 resolutions down to 10- - 10- arc sec, which for the first time has rendered possible radio investigations of the structures of the very compact radio sources found in galactic nuclei and quasars. The observations obtained with this new instrumentation, combined with the work in the optical, infrared and X-ray domain, have made great strides in recent years toward improved testing of radio source models. We feel that the material presented at the Institute represents a rather complete and comprehensive coverage of the present status of studies of non-thermal radio sources, though one is aware of rapid developments in this field of research. The various aspects of this exciting subject were covered in a series of lectures, pre sented in this volume, totaling 44 hours and in 14 topical seminars given by the participants. I wish to express my gratitude to the Scientific Affairs Divi sion of the North Atlantic Treaty Organization for the generous support given to the course. Sincere thanks are also due to Mr. L. Baldeschi for helping with the organization of the meeting and for drawing a number of figures contained in this volume; to Mrs. B. Mandel for the patient typing and help in the editing; and to Mr. R. Primavera for the photographic reproduction of part of the figures."

This volume contains a series of lectures presented at the 5th Course of the International School of Astrophysics held in Erice (Sicily) from July 1st to July 14, 1979 at the "E. l1ajorana" Centre for Scientific Culture. The course was fully supported by a grant from the NATO Advanced Institute Programme. It was attended by about one hundred participants from ten countries. Since the discovery of the first extra-solar X-ray source in the early 1960's, X-ray astronomy has played an increasingly im portant role in the study of the Universe, bringing new insight to almost every field of modern astrophysics from stellar evolution to cosmology. Generally speaking, this branch of astronomy is concerned with the discovery, classification and study of "hot matter" in the universe, including high energy non-thermal pheno mena. In particular, X-ray observations appear to provide the main, if not the only, probe to inspect regions where collapsed objects are formed, such as the environment of neutron stars and of black holes in the presence of matter accretion onto the ob jects themselves. It is significant that the first candidate black hole (Cyg X-I) has been primarily singled out by its X-ray emission. In the same context, it is well known that one of the fundamental problems in modern astrophysics is the understanding of the strong activity taking place in galactic nuclei."

I.A.U. symposium No. 110 on VLBI and Compact Radio Sources was held in Bologna, Italy from June 27 to July 1, 1983. 166 participants from 19 countries were registered and 106 invited and contributed papers were registered. The scientific presentations and discussion concentrated on VLBI observation and interpretation of galactic and extragalactic radio sour ces, including topics as diverse as quasars and galactic nuclei, inter stellar masers, pulsars, and astrometry. Geodetic applications and tech nical development were treated only briefly, as these topics have been the subject of other recent international symposia. Since the first VLBI observations in 1967, sensitivity, resolution, and image quality have improved dramatically. Radio maps shown at the symposium were of comparable quality to conventional synthesis maps be ing made at the time of the first VLBI experiments 15 years ago, but with a resolution more than a factor of 1000 better. We wanted to accommodate the large number of contributed papers in this rapidly developing field, but there was inadequate time for normal oral presentations and discussion. We therefore asked that all contrib uted papers be put on display for at least 24 hours prior to a brief oral summary. A question and discussion period followed groups of oral pres entations on the same or similar topic. In this way the opportunity for interactive discussion, not available in conventional poster displays, was preserved."

The cosmic radiation was discovered by Hess in 1912 but its origin is still the subject of much controversy and considerable study. For several decades most workers in the cosmic ray field were interested in the Nuclear Physical aspect of the particle beam and many important dis coveries were made, notably the identification of the positron, the muon, the pion and the strange particles. More recently however, em phasis has changed to the Astrophysical aspect both with re8ard to the origin of the radiation and to its relation with the other radiation fields. Mindful of the increasing importance of the Astrophysical facets of the subject the Cosmic Ray Commission of IUPAP approached the High Energy Astrophysics Commission of the IAU with the suggestion of a joint Symposium on Cosmic Ray Origin. The plan was to bring together workers in all the various astronomical fields - from Radio, through Optical to Gamma Rays - with Cosmic Ray physicists and to fully explore the various interrelations. The approach was received with enthusiasm and this book contains the proceedings of the ensuing Symposium, (styled IUPAP/ IAU Symposium No. 94) which was held in Bologna from 11th - 14th June 1980. Virtually all the papers presented are reproduced here. The Scientific Organizing Committee chose the invited speakers whose papers formed the major scientific component of the meeting. Additional short contributed papers were also called for and very brief 2-page resumes (which have not been refereed) are given in the proceedings."