Matematikkens fundament

Modeling and Control of Dynamic Spatially Distributed Systems: Pharmaceutical Processes provides a balanced approach to help readers to get started quickly in the field of biochemical pharmaceuticals. From a theoretical perspective, dynamic spatially distributed systems are introduced to address their industrial applications. After identifying problems, the book provides readers with modeling and control system design techniques via a novel fuzzy set (class of objects with a continuum of grades of membership, to describe the grade of the object belonging to this fuzzy set) and intelligent computation methods. From an application perspective, the book provides a thorough understanding of Good Manufacture Practices (GMP) and the importance of identification, modelling, and intelligent control of such systems, reducing the test-and-error cost, and the R&D design time cycle of original drug development.

This book is motivated by the fascinating interrelations between ergodic theory and number theory (as established since the 1950s). It examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions.

Some of the hottest topics today involve cryptocurrency, and FinTech. With all these trends, businesses need to become better informed. This book provides an easy-to-read, yet comprehensive, view of cryptocurrency in the U.S. and international markets, as well as the key issues, technologies, applications, challenges, and trends.

Whether the source is more industry-based or academic research, there certainly appears to be a growing interest in the field of cryptocurrency. The New York Times had a cover story on March 24, 2022, titled "Time to Enter the Crypto Zone?," and they talked about institutional investors pouring billions into digital tokens, salaries being taken in Bitcoins, and even Bitcoin ATMs in grocery stores. Certainly, there have been ups and downs in crypto, but it has a kind of alluring presence that tempts one to include crypto as part of one's portfolio. Those who are "prime crypto-curious" investors are usually familiar with the tech/pop culture and feel they want to diversify a bit in this fast-moving market. Even universities are beginning to offer more courses and create "Centers on Cryptocurrency." Some universities are even requiring their students who take a crypto course to pay the course tuition via cryptocurrency.In response to the growing interest and fascination about the crypto industry and cryptocurrency in general, Cryptocurrency Concepts, Technology, and Applications brings together many leading worldwide contributors to discuss a broad range of issues associated with cryptocurrency. The book covers a wide array of crypto-related topics, including:BlockchainNFTsData analytics and AICrypto crimeCrypto industry and regulationCrypto and public choiceConsumer confidenceBitcoin and other cryptocurrencies.Presenting various viewpoints on where the crypto industry is heading, this timely book points out both the advantages and limitations of this emerging field. It is an easy-to-read, yet comprehensive, overview of cryptocurrency in the U.S. and international markets.

¿This book deals with the rise of mathematics in physical sciences, beginning with Galileo and Newton and extending to the present day. The book is divided into two parts. The first part gives a brief history of how mathematics was introduced into physics¿despite its "unreasonable effectiveness" as famously pointed out by a distinguished physicist¿and the criticisms it received from earlier thinkers. The second part takes a more philosophical approach and is intended to shed some light on that mysterious effectiveness. For this purpose, the author reviews the debate between classical philosophers on the existence of innate ideas that allow us to understand the world and also the philosophically based arguments for and against the use of mathematics in physical sciences. In this context, Schopenhauer¿s conceptions of causality and matter are very pertinent, and their validity is revisited in light of modern physics. The final question addressed is whether the effectiveness of mathematics can be explained by its ¿existence¿ in an independent platonic realm, as Gödel believed.The book aims at readers interested in the history and philosophy of physics. It is accessible to those with only a very basic (not professional) knowledge of physics.

CoPart 3 is a dual complement to Visual Category Theory Brick by Brick Part 3. It covers adjoint functors, diagram shapes and categories, cones and cocones, limits and colimits, pullbacks, pushouts.

This book helps readers easily learn basic model checking by presenting examples, exercises and case studies. The toolset mCRL2 provides a language to specify the behaviour of distributed systems, in particular where there is concurrency with inter-process communication. This language allows us to analyse a distributed system with respect to its functional requirements. For example, biological cells, supply chain management systems, patient support platforms, and communication protocols.The underlying technique is based on verifying requirements through model checking. The book explains the syntax of mCRL2 and offers modelling tips and tricks.

A set in mathematics is just a collection of elements; an example is the set of natural numbers {1, 2, 3, ...}. Simplifying somewhat, the theory of sets can be regarded as the foundation on which the whole of mathematics is built; and the founder of set theory is the German logician and mathematician Georg Cantor (1845¿1918). However, the aspect of Cantor's work that's most widely known-or most controversial, at any rate-isn't so much set theory in general, but rather those parts of that theory that have to do with infinite sets in particular. Cantor claimed among other things that the infinite set of real numbers contains strictly more elements than the infinite set of natural numbers. From this result, he concluded that there's more than one kind of infinity; in fact, he claimed that there are an infinite number of different infinities, or transfinite numbers. (He also believed these results had been communicated to him by God.) The aim of this book is to explain and investigate these claims of Cantor's in depth (and question them, where appropriate). It's not a textbook, though; instead, it's a popular account-it tells a story-and the target audience is interested lay readers, not mathematicians or logicians. What little mathematics is needed to understand the story is explained in the book itself.

Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this book constitutes the refereed proceedings of the Third Tsinghua Interdisciplinary Workshop on Logic, Language, and Meaning, TLLM 2022, which was held virtually in April 2022.The 9 full papers presented in this volume were carefully reviewed and selected from 13 submissions. The workshop covers a wide range of topics such as dynamic semantics, logical dynamics, Dynamic Epistemic Logic, Discourse Representation Theory, formal semantics, free choice inference, update semantics, and donkey sentences.

This monograph is a defence of the Fregean take on logic. The author argues that Frege¿s projects, in logic and philosophy of language, are essentially connected and that the formalist shift produced by the work of Peano, Boole and Schroeder and continued by Hilbert and Tarski is completely alien to Frege's approach in the Begriffsschrift. A central thesis of the book is that judgeable contents, i.e. propositions, are the primary bearers of logical properties, which makes logic embedded in our conceptual system. This approach allows coherent and correct definitions of logical constants, logical consequence, and truth and connects their use to the practices of rational agents in science and everyday life.

This 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.

Edited 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.

This 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.

This book is for readers who have learned about first order logic; Gödel's completeness theorem; the Löwenheim-Skolem theorem; the Tarski-Vaught criterion for being elementary sub-model; and who know naive set theory. A graduate course in model theory will be helpful. The thesis of the book is that we can find worthwhile dividing lines among complete first order theories T; mainly countable. That is, properties dividing them in some sense between understandable and complicated ones.The main test problem is the number of models of T of the infinite cardinal l as a function of l. This culminates in the so-called main gap theorem saying the number is either maximal or quite small in suitable sense. Toward this, other properties are introduced and investigated, such as being stable or being super stable, where can we define dimension and weight, particularly for super stable theories.

The Journal of Applied Logics - IfCoLog Journal of Logics and their Applications (FLAP) covers all areas of pure and applied logic, broadly construed. All papers published are free open access, and available via the College Publications website. This Journal is open access, puts no limit on the number of pages of any article, puts no limit on the number of papers in an issue and puts no limit on the number of issues per year. We insist only on a very high academic standard, and will publish issues as they come.

Paradoxes seized the attention of logicians in the middle ages, and were used both as tests for the viability of theories of logic, language, epistemology, and possibly every philosophical issue, and also in the specific genre of insolubles as needing a theoretical solution, usually involving issues about signification, truth, knowledge and modality. Numerous theories were developed, not only in the Latin West, but also in the Islamic world and in the Byzantine tradition. Some of these theories are well known, others barely investigated, if at all. The papers in this volume discuss and contrast a range of these theories and consider their advantages and drawbacks, and their relation to more recent theories of paradox and antinomy. Several of the papers were presented at a workshop organised at the University of St Andrews, Scotland, as part of the Leverhulme-funded project 'Theories of Paradox in Fourteenth-Century Logic: Edition and Translation of Key Texts'.

Oiva Ketonen (1913--2000) was the closest to a student the creator of modern proof theory Gerhard Gentzen ever had. Their encounter took place in 1938--39 in Göttingen, with Ketonen hoping to receive a suitable topic for a doctoral dissertation and Gentzen instead deeply immersed in attempts at proving the consistency of analysis. Ketonen's thesis of 1944, his only work in logic, introduced what is today called the G3-sequent calculus. It is his best-known discovery, a sequent calculus for classical propositional logic the logical rules of which are all invertible. Few read his thesis, the results of which were instead made available through a long review by Paul Bernays. Ketonen's calculus is the basis of Evert Beth's tableau method and of the sequent calculi in Stephen Kleene's influential {\it Introduction to Metamathematics}. A second result was a sharpening of the midsequent theorem, by which the number of quantifier inferences with eigenvariables could be minimized. The existence of a weakest possible midsequent followed, in the sense that if any midsequent is derivable, a weakest one is. Turning this into a contrapositive, Ketonen found a purely syntactic method for proofs of underivability that he applied to affine plane geometry. His result, in modern terms, was a positive solution to the word problem for the universal fragment of plane affine geometry, with a syntactic proof of underivability of the parallel postulate from the rest of the affine axioms as a corollary.

Parte della cultura si limitava alla conoscenza di calendari nel tempo antico. Per questo chi vuole indagare su qualsiasi civiltà trova in questo libro gli strumenti utili.L'autore introduce l'uguaglianza 3402= =(30.24)112.5.Si isola (3024) che interviene col significato di 360° lungo un cerchio.Inoltre 34.02 interviene in misure di ore alla ziggurat babilonese di Uruk (4000a.C.). ¿ Come si possa ricavare un calendario viene spiegato con ricostruzione pratica di quello di Dionigi il ¿ Piccolo,ordinato da un papa nei primi secoli di cristianesimo. Importante risultato è la decifrazione di una tavoletta aritmetica egiziana le cui cifre appaio no in luoghi differenti,dando possibilità di unificare le misure di angoli. Nella riforma di Ezechiele si rileva un angolo di 72°; ma lo stesso angolo è necessario per ogni galassia.In un libro di E.Fermi(Notes On Quantum Mechanics) è dimostrato che ogni( quantità )osservabile si ricava con sistema di equazioni. ¿Così,abbiamo qui un metodo per controllare le distanze di alcune galassie e stelle da (punti=)sorgenti di forza a cui sono collegate.

Parte della scienza si limitava alla conoscenza dei calendari nel tempo antico. Per questa ragione, chi volesse indagare su qualunque civiltà del passato, trova qui tutti gli strumenti utili. L'autore ha introdotto l'uguaglianza 3402 = (30.24) 112.5, utile per isolare (30.24), che interviene col significato di rotazione di 360° lungo un cerchio. Inoltre 34.02 è anche impiegato per misurare ore alla ziggurat babilonese di Uruk (4000 a.C). Come si possa ricavare un calendario viene spiegato con la costruzione pratica di quello di Dionigi, scritto per ordine di un papa nei primi secoli del cristianesimo. Risulta importante la decifrazione di una tabella aritmetica egiziana, le cui cifre intervengono in vari contesti, dando la possibilità di unificare tutte le misure di angoli. Nella riforma di Ezechiele compare un angolo di 72°; e lo stesso an golo è indispensabile anche per le galassie. In un libro di E. Fermi (No tes on Quantum Mechanics) è dimostrato che ogni (quantità) osservabile si descrive con un sistema di equazioni; così con la soluzione di un sistema abbiamo qui un metodo per controllare distanze di alcune galassie ad anche di stelle da punti considerati come sorgenti di forza, da cui dipendono. Se il lettore non vuol leggere qualche argomento può evitarlo passando oltre.