Semiorder is probably one of the most frequently ordered structures in science. It naturally appears in fields like psychometrics, economics, decision sciences, linguistics and archaeology. It explicitly takes into account the inevitable imprecisions of scientific instruments by allowing the replacement of precise numbers by intervals. The purpose of this book is to dissect this structure and to study its fundamental properties. The main subjects treated are the numerical representations of semiorders, the generalizations of the concept to valued relations, the aggregation of semiorders and their basic role in a general theoretical framework for multicriteria decision-aid methods. Audience: This volume is intended for students and researchers in the fields of decision analysis, management science, operations research, discrete mathematics, classification, social choice theory, and order theory, as well as for practitioners in the design of decision tools.

Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures

In this monograph, questions of extensions and relaxations are consid ered. These questions arise in many applied problems in connection with the operation of perturbations. In some cases, the operation of "small" per turbations generates "small" deviations of basis indexes; a corresponding stability takes place. In other cases, small perturbations generate spas modic change of a result and of solutions defining this result. These cases correspond to unstable problems. The effect of an unstability can arise in extremal problems or in other related problems. In this connection, we note the known problem of constructing the attainability domain in con trol theory. Of course, extremal problems and those of attainability (in abstract control theory) are connected. We exploit this connection here (see Chapter 5). However, basic attention is paid to the problem of the attainability of elements of a topological space under vanishing perturba tions of restrictions. The stability property is frequently missing; the world of unstable problems is of interest for us. We construct regularizing proce dures. However, in many cases, it is possible to establish a certain property similar to partial stability. We call this property asymptotic nonsensitivity or roughness under the perturbation of some restrictions. The given prop erty means the following: in the corresponding problem, it is the same if constraints are weakened in some "directions" or not. On this basis, it is possible to construct a certain classification of constraints, selecting "di rections of roughness" and "precision directions.""

The notion of complexity is an important contribution of logic to theoretical computer science and mathematics. This volume attempts to approach complexity in a holistic way, investigating mathematical properties of complexity hierarchies at the same time as discussing algorithms and computational properties. A main focus of the volume is on some of the new paradigms of computation, among them Quantum Computing and Infinitary Computation. The papers in the volume are tied together by an introductory article describing abstract properties of complexity hierarchies.

This volume will be of great interest to both mathematical logicians and theoretical computer scientists, providing them with new insights into the various views of complexity and thus shedding new light on their own research.

During the last few decades the ideas, methods, and results of the theory of Boolean algebras have played an increasing role in various branches of mathematics and cybernetics. This monograph is devoted to the fundamentals of the theory of Boolean constructions in universal algebra. Also considered are the problems of presenting different varieties of universal algebra with these constructions, and applications for investigating the spectra and skeletons of varieties of universal algebras. For researchers whose work involves universal algebra and logic.

Quantum Structures and the Nature of Reality is a collection of papers written for an interdisciplinary audience about the quantum structure research within the International Quantum Structures Association. The advent of quantum mechanics has changed our scientific worldview in a fundamental way. Many popular and semi-popular books have been published about the paradoxical aspects of quantum mechanics. Usually, however, these reflections find their origin in the standard views on quantum mechanics, most of all the wave-particle duality picture. Contrary to relativity theory, where the meaning of its revolutionary ideas was linked from the start with deep structural changes in the geometrical nature of our world, the deep structural changes about the nature of our reality that are indicated by quantum mechanics cannot be traced within the standard formulation. The study of the structure of quantum theory, its logical content, its axiomatic foundation, has been motivated primarily by the search for their structural changes. Due to the high mathematical sophistication of this quantum structure research, no books have been published which try to explain the recent results for an interdisciplinary audience. This book tries to fill this gap by collecting contributions from some of the main researchers in the field. They reveal the steps that have been taken towards a deeper structural understanding of quantum theory.

Multi-Criteria Decision Making (MCDM) has been one of the fastest growing problem areas in many disciplines. The central problem is how to evaluate a set of alternatives in terms of a number of criteria. Although this problem is very relevant in practice, there are few methods available and their quality is hard to determine. Thus, the question Which is the best method for a given problem?' has become one of the most important and challenging ones. This is exactly what this book has as its focus and why it is important. The author extensively compares, both theoretically and empirically, real-life MCDM issues and makes the reader aware of quite a number of surprising abnormalities' with some of these methods. What makes this book so valuable and different is that even though the analyses are rigorous, the results can be understood even by the non-specialist. Audience: Researchers, practitioners, and students; it can be used as a textbook for senior undergraduate or graduate courses in business and engineering.

The Institute Vienna Circle held a conference in Vienna in 2003, Cambridge and Vienna Frank P. Ramsey and the Vienna Circle, to commemorate the philosophical and scientific work of Frank Plumpton Ramsey (1903 1930). This Ramsey conference provided not only historical and biographical perspectives on one of the most gifted thinkers of the Twentieth Century, but also new impulses for further research on at least some of the topics pioneered by Ramsey, whose interest and potential are greater than ever.

Ramsey did pioneering work in several fields, practitioners of which rarely know of his important work in other fields: philosophy of logic and theory of language, foundations of mathematics, mathematics, probability theory, methodology of science, philosophy of psychology, and economics. There was a focus on the one topic which was of strongest mutual concern to Ramsey and the Vienna Circle, namely the question of foundations of mathematics, in particular the status of logicism.

Although the major scientific connection linking Ramsey with Austria is his work on logic, to which the Vienna Circle dedicated several meetings, certainly the connection which is of greater general interest concerns Ramsey's visits and discussions with Wittgenstein. Ramsey was the only important thinker to actually visit Wittgenstein during his school-teaching career in Puchberg and Ottertal in the 1920s, in Lower Austria; and later, Ramsey was instrumental in getting Wittgenstein positions at Cambridge. "

This book reflects the progress made in the forty years since the appearance of Abraham Robinson 's revolutionary book Nonstandard Analysis in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained.

This biography attempts to shed light on all facets of Zermelo's life and achievements. Personal and scientific aspects are kept separate as far as coherence allows, in order to enable the reader to follow the one or the other of these threads. The presentation of his work explores motivations, aims, acceptance, and influence. Selected proofs and information gleaned from unpublished notes and letters add to the analysis.

Both modern mathematical music theory and computer science are strongly influenced by the theory of categories and functors. One outcome of this research is the data format of denotators, which is based on set-valued presheaves over the category of modules and diaffine homomorphisms. The functorial approach of denotators deals with generalized points in the form of arrows and allows the construction of a universal concept architecture. This architecture is ideal for handling all aspects of music, especially for the analysis and composition of highly abstract musical works.

This book presents an introduction to the theory of module categories and the theory of denotators, as well as the design of a software system, called Rubato Composer, which is an implementation of the category-theoretic concept framework. The application is written in portable Java and relies on plug-in components, so-called rubettes, which may be combined in data flow networks for the generation and manipulation of denotators.

The Rubato Composer system is open to arbitrary extension and is freely available under the GPL license. It allows the developer to build specialized rubettes for tasks that are of interest to composers, who in turn combine them to create music. It equally serves music theorists, who use them to extract information from and manipulate musical structures. They may even develop new theories by experimenting with the many parameters that are at their disposal thanks to the increased flexibility of the functorial concept architecture.

Two contributed chapters by Guerino Mazzola and Florian Thalmann illustrate the application of the theory as well as the software in the development of compositional tools and the creation of a musical work with the help of the Rubato framework.

Can a line be analysed mathematically such a way that it does not fall apart into a set of discrete points? Are there objects of pure mathematics that can change through time? L. E. J. Brouwer argued that the two questions are related and that the answer to both is "yes," introducing the concept of choice sequences. This book subjects Brouwer's choice sequences to a phenomenological critique in the style of Husserl.

Over the last decade and particularly in recent years, the macroscopic porous media theory has made decisive progress concerning the fundamentals of the theory and the development of mathematical models in various fields of engineering and biomechanics. This progress has attracted some attention, and therefore conferences devoted almost exclusively to the macrosopic porous media theory have been organized in order to collect all findings, to present new results, and to discuss new trends. Many important contributions have also been published in national and international journals, which have brought the porous media theory, in some parts, to a close. Therefore, the time seems to be ripe to review the state of the art and to show new trends in the continuum mechanical treatment of saturated and unsaturated capillary and non-capillary porous solids.

This book addresses postgraduate students and scientists working in engineering, physics, and mathematics. It provides an outline of modern theory of porous media and shows some trends in theory and in applications.

The present anthology has its origin in two international conferences that were arranged at Uppsala University in August 2004: "Logicism, Intuitionism and F- malism: What has become of them?" followed by "Symposium on Constructive Mathematics." The rst conference concerned the three major programmes in the foundations of mathematics during the classical period from Frege's Begrif- schrift in 1879 to the publication of Godel' ] s two incompleteness theorems in 1931: The logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. The main purpose of the conf- ence was to assess the relevance of these foundational programmes to contemporary philosophy of mathematics. The second conference was announced as a satellite event to the rst, and was speci cally concerned with constructive mathematics-an activebranchofmathematicswheremathematicalstatements-existencestatements in particular-are interpreted in terms of what can be effectively constructed. C- structive mathematics may also be characterized as mathematics based on intuiti- isticlogicand, thus, beviewedasadirectdescendant ofBrouwer'sintuitionism. The two conferences were successful in bringing together a number of internationally renowned mathematicians and philosophers around common concerns. Once again it was con rmed that philosophers and mathematicians can work together and that real progress in the philosophy and foundations of mathematics is possible only if they do. Most of the papers in this collection originate from the two conferences, but a few additional papers of relevance to the issues discussed at the Uppsala c- ferences have been solicited especially for this volume."

The International Congresses of Logic, Methodology and Philosophy of Science, which are held every fourth year, give a cross-section of ongoing research in logic and philosophy of science. Both the invited lectures and the many contributed papers are conductive to this end. At the 9th Congress held in Uppsala in 1991 there were 54 invited lectures and around 650 contributed papers divided into 15 different sections. Some of the speakers who presented contributed papers that attracted special interest were invited to submit their papers for publication, and the result is the present volume. A few papers appear here more or less as they were presented at the Congress whereas others are expansions or elaborations of the talks given at the Congress. A selection of this kind, containing 38 papers drawn from the 650 contributed papers presented at the Uppsala Congress, cannot do justice to all facets of the field as it appeared at the Congress. But it should allow the reader to get a representative survey of contemporary research in large areas of philosophical logic and philosophy of science. About half of the papers of the volume appear in sections listed at the Congress under the heading Philosophical and Foundational Problems about the Sciences. The section Foundations of Logic, Mathematics and Computer Science is represented by three papers, Foundations of Physical Sciences by six papers, Foundations of Biological Sciences by three papers, Foundations of Cognitive Science and AI by one paper, and Foundations of Linguistics by three papers.

From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein's Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane's work in the early 1940's and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics.

From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.

The past fifteen years has witnessed an explosive growth in the fundamental research and applications of artificial neural networks (ANNs) and fuzzy logic (FL). The main impetus behind this growth has been the ability of such methods to offer solutions not amenable to conventional techniques, particularly in application domains involving pattern recognition, prediction and control. Although the origins of ANNs and FL may be traced back to the 1940s and 1960s, respectively, the most rapid progress has only been achieved in the last fifteen years. This has been due to significant theoretical advances in our understanding of ANNs and FL, complemented by major technological developments in high-speed computing. In geophysics, ANNs and FL have enjoyed significant success and are now employed routinely in the following areas (amongst others): 1. Exploration Seismology. (a) Seismic data processing (trace editing; first break picking; deconvolution and multiple suppression; wavelet estimation; velocity analysis; noise identification/reduction; statics analysis; dataset matching/prediction, attenuation), (b) AVO analysis, (c) Chimneys, (d) Compression I dimensionality reduction, (e) Shear-wave analysis, (f) Interpretation (event tracking; lithology prediction and well-log analysis; prospect appraisal; hydrocarbon prediction; inversion; reservoir characterisation; quality assessment; tomography). 2. Earthquake Seismology and Subterranean Nuclear Explosions. 3. Mineral Exploration. 4. Electromagnetic I Potential Field Exploration. (a) Electromagnetic methods, (b) Potential field methods, (c) Ground penetrating radar, (d) Remote sensing, (e) inversion.

In the 20th century philosophy of mathematics has to a great extent been dominated by views developed during the so-called foundational crisis in the beginning of that century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justi?cation and consistency of mathematics. Paradigmatic in this - spect is Hilbert's program which inherits from Frege and Russell the project to formalize all areas of ordinary mathematics and then adds the requi- ment of a proof, by epistemically privileged means (?nitistic reasoning), of the consistency of such formalized theories. While interest in modi?ed v- sions of the original foundational programs is still thriving, in the second part of the twentieth century several philosophers and historians of mat- matics have questioned whether such foundational programs could exhaust the realm of important philosophical problems to be raised about the nature of mathematics. Some have done so in open confrontation (and hostility) to the logically based analysis of mathematics which characterized the cl- sical foundational programs, while others (and many of the contributors to this book belong to this tradition) have only called for an extension of the range of questions and problems that should be raised in connection with an understanding of mathematics. The focus has turned thus to a consideration of what mathematicians are actually doing when they produce mathematics. Questions concerning concept-formation, understanding, heuristics, changes instyle of reasoning, the role of analogies and diagrams etc.

Mathematical Linguistics introduces the mathematical foundations of linguistics to computer scientists, engineers, and mathematicians interested in natural language processing. The book presents linguistics as a cumulative body of knowledge from the ground up: no prior knowledge of linguistics is assumed. As the first textbook of its kind, this book is useful for those in information science and in natural language technologies.

Information is precious. It reduces our uncertainty in making decisions. Knowledge about the outcome of an uncertain event gives the possessor an advantage. It changes the course of lives, nations, and history itself. Information is the food of Maxwell's demon. His power comes from know ing which particles are hot and which particles are cold. His existence was paradoxical to classical physics and only the realization that information too was a source of power led to his taming. Information has recently become a commodity, traded and sold like or ange juice or hog bellies. Colleges give degrees in information science and information management. Technology of the computer age has provided access to information in overwhelming quantity. Information has become something worth studying in its own right. The purpose of this volume is to introduce key developments and results in the area of generalized information theory, a theory that deals with uncertainty-based information within mathematical frameworks that are broader than classical set theory and probability theory. The volume is organized as follows."

The idea about this book has evolved during the process of its preparation as some of the results have been achieved in parallel with its writing. One reason for this is that in this area of research results are very quickly updated. Another is, possibly, that a strong, unchallenged theoretical basis in this field still does not fully exist. From other hand, the rate of innovation, competition and demand from different branches of industry (from biotech industry to civil and building engineering, from market forecasting to civil aviation, from robotics to emerging e-commerce) is increasingly pressing for more customised solutions based on learning consumers behaviour. A highly interdisciplinary and rapidly innovating field is forming which focus is the design of intelligent, self-adapting systems and machines. It is on the crossroads of control theory, artificial and computational intelligence, different engineering disciplines borrowing heavily from the biology and life sciences. It is often called intelligent control, soft computing or intelligent technology. Some other branches have appeared recently like intelligent agents (which migrated from robotics to different engineering fields), data fusion, knowledge extraction etc., which are inherently related to this field. The core is the attempts to enhance the abilities of the classical control theory in order to have more adequate, flexible, and adaptive models and control algorithms.

When we learn from books or daily experience, we make associations and draw inferences on the basis of information that is insufficient for under standing. One example of insufficient information may be a small sample derived from observing experiments. With this perspective, the need for de veloping a better understanding of the behavior of a small sample presents a problem that is far beyond purely academic importance. During the past 15 years considerable progress has been achieved in the study of this issue in China. One distinguished result is the principle of in formation diffusion. According to this principle, it is possible to partly fill gaps caused by incomplete information by changing crisp observations into fuzzy sets so that one can improve the recognition of relationships between input and output. The principle of information diffusion has been proven suc cessful for the estimation of a probability density function. Many successful applications reflect the advantages of this new approach. It also supports an argument that fuzzy set theory can be used not only in "soft" science where some subjective adjustment is necessary, but also in "hard" science where all data are recorded."

In the mid-1960's I had the pleasure of attending a talk by Lotfi Zadeh at which he presented some of his basic (and at the time, recent) work on fuzzy sets. Lotfi's algebra of fuzzy subsets of a set struck me as very nice; in fact, as a graduate student in the mid-1950's, I had suggested similar ideas about continuous-truth-valued propositional calculus (inffor "and," sup for "or") to my advisor, but he didn't go for it (and in fact, confused it with the foundations of probability theory), so I ended up writing a thesis in a more conventional area of mathematics (differential algebra). I especially enjoyed Lotfi's discussion of fuzzy convexity; I remember talking to him about possible ways of extending this work, but I didn't pursue this at the time. I have elsewhere told the story of how, when I saw C. L. Chang's 1968 paper on fuzzy topological spaces, I was impelled to try my hand at fuzzi fying algebra. This led to my 1971 paper "Fuzzy groups," which became the starting point of an entire literature on fuzzy algebraic structures. In 1974 King-Sun Fu invited me to speak at a U. S. -Japan seminar on Fuzzy Sets and their Applications, which was to be held that summer in Berkeley."

This edition has been called startlingly up-to-date, and in this corrected second printing you can be sure that it 's even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.

The capabilities of modern technology are rapidly increasing, spurred on to a large extent by the tremendous advances in communications and computing. Automated vehicles and global wireless connections are some examples of these advances. In order to take advantage of such enhanced capabilities, our need to model and manipulate our knowledge of the geophysical world, using compatible representations, is also rapidly increasing. In response to this one fundamental issue of great concern in modern geographical research is how to most effectively capture the physical world around us in systems like geographical information systems (GIS). Making this task even more challenging is the fact that uncertainty plays a pervasive role in the representation, analysis and use of geospatial information. The types of uncertainty that appear in geospatial information systems are not the just simple randomness of observation, as in weather data, but are manifested in many other forms including imprecision, incompleteness and granularization. Describing the uncertainty of the boundaries of deserts and mountains clearly require different tools than those provided by probability theory. The multiplicity of modalities of uncertainty appearing in GIS requires a variety of formalisms to model these uncertainties. In light of this it is natural that fuzzy set theory has become a topic of intensive interest in many areas of geographical research and applications This volume, Fuzzy Modeling with Spatial Information for Geographic Problems, provides many stimulating examples of advances in geographical research based on approaches using fuzzy sets and related technologies.