Includes entries for maps and atlases.

Papers on Smarandache Lattice and Pseudo Complement, Smarandache’s Conjecture on Consecutive Primes, Signed Domatic Number of Directed Circulant Graphs, Generalized Quasi-Kenmotsu Manifolds, Geometry on Non-Solvable Equations-A Review on Contradictory Systems, and other topics. Contributors: Octavian Cira, Linfan Mao, N. Kannappa, K. Suresh, F. Smarandache, M. Ali, A. Raheem, A. Q. Baig, M. Javaid, Barnali Laha, Arindam Bhattacharyya, and others.

The Fourth Scandinavian Logic Symposium and the First Soviet-Finnish Logic Conference were held in JyvaskyIa, Finland, June 29-July 6, 1976. The Conferences were organized by a committee which consisted of the editors of the present volume. The Conferences were supported financially by the Ministry of Education of Finland, by the Academy of Finland, and by the Division of Logic, Methodology, and Philosophy of Science of the International Union of History of Science. The Philosophical Society of Finland and the Jyvaskyla Summer Festival gave valuable help in various practicalities. 35 papers by authors representing 10 countries were presented at the two meetings. Of those papers 24 appear here. THE EDITORS v TABLE OF CONTENTS PREFACE v PART 1/ PROOF THEORY GEORG KREISEL / Some Facts from the Theory of Proofs and Some Fictions from General Proof Theory 3 DAG PRAWITZ / Proofs and the Meaning and Completeness of the Logical Constants 25 v. A. SMIRNOV / Theory of Quantification and tff-calculi 41 LARS SVENONIUS/Two Kinds of Extensions of Primitive Recursive Arithmetic 49 DIRK VAN DALEN and R. STATMAN / Equality in the Presence of Apartness 95 PART II / INFINITARY LANGUAGES VEIKKO RANTALA / Game-Theoretical Semantics and Back-and- Forth 119 MAARET KAR TTUNEN / Infinitary Languages N oo~.

The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly, which publishes original research papers and survey articles in all aspects of mathematical combinatorics, Smarandache multi-spaces, Smarandache geometries, non-Euclidean geometry, topology and their applications to other sciences.

The Baltic Sea oceanographic research community is wide and the research history is over 100 years old. Nevertheless, there is still no single, coherent book on the physical oceanography of the Baltic Sea as a whole. There is a strong need for such a book, coming from working oceanographers as well as the university teaching programmes in advanced undergraduate to graduate levels. In the regional conference series in physical oceanography (Baltic Sea Science Conference, Baltic Sea Oceanographers' conference, Baltex-conferences) about 500 scientists take part regularly. Even more scientists work in the fields of marine biology, chemistry and the environment, and they need information on the physics of the Baltic Sea as well. There are nine countries bordering on the Baltic Sea and five more in the runoff area. The Baltic Sea as a source of fish, means of transportation and leisure activities is highly important to the regional society. In the runoff area there are a total of 85 million people. Research and protection strategies need to be developed, as the Baltic Sea is probably the most polluted sea in the world. Since the Baltic Sea has become an inner sea of the EU (apart from small shore parts of Russia in Petersburg and Kaliningrad), it is anticipated that the importance of the region will consequently rise. The book will arouse interest among students, scientists and decision makers involved with the Baltic problems. It will also give important background information for those working with biogeochemical processes in the Baltic Sea, because the physical forcing for those processes is of vital importance.

At its meeting in April 1990 at the University of Cambridge, the Executive Committee of the International Mathematical Union (IMU) decided that the largely unorganized archives of the Union should be properly arranged and catalogued. Simultaneously, the Executive Committee expressed the wish that a history of the Union should be written [1). As Secretary of the Union, I had proposed that these issues be dis cussed at the Cambridge meeting, but without having had in mind any personal role in the practical execution of such projects. At that time, the papers of the IMU were stored in Zurich, at the Eidgenossische Technische Hochschule, and I saw no reason why they could not remain there. At about this time, Professor K. Chandrasekharan produced a handwritten article titled "The Prehistory of the International Mathematical Union" [2), and it seemed to me that this might serve as the beginning of a more compre hensive history. I had first thought that Tuulikki MakeUiinen, who during eight years as the Office Secretary ofthe IMU had become well acquainted with the Union, would do the arranging of the archives in Zurich. She had a preliminary look at the material there, but it soon became clear that the amount of work required to bring order to it was too great to be accomplished in a few short visits from Helsinki. The total volume of material was formidable.

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.