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Author: Anatolij Dvurecenskij
Publisher: Springer Science & Business Media
ISBN: 9401724229
Category : Mathematics
Languages : en
Pages : 551
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Book Description
D. Hilbert, in his famous program, formulated many open mathematical problems which were stimulating for the development of mathematics and a fruitful source of very deep and fundamental ideas. During the whole 20th century, mathematicians and specialists in other fields have been solving problems which can be traced back to Hilbert's program, and today there are many basic results stimulated by this program. It is sure that even at the beginning of the third millennium, mathematicians will still have much to do. One of his most interesting ideas, lying between mathematics and physics, is his sixth problem: To find a few physical axioms which, similar to the axioms of geometry, can describe a theory for a class of physical events that is as large as possible. We try to present some ideas inspired by Hilbert's sixth problem and give some partial results which may contribute to its solution. In the Thirties the situation in both physics and mathematics was very interesting. A.N. Kolmogorov published his fundamental work Grundbegriffe der Wahrschein lichkeitsrechnung in which he, for the first time, axiomatized modern probability theory. From the mathematical point of view, in Kolmogorov's model, the set L of ex perimentally verifiable events forms a Boolean a-algebra and, by the Loomis-Sikorski theorem, roughly speaking can be represented by a a-algebra S of subsets of some non-void set n.
Author: Anatolij Dvurecenskij
Publisher: Springer Science & Business Media
ISBN: 9401724229
Category : Mathematics
Languages : en
Pages : 551
Get Book Here
Book Description
D. Hilbert, in his famous program, formulated many open mathematical problems which were stimulating for the development of mathematics and a fruitful source of very deep and fundamental ideas. During the whole 20th century, mathematicians and specialists in other fields have been solving problems which can be traced back to Hilbert's program, and today there are many basic results stimulated by this program. It is sure that even at the beginning of the third millennium, mathematicians will still have much to do. One of his most interesting ideas, lying between mathematics and physics, is his sixth problem: To find a few physical axioms which, similar to the axioms of geometry, can describe a theory for a class of physical events that is as large as possible. We try to present some ideas inspired by Hilbert's sixth problem and give some partial results which may contribute to its solution. In the Thirties the situation in both physics and mathematics was very interesting. A.N. Kolmogorov published his fundamental work Grundbegriffe der Wahrschein lichkeitsrechnung in which he, for the first time, axiomatized modern probability theory. From the mathematical point of view, in Kolmogorov's model, the set L of ex perimentally verifiable events forms a Boolean a-algebra and, by the Loomis-Sikorski theorem, roughly speaking can be represented by a a-algebra S of subsets of some non-void set n.
Author: Andrei Y. Khrennikov
Publisher: Springer Science & Business Media
ISBN: 3642051014
Category : Science
Languages : en
Pages : 226
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Book Description
Quantum-like structure is present practically everywhere. Quantum-like (QL) models, i.e. models based on the mathematical formalism of quantum mechanics and its generalizations can be successfully applied to cognitive science, psychology, genetics, economics, finances, and game theory. This book is not about quantum mechanics as a physical theory. The short review of quantum postulates is therefore mainly of historical value: quantum mechanics is just the first example of the successful application of non-Kolmogorov probabilities, the first step towards a contextual probabilistic description of natural, biological, psychological, social, economical or financial phenomena. A general contextual probabilistic model (Växjö model) is presented. It can be used for describing probabilities in both quantum and classical (statistical) mechanics as well as in the above mentioned phenomena. This model can be represented in a quantum-like way, namely, in complex and more general Hilbert spaces. In this way quantum probability is totally demystified: Born's representation of quantum probabilities by complex probability amplitudes, wave functions, is simply a special representation of this type.
Author: Kurt Engesser
Publisher: Elsevier
ISBN: 008055038X
Category : Mathematics
Languages : en
Pages : 821
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Book Description
Since its inception in the famous 1936 paper by Birkhoff and von Neumann entitled "The logic of quantum mechanics quantum logic, i.e. the logical investigation of quantum mechanics, has undergone an enormous development. Various schools of thought and approaches have emerged and there are a variety of technical results.Quantum logic is a heterogeneous field of research ranging from investigations which may be termed logical in the traditional sense to studies focusing on structures which are on the border between algebra and logic. For the latter structures the term quantum structures is appropriate. The chapters of this Handbook, which are authored by the most eminent scholars in the field, constitute a comprehensive presentation of the main schools, approaches and results in the field of quantum logic and quantum structures. Much of the material presented is of recent origin representing the frontier of the subject. The present volume focuses on quantum structures. Among the structures studied extensively in this volume are, just to name a few, Hilbert lattices, D-posets, effect algebras MV algebras, partially ordered Abelian groups and those structures underlying quantum probability.- Written by eminent scholars in the field of logic- A comprehensive presentation of the theory, approaches and results in the field of quantum logic- Volume focuses on quantum structures
Author: Luigi Cocco
Publisher: BoD – Books on Demand
ISBN: 9535124773
Category : Technology & Engineering
Languages : en
Pages : 294
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Book Description
Investigating the incessant technology growth and the even higher complexity of engineering systems, one of the crucial requirements to confidently steer both scientific and industrial challenges is to identify an appropriate measurement approach. A general process can be considered effective and under control if the following elements are consciously and cyclically managed: numeric target, adequate tools, output analysis, and corrective actions. The role of metrology is to rigorously harmonize this virtuous circle, providing guidance in terms of instruments, standards, and techniques to improve the robustness and the accuracy of the results. This book is designed to offer an interdisciplinary experience into the science of measurement, not only covering high-level measurement strategies but also supplying analytical details and experimental setups.
Author: Vincent F. Hendricks
Publisher: Springer Science & Business Media
ISBN: 9401735980
Category : Philosophy
Languages : en
Pages : 387
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Book Description
In 1953, exactly 50 years ago to this day, the first volume of Studia Logica appeared under the auspices of The Philosophical Committee of The Polish Academy of Sciences. Now, five decades later the present volume is dedicated to a celebration of this 50th Anniversary of Studia Logica. The volume features a series of papers by distinguished scholars reflecting both the aim and scope of this journal for symbolic logic.
Author: Maria Luisa Dalla Chiara
Publisher: Springer
ISBN: 3030044718
Category : Philosophy
Languages : en
Pages : 192
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Book Description
This book provides a general survey of the main concepts, questions and results that have been developed in the recent interactions between quantum information, quantum computation and logic. Divided into 10 chapters, the books starts with an introduction of the main concepts of the quantum-theoretic formalism used in quantum information. It then gives a synthetic presentation of the main “mathematical characters” of the quantum computational game: qubits, quregisters, mixtures of quregisters, quantum logical gates. Next, the book investigates the puzzling entanglement-phenomena and logically analyses the Einstein–Podolsky–Rosen paradox and introduces the reader to quantum computational logics, and new forms of quantum logic. The middle chapters investigate the possibility of a quantum computational semantics for a language that can express sentences like “Alice knows that everybody knows that she is pretty”, explore the mathematical concept of quantum Turing machine, and illustrate some characteristic examples that arise in the framework of musical languages. The book concludes with an analysis of recent discussions, and contains a Mathematical Appendix which is a survey of the definitions of all main mathematical concepts used in the book.
Author: Hanne Andersen
Publisher: Springer Science & Business Media
ISBN: 9400758456
Category : Science
Languages : en
Pages : 499
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Book Description
This volume is a serious attempt to open up the subject of European philosophy of science to real thought, and provide the structural basis for the interdisciplinary development of its specialist fields, but also to provoke reflection on the idea of ‘European philosophy of science’. This efforts should foster a contemporaneous reflection on what might be meant by philosophy of science in Europe and European philosophy of science, and how in fact awareness of it could assist philosophers interpret and motivate their research through a stronger collective identity. The overarching aim is to set the background for a collaborative project organising, systematising, and ultimately forging an identity for, European philosophy of science by creating research structures and developing research networks across Europe to promote its development.
Author: M. M. Rao
Publisher: Springer Science & Business Media
ISBN: 1461220548
Category : Mathematics
Languages : en
Pages : 411
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Book Description
As in the case of the two previous volumes published in 1986 and 1997, the purpose of this monograph is to focus the interplay between real (functional) analysis and stochastic analysis show their mutual benefits and advance the subjects. The presentation of each article, given as a chapter, is in a research-expository style covering the respective topics in depth. In fact, most of the details are included so that each work is essentially self contained and thus will be of use both for advanced graduate students and other researchers interested in the areas considered. Moreover, numerous new problems for future research are suggested in each chapter. The presented articles contain a substantial number of new results as well as unified and simplified accounts of previously known ones. A large part of the material cov ered is on stochastic differential equations on various structures, together with some applications. Although Brownian motion plays a key role, (semi-) martingale theory is important for a considerable extent. Moreover, noncommutative analysis and probabil ity have a prominent role in some chapters, with new ideas and results. A more detailed outline of each of the articles appears in the introduction and outline to assist readers in selecting and starting their work. All chapters have been reviewed.
Author: Diederik Aerts
Publisher: Frontiers Media SA
ISBN: 2889198766
Category : Psychology
Languages : en
Pages : 170
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Book Description
Traditional approaches to cognitive psychology correspond with a classical view of logic and probability theory. More specifically, one typically assumes that cognitive processes of human thought are founded on the Boolean structures of classical logic, while the probabilistic aspects of these processes are based on the Kolmogorovian structures of classical probability theory. However, growing experimental evidence indicates that the models founded on classical structures systematically fail when human decisions are at stake. These experimental deviations from classical behavior have been called `paradoxes’, `fallacies’, `effects’ or `contradictions’, depending on the specific situation where they appear. But, they involve a broad spectrum of cognitive and social science domains, ranging from conceptual combination to decision making under uncertainty, behavioral economics, and linguistics. This situation has constituted a serious drawback to the development of various disciplines, like cognitive science, linguistics, artificial intelligence, economic modeling and behavioral finance. A different approach to cognitive psychology, initiated two decades ago, has meanwhile matured into a new domain of research, called ‘quantum cognition’. Its main feature is the use of the mathematical formalism of quantum theory as modeling tool for these cognitive situations where traditional classically based approaches fail. Quantum cognition has recently attracted the interest of important journals and editing houses, academic and funding institutions, popular science and media. Specifically, within a quantum cognition approach, one assumes that human decisions do not necessarily obey the rules of Boolean logic and Kolmogorovian probability, and can on the contrary be modeled by the quantum-mechanical formalism. Different concrete quantum-theoretic models have meanwhile been developed that successfully represent the cognitive situations that are classically problematical, by explaining observed deviations from classicality in terms of genuine quantum effects, such as `contextuality’, `emergence’, `interference’, `superposition’, `entanglement’ and `indistinguishability’. In addition, the validity of these quantum models is convincingly confirmed by new experimental tests. We also stress that, since the use of a quantum-theoretic framework is mainly for modeling purposes, the identification of quantum structures in cognitive processes does not presuppose (without being incompatible with it) the existence of microscopic quantum processes in the human brain. In this Research Topic, we review the major achievements that have been obtained in quantum cognition, by providing an accurate picture of the state-of-the-art of this emerging discipline. Our overview does not pretend to be either complete or exhaustive. But, we aim to introduce psychologists and social scientists to this challenging new research area, encouraging them, at the same time, to consider its promising results. It is our opinion that, if continuous progress in this domain can be realized, quantum cognition can constitute an important breakthrough in cognitive psychology, and potentially open the way towards a new scientific paradigm in social science.
Author: Friedrich Wehrung
Publisher: Springer
ISBN: 3319615998
Category : Mathematics
Languages : en
Pages : 245
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Book Description
Adopting a new universal algebraic approach, this book explores and consolidates the link between Tarski's classical theory of equidecomposability types monoids, abstract measure theory (in the spirit of Hans Dobbertin's work on monoid-valued measures on Boolean algebras) and the nonstable K-theory of rings. This is done via the study of a monoid invariant, defined on Boolean inverse semigroups, called the type monoid. The new techniques contrast with the currently available topological approaches. Many positive results, but also many counterexamples, are provided.
