Matematikkens fundament

Einsatzfertige Lerneinheiten vermitteln fundamentale mathematische Techniken, die über die Oberstufe hinaus von Bedeutung sind. Die Lerninhalte eignen sich auch zur gezielten Vorbereitung auf Mathematikwettbewerbe. Den Anfang machen universelle Beweistechniken, die in unterschiedlichen Kontexten angewandt werden. Es folgen lineare Kongruenzen, die Eulersche ¿-Funktion und der Satz von Euler. Abwechslungsreiche Aufgaben wiederholen und vertiefen den Umgang mit Ungleichungen. Die Schüler*innen führen Beweise in unterschiedlichen Gebieten. Die Aufgaben fördern die mathematische Denkfähigkeit, Phantasie und Kreativität.

In 1922, Curry started reading Principia Mathematica and was intrigued by the complications of its substitution rule. As a result of trying to analyze substitution, Curry conceived the combinators in 1926. This collection is dedicated to Jonathan Seldin's 80th anniversary. Seldin is the penultimate PhD student of Curry and the guardian of Curry's paradigm.The search at the beginning of the 20th century for powerful systems that combine computations and deductions (functions and logic) and that are able to formalise mathematics has led to the birth of the mighty ¿-calculus of Church, Combinatory Logic of Curry and Category Theory of Eilenberg and Mac Lane, all of which are well represented in this collection. The struggle for internalising as much as possible while keeping the system consistent is clear in the evolution of the ¿-calculus and combinatory logic and can be felt again in the articles in this volume. Similarly, the struggle for elegant theories that minimise the number of basic concepts while remaining as close as possible to the language's structure is clear. Generalising concepts, connecting areas that may seem far apart and applying useful techniques from one area to the other is also represented well in this volume where for example notions like coherence, confluence, commuting diagrams, are extended between ¿-calculus, rewriting systems and category theory, and where embedding relations are given to allow a lot of disciplines from logic to mathematics to computer science to meet.

In recent years, the interaction between harmonic analysis and convex geometry has increased which has resulted in solutions to several long-standing problems. This collection is based on the topics discussed during the Research Semester on Harmonic Analysis and Convexity at the Institute for Computational and Experimental Research in Mathematics in Providence RI in Fall 2022. The volume brings together experts working in related fields to report on the status of major problems in the area including the isomorphic Busemann-Petty and slicing problems for arbitrary measures, extremal problems for Fourier extension and extremal problems for classical singular integrals of martingale type, among others.

Både studerende og lærere vil have glæde over denne matematiske bevissamling.Bogen gør både undervisningen i matematik samt eksamenslæsningen op til de mundtlige eksamener lettere. Endvidere findes der utallige grafer, figurer ligesom der til værket findes et resume og bemærkninger ved hvert matematiske bevis, således at indholdet i beviset bliver komprimeret. Bemærkningerne indeholder ekstra overvejelser som den studerende kan bruge til de svære spørgsmål i en eksamenssituation. Bevissamlingen indeholder også beviser, som går ud over det almindelige pensum, så der er spændende beviser til den flittige studerende.

Zu seinen Lebzeiten war Kurt Gödel außerhalb der Fachwelt der Mathematiker, Philosophen und theoretischen Physiker kaum bekannt. Zu Beginn seiner Karriere schuf er beeindruckende Arbeiten zur Vollständigkeit und Beweisbarkeit formaler logischer Systeme, die zu seiner Dissertation und seiner Habilitations-schrift wurden und ihn unter Fachleuten weltberühmt machten. Seine Unvoll-ständigkeitssätze läuteten das Ende der formal-logischen Programme der Logizisten (Russell et al.) und der Formalisten (Hilbert et al.) ein. Später erzielte er auch signifikante Ergebnisse in der Mengenlehre. Nach seiner Emigration in die USA (Princeton), widmete er sich mehr der Philosophie, dem Leitmotiv seines Lebens, und er fand auch eine einzigartige Lösung zu Einsteins Feld-gleichungen der Gravitation, sein ¿Gödel-Universum¿. Dieses Buch beschreibt sowohl den Gödel, der ein genialer Wissenschaftler war, und der gewagte und neuartige Hypothesen zu den Fundamenten der Mathe-matik und Physik hervorbrachte, ¿ als auch den Gödel, der ein perfekter Rationalist war, aber sein Alltagsleben nur mit Mühe meistern konnte und zeitlebens unter Depressionen, Angstneurosen und Hypochondrie litt. Ein Leben voller Paradoxen, in dem er trotz all seiner psychischen Probleme Beachtliches leistete und zu einem Vorbild für viele jüngere Wissenschaftler wurde. Das Buch liefert den Kontext zu seinen Errungenschaften, die ein verblüffend breites Spektrum intellektueller Unternehmungen darstellen, und zu seiner zunehmenden Geisteskrankheit; und es zeigt, wie er eine lange und erfolgreiche Karriere mit Hilfe seiner loyalen Ehefrau Adele und einigen seiner Freunde durchlaufen konnte. Dies ist eine faszinierende Geschichte der wissen-schaftlichen Genialität und der menschlichen Natur.

Samson Abramsky¿s wide-ranging contributions to logical and structural aspects of Computer Science have had a major influence on the field. This book is a rich collection of papers, inspired by and extending Abramsky¿s work. It contains both survey material and new results, organised around six major themes: domains and duality, game semantics, contextuality and quantum computation, comonads and descriptive complexity, categorical and logical semantics, and probabilistic computation. These relate to different stages and aspects of Abramsky¿s work, reflecting its exceptionally broad scope and his ability to illuminate and unify diverse topics.Chapters in the volume include a review of his entire body of work, spanning from philosophical aspects to logic, programming language theory, quantum theory, economics and psychology, and relating it to a theory of unification of sciences using dual adjunctions. The section on game semantics shows how Abramsky¿s work hasled to a powerful new paradigm for the semantics of computation. The work on contextuality and categorical quantum mechanics has been highly influential, and provides the foundation for increasingly widely used methods in quantum computing. The work on comonads and descriptive complexity is building bridges between currently disjoint research areas in computer science, relating Structure to Power.The volume also includes a scientific autobiography, and an overview of the contributions. The outstanding set of contributors to this volume, including both senior and early career academics, serve as testament to Samson Abramsky¿s enduring influence. It will provide an invaluable and unique resource for both students and established researchers.

Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights.This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features:Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model constructionOffers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other resultsProvides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic argumentsIncludes a large number of exercises of varying levels of difficulty, supplementing each chapterThe text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas.Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.

"The paradoxes about truth are the subject of extensive research. Developing an original approach, this book argues that we should diverge from classical logic and presents a number of formal theories of truth. Also included is a beginner-friendly introduction to semantic paradoxes, and a discussion of alternative non-classical theories"--

Math Mammoth Foundational Word Problems is a workbook intended for students in 3rd grade and above that need more practice with word problems. I especially aim to help students in 4th-7th grades who have trouble with word problems and who need to "go back to the beginning", so to speak, to learn to solve them.The problems are generally based on a knowledge of third grade math, such as regrouping in addition and subtraction, the multiplication tables, concept of division, and calculating change. The problems do not require long multiplication, long division, or calculations with fractions or decimals (with the exception of money amounts).The worksheets have examples and some instruction about common strategies for word problems. Students are often encouraged to write down the calculations they do. They are also encouraged to make a drawing or a sketch of the situation in the problem, to help them understand the situation and to see what math operation is involved.Overall, most of the problems require two or more steps; however, some problem sets focus on differentiating what math operation should be used, and those only require a single step.Several problem sets are designed to help students to master problems that involve equal-size groups. These questions either ask for the total (and are thus solved by multiplying), or they ask for the size of the group or for the quantity of the groups (one can write a missing number multiplication sentence, such as 3 × ___ = 18 or use division).The book also includes word problems with fractions, with large numbers, with money, geometry word problems, and challenges in the form of puzzle corners.You can download sample pages from this link: https://www.mathmammoth.com/word_problems/Foundational_Word_Problems_samples.pdf

Kurt Gödel (1906-1978) gained world-wide fame by his incompleteness theorem of 1931. Later, he set as his aim to solve what are known as Hilbert's first and second problems, namely Cantor's continuum hypothesis about the cardinality of real numbers, and secondly the consistency of the theory of real numbers and functions. By 1940, he was halfway through the first problem, in what was his last published result in logic and foundations. His intense attempts thereafter at solving these two problems have remained behind the veil of a forgotten German shorthand he used in all of his writing. Results on Foundations is a set of four shorthand notebooks written in 1940-42 that collect results Gödel considered finished. Its main topic is set theory in which Gödel anticipated several decades of development. Secondly, Gödel completed his 1933 program of establishing the connections between intuitionistic and modal logic, by methods and results that today are at the same time new and 80 years old.The present edition of Gödel's four notebooks encompasses the 368 numbered pages and 126 numbered theorems of the Results on Foundations, together with a list of 74 problems on set theory Gödel prepared in 1946, and a list of an unknown date titled "The grand program of my research in ca. hundred questions.''

This book revises and expands upon the prior edition, The Navier-Stokes Problem. The focus of this book is to provide a mathematical analysis of the Navier-Stokes Problem (NSP) in R^3 without boundaries. Before delving into analysis, the author begins by explaining the background and history of the Navier-Stokes Problem. This edition includes new analysis and an a priori estimate of the solution. The estimate proves the contradictory nature of the Navier-Stokes Problem. The author reaches the conclusion that the solution to the NSP with smooth and rapidly decaying data cannot exist for all positive times. By proving the NSP paradox, this book provides a solution to the millennium problem concerning the Navier-Stokes Equations and shows that they are physically and mathematically contradictive.

Fuzzy sets have experienced multiple expansions since their conception to enhance their capacity to convey complex information. Intuitionistic fuzzy sets, image fuzzy sets, q-rung orthopair fuzzy sets, and neutrosophic sets are a few of these extensions. Researchers and academics have acquired a lot of information about their theories and methods for making decisions. However, only a small number of research findings discuss how neutrosophic sets theory and their extensions (NSTEs) are used in education. The Handbook of Research on the Applications of Neutrosophic Sets Theory and Their Extensions in Education implements fresh scientific approaches to enhance the quality of decisions under neutrosophic environments, particularly within education. Covering key topics such as data modeling, educational technologies, decision making, and learning management systems, this major reference work is ideal for instructional designers, researchers, academicians, scholars, practitioners, instructors, and students.

This book uses algebraic tools to study the elementary properties of classes of fields and related algorithmic problems. The first part covers foundational material on infinite Galois theory, profinite groups, algebraic function fields in one variable and plane curves. It provides complete and elementary proofs of the Chebotarev density theorem and the Riemann hypothesis for function fields, together with material on ultraproducts, decision procedures, the elementary theory of algebraically closed fields, undecidability and nonstandard model theory, including a nonstandard proof of Hilbert's irreducibility theorem. The focus then turns to the study of pseudo algebraically closed (PAC) fields, related structures and associated decidability and undecidability results. PAC fields (fields K with the property that every absolutely irreducible variety over K has a rational point) first arose in the elementary theory of finite fields and have deep connections with number theory.Thisfourth edition substantially extends, updates and clarifies the previous editions of this celebrated book, and includes a new chapter on Hilbertian subfields of Galois extensions. Almost every chapter concludes with a set of exercises and bibliographical notes. An appendix presents a selection of open research problems. Drawing from a wide literature at the interface of logic and arithmetic, this detailed and self-contained text can serve both as a textbook for graduate courses and as an invaluable reference for seasoned researchers.

»Philosophy of Mathematics« is understood, in this book, as an effort to clarify such questions that mathematics itself raises but cannot answer with its own methods. These include, for example, questions about the ontological status of mathematical objects (e.g., what is the nature of mathematical objects?) and the epistemological status of mathematical theorems (e.g., from what sources do we draw when we prove mathematical theorems?). The answers given by Plato, Aristotle, Euclid, Descartes, Locke, Leibniz, Kant, Cantor, Frege, Dedekind, Hilbert and others will be studied in detail. This will lead us to deep insights, not only into the history of mathematics, but also into the conception of mathematics as it is commonly held in the present time.The book is a translation from the German, however revised and considerably expanded. Various chapters have been completely rewritten.

This contributed volume overviews recently presented approaches for carrying out QSPR/QSAR analysis by using a simplifying molecular input-line entry system (SMILES) to represent the molecular structure. In contrast to traditional SMILES, quasi-SMILES is a sequence of special symbols-codes that reflect molecular features and codes of experimental conditions. SMILES and quasi-SMILES serve as a basis to develop QSPR/QSAR as well Nano-QSPR/QSAR via the Monte Carlo calculation that provides the so-called optimal descriptors for QSPR/QSAR models. The book presents a reliable technology for developing Nano-QSPR/QSAR while it also includes the description of the algorithms of the Monte Carlo optimization. It discusses the theory and practice of the technique of variational authodecoders (VAEs) based on SMILES and analyses in detail the index of ideality of correlation (IIC) and the correlation intensity index (CII) which are new criteria for the predictive potential of the model. The mathematical apparatus used is simple so that students of relevant specializations can easily follow. This volume is a valuable contribution to the field and will be of great interest to developers of models of physicochemical properties and biological activity, chemical technologists, and toxicologists involved in the area of drug design.

Prophetess Dr. Kemba Jarena Lucas (PK), founder and chief visionary officer of Daughters of Destiny International (DOD), is celebrating the organization's jade anniversary by supporting her tribe and any Christian woman willing to help children ages 4-8 overcome their math learning gaps. As a Christian educator who began preaching at the age of 12, she offers a one-of-a-kind, Christ-centered approach to assisting her brilliant crew in growing closer to God, overcoming challenges, and progressing in their math skill sets without the pressure of completing a task in a specific time frame because everyone grows differently. Her brilliant mathematicians are guided by their Auntie-Prophetess, a fifteen-year veteran homeschool parent, to gain confidence while conquering the learning obstacles that they will face from time to time without shame and by asking for help. In "All That Math!: A Kembativity Mathematics Workbook for Ages 4-8 featuring 'The Parsonage Tales of Kemba Jarena, God's Little Prophet Series' Fam," PK, along with a few characters from her children's series, inspires her brilliant nieces and nephews to close learning gaps day by day, starting today. By purchasing this book, you are financially supporting Daughters of Destiny International's Build Her Global Project, which is co-sponsored by KJL Design + Build, KEMBA IC partners, and other faith-based organizations around the world. This global project will provide borehole wells, butterfly meditation gardens, Gchapels, prayer walls, exterior remodeling, landscaped campuses, and hospitable built environments to foster community within a harmonized sacred space of divinity, humanity, and nature, thereby catalyzing educational, economic, and communal growth, particularly for women and girls.

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.

This book analyzes the generation of the arrow-categories of a given category, which is a foundational and distinguishable Category Theory phenomena, in analogy to the foundational role of sets in the traditional set-based Mathematics, for defi nition of natural numbers as well. This inductive transformation of a category into the infinite hierarchy of the arrowcategories is extended to the functors and natural transformations. The author considers invariant categorial properties (the symmetries) under such inductive transformations. The book focuses in particular on Global symmetry (invariance of adjunctions) and Internal symmetries between arrows and objects in a category (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Internal symmetry to Computer Science: for Intuitionistic Logic, Untyped Lambda Calculus with Fixpoint Operators, Labeled Transition Systems in Process Algebras and Modal logics as well as Data Integration Theory.

This book demonstrates how to formally model various mathematical domains (including algorithms operating in these domains) in a way that makes them amenable to a fully automatic analysis by computer software.The presented domains are typically investigated in discrete mathematics, logic, algebra, and computer science; they are modeled in a formal language based on first-order logic which is sufficiently rich to express the core entities in whose correctness we are interested: mathematical theorems and algorithmic specifications. This formal language is the language of RISCAL, a ¿mathematical model checker¿ by which the validity of all formulas and the correctness of all algorithms can be automatically decided. The RISCAL software is freely available; all formal contents presented in the book are given in the form of specification files by which the reader may interact with the software while studying the corresponding book material.