Mathematical foundations Books

Book SynopsisThis book constitutes revised selected papers from the refereed proceedings of the 4th International Workshop on Dynamic Logic, DaLí 2022, held in Haifa, Israel, in July/August 2022.The 8 full papers presented in this volume were carefully reviewed and selected from 22 submissions. They deal with new trends and applications in the area of Dynamic Logic. Table of ContentsFirst steps in updating knowing how.- Parametrized modal logic II: the unidimensional case.- Relating Kleene algebras.- Dynamic epistemic logic for budget-constrained agents.- Action models for coalition logic.- Quantum logic for observation of physical quantities.- Cautious distributed belief.- A STIT logic of intentionality.

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Book SynopsisEdited in collaboration with FoLLI, this book constitutes the refereed proceedings of the 10th Indian Conference on Logic and Its Applications, ICLA 2023, which was held in Indore, India, in March 2023.Besides 6 invited papers presented in this volume, there are 9 contributed full papers which were carefully reviewed and selected from 18 submissions. The volume covers a wide range of topics. These topics are related to modal and temporal logics, intuitionistic connexive and imperative logics, systems for reasoning with vagueness and rough concepts, topological quasi-Boolean logic and quasi-Boolean based rough set models, and first-order definability of path functions of graphs.Table of ContentsA Note on the Ontology of Mathematics.- Boolean Functional Synthesis: From Under the Hood of Solvers.- Labelled Calculi for Lattice-based Modal Logics.- Two Ways to Scare a Gruffalo.- Determinacy Axioms and Large Cardinals.- Big ideas from logic for mathematics and computing education.- Modal Logic of Generalized Separated Topological Spaces.- Multiple-valued Semantics for Metric Temporal Logic.- Segment transit function of the induced path function of graphs and its first-order definability.- Fuzzy Free Logic with Dual Domain Semantics.- A New Dimension of Imperative Logic. -Quasi-Boolean based models in Rough Set theory: A case of Covering.- Labelled calculi for the logics of rough concepts.- An Infinity of Intuitionistic Connexive Logics.- Relational Semantics for Normal Topological Quasi-Boolean Logic.

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Book SynopsisThis book constitutes the proceedings of the 5th International Workshop on Formal Methods Teaching, FMTea 2023, which was held in Lübeck, Germany, in March 2023.The 7 full papers presented in this volume were carefully reviewed and selected from 10 submissions. FMTea 2023 aim is to support a worldwide improvement in learning Formal Methods, mainly by teaching but also via self-learning.Table of ContentsAutomated Exercise Generation for Satisfiability Checking.- Graphical Loop Invariant Based Programming.- A Gentle Introduction to Verification of Parameterized Reactive Systems.- Model Checking Concurrent Programs for Autograding in pseuCo Book.- Teaching TLA+ to Engineers at Microsoft.- Teaching and Training in Formalisation with B.- Teaching low-code Formal Methods with Coloured Petri Nets.

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Book SynopsisThis exploration of a selection of fundamental topics and general purpose tools provides a roadmap to undergraduate students who yearn for a deeper dive into many of the concepts and ideas they have been encountering in their classes whether their motivation is pure curiosity or preparation for graduate studies. The topics intersect a wide range of areas encompassing both pure and applied mathematics. The emphasis and style of the book are motivated by the goal of developing self-reliance and independent mathematical thought. Mathematics requires both intuition and common sense as well as rigorous, formal argumentation. This book attempts to showcase both, simultaneously encouraging readers to develop their own insights and understanding and the adoption of proof writing skills. The most satisfying proofs/arguments are fully rigorous and completely intuitive at the same time.

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Book SynopsisThe two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability.Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs.Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.Trade Review“This monograph by the outstanding Czech logician Pavel Pudlák provides a broad but also deep survey of work in logic and computer science relevant to foundational issues, interpreted in a wide sense. … This is a fine overview of logic and complexity theory that can be confidently recommended to anybody who would like to orient themselves in an increasingly intricate and difficult field.” (Alasdair Urquhart, Philosophia Mathematica, Vol. 23 (3), October, 2015)“For the non-expert it offers indeed a ‘gentle introduction’ to logic that is well selected and excellently explained. And for the logician it certainly offers some of the best introductions to those topics outside their area of direct expertise. … it contains plenty of informal explanations, intuition and motivation. … It is truly a gift to the logic and wider communities … . This book is very enjoyable to read and I wish it all success.” (Olaf Beyersdorff, Mathematical Reviews, August, 2014)“It spans the historical, logical, and at times philosophical underpinnings of the theory of computational complexity. Students of mathematics seeking a transition to higher mathematics will find it helpful, as will mathematicians with expertise in other areas. … an excellent choice for a first text in studying complexity, or as a clarifying adjunct to any assigned text in this area. … a compact guide for graduate students with a need for or interest in computational complexity and its foundations.” (Tom Schulte, MAA Reviews, July, 2014)“This book, exactly as indicated by its title, deals with the main philosophical, historical, logical and mathematical aspects … in a quite approachable and attractive way. … the prospective readers of this book are mathematicians with an interest in the foundations, philosophers with a good background in mathematics, and also philosophically minded scientists. Due to the author’s nice style, the book will be a very good choice for the first text in studying this subject.” (Branislav Boričić, zbMATH, Vol. 1270, 2013)Table of Contents​​​​​​​​​​Mathematician’s world.- Language, logic and computations.- Set theory.- Proofs of impossibility.- The complexity of computations.- Proof complexity.- Consistency, Truth and Existence.- References.

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Book SynopsisWhile most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself.By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics.Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.Trade Review“This is a book of both analysis and set theory, and the analysis begins at an elementary level with the necessary treatment of completeness of the reals. … the analysis makes it valuable to the serious student, say a senior or first-year graduate student. … Stillwell’s book can work well as a text for the course in foundations, with its good treatment of the cardinals and ordinals. … This enjoyable book makes the connection clear.” (James M. Cargal, The UMAP Journal, Vol. 38 (1), 2017)“This book is an interesting introduction to set theory and real analysis embedded in properties of the real numbers. … The 300-plus problems are frequently challenging and will interest both upper-level undergraduate students and readers with a strong mathematical background. … A list of approximately 100 references at the end of the book will help students to further explore the topic. … Summing Up: Recommended. Lower-division undergraduates.” (D. P. Turner, Choice, Vol. 51 (11), August, 2014)“This is an informal look at the nature of the real numbers … . There are extensive historical notes about the evolution of real analysis and our understanding of real numbers. … Stillwell has deliberately set out to provide a different sort of construction where you understand what the foundation is supporting and why it is important. I think this is very successful, and his book … is much more informative and enjoyable.” (Allen Stenger, MAA Reviews, February, 2014)“This book will be fully appreciated by either professional mathematicians or those students, who already have passed a course in analysis or set theory. … The book contains a quantity of motivation examples, worked examples and exercises, what makes it suitable also for self-study.” (Vladimír Janiš, zbMATH, 2014)“The book offers a rigorous foundation of the real number system. It is intended for senior undergraduates who have already studied calculus, but a wide range of readers will find something interesting, new, or instructive in it. … This is an extremely reader-friendly book. It is full of interesting examples, very clear explanations, historical background, applications. Each new idea comes after proper motivation.” (László Imre Szabó, Acta Scientiarum Mathematicarum (Szeged), Vol. 80 (1-2), 2014)Table of ContentsThe Fundamental Questions.- From Discrete to Continuous.- Infinite Sets.- Functions and Limits.- Open Sets and Continuity.- Ordinals.- The Axiom of Choice.- Borel Sets.- Measure Theory.- Reflections.- Bibliography.- Index.

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Book SynopsisThis is a comprehensive book on the life and works of Leon Henkin (1921–2006), an extraordinary scientist and excellent teacher whose writings became influential right from the beginning of his career with his doctoral thesis on “The completeness of formal systems” under the direction of Alonzo Church. Upon the invitation of Alfred Tarski, Henkin joined the Group in Logic and the Methodology of Science in the Department of Mathematics at the University of California Berkeley in 1953. He stayed with the group until his retirement in 1991. This edited volume includes both foundational material and a logic perspective. Algebraic logic, model theory, type theory, completeness theorems, philosophical and foundational studies are among the topics covered, as well as mathematical education. The work discusses Henkin’s intellectual development, his relation to his predecessors and contemporaries and his impact on the recent development of mathematical logic. It offers a valuable reference work for researchers and students in the fields of philosophy, mathematics and computer science.Table of ContentsPart I Biographical Studies.- Leon Henkin.- Lessons from Leon.- Tracing back “Logic in Wonderland” to my work with Leon Henkin.- Henkin and the Suit.- A Fortuitous Year with Leon Henkin.- Leon Henkin and a Life of Service.- Part II Henkin‘s Contribution to XX Century Logic.- Leon Henkin and Cylindric Algebras.- A Bit of History Related to Logic Based on Equality.- Pairing Logical and Pedagogical Foundations for the Theory of Positive Rational Numbers. Henkin‘s unfinished work.- Leon Henkin the Reviewer.- Henkin‘s Theorem in Textbooks.- Henkin on Completeness.- Part III Extensions and Perspectives in Henkin‘s Work.- The Countable Henkin Principle.- Reflections on a Theorem of Henkin.- Henkin‘s Completeness Proof and Glivenko‘s Theorem.- From Classical to Fuzzy Type Theory.- The Henkin Sentence.- April the 19th.- Henkin and Hybrid Logic.- Changing a Semantics: Oportunism or Courage?.- Appendix Curriculum Vitae: Leon Henkin.

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Book SynopsisThis volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.Trade Review“Swedish logician and philosopher Dag Prawitz and his distinguished contributions to philosophical and mathematical logic are the focus of this book. … This is an excellent book, celebrating not only Prawitz’s career, but also a movement in the contrary direction of W. V. O Quine’s views against the so-called (somehow prejudicially) ‘deviant’ logics, and I cannot forbear from congratulating the editor for the distinctive choice of topics and for the general tone of the book.” (Walter Carnielli, Computing Reviews, May, 2015)Table of ContentsPrawitz, proofs, and meaning; Wansing, Heinrich.- A short scientific autobiography; Prawitz, Dag.- Explaining deductive inference; Prawitz, Dag.- Necessity of Thought; Cozzo, Cesare.- On the Motives for Proof Theory; Detlefsen, Michael.- Inferential Semantics; Došen, Kosta.- Cut elimination, substitution and normalization; Dyckhoff, Roy.- Inversion principles and introduction rules; Milne, Peter.- Intuitionistic Existential Instantiation and Epsilon Symbol; Mints, Grigori.- Meaning in Use; Negri, Sara and von Plato, Jan.- Fusing Quantifiers and Connectives: Is Intuitionistic Logic Different?; Pagin, Peter.- On constructive fragments of Classical Logic; Pereira; Luiz Carlos and Haeusler, Edward Hermann.- General-Elimination Harmony and Higher-Level Rules; Read, Stephen.- Hypothesis-discharging rules in atomic bases; Sandqvist, Tor.- Harmony in proof-theoretic semantics: A reductive analysis; Schroeder-Heister, Peter.- First-order Logic without bound variables: Compositional Semantics; Tait, William W.- On Gentzen’s Structural Completeness Proof; Tennant, Neil.- A Notion of C-Justification for Empirical Statements; Usberti, Gabriele.

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Book SynopsisGerhard Gentzen is best known for his development of the proof systems of natural deduction and sequent calculus, central in many areas of logic and computer science today. Another noteworthy achievement is his resolution of the embarrassing situation created by Gödel's incompleteness results, especially the second one about the unprovability of consistency of elementary arithmetic. After these successes, Gentzen dedicated the rest of his short life to the main problem of Hilbert's proof theory, the question of the consistency of analysis. He was arrested in the summer of 1945 with other professors of the German University of Prague and died soon afterward of starvation in a prison cell. Attempts at locating his lost manuscripts failed at the time, but several decades later, two slim folders of shorthand notes were found. In this volume, Jan von Plato gives an overview of Gentzen's life and scientific achievements, based on detailed archival and systematic studies, and essential for placing the translations of shorthand manuscripts that follow in the right setting. The materials in this book are singular in the way they show the birth and development of Gentzen's central ideas and results, sometimes in a well-developed form, and other times as flashes into the anatomy of the workings of a unique mind.Trade Review“This book is obviously indispensable to historians of logic in the immediate wake of Gödel’s 1931 incompleteness theorems. … Saved from the Cellar is also valuable for less specialist readers (like myself ) who wish to understand the broader outlines of what proof theory has meant to several of its leading creators.” (Colin McLarty, Isis, Vol. 111 (1), 2020)“The book contains translations of shorthand notes which survived in the Nachlass of the mathematical logician Gerhard Gentzen. ... The book is valuable source for the history of modern logic; the editor did an excellent work in getting the shorthand notes, first transcribed in normal German text, and then translating it to English.” (Reinhard Kahle, zbMath 1414.03002, 2019)“Every general reader interested in modern logic and its history, … may find a source of inspiration in Genzen’s unpublished notes of the thirties, as well as for the philosopher concerned with epistemological aspects of modern logic.” (Adrian Rezus, Studia Logica, Vol. 107, 2019)“This is an account and transcription of two slim folders of stenographic material in Gerhard Gentzen's handwriting that were found in 1984. … this book is a valuable contribution to the history of the development of mathematical logic in the first half of the twentieth century.” (Henry Africk, Mathematical Reviews, December, 2017)Table of ContentsPart I: A Sketch of Gentzen's Life and Work.- 1. Overture.- 2. Gentzen's years of study.- Dr. Gentzen's arduous years in Nazi Germany.- 4. The scientific accomplishments.- 5. Loose ends.- 6. Gentzen's genuis.- Part II: Overview of the Shorthand Notes.- 1. Gentzen's series of stenographic manuscripts.- 2. The items in this collection.- Practical remarks on the manuscripts.- Manuscript illustrations.- The German alphabet in Latin, Sutterlin, and Fraktur Type.- Bibliography for parts I and II.- Index of names for Parts I and II.- Part III: The Original Writings.- 1. Reduction of number-theoretic problems to predicate logic.- 2. Replacement of functions by predicates.- 3. The formation of abstract concepts.- 4. Five different forms of natural calculi.- 5. Formal conception of correctness in arithmetic I.- 6. Investigations into logical inferences.- 7. Reduction of classical to intuitionistic logic.- 8. CV of the candidate Gerhard Gentzen.-0 9. Letters to Heyting.- 10. Formal conception of correctness in arithmetic II.- 11. Proof theory of number theory.- 12. Consistency of artihmetic, for publication.- 13. Correspondence with Paul Bernays.- 14. Forms of type theory.- 15. Predicate logic.- 16. Propositional logic.- 17. Foundational research in mathematics.- Table of cross-references in the Gentzen papers.- Index of names in the Gentzen papers.- Index of subjects in the Gentzen papers.

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Book SynopsisNow in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing.Key topics and features include: A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals Discussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers The user-friendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make Number Theory: An Introduction via the Density of Primes ideal for both self-study and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed.Trade Review“In this text, Fine (mathematics, Fairfield Univ.) and Rosenberger (Univ. of Hamburg, Germany) successfully present number theory from the inception of primes to recent developments in algebraic and analytic number theory and cryptography. … Numerous exercises and open problems are provided. The breadth and depth of topics covered are impressive, making this an excellent text for those interested in the field of number theory. Summing Up: Recommended. Upper-division undergraduates and graduate students.” (J. T. Zerger, Choice, Vol. 54 (9), May, 2017)“The book is chatty and leisurely, with lots of historical notes and lots of worked examples. The exercises at the end of each chapter are good and there are a reasonable number of them. … a good text for an introductory course … .” (Allen Stenger, MAA Reviews, maa.org, November, 2016)Table of ContentsIntroduction and Historical Remarks.- Basic Number Theory.- The Infinitude of Primes.- The Density of Primes.- Primality Testing: An Overview.- Primes and Algebraic Number Theory.- The Fields Q_p of p-adic Numbers: Hensel's Lemma.- References.- Index.

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Book SynopsisThis is a two-volume collection presenting the selected works of Herbert Busemann, one of the leading geometers of the twentieth century and one of the main founders of metric geometry, convexity theory and convexity in metric spaces. Busemann also did substantial work (probably the most important) on Hilbert’s Problem IV. These collected works include Busemann’s most important published articles on these topics. Volume I of the collection features Busemann’s papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces. Volume II includes Busemann’s papers on convexity and integral geometry, on Hilbert’s Problem IV, and other papers on miscellaneous subjects. Each volume offers biographical documents and introductory essays on Busemann’s work, documents from his correspondence and introductory essays written by leading specialists on Busemann’s work. They are a valuable resource for researchers in synthetic and metric geometry, convexity theory and the foundations of geometry. Table of ContentsPreface.- Introduction to Volume I.- List of publications of Herbert Busemann.- Acknowledgements.- Essays.- A. Papadpoulos: Herbert Busemann (1905-1994).- A. Papadopoulos and M. Troyanov: On three early papers by Herbert Busemann on the foundations of geometry.- M. Troyanov: On Pasch's Axiom and Desargues' Theorem in Busemann's work.- V. N. Berestovskiy: Busemann's results, ideas, questions on locally compact homogeneous geodesic spaces.- A. Papadopoulos and S. Yamada: Busemann's problems on G-spaces.- Busemann's metric theory of timelike spaces.- A. Papadopoulos: Chronogeometry.- W. M. Boothby: Review of Busemann's book The geometry of Geodesics.- F. A. Ficken: Review of Busemann's book Metric Methods in Finsler Spaces and in the Foundations of Geometry.- Busemann's papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces.

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Book SynopsisThis text presents six mini-courses, all devoted to interactions between representation theory of algebras, homological algebra, and the new ever-expanding theory of cluster algebras. The interplay between the topics discussed in this text will continue to grow and this collection of courses stands as a partial testimony to this new development. The courses are useful for any mathematician who would like to learn more about this rapidly developing field; the primary aim is to engage graduate students and young researchers. Prerequisites include knowledge of some noncommutative algebra or homological algebra. Homological algebra has always been considered as one of the main tools in the study of finite-dimensional algebras. The strong relationship with cluster algebras is more recent and has quickly established itself as one of the important highlights of today’s mathematical landscape. This connection has been fruitful to both areas—representation theory provides a categorification of cluster algebras, while the study of cluster algebras provides representation theory with new objects of study.The six mini-courses comprising this text were delivered March 7–18, 2016 at a CIMPA (Centre International de Mathématiques Pures et Appliquées) research school held at the Universidad Nacional de Mar del Plata, Argentina. This research school was dedicated to the founder of the Argentinian research group in representation theory, M.I. Platzeck.The courses held were: Advanced homological algebra Introduction to the representation theory of algebras Auslander-Reiten theory for algebras of infinite representation type Cluster algebras arising from surfaces Cluster tilted algebras Cluster characters Introduction to K-theory Brauer graph algebras and applications to cluster algebras Table of ContentsIntroduction to the Representation Theory of Finite-Dimensional Algebras: The Functorial Approach (M. I. Platzeck).- Auslander–Reiten Theory for Finite-Dimensional Algebras (P. Malicki).- Cluster Algebras From Surfaces (R. Schiffler).- Cluster Characters (P.-G. Plamondon).- A Course on Cluster Tilted Algebras (I. Assem).- Brauer Graph Algebras (S. Schroll).

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Book SynopsisTable of ContentsEinführung.- Erster Teil: Die Elemente.- I. Logik.- 1. Quantifizierung und Identität.- 2. Virtuelle Klassen.- 3. Virtuelle Relationen.- II. Reale Klassen.- 4. Realität, Extensionalität und Individuen.- 5. Das Virtuelle unter dem Realen.- 6. Identität und Einsetzung.- III. Klassen von Klassen.- 7. Einerklassen.- 8. Vereinigungen, Durchschnitte, Kennzeichnungen.- 9. Relationen als Klassen von Klassen.- 10. Funktionen.- IV. Natürliche Zahlen.- 11. Zahlen — naiv.- 12. Zahlen — konstituiert.- 13. Induktion.- V. Iteration und Arithmetik.- 14. Folgen und Iterierte.- 15. Die Vorfahrenrelation.- 16. Summe, Produkt, Potenz.- Zweiter Teil: Höhere Zahlformen.- VI. Reelle Zahlen.- 17. Programm; Zahlenpaare.- 18. Rationale und reelle Zahlen — konstituiert.- 19. Existenzforderungen. Operationen und Erweiterungen.- VII. Ordnung und Ordinalzahlen.- 20. Transfinite Induktion.- 21. Ordnung.- 22. Ordinalzahlen.- 23. Sätze über Ordinalzahlen.- 24. Die Ordnung der Ordinalzahlen.- VIII. Transfinite Rekursion.- 25. Transfinite Rekursion.- 26. Sätze über transfinite Rekursion.- 27. Aufzählung.- IX. Kardinalzahlen.- 28. Relative Größe von Klassen.- 29. Das Schröder-Bernsteinsche Theorem.- 30. Unendliche Kardinalzahlen.- X. Das Auswahlaxiom.- 31. Selektionen und Selektoren.- 32. Weitere äquivalente Formulierungen des Axioms.- 33. Die Stellung des Axioms.- Dritter Teil: Axiomensysteme.- XI. Die Russellsche Typentheorie.- 34. Der konstruktive Teil.- 35. Klassen und das Reduzibilitätsaxiom.- 36. Die moderne Typentheorie.- XII. Universelle Variablen und Zermelo.- 37. Die Typentheorie mit universellen Variablen.- 38. Kumulative Typen und Zermelo.- 39. Unendlichkeitsaxiome und andere.- XIII. Stratifizierung und äußerste Klassen.- 40. New foundations.- 41. Nicht-Cantorsche Klassen. Noch einmal Induktion.- 42. Hinzufügen äußerster Klassen.- XIV. Das System von von Neumann und andere Systeme.- 43. Das System von von Neumann-Bernays.- 44. Abweichungen und Vergleiche.- 45. Die Stärke der verschiedenen Systeme.- Vierter Teil: Anhang.- I. Zusammenstellung von fünf Axiomensystemen.- II. Liste durchnumerierter Formeln.- III. Bibliographie.- Sachwortverzeichnis.

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Table of ContentsImplicit definability and compactness in infinitary languages.- Some remarks on the model theory of infinitary languages.- Remarks on the theory of geometrical constructions.- Note on admissible ordinals.- An algebraic proof of the barwise compactness theorem.- Formulas with linearly ordered quantifiers.- Some problems in group theory.- Choice of infinitary languages by means of definability criteria; Generalized recursion theory.- Definability, automorphisms, and infinitary languages.- The hanf number for complete sentences.- Quantified algebras.- Normal derivability in classical logic.- A determinate logic.- (?1, ?) properties of unions of models.

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Table of ContentsInductive definitions and subsystems of analysis.- Proof theoretic equivalences between classical and constructive theories for analysis.- Inductive definitions, constructive ordinals, and normal derivations.- The ??+1-Rule.- Ordinal analysis of ID?.- Proof-theoretical analysis of ID? by the method of local predicativity.

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Book SynopsisThis is a softcover reprint of the English translation of 1968 of N. Bourbaki's, Theorie des Ensembles (1970).Table of ContentsI. Description of Formal Mathematics.- § 1. Terms and relations.- 1. Signs and assemblies.- 2. Criteria of substitution.- 3. Formative constructions.- 4. Formative criteria.- § 2. Theorems.- 1. The axioms.- 2. Proofs.- 3. Substitutions in a theory.- 4. Comparison of theories.- § 3. Logical theories.- 1. Axioms.- 2. First consequences.- 3. Methods of proof.- 4. Conjunction.- 5. Equivalence.- § 4. Quantified theories.- 1. Definition of quantifiers.- 2. Axioms of quantified theories.- 3. Properties of quantifiers.- 4. Typical quantifiers.- § 5. Equalitarian theories.- 1. The axioms.- 2. Properties of equality.- 3. Functional relations.- Appendix. Characterization of terms and relations.- 1. Signs and words.- 2. Significant words.- 3. Characterization of significant words.- 4. Application to assemblies in a mathematical theory.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for the Appendix.- II. Theory of Sets.- § 1. Collectivizing relations.- 1. The theory of sets.- 2. Inclusion.- 3. The axiom of extent.- 4. Collectivizing relations.- 5. The axiom of the set of two elements.- 6. The scheme of selection and union.- 7. Complement of a set. The empty set.- § 2. Ordered pairs.- 1. The axiom of the ordered pair.- 2. Product of two sets.- § 3. Correspondences.- 1. Graphs and correspondences.- 2. Inverse of a correspondence.- 3. Composition of two correspondences.- 4. Functions.- 5. Restrictions and extensions of functions.- 6. Definition of a function by means of a term.- 7. Composition of two functions. Inverse function.- 8. Retractions and sections.- 9. Functions of two arguments.- § 4. Union and intersection of a family of sets.- 1. Definition of the union and the intersection of a family of sets.- 2. Properties of union and intersection.- 3. Images of a union and an intersection.- 4. Complements of unions and intersections.- 5. Union and intersection of two sets.- 6. Coverings.- 7. Partitions.- 8. Sum of a family of sets.- § 5. Product of a family of sets.- 1. The axiom of the set of subsets.- 2. Set of mappings of one set into another.- 3. Definitions of the product of a family of sets.- 4. Partial products.- 5. Associativity of products of sets.- 6. Distributivity formulae.- 7. Extension of mappings to products.- § 6. Equivalence relations.- 1. Definition of an equivalence relation.- 2. Equivalence classes; quotient set.- 3. Relations compatible with an equivalence relation.- 4. Saturated subsets.- 5. Mappings compatible with equivalence relations.- 6. Inverse image of an equivalence relation; induced equivalence relation.- 7. Quotients of equivalence relations.- 8. Product of two equivalence relations.- 9. Classes of equivalent objects.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- III. Ordered Sets, Cardinals, Integers.- § 1. Order relations. Ordered sets.- 1. Definition of an order relation.- 2. Preorder relations.- 3. Notation and terminology.- 4. Ordered subsets. Product of ordered sets.- 5. Increasing mappings.- 6. Maximal and minimal elements.- 7. Greatest element and least element.- 8. Upper and lower bounds.- 9. Least upper bound and greatest lower bound.- 10. Directed sets.- 11. Lattices.- 12. Totally ordered sets.- 13. Intervals.- § 2. Well-ordered sets.- 1. Segments of a well-ordered set.- 2. The principle of transfinite induction.- 3. Zermelo’s theorem.- 4. Inductive sets.- 5. Isomorphisms of well-ordered sets.- 6. Lexicographic products.- § 3. Equipotent sets. Cardinals.- 1. The cardinal of a set.- 2. Order relation between cardinals.- 3. Operations on cardinals.- 4. Properties of the cardinals 0 and 1.- 5. Exponentiation of cardinals.- 6. Order relation and operations on cardinals.- § 4. Natural integers. Finite sets.- 1. Definition of integers.- 2. Inequalities between integers.- 3. The principle of induction.- 4. Finite subsets of ordered sets.- 5. Properties of finite character.- § 5. Properties of integers.- 1. Operations on integers and finite sets.- 2. Strict inequalities between integers.- 3. Intervals in sets of integers.- 4. Finite sequences.- 5. Characteristic functions of sets.- 6. Euclidean division.- 7. Expansion to base b.- 8. Combinatorial analysis.- § 6. Infinite sets.- 1. The set of natural integers.- 2. Definition of mappings by induction.- 3. Properties of infinite cardinals.- 4. Countable sets.- 5. Stationary sequences.- § 7. Inverse limits and direct limits.- 1. Inverse limits.- 2. Inverse systems of mappings.- 3. Double inverse limit.- 4. Conditions for an inverse limit to be non-empty.- 5. Direct limits.- 6. Direct systems of mappings.- 7. Double direct limit. Product of direct limits.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- Exercises for § 7.- Historical Note on § 5.- IV. Structures.- § 1. Structures and isomorphisms.- 1. Echelons.- 2. Canonical extensions of mappings.- 3. Transportable relations.- 4. Species of structures.- 5. Isomorphisms and transport of structures.- 6. Deduction of structures.- 7. Equivalent species of structures.- § 2. Morphisms and derived structures.- 1. Morphisms.- 2. Finer structures.- 3. Initial structures.- 4. Examples of initial structures.- 5. Final structures.- 6. Examples of final structures.- § 3. Universal mappings.- 1. Universal sets and mappings.- 2. Existence of universal mappings.- 3. Examples of universal mappings.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Historical Note on Chapters I-IV.- Summary of Results.- § 1. Elements and subsets of a set.- § 2. Functions.- § 3. Products of sets.- § 4. Union, intersection, product of a family of sets.- § 5. Equivalence relations and quotient sets.- § 6. Ordered sets.- § 7. Powers. Countable sets.- § 8. Scales of sets. Structures.- Index of notation.- Index of terminology.- Axioms and schemes of the theory of sets.

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Book SynopsisThis is a thoroughly revised and enlarged second edition that presents the main results of descriptive complexity theory, that is, the connections between axiomatizability of classes of finite structures and their complexity with respect to time and space bounds. The logics that are important in this context include fixed-point logics, transitive closure logics, and also certain infinitary languages; their model theory is studied in full detail. The book is written in such a way that the respective parts on model theory and descriptive complexity theory may be read independently.Table of ContentsPreliminaries.- The Ehrenfeucht-Fraïssé Method.- More on Games.- 0-1 Laws.- Satisfiability in the Finite.- Finite Automata and Logic: A Microcosm of Finite Model Theory.- Descriptive Complexity Theory.- Logics with Fixed-Point Operators.- Logic Programs.- Optimization Problems.- Logics for PTIME.- Quantifiers and Logical Reductions.

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Book SynopsisLe Livre de Théorie des ensembles qui vient en tête du traité présente les fondements axiomatiques de la théorie des ensembles. Il comprend les chapitres : 1. Description de la mathématique formelle ; 1. Théorie des ensembles ; 2. Ensembles ordonnés. Cardinaux. 3. nombres entiers ; 4. Structures.Table of ContentsDescription de la mathématique formelle.- Théorie des ensembles.- Ensembles ordonnés, cardinaux, nombres entiers.- Structures.

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Book SynopsisThis monograph covers the recent major advances in various areas of set theory. From the reviews: "One of the classical textbooks and reference books in set theory....The present ‘Third Millennium’ edition...is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master....This well-written book promises to influence the next generation of set theorists, much as its predecessor has done." --MATHEMATICAL REVIEWSTrade ReviewFrom the reviews of the third edition: "Thomas Jech’s text has long been considered a classic survey of the state of the set theory … . As every logician will know, this is a work of extraordinary scholarship, essential for any graduate logician who needs to know where the current boundaries of research are situated. Each chapter ends with a valuable historical survey and there is an extensive bibliography. This will continue to be the bible for set theorists in the new century." (Gerry Leversha, The Mathematical Gazette, March, 2005) "The book does masterly what it is supposed to do. … every mathematician who wishes to refresh his knowledge of set theory will read it with pleasure. … They will also find historical notes, and precise references … . A very comprehensive bibliography, and detailed indexes complete the work. This book fills a serious gap in the literature and there is no doubt that it will become a standard reference … . One can strongly recommend its acquisition for any mathematical library." (Jean-Roger Roisin, Bulletin of the Belgian Mathematical Society, Vol. 11 (3), 2004) "One of the classical textbooks and reference books in set theory is Jech’s Set Theory. … The present ‘Third Millennium’ edition … is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master. … This well-written book promises to influence the next generation of set theorists, much as its predecessor has done over the last quarter of a century." (Eva Coplakova, Mathematical Reviews, 2004 g) "Jech’s book, ‘Set Theory’ has been a standard reference for over 25 years. This ‘Third Millennium Edition’, not only includes all the materials in the first two editions, but also covers recent developments of set theory during the last 25 years. We believe that this new version will become a standard reference on set theory for the next few years." (Guohua Wu, New Zealand Mathematical Society Newsletter, April, 2004) "Jech’s classic monograph has been a standard reference for a generation of set theorists. Though … labeled ‘The Third Millennium Edition’, the present work is in fact a new book. ... Even sections presenting older results have been rewritten and modernized. Exercises have been moved to the end of each section. The bibliography, the section on notation, and the index have been considerably expanded as well. This new edition will certainly become a standard reference on set theory for years to come." (Jörg D. Brendle, Zentralblatt MATH, Vol. 1007, 2003) "Thomas Jech’s Set Theory contains the most comprehensive treatment of the subject in any one volume. The present third edition is a revised and expanded version … . The third edition has three parts. The first, Jech says, every student of set theory should learn, the second every set theorist should master and the third consists of various results reflecting ‘the state of the art of set theory at the turn of the new millennium’. This last part especially contains a lot of new material." (Martin Bunder, The Australian Mathematical Society Gazette, Vol. 30 (2), 2003)Table of ContentsBasic Set Theory.- Axioms of Set Theory.- Ordinal Numbers.- Cardinal Numbers.- Real Numbers.- The Axiom of Choice and Cardinal Arithmetic.- The Axiom of Regularity.- Filters, Ultrafilters and Boolean Algebras.- Stationary Sets.- Combinatorial Set Theory.- Measurable Cardinals.- Borel and Analytic Sets.- Models of Set Theory.- Advanced Set Theory.- Constructible Sets.- Forcing.- Applications of Forcing.- Iterated Forcing and Martin’s Axiom.- Large Cardinals.- Large Cardinals and L.- Iterated Ultrapowers and L[U].- Very Large Cardinals.- Large Cardinals and Forcing.- Saturated Ideals.- The Nonstationary Ideal.- The Singular Cardinal Problem.- Descriptive Set Theory.- The Real Line.- Selected Topics.- Combinatorial Principles in L.- More Applications of Forcing.- More Combinatorial Set Theory.- Complete Boolean Algebras.- Proper Forcing.- More Descriptive Set Theory.- Determinacy.- Supercompact Cardinals and the Real Line.- Inner Models for Large Cardinals.- Forcing and Large Cardinals.- Martin’s Maximum.- More on Stationary Sets.

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Book SynopsisOver the years, this book has become a standard reference and guide in the set theory community. It provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research, with open questions and speculations throughout.Table of ContentsPreliminaries.- Beginnings.- Partition Properties.- Forcing and Sets of Reals.- Aspects of Measurability.- Strong Hypotheses.- Determinacy.

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Book SynopsisStochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the book.Table of Contents1 Infinitesimal calculus, consistently and accessibly.- 2 Radically elementary probability theory.- 3 Radically elementary stochastic integrals.- 4 The radically elementary Girsanov theorem and the diffusion invariance principle.- 5 Excursion to nancial economics: A radically elementary approach to the fundamental theorems of asset pricing.- 6 Excursion to financial engineering: Volatility invariance in the Black-Scholes model.- 7 A radically elementary theory of Itô diffusions and associated partial differential equations.- 8 Excursion to mathematical physics: A radically elementary definition of Feynman path integrals.- 9 A radically elementary theory of Lévy processes.- 10 Final remarks.

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Book SynopsisTable of ContentsEinleitung: Überblick über den Inhalt des zweiten Halbbandes.- III. Die logischen Grundlagen des statistischen Schließens.- 1. ,Jenseits von Popper und Carnap‘.- 1.a Programm und Abgrenzung vom Projekt einer induktiven Logik.- 1.b Die relative Häufigkeit auf lange Sicht und die Häufigkeitsdefinition der statistischen Wahrscheinlichkeit.- 1.c Der Vorschlag von Braithwaite, die statistische Wahrscheinlichkeit als theoretischen Begriff einzuführen.- 1.d Vorbereitende Betrachtungen zur Testproblematik statistischer Hypothesen.- 1.e Zusammenfassung und Ausblick.- 2. Präludium: Der intuitive Hintergrund.- 3. Die Grundaxiome. Statistische Unabhängigkeit.- 3.a Die Kolmogoroff-Axiome.- 3.b Unabhängigkeit im statistischen Sinn.- 3.c Hypothesen und Oberhypothesen.- 4. Die komparative Stützungslogik.- 4.a Vorbetrachtungen.- 4.b Einige zusätzliche Zwischenbetrachtungen.- 4.c Die Axiome der Stützungslogik.- 5. Die Likelihood-Regel.- 5.a Kombinierte statistische Aussagen.- 5.b Likelihood und Likelihood-Regel.- 6. Die Leistungsfähigkeit der Likelihood-Regel.- 6.a Die Einzelfall-Regel und ihre Begründung.- 6.b Der statistische Stützungsschluß im diskreten Fall und seine Rechtfertigung.- 6.c Übergang zum stetigen Fall.- 6.d Wahrscheinlichkeitsverteilung und Likelihoodfunktion (,Plausibilitätsverteilung‘).- 6.e Denken in Likelihoods und Bayesianismus.- 7. Vorläufiges Postludium: Ergänzende Betrachtungen zu den statistischen Grundbegriffen.- 7.a Der Begriff des statistischen Datums.- 7.b Chance und Häufigkeit auf lange Sicht.- 7.c Versuchstypen.- 8. Zufall, Grundgesamtheit und Stichprobenauswahl.- 9. Die Problematik der statistischen Testtheorie, erläutert am Beispiel zweier konkurrierender Testtheorien.- 9.a Vorbetrachtungen. Ein warnendes historisches Beispiel.- 9.b Macht und Umfang eines Tests. Die Testtheorie von Neyman-Pearson.- 9.c Die Mehrdeutigkeit der Begriffe „Annahme“ und „Verwerfung“ 159 9.d Einige kritische Bemerkungen zu den Begriffen Umfang und Macht 160 9.e Die Likelihood-Testtheorie.- 10. Probleme der Schätzungstheorie.- 10.a Vorbemerkungen.- 10.b Was ist Schätzung? Klassifikation von Schätzungen.- 10.c Einige spezielle Begriffe der statistischen Schätzungstheorie.- 10.d Die Doppeldeutigkeit von „Schätzung“ und die Mehrdeutigkeit von „Güte einer Schätzung“.- 10.e Theoretische Schätzungen und Schätzhandlungen.- 10.f Das Skalendilemma. Zwecke von Schätzungen.- 10.g Schätzungen im engeren und Schätzungen im weiteren Sinn.- 10.h Kritisches zu den Optimalitätsmerkmalen auf lange Sicht, zur Minimax-Theorie und zur Intervallschätzung.- 10.i Ein Präzisierungsversuch des Begriffes der besser gestützten Schätzung.- 10.j Ist die Schätzungstheorie von Savage das Analogon zur Testtheorie von Neyman-Pearson?.- 11. Kritische Betrachtungen zur Likelihood-Stützungs-und-Testtheorie.- 11.a Ist der Likelihood-Test schlechter als nutzlos ?.- 11.b Das Karten-Paradoxon von Kerridge.- 11.c Die logische Struktur des Stützungsbegriffs.- 12. Subjektivismus oder Objektivismus ?.- 12.a Die subjektivistische (personalistische) Kritik: de Finetti und Savage kontra Objektivismus.- 12.b Die Propensity-Interpretation der statistischen Wahrscheinlichkeit: Popper, Giere und Suppes.- 13. Versuch einer Skizze der logischen Struktur des Fiduzial-Argumentes von R. A. Fisher.- Bibliographie.- IV. ,Statistisches Schließen — Statistische Begründung — Statistische Analyse‘statt,Statistische Erklärung‘.- 1. Elf Paradoxien und Dilemmas.- (I) Die Paradoxie der Erklärung des Unwahrscheinlichen.- (II) Das Paradoxon der irrelevanten Gesetzesspezialisierung.- (III) Das Informationsdilemma.- (IV) Das Erklärungs-Bestätigungs-Dilemma.- (V) Das Paradoxon der reinen ex post facto Kausalerklärung.- (VI) Das Verzahnungsparadoxon.- (VII) Das Erklärungs-Begründungs-Dilemma.- (VIII) Das Dilemma der nomologischen Implikation.- (IX) Das ,Weltanschauungsdilemma‘.- (X) Das Argumentationsdilemma.- (XI) Das Gesetzesparadoxon.- 2. Diskussion.- 2.a Problemreduktionen.- 2.b Das Problem der nomologischen Implikation. Statistisches Schließen und statistische Begründungen.- 2.c Verzahnungen von Erklärungs- und Bestätigungsproblemen.- 2.d Die Leibniz-Bedingung. Unbehebbare intuitive Konflikte.- 3. Statistische Begründungen statt statistische Erklärungen. Der statistische Begründungsbegriff als Explikat der Einzelfall-Regel.- 4. Statistische Analysen.- 4.a Kausale Relevanz und Abschirmung.- 4.b Statistische Oberflächenanalyse und statistisch-kausale Tiefenanalyse von Minimalform.- 4.c Statistische Analyse und statistisches Situationsverständnis.- 4.d Was könnte unter „Statistische Erklärung“ verstanden werden?.- Bibliographie.- Anhang I: Indeterminismus vom zweiten Typ.- Anhang II: Das Repräsentationstheorem von B. de Finetti.- 1. Intuitiver Zugang.- 1.a Bernoulli-Wahrscheinlichkeiten und Mischungen von Bernoulli-Wahrscheinlichkeiten.- 1.b Das Problem des Lernens aus der Erfahrung.- 1.c Die Bedeutung des Begriffs der Vertauschbarkeit.- 2. Formale Skizze. Übergang zum kontinuierlichen Fall.- 2.a Vertauschbarkeit und Symmetrie.- 2.b Mischungen und Lernen aus der Erfahrung: Der Riemannsche Fall..- 2.c Mischungen im abstrakten maßtheoretischen Fall. Das Repräsentationstheorem.- 2.d Diskussion.- Bibliographie.- Anhang III: Metrisierung qualitativer Wahrscheinlichkeitsfelder.- 1. Axiomatische Theorien der Metrisierung. Extensive Größen.- 2. Metrisierung von Wahrscheinlichkeitsfeldern.- 2.a Metrisierung klassischer absoluter Wahrscheinlichkeitsfelder im endlichen und abzählbaren Fall.- 2.b Metrisierung quantenmechanischer Wahrscheinlichkeitsfelder.- 2.c Metrisierung qualitativer bedingter Wahrscheinlichkeitsfelder.- Bibliographie.- Autorenregister.- Verzeichnis der Symbole und Abkürzungen.

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Book SynopsisIn this book, Paulo Guilherme Santos studies diagonalization in formal mathematics from logical aspects to everyday mathematics. He starts with a study of the diagonalization lemma and its relation to the strong diagonalization lemma. After that, Yablo’s paradox is examined, and a self-referential interpretation is given. From that, a general structure of diagonalization with paradoxes is presented. Finally, the author studies a general theory of diagonalization with the help of examples from mathematics.Table of ContentsDiagonalization in Mathematics.- Diagonalization Lemma.- Fixed Point Theorems.- Paradoxes: Liar, Yablo’s Paradox, Curry’s Paradox.

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Book SynopsisParameterized Complexity in the Polynomial Hierarchy was co-recipient of the E.W. Beth Dissertation Prize 2017 for outstanding dissertations in the fields of logic, language, and information. This work extends the theory of parameterized complexity to higher levels of the Polynomial Hierarchy (PH). For problems at higher levels of the PH, a promising solving approach is to develop fixed-parameter tractable reductions to SAT, and to subsequently use a SAT solving algorithm to solve the problem. In this dissertation, a theoretical toolbox is developed that can be used to classify in which cases this is possible. The use of this toolbox is illustrated by applying it to analyze a wide range of problems from various areas of computer science and artificial intelligence.Table of ContentsComplexity Theory and Non-determinism.- Parameterized Complexity Theory.- Fpt-Reducibility to SAT.- The Need for a New Completeness Theory.- A New Completeness Theory.- Fpt-algorithms with Access to a SAT Oracle.- Problems in Knowledge Representation and Reasoning.- Model Checking for Temporal Logics.- Problems Related to Propositional Satisfiability.- Problems in Judgment Aggregation.- Planning Problems.- Graph Problems.- Relation to Other Topics in Complexity Theory.- Subexponential-Time Reductions.- Non-Uniform Parameterized Complexity.- Open Problems and Future Research Directions.- Conclusion.- Compendium of Parameterized Problems.- Generalization to Higher Levels of the Polynomial Hierarchy.

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Book SynopsisDas Lehrbuch diskutiert gängige Fragen der Analysis und linearen Algebra und verwendet für die rechnergestützten Antworten die Software Matlab und Mathematica. Es stellt mathematische Standard-Algorithmen im Detail vor und zeigt deren Umsetzung in die Programme. Zusätzlich erläutert es, wie deren Funktionen Probleme lösen. Die Inhalte sind nach Datentypen (Polynome, reelle Funktionen, Matrizen) gegliedert. Im Vordergrund: die Objekte am Rechner, Grundoperationen an diesen Objekten und typische Fragen. Mit Algorithmen in Pseudocode. Plus zum Download: Programme für Mathematica und Matlab, alle Beispiele, Grafiken, interaktive Elemente.Trade ReviewFrom the reviews:“Algorithmic methods are useful tools for solving technical, scientific, or industrial problems. … This book for students in the first or second year presents the basic knowledge necessary for applying the methods to different problems. … The presentation is given in mathematical rigorous style and no further reading … to be required to read the text or design a course. … underlying course was designed for students of technical mathematics, but is as well suited for computer science students in the area of scientific computing.” (Thomas Rauber, Zentralblatt MATH, Vol. 1181, 2010)Table of ContentsEinleitung.- I. Grundbegriffe und Grundfragen einer algorithmischen Mathematik:1. Probleme, Lösungen und Algorithmen.- 2 Einführende Beispiele zur algorithmischen Lösung am Computer.- 3. Kondition eines Problems.- 4. Eigenschaften von Algorithmen.- II. Zahlbereiche: 5. Natürliche und ganze Zahlen.- 6. Kongruenzklassen modulo m.- 7. Rationale Zahlen.- 8. Reelle Zahlen.- III. Vektoren: 9. Mathematische Grundlagen.- 10. Vektoren am Computer.- 11. Euklidsches Skalarprodukt in Rm.- 12. Orthonormalisierung in Rm.- IV. Univariate Polynome: 13. Mathematische Grundlagen.- 14. Polynome am Computer.- 15. Polynomdivision und größter gemeinsamer Teiler.- 16. Polynomauswertung in R.- 17. Polynominterpolation in R.

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Book Synopsis

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Book SynopsisThis is a problem-based book aimed at high-school students interested in mathematical topics related to the ISI and CMI entrance tests as well as Mathematics Olympiads. This book will help students in designing a well-planned pathway to tackle complicated problems from topics such as number theory, combinatorics, algebra, calculus, Euclidean and coordinate geometry, probability and statistics.

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Book SynopsisMathematics is often considered as a body of knowledge that is essen­ tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of clearly formulating and correctly proving it. However, the question is not so simple, and P. Weingartner's paper (Language and Coding-Dependency of Results in Logic and Mathe­ matics) deals with some results in logic and mathematics which reveal that certain notions are in general not invariant with respect to different choices of language and of coding processes. Five example are given: 1) The validity of axioms and rules of classical propositional logic depend on the interpretation of sentential variables; 2) The language­ dependency of verisimilitude; 3) The proof of the weak and strong anti­ inductivist theorems in Popper's theory of inductive support is not invariant with respect to limitative criteria put on classical logic; 4) The language-dependency of the concept of provability; 5) The language­ dependency of the existence of ungrounded and paradoxical sentences (in the sense of Kripke). The requirements of logical rigour and consistency are not the only criteria for the acceptance and appreciation of mathematical proposi­ tions and theories.Table of ContentsGeneral Philosophical Perspectives.- Logic, Mathematics, Ontology.- From Certainty to Fallibility in Mathematics?.- Moderate Mathematical Fictionism.- Language and Coding-Dependency of Results in Logic and Mathematics.- What is a Profound Result in Mathematics?.- The Hylemorphic Schema in Mathematics.- Foundational Approaches.- Categorical Foundations of the Protean Character of Mathematics.- Category Theory and Structuralism in Mathematics: Syntactical Considerations.- Reflection in Set Theory. The Bernays-Levy Axiom System.- Structuralism and the Concept of Set.- Aspects of Mathematical Experience.- Logicism Revisited in the Propositional Fragment of Le?niewski’s Ontology.- The Applicability of Mathematics.- The Relation of Mathematics to the Other Sciences.- Mathematics and Physics.- The Mathematical Overdetermination of Physics.- Gödel’s Incompleteness Theorem and Quantum Thermodynamic Limits.- Mathematical Models in Biology.- The Natural Numbers as a Universal Library.- Mathematical Symmetry Principles in the Scientific World View.- Historical Considerations.- Mathematics and Logics. Hungarian Traditions and the Philosophy of Non-Classical Logic.- Umfangslogik, Inhaltslogik, Theorematic Reasoning.

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Book SynopsisIn this book, trigonometry is presented mainly through the solution of specific problems. The problems are meant to help the reader consolidate their knowledge of the subject. In addition, they serve to motivate and provide context for the concepts, definitions, and results as they are presented. In this way, it enables a more active mastery of the subject, directly linking the results of the theory with their applications. Some historical notes are also embedded in selected chapters.The problems in the book are selected from a variety of disciplines, such as physics, medicine, architecture, and so on. They include solving triangles, trigonometric equations, and their applications. Taken together, the problems cover the entirety of material contained in a standard trigonometry course which is studied in high school and college.We have also added some interesting, in our opinion, entertainment problems. To solve them, no special knowledge is required. While they are not directly related to the subject of the book, they reflect its spirit and contribute to a more lighthearted reading of the material.

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Book SynopsisBritish-Israeli recreational mathematician, communicator and educator, Yossi Elran explores in-depth six of the most ingenious math puzzles, exposing their long 'tails': the stories, trivia, quirks and oddities of their history and, of course, the math and mathematicians behind them. In his unique 'talmudic', associative way, Elran shows the hidden connections between Lewis Carroll's 'Cats and Rats' puzzle and the math of taxi driving, a number pyramid magic trick and Hollywood movie fractals, and even how packing puzzles are related to COVID-19!Elran has a great talent for explaining difficult topics — including quantum mechanics, a topic he relates to some original 'operator' puzzles — making the book very accessible for all audiences.With over 40 additional, original puzzles, and touching on dozens of hot math topics, this is a perfect book for math lovers, educators, kids and adults, and anyone who loves a great read.Yossi Elran is co-author of our bestselling The Paper Puzzle Book, and heads the Innovation Center at the Davidson Institute of Science Education, the educational arm of the world-renowned Weizmann Institute of Science in Israel.

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Book Synopsis​Gaisi Takeuti was one of the most brilliant, genius, and influential logicians of the 20th century. He was a long-time professor and professor emeritus of mathematics at the University of Illinois at Urbana-Champaign, USA, before he passed away on May 10, 2017, at the age of 91. Takeuti was one of the founders of Proof Theory, a branch of mathematical logic that originated from Hilbert's program about the consistency of mathematics. Based on Gentzen's pioneering works of proof theory in the 1930s, he proposed a conjecture in 1953 concerning the essential nature of formal proofs of higher-order logic now known as Takeuti's fundamental conjecture and of which he gave a partial positive solution. His arguments on the conjecture and proof theory in general have had great influence on the later developments of mathematical logic, philosophy of mathematics, and applications of mathematical logic to theoretical computer science. Takeuti's work ranged over the whole spectrum of mathematical logic, including set theory, computability theory, Boolean valued analysis, fuzzy logic, bounded arithmetic, and theoretical computer science. He wrote many monographs and textbooks both in English and in Japanese, and his monumental monograph Proof Theory, published in 1975, has long been a standard reference of proof theory. He had a wide range of interests covering virtually all areas of mathematics and extending to physics. His publications include many Japanese books for students and general readers about mathematical logic, mathematics in general, and connections between mathematics and physics, as well as many essays for Japanese science magazines. This volume is a collection of papers based on the Symposium on Advances in Mathematical Logic 2018. The symposium was held September 18–20, 2018, at Kobe University, Japan, and was dedicated to the memory of Professor Gaisi Takeuti. Table of ContentsS. Fuchino and A. Ottenbreit Ottenbreit Maschio Rodrigues, Reflection principles, generic large cardinals, and the Continuum Problem.- D. Ikegami and N. Trang, On supercompactness of ω1.- S. Iwata, Interpolation properties for Sacchetti’s logics.- T. Kurahashi, Rosser provability and the second incompleteness theorem.- H. Kurokawa, On Takeuti’s early view of the concept of set.- Yo Matsubara and T. Usuba, On Countable Stationary Towers.- M. Ozawa, Reforming Takeuti’s Quantum Set Theory to Satisfy De Morgan’s Laws.- T. Usuba, Choiceless Lowenheim-Skolem property and uniform definability of grounds.- M. Yasugi, Y. Tsujii, T. Mori, Irrational-based computability of functions.- M. Yasugi, “Gaisi Takeuti’s finitist standpoint” and its mathematical embodiment.- Y. Yoshinobu, Properness under closed forcing.

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