# Realizability An Introduction To Its Categorical Side

Download **Realizability An Introduction To Its Categorical Side** or read online books in PDF, EPUB, Tuebl, and kindle. Click Get Book button to get *Realizability An Introduction To Its Categorical Side* book now. We cannot guarantee every books is in the library. Use search box to get ebook that you want.

## Realizability

Author | : Jaap van Oosten |

Publsiher | : Elsevier |

Total Pages | : 328 |

Release | : 2008-04-10 |

ISBN | : 9780080560069 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Realizability Book Excerpt:**

Aimed at starting researchers in the field, Realizability gives a rigorous, yet reasonable introduction to the basic concepts of a field which has passed several successive phases of abstraction. Material from previously unpublished sources such as Ph.D. theses, unpublished papers, etc. has been molded into one comprehensive presentation of the subject area. - The first book to date on this subject area - Provides an clear introduction to Realizability with a comprehensive bibliography - Easy to read and mathematically rigorous - Written by an expert in the field

## Theory and Applications of Models of Computation

Author | : T.V. Gopal,Gerhard Jäger,Silvia Steila |

Publsiher | : Springer |

Total Pages | : 698 |

Release | : 2017-04-13 |

ISBN | : 3319559117 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Theory and Applications of Models of Computation Book Excerpt:**

This book constitutes the refereed proceedings of the 14th Annual Conference on Theory and Applications of Models of Computation, TAMC 2017, held in Bern, Switzerland, in April 2017. The 45 revised full papers presented together with 4 invited papers were carefully reviewed and selected from 103 submissions. The main themes of TAMC 2017 have been computability, computer science logic, complexity, algorithms, and models of computation and systems theory.

## Joachim Lambek The Interplay of Mathematics Logic and Linguistics

Author | : Claudia Casadio,Philip J. Scott |

Publsiher | : Springer Nature |

Total Pages | : 432 |

Release | : 2021-04-21 |

ISBN | : 3030665453 |

Category | : Philosophy |

Language | : EN, FR, DE, ES & NL |

**Joachim Lambek The Interplay of Mathematics Logic and Linguistics Book Excerpt:**

This book is dedicated to the life and work of the mathematician Joachim Lambek (1922–2014). The editors gather together noted experts to discuss the state of the art of various of Lambek’s works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist (notably in ring theory). In the 1960s, he began to work in category theory, categorical algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard’s linear logic. In a surprising development (2000), he introduced a novel and deeper algebraic framework (which he called pregroup grammars) for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists.

## Objects Structures and Logics

Author | : Gianluigi Oliveri |

Publsiher | : Springer Nature |

Total Pages | : 135 |

Release | : 2022 |

ISBN | : 3030847063 |

Category | : Electronic Book |

Language | : EN, FR, DE, ES & NL |

**Objects Structures and Logics Book Excerpt:**

## Concepts of Proof in Mathematics Philosophy and Computer Science

Author | : Dieter Probst,Peter Schuster |

Publsiher | : Walter de Gruyter GmbH & Co KG |

Total Pages | : 384 |

Release | : 2016-07-25 |

ISBN | : 150150262X |

Category | : Philosophy |

Language | : EN, FR, DE, ES & NL |

**Concepts of Proof in Mathematics Philosophy and Computer Science Book Excerpt:**

A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

## A Computable Universe

Author | : Hector Zenil |

Publsiher | : World Scientific |

Total Pages | : 810 |

Release | : 2013 |

ISBN | : 981437430X |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**A Computable Universe Book Excerpt:**

This volume, with a foreword by Sir Roger Penrose, discusses the foundations of computation in relation to nature.It focuses on two main questions: What is computation? How does nature compute?The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and philosophical implications.The volume provides a state-of-the-art collection of technical papers and non-technical essays, representing a field that assumes information and computation to be key in understanding and explaining the basic structure underpinning physical reality. It also includes a new edition of Konrad Zuse''s OC Calculating SpaceOCO (the MIT translation), and a panel discussion transcription on the topic, featuring worldwide experts in quantum mechanics, physics, cognition, computation and algorithmic complexity.The volume is dedicated to the memory of Alan M Turing OCo the inventor of universal computation, on the 100th anniversary of his birth, and is part of the Turing Centenary celebrations.

## A Computable Universe

Author | : Hector Zenil |

Publsiher | : World Scientific |

Total Pages | : 856 |

Release | : 2012-10-30 |

ISBN | : 9814447781 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**A Computable Universe Book Excerpt:**

This volume, with a Foreword writer Sir Roger Penrose, discusses the foundations of computation in relation to nature. It focuses on two main questions: What is computation?How does nature compute? The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and philosophical implications. The volume provides a state-of-the-art collection of technical papers and non-technical essays, representing a field that assumes information and computation to be key in understanding and explaining the basic structure underpinning physical reality. It also includes a new edition of Konrad Zuse's “Calculating Space” (the MIT translation), and a panel discussion transcription on the topic, featuring worldwide experts in quantum mechanics, physics, cognition, computation and algorithmic complexity. The volume is dedicated to the memory of Alan M Turing — the inventor of universal computation, on the 100th anniversary of his birth, and is part of the Turing Centenary celebrations. Contents:Foreword (R Penrose)PrefaceAcknowledgementsIntroducing the Computable Universe (H Zenil)Historical, Philosophical & Foundational Aspects of Computation:Origins of Digital Computing: Alan Turing, Charles Babbage, & Ada Lovelace (D Swade)Generating, Solving and the Mathematics of Homo Sapiens. E Post's Views on Computation (L De Mol)Machines (R Turner)Effectiveness (N Dershowitz & E Falkovich)Axioms for Computability: Do They Allow a Proof of Church's Thesis? (W Sieg)The Mathematician's Bias — and the Return to Embodied Computation (S B Cooper)Intuitionistic Mathematics and Realizability in the Physical World (A Bauer)What is Computation? Actor Model versus Turing's Model (C Hewitt)Computation in Nature & the Real World:Reaction Systems: A Natural Computing Approach to the Functioning of Living Cells (A Ehrenfeucht, J Kleijn, M Koutny & G Rozenberg)Bacteria, Turing Machines and Hyperbolic Cellular Automata (M Margenstern)Computation and Communication in Unorganized Systems (C Teuscher)The Many Forms of Amorphous Computational Systems (J Wiedermann)Computing on Rings (G J Martínez, A Adamatzky & H V McIntosh)Life as Evolving Software (G J Chaitin)Computability and Algorithmic Complexity in Economics (K V Velupillai & S Zambelli)Blueprint for a Hypercomputer (F A Doria)Computation & Physics & the Physics of Computation:Information-Theoretic Teleodynamics in Natural and Artificial Systems (A F Beavers & C D Harrison)Discrete Theoretical Processes (DTP) (E Fredkin)The Fastest Way of Computing All Universes (J Schmidhuber)The Subjective Computable Universe (M Hutter)What Is Ultimately Possible in Physics? (S Wolfram)Universality, Turing Incompleteness and Observers (K Sutner)Algorithmic Causal Sets for a Computational Spacetime (T Bolognesi)The Computable Universe Hypothesis (M P Szudzik)The Universe is Lawless or “Pantôn chrêmatôn metron anthrôpon einai” (C S Calude, F W Meyerstein & A Salomaa)Is Feasibility in Physics Limited by Fantasy Alone? (C S Calude & K Svozil)The Quantum, Computation & Information:What is Computation? (How) Does Nature Compute? (D Deutsch)The Universe as Quantum Computer (S Lloyd)Quantum Speedup and Temporal Inequalities for Sequential Actions (M Żukowski)The Contextual Computer (A Cabello)A Gödel-Turing Perspective on Quantum States Indistinguishable from Inside (T Breuer)When Humans Do Compute Quantum (P Zizzi)Open Discussion Section:Open Discussion on A Computable Universe (A Bauer, T Bolognesi, A Cabello, C S Calude, L De Mol, F Doria, E Fredkin, C Hewitt, M Hutter, M Margenstern, K Svozil, M Szudzik, C Teuscher, S Wolfram & H Zenil)Live Panel Discussion (transcription):What is Computation? (How) Does Nature Compute? (C S Calude, G J Chaitin, E Fredkin, A J Leggett, R de Ruyter, T Toffoli & S Wolfram)Zuse's Calculating Space:Calculating Space (Rechnender Raum) (K Zuse)Afterword to Konrad Zuse's Calculating Space (A German & H Zenil) Readership: Graduate students who are specialized researchers in computer science, information theory, quantum theory and modern philosophy and the general public who are interested in these subject areas. Keywords:Digital Physics;Computational Universe;Digital Philosophy;Reality Theories of the Universe;Models of the World;Thring Computation RandomnessKey Features:The authors are all prominent researchersNo competing titlesState-of-the-art collection of technical papers and non-technical essays

## Algebraic Computability and Enumeration Models

Author | : Cyrus F. Nourani |

Publsiher | : CRC Press |

Total Pages | : 310 |

Release | : 2016-02-24 |

ISBN | : 1771882484 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Algebraic Computability and Enumeration Models Book Excerpt:**

This book, Algebraic Computability and Enumeration Models: Recursion Theory and Descriptive Complexity, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic viewpoint. The reader is first introduced to categories and functorial models, with Kleene algebra examples for languages. Functorial models for Peano arithmetic are described toward important computational complexity areas on a Hilbert program, leading to computability with initial models. Infinite language categories are also introduced to explain descriptive complexity with recursive computability with admissible sets and urelements. Algebraic and categorical realizability is staged on several levels, addressing new computability questions with omitting types realizably. Further applications to computing with ultrafilters on sets and Turing degree computability are examined. Functorial models computability is presented with algebraic trees realizing intuitionistic types of models. New homotopy techniques are applied to Marin Lof types of computations with model categories. Functorial computability, induction, and recursion are examined in view of the above, presenting new computability techniques with monad transformations and projective sets. This informative volume will give readers a complete new feel for models, computability, recursion sets, complexity, and realizability. This book pulls together functorial thoughts, models, computability, sets, recursion, arithmetic hierarchy, filters, with real tree computing areas, presented in a very intuitive manner for university teaching, with exercises for every chapter. The book will also prove valuable for faculty in computer science and mathematics.

## Revolutions and Revelations in Computability

Author | : Ulrich Berger,Johanna N. Y. Franklin,Florin Manea,Arno Pauly |

Publsiher | : Springer Nature |

Total Pages | : 327 |

Release | : 2022-06-25 |

ISBN | : 3031087402 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Revolutions and Revelations in Computability Book Excerpt:**

This book constitutes the proceedings of the 18th Conference on Computability in Europe, CiE 2022, in Swansea, UK, in July 2022. The 19 full papers together with 7 invited papers presented in this volume were carefully reviewed and selected from 41 submissions. The motto of CiE 2022 was “Revolutions and revelations in computability”. This alludes to the revolutionary developments we have seen in computability theory, starting with Turing's and Gödel's discoveries of the uncomputable and the unprovable and continuing to the present day with the advent of new computational paradigms such as quantum computing and bio-computing, which have dramatically changed our view of computability and revealed new insights into the multifarious nature of computation.

## Theory and Applications of Models of Computation

Author | : T-H. Hubert Chan,Lap Chi Lau,Luca Trevisan |

Publsiher | : Springer |

Total Pages | : 356 |

Release | : 2013-04-15 |

ISBN | : 3642382363 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Theory and Applications of Models of Computation Book Excerpt:**

This book constitutes the refereed proceedings of the 10th International Conference on Theory and Applications of Models of Computation, TAMC 2013, held in Hong Kong, China, in May 2013. The 31 revised full papers presented were carefully reviewed and selected from 70 submissions. Bringing together a wide range of researchers with interests in computational theory and applications, the papers address the three main themes of the conference which were computability, complexity, and algorithms and present current research in these fields with aspects to theoretical computer science, algorithmic mathematics, and applications to the physical sciences.

## Logic Language Information and Computation

Author | : Alexandra Silva,Renata Wassermann,Ruy de Queiroz |

Publsiher | : Springer Nature |

Total Pages | : 427 |

Release | : 2021-10-05 |

ISBN | : 3030888533 |

Category | : Philosophy |

Language | : EN, FR, DE, ES & NL |

**Logic Language Information and Computation Book Excerpt:**

Edited in collaboration with FoLLI, the Association of Logic, Language and Information this book constitutes the refereed proceedings of the 27th Workshop on Logic, Language, Information and Communication, WoLLIC 2021, Virtual Event, in October 2021. The 25 full papers presented included 6 invited lectures were fully reviewed and selected from 50 submissions. The idea is to have a forum which is large enough in the number of possible interactions between logic and the sciences related to information and computation.

## Foundations of Software Science and Computation Structures

Author | : Javier Esparza,Andrzej S. Murawski |

Publsiher | : Springer |

Total Pages | : 568 |

Release | : 2017-03-15 |

ISBN | : 366254458X |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Foundations of Software Science and Computation Structures Book Excerpt:**

This book constitutes the proceedings of the 20th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2017, which took place in Uppsala, Sweden in April 2017, held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2017. The 32 papers presented in this volume were carefully reviewed and selected from 101 submissions. They were organized in topical sections named: coherence spaces and higher-order computation; algebra and coalgebra; games and automata; automata, logic and formal languages; proof theory; probability; concurrency; lambda calculus and constructive proof; and semantics and category theory.

## Higher Order Computability

Author | : John Longley,Dag Normann |

Publsiher | : Springer |

Total Pages | : 571 |

Release | : 2015-11-06 |

ISBN | : 3662479923 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Higher Order Computability Book Excerpt:**

This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages. In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them. The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers

## Logical Foundations of Computer Science

Author | : Sergei Artemov,Anil Nerode |

Publsiher | : Springer |

Total Pages | : 369 |

Release | : 2017-12-22 |

ISBN | : 3319720562 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Logical Foundations of Computer Science Book Excerpt:**

This book constitutes the refereed proceedings of the International Symposium on Logical Foundations of Computer Science, LFCS 2018, held in Deerfield Beach, FL, USA, in January 2018. The 22 revised full papers were carefully reviewed and selected from 22 submissions. The scope of the Symposium is broad and includes constructive mathematics and type theory; homotopy type theory; logic, automata, and automatic structures; computability and randomness; logical foundations of programming; logical aspects of computational complexity; parameterized complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logics; logical foundations of database theory; equational logic and term rewriting; lambda andcombinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple-agent system logics; logics of proof and justification; non-monotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; mathematical fuzzy logic; system design logics; and other logics in computer science.

## Sets and Extensions in the Twentieth Century

Author | : Anonim |

Publsiher | : Elsevier |

Total Pages | : 880 |

Release | : 2012-01-24 |

ISBN | : 0080930662 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Sets and Extensions in the Twentieth Century Book Excerpt:**

Set theory is an autonomous and sophisticated field of mathematics that is extremely successful at analyzing mathematical propositions and gauging their consistency strength. It is as a field of mathematics that both proceeds with its own internal questions and is capable of contextualizing over a broad range, which makes set theory an intriguing and highly distinctive subject. This handbook covers the rich history of scientific turning points in set theory, providing fresh insights and points of view. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in mathematics, the history of philosophy, and any discipline such as computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration Serves as a singular contribution to the intellectual history of the 20th century Contains the latest scholarly discoveries and interpretative insights

## Logic Language Information and Computation

Author | : Anuj Dawar,Ruy de Queiroz |

Publsiher | : Springer Science & Business Media |

Total Pages | : 259 |

Release | : 2010-06-17 |

ISBN | : 3642138233 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Logic Language Information and Computation Book Excerpt:**

This volume contains the papers presented at WoLLIC 2010: 17th Workshop on Logic, Language, Information and Computation held during July 6–9, 2010, on the campus of Universidade de Bras ́ ?lia (UnB), Brazil. The Workshop on Logic, Language, Information and Computation (WoL- LIC) is an annual event, meeting every year since 1994, which aims at fostering interdisciplinary research in pure and applied logic. The idea is to have a forum which is large enough in the number of possible interactions between logic and the sciences related to information and computation, and yet is small enough to allow for concrete and useful interaction among participants. The present volume contains 13 contributed papers that were selected from among 32 submissions after a rigorous review by the Program Committee. Each submission was reviewed by at least two, and on average three, Program C- mittee members. This volume also containspapersor abstractsthat relateto the seven invited talks presented at the workshop. Between them, these papers give a snapshot of some fascinating work taking place at the frontiers between computation, logic, and linguistics. We are grateful to all the people who made this meeting possible and are responsible for its success: the members of the Program Committee and the external reviewers, the invited speakers, the contributors, and the people who were involved in organizing the workshop.

## Handbook of Computability and Complexity in Analysis

Author | : Vasco Brattka,Peter Hertling |

Publsiher | : Springer Nature |

Total Pages | : 427 |

Release | : 2021-06-04 |

ISBN | : 3030592340 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Handbook of Computability and Complexity in Analysis Book Excerpt:**

Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s. This was motivated by questions such as: which real numbers and real number functions are computable, and which mathematical tasks in analysis can be solved by algorithmic means? Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity, dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics. In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology. This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field. It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research literature in this field.

## Extended Abstracts Fall 2013

Author | : Maria del Mar González,Paul C. Yang,Nicola Gambino,Joachim Kock |

Publsiher | : Birkhäuser |

Total Pages | : 110 |

Release | : 2015-11-12 |

ISBN | : 3319212842 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Extended Abstracts Fall 2013 Book Excerpt:**

The two parts of the present volume contain extended conference abstracts corresponding to selected talks given by participants at the "Conference on Geometric Analysis" (thirteen abstracts) and at the "Conference on Type Theory, Homotopy Theory and Univalent Foundations" (seven abstracts), both held at the Centre de Recerca Matemàtica (CRM) in Barcelona from July 1st to 5th, 2013, and from September 23th to 27th, 2013, respectively. Most of them are brief articles, containing preliminary presentations of new results not yet published in regular research journals. The articles are the result of a direct collaboration between active researchers in the area after working in a dynamic and productive atmosphere. The first part is about Geometric Analysis and Conformal Geometry; this modern field lies at the intersection of many branches of mathematics (Riemannian, Conformal, Complex or Algebraic Geometry, Calculus of Variations, PDE's, etc) and relates directly to the physical world, since many natural phenomena posses an intrinsic geometric content. The second part is about Type Theory, Homotopy Theory and Univalent Foundations. The book is intended for established researchers, as well as for PhD and postdoctoral students who want to learn more about the latest advances in these highly active areas of research.

## Foundations of Secure Computation

Author | : Friedrich L. Bauer,Ralf Steinbrüggen |

Publsiher | : IOS Press |

Total Pages | : 319 |

Release | : 2000 |

ISBN | : 9781586030155 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Foundations of Secure Computation Book Excerpt:**

The final quarter of the 20th century has seen the establishment of a global computational infrastructure. This and the advent of programming languages such as Java, supporting mobile distributed computing, has posed a significant challenge to computer sciences. The infrastructure can support commerce, medicine and government, but only if communications and computing can be secured against catastrophic failure and malicious interference.

## Encyclopaedia of Mathematics

Author | : Michiel Hazewinkel |

Publsiher | : Springer Science & Business Media |

Total Pages | : 536 |

Release | : 2012-12-06 |

ISBN | : 9401512337 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Encyclopaedia of Mathematics Book Excerpt:**

This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fme subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.