Novice Mathematics Teachers' Professional Learning
This book investigates three novice high school mathematics teachers' professional learning processes in the early stages of their careers at schools in Shanghai, China. Teacher professional learning is examined as a complex and dynamic system that connects both cognitive and situated perspectives on learning theory. Inspiring mathematics teachers to adopt student-focused pedagogies is challenging, particularly in China where tensions in teacher-centred, content-focused and examination-oriented practices are predominant. The three novice teachers who participated in this study brought different beliefs and knowledge derived from their different individual experiences to bear on their teaching practices. However, they were strongly influenced by the environments in which they taught and mainly adopted a professional learning approach to teacher-centred practices, despite reporting that they favoured student-centred teaching practices. The study also observed professional learning towards student-centred pedagogical aspects in a single teacher case with mentorship support, indicating that student-centred pedagogies may be promoted within the constraints of the existing dominant teaching practice.
Research in Computational Topology 2
This second volume of Research in Computational Topology is a celebration and promotion of research by women in applied and computational topology, containing the proceedings of the second workshop for Women in Computational Topology (WinCompTop) as well as papers solicited from the broader WinCompTop community. The multidisciplinary and international WinCompTop workshop provided an exciting and unique opportunity for women in diverse locations and research specializations to interact extensively and collectively contribute to new and active research directions in the field. The prestigious senior researchers that signed on to head projects at the workshop are global leaders in the discipline, and two of them were authors on some of the first papers in the field. Some of the featured topics include topological data analysis of power law structure in neural data; a nerve theorem for directional graph covers; topological or homotopical invariantsfor directed graphs encoding connections among a network of neurons; and the issue of approximation of objects by digital grids, including precise relations between the persistent homology of dual cubical complexes.
Lessons in Enumerative Combinatorics
This textbook introduces enumerative combinatorics through the framework of formal languages and bijections. By starting with elementary operations on words and languages, the authors paint an insightful, unified picture for readers entering the field. Numerous concrete examples and illustrative metaphors motivate the theory throughout, while the overall approach illuminates the important connections between discrete mathematics and theoretical computer science. Beginning with the basics of formal languages, the first chapter quickly establishes a common setting for modeling and counting classical combinatorial objects and constructing bijective proofs. From here, topics are modular and offer substantial flexibility when designing a course. Chapters on generating functions and partitions build further fundamental tools for enumeration and include applications such as a combinatorial proof of the Lagrange inversion formula. Connections to linear algebra emerge in chapters studying Cayley trees, determinantal formulas, and the combinatorics that lie behind the classical Cayley-Hamilton theorem. The remaining chapters range across the Inclusion-Exclusion Principle, graph theory and coloring, exponential structures, matching and distinct representatives, with each topic opening many doors to further study. Generous exercise sets complement all chapters, and miscellaneous sections explore additional applications. Lessons in Enumerative Combinatorics captures the authors' distinctive style and flair for introducing newcomers to combinatorics. The conversational yet rigorous presentation suits students in mathematics and computer science at the graduate, or advanced undergraduate level. Knowledge of single-variable calculus and the basics of discrete mathematics is assumed; familiarity with linear algebra will enhance the study of certain chapters.
Motion and Genetic Definitions in the Sixteenth-Century Euclidean Tradition
1. Introduction.- 2. Oronce Fine.- 3. Jacques Peletier.- 4. Fran癟ois de Foix-Candale.- 5. Henry Billingsley.- 6. John Dee.- 7. Federico Commandino.- 8. Christoph Clavius.- 9. Synthesis: Continuities and Transformations in the Status of Geometrical Motion and Genetic Definitions from Fine to Clavius.- 10. Later Developments in the Seventeenth Century: a Cartesian Epilogue.
Researches
A Compilation of Original Mathematical and StatisticalResearch Reports on Diverse Themes. The Treatments presented contain full Equations, Diagrams, Tabulations, and, where appropriate, Computer Programs. Topics explored in the First Volume include: - - Mathematical Aspects of Coding - The Theory of Mixtures - The Desmotropic Disk System - Spacial Applications - Studies in Spacial Association
Framing Global Mathematics
This open access book is about the shaping of international relations in mathematics over the last two hundred years. It focusses on institutions and organizations that were created to frame the international dimension of mathematical research. Today, striking evidence of globalized mathematics is provided by countless international meetings and the worldwide repository ArXiv. The text follows the sinuous path that was taken to reach this state, from the long nineteenth century, through the two wars, to the present day. International cooperation in mathematics was well established by 1900, centered in Europe. The first International Mathematical Union, IMU, founded in 1920 and disbanded in 1932, reflected above all the trauma of WW I. Since 1950 the current IMU has played an increasing role in defining mathematical excellence, as is shown both in the historical narrative and by analyzing data about the International Congresses of Mathematicians. For each of the three periods discussed, interactions are explored between world politics, the advancement of scientific infrastructures, and the inner evolution of mathematics. Readers will thus take a new look at the place of mathematics in world culture, and how international organizations can make a difference. Aimed at mathematicians, historians of science, scientists, and the scientifically inclined general public, the book will be valuable to anyone interested in the history of science on an international level.
Framing Global Mathematics
This open access book is about the shaping of international relations in mathematics over the last two hundred years. It focusses on institutions and organizations that were created to frame the international dimension of mathematical research. Today, striking evidence of globalized mathematics is provided by countless international meetings and the worldwide repository ArXiv. The text follows the sinuous path that was taken to reach this state, from the long nineteenth century, through the two wars, to the present day. International cooperation in mathematics was well established by 1900, centered in Europe. The first International Mathematical Union, IMU, founded in 1920 and disbanded in 1932, reflected above all the trauma of WW I. Since 1950 the current IMU has played an increasing role in defining mathematical excellence, as is shown both in the historical narrative and by analyzing data about the International Congresses of Mathematicians. For each of the three periods discussed, interactions are explored between world politics, the advancement of scientific infrastructures, and the inner evolution of mathematics. Readers will thus take a new look at the place of mathematics in world culture, and how international organizations can make a difference. Aimed at mathematicians, historians of science, scientists, and the scientifically inclined general public, the book will be valuable to anyone interested in the history of science on an international level.
Chapters from G繹del's Unfinished Book on Foundational Research in Mathematics
Basic Topology 1
This first of the three-volume book is targeted as a basic course in topology for undergraduate and graduate students of mathematics. It studies metric spaces and general topology. It starts with the concept of the metric which is an abstraction of distance in the Euclidean space. The special structure of a metric space induces a topology that leads to many applications of topology in modern analysis and modern algebra, as shown in this volume. This volume also studies topological properties such as compactness and connectedness. Considering the importance of compactness in mathematics, this study covers the Stone-Cech compactification and Alexandroff one-point compactification. This volume also includes the Urysohn lemma, Urysohn metrization theorem, Tietz extension theorem, and Gelfand-Kolmogoroff theorem. The content of this volume is spread into eight chapters of which the last chapter conveys the history of metric spaces and the history of the emergence of the conceptsleading to the development of topology as a subject with their motivations with an emphasis on general topology. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power, and active learning of the subject, all the while covering a wide range of theories and applications in a balanced unified way.
Comparison of Mathematics and Physics Education II
Interdisciplinary teaching is considered as one of the main goals of education worldwide. At the same time, it poses an immense challenge to teachers who have been trained in only one of the combines subjects. This is true even for closely related disciplines such as mathematics and physics. In this volume, practice-oriented educational comparisons are made across various topics that are highly relevant in both subjects. Furthermore, practical examples are presented in the form of lesson plans in which exemplary implementation in class is presented, considering both educational perspectives.
Universal Algebraic Logic
This book gives a comprehensive introduction to Universal Algebraic Logic. The three main themes are (i) universal logic and the question of what logic is, (ii) duality theories between the world of logics and the world of algebra, and (iii) Tarskian algebraic logic proper including algebras of relations of various ranks, cylindric algebras, relation algebras, polyadic algebras and other kinds of algebras of logic. One of the strengths of our approach is that it is directly applicable to a wide range of logics including not only propositional logics but also e.g. classical first order logic and other quantifier logics. Following the Tarskian tradition, besides the connections between logic and algebra, related logical connections with geometry and eventually spacetime geometry leading up to relativity are also part of the perspective of the book. Besides Tarskian algebraizations of logics, category theoretical perspectives are also touched upon. This book, apart from being a monograph containing state of the art results in algebraic logic, can be used as the basis for a number of different courses intended for both novices and more experienced students of logic, mathematics, or philosophy. For instance, the first two chapters can be used in their own right as a crash course in Universal Algebra.
Reachability Problems
This book constitutes the refereed proceedings of the 15th International Conference on Reachability Problems, RP 2022, held in Kaiserslautern, Germany, in October 2022.The 8 full papers presented were carefully reviewed and selected from 14 submissions. In addition, 3 invited papers were included in this volume. The RP proceedings cover topics such as reachability for infinite state systems; rewriting systems; reachability analysis in counter/timed/cellular/communicating automata; Petri nets; computational aspects of semigroups, groups, and rings; reachability in dynamical and hybrid systems; frontiers between decidable and undecidable reachability problems; complexity and decidability aspects; predictability in iterative maps; and new computational paradigms.
Casual Calculus: A Friendly Student Companion - Volume 2
Yes, this is another Calculus book. However, I think it fits in a niche between the two predominant types of such texts. It could be used as a textbook, albeit a streamlined one -- it contains exposition on each topic, with an introduction, rationale, train of thought, and solved examples with accompanying suggested exercises. It could be used as a solution guide -- because it contains full written solutions to each of the hundreds of exercises posed inside. But its best position is right in between these two extremes. It is best used as a companion to a traditional text or as a refresher -- with its conversational tone, its 'get right to it' content structure, and its inclusion of complete solutions to many problems, it is a friendly partner for students who are learning Calculus, either in class or via self-study.Exercises are structured in three sets to force multiple encounters with each topic. Solved examples in the text are accompanied by 'You Try It' problems, which are similar to the solved examples; the students use these to see if they're ready to move forward. Then at the end of the section, there are 'Practice Problems': more problems similar to the You Try It problems, but given all at once. Finally, each section has Challenge Problems -- these lean to being equally or a bit more difficult than the others, and they allow students to check on what they've mastered.My goal is to keep the students engaged with the text, and so the writing style is very informal, with attempts at humor along the way. Because we have large engineering and meteorology programs at my institution, and they make up the largest portion of our Calculus students; naturally, then, these sorts of STEM students are the target audience.
A Brief History of Analysis
This book explores the origins of mathematical analysis in an accessible, clear, and precise manner. Concepts such as function, continuity, and convergence are presented with a unique historical point of view. In part, this is accomplished by investigating the impact of and connections between famous figures, like Newton, Leibniz, Johann Bernoulli, Euler, and more. Of particular note is the treatment of Karl Weierstra?, whose concept of real numbers has been frequently overlooked until now. By providing such a broad yet detailed survey, this book examines how analysis was formed, how it has changed over time, and how it continues to evolve today. A Brief History of Analysis will appeal to a wide audience of students, instructors, and researchers who are interested in discovering new historical perspectives on otherwise familiar mathematical ideas.
Casual Calculus: A Friendly Student Companion - Volume 2
Yes, this is another Calculus book. However, it fits in a niche between the two predominant types of such texts. It could be used as a textbook, albeit a streamlined one - it contains exposition on each topic, with an introduction, rationale, train of thought, and solved examples with accompanying suggested exercises. It could be used as a solution guide - because it contains full written solutions to each of the hundreds of exercises posed inside. But its best position is right in between these two extremes. It is best used as a companion to a traditional text or as a refresher - with its conversational tone, its "get right to it" content structure, and its inclusion of complete solutions to many problems, it is a friendly partner for students who are learning Calculus, either in class or via self-study.Exercises are structured in three sets to force multiple encounters with each topic. Solved examples in the text are accompanied by "You Try It" problems, which are similar to the solved examples; the students use these to see if they're ready to move forward. Then at the end of the section, there are "Practice Problems" more problems similar to the "You Try It" problems, but given all at once. Finally, each section has Challenge Problems - these lean to being equally or a bit more difficult than the others, and they allow students to check on what they've mastered.The goal is to keep the students engaged with the text, and so the writing style is very informal, with attempts at humor along the way. The target audience is STEM students including those in engineering and meteorology programs.
The Riordan Group and Applications
The ever-growing applications and richness of approaches to the Riordan group is captured in this comprehensive monograph, authored by those who are among the founders and foremost world experts in this field. The concept of a Riordan array has played a unifying role in enumerative combinatorics over the last three decades. The Riordan arrays and Riordan group is a new growth point in mathematics that is both being influenced by, and continuing its contributions to, other fields such as Lie groups, elliptic curves, orthogonal polynomials, spline functions, networks, sequences and series, Beal conjecture, Riemann hypothesis, to name several. In recent years the Riordan group has made links to quantum field theory and has become a useful tool for computer science and computational chemistry. We can look forward to discovering further applications to unexpected areas of research. Providing a baseline and springboard to further developments and study, this book may also serve asa text for anyone interested in discrete mathematics, including combinatorics, number theory, matrix theory, graph theory, and algebra.
The Rise and Fall of the German Combinatorial Analysis
This text presents the ideas of a particular group of mathematicians of the late 18th century known as "the German combinatorial school" and its influence. The book tackles several questions concerning the emergence and historical development of the German combinatorial analysis, which was the unfinished scientific research project of that group of mathematicians. The historical survey covers the three main episodes in the evolution of that research project: its theoretical antecedents (which go back to the innovative ideas on mathematical analysis of the late 17th century) and first formulation, its consolidation as a foundationalist project of mathematical analysis, and its dissolution at the beginning of the 19th century. In addition, the book analyzes the influence of the ideas of the combinatorial school on German mathematics throughout the 19th century.
Mathematical Surprises
This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss'spurely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems. Supplementary material to the book can be found at https: //github.com/motib/surprises.
Kurt G繹del
Preface.- Prologue.- La Belle ?poque in Br羹nn. Beginnings.- School Days. A New Nation.- Moving to the Capital. Student Life in 1920's Vienna.- The Wiener Kreis and the Mathematical Colloquium. Graduate work.- Private Life in Vienna.- G繹del's Doctoral Thesis. The Incompleteness Theorems.- The Mathematician in Vienna. Habilitation.- Matters of Health.- A Sojourn Abroad: 1933/34 - Princeton.- Back to Vienna. The First Breakdown.- 'Commuting' between Vienna and Princeton - The Late 1930's. Marriage.- The Continuum Hypothesis.- Professional Uncertainty. A Long Journey Eastwards.- Princeton and the IAS - Philosophy, Einstein and von Neumann. A Bizarre Birthday Present: G繹del's Universe.- Reception, Recognition, Honors. Einstein's Loss. The Professor at Princeton.- The 1960's: Fame and Seclusion.- Later Years. Philosophy, Cosmology, Logic, Computability.- G繹del's Legacy - The Lessons of an Unusual Life.- Epilogue.
Research Trends in Graph Theory and Applications
The Workshop for Women in Graph Theory and Applications was held at the Institute for Mathematics and Its Applications (University of Minnesota, Minneapolis) on August 19-23, 2019. During this five-day workshop, 42 participants performed collaborative research, in six teams, each focused on open problems in different areas of graph theory and its applications. The research work of each team was led by two experts in the corresponding area, who prior to the workshop, carefully selected relevant and meaningful open problems that would yield high-quality research and results of strong impact. As a result, all six teams have made significant contributions to several open problems in their respective areas. The workshop led to the creation of the Women in Graph Theory and Applications Research Collaboration Network, which provided the framework to continue collaborating and to produce this volume.This book contains six chapters, each of them on one of the different areas of research at the Workshop for Women in Graph Theory and Applications, and written by participants of each team.
Quaternions for Computer Graphics
If you have ever wondered what quaternions are - then look no further, John Vince will show you how simple and useful they are. This 2nd edition has been completely revised and includes extra detail on the invention of quaternions, a complete review of the text and equations, all figures are in colour, extra worked examples, an expanded index, and a bibliography arranged for each chapter. Quaternions for Computer Graphics includes chapters on number sets and algebra, imaginary and complex numbers, the complex plane, rotation transforms, and a comprehensive description of quaternions in the context of rotation. The book will appeal to students of computer graphics, computer science and mathematics, as well as programmers, researchers, academics and professional practitioners interested in learning about quaternions. John Vince explains in an easy-to-understand language, with the aid of useful figures, how quaternions emerged, gave birth to modern vector analysis, disappeared, and reemerged to be adopted by the flight simulation industry and computer graphics. This book will give you the confidence to use quaternions within your every-day mathematics, and explore more advanced texts.
How to succeed in EPSO numerical reasoning tests
This book will help you to prepare numerical reasoning tests for any competition or selection procedure, especially to work in the EU institutions - EPSO Administrators (AD), Assistants (AST), and calls for contract agents (CAST). It includes the tools that you need to face a numerical reasoning test, many tips, recommendations, and shortcuts, as well as an explanation about the particular conditions of EPSO tests. We will also analyse in detail a series of sample exercises, very similar to the ones that you are going to find in the real exam, guiding you step-by-step along the process -mental and technical- that will allow you to get to the right answer of every question. The author is an economist and successful candidate in the reasoning tests of the highly competitive EPSO AD cycle, several EPSO CAST profiles, and other tests and interviews at the assessment centre, being also an official at the public administration in Spain and having held various positions as a head of unit from 2009 to the present. He has also worked as a national expert at the European Commission, provided training to civil servants, coached many candidates for open competitions and temporary/contract agents' positions, and written books for this purpose.=========== Author's note: I would like to clarify to all my potential readers that this book is structured as a clear training guide, easy to read (by using font 12 and spacing 1.5), to facilitate making your own notes on the pages themselves and solving the practices presented throughout the book.
Invertible Fuzzy Topological Spaces
Motivation and Preliminaries.- H-fuzzy topological spaces.- Invertible fuzzy topological spaces.- Types of invertible fuzzy topological spaces.- Properties of invertible fuzzy topological spaces.- Invertibility of the related spaces.- Invertible R-topological spaces.
The Language of Symmetry
This book is a re-assessment of the structure and reach of symmetry, by an interdisciplinary group of specialists from the arts, humanities, and sciences at Oxford University. This book aims to open up the scope of interdisciplinary work in the study of symmetry and is intended for scholars of any background.
The Language of Symmetry
This book is a re-assessment of the structure and reach of symmetry, by an interdisciplinary group of specialists from the arts, humanities, and sciences at Oxford University. This book aims to open up the scope of interdisciplinary work in the study of symmetry and is intended for scholars of any background.
Mathematical Stories II - Recursion, Divisibility and Proofs
Using field-tested, carefully crafted units of study, the authors in this essential teach fundamental mathematical techniques that are relevant well beyond the elementary school years. In this Volume II, the Gaussian summation formula and a recursion formula are derived and applied. Tasks on divisibility, prime factors and divisors follow. For calculating with remainders, the modulo calculation is introduced and applied. Students learn to perform proofs in a variety of contexts. As in Volume I, "Graphs, Games, and Proofs," the tasks encourage mathematical thinking skills, imagination, and creativity. The detailed sample solutions are designed for non-mathematicians.This book is a translation of the original German 1st edition Mathematische Geschichten II - Rekursion, Teilbarkeit und Beweise by Susanne Schindler-Tschirner and Werner Schindler, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2019. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.
Analysis of Reaction-Diffusion Models with the Taxis Mechanism
Chapter 1. Large time behavior of solutions to the chemotaxis-fluid.- Chapter 2. Global existence in Keller--Segel-- Navier--Stokes system involving tensor-valued sensitivity.- Chapter 3. Large time behavior of solutions to chemotaxis--haptotaxis models.- Chapter 4. Large time behavior of Keller--Segel--(Navier)--Stokes system modeling coral fertilization.- Chapter 5. Qualitative properties to density-suppressed motility models.- Chapter 6. Large time behavior of multi-taxis cross-diffusion system.
Advances in Optimization and Nonlinear Analysis
The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics.
Analysis of Reaction-Diffusion Models with the Taxis Mechanism
Chapter 1. Large time behavior of solutions to the chemotaxis-fluid.- Chapter 2. Global existence in Keller--Segel-- Navier--Stokes system involving tensor-valued sensitivity.- Chapter 3. Large time behavior of solutions to chemotaxis--haptotaxis models.- Chapter 4. Large time behavior of Keller--Segel--(Navier)--Stokes system modeling coral fertilization.- Chapter 5. Qualitative properties to density-suppressed motility models.- Chapter 6. Large time behavior of multi-taxis cross-diffusion system.
Who's Counting?
For decades, New York Times best-selling author John Allen Paulos has enlightened readers by showing how to make sense of the numbers and probabilities behind real-world events, political calculations, and everyday personal decisions. Who's Counting? features dozens of his insightful essays--original writings on contemporary issues like the COVID-19 pandemic, online conspiracy theories, "fake news," and climate change, as well as a selection of enduring columns from his popular ABC News column of the same name. With an abiding respect for reason, a penchant for puzzles with societal implications, and a disarming sense of humor, Paulos does in this collection what he's famous for: clarifies mathematical ideas for everyone and shows how they play a role in government, media, popular culture, and life. He argues that if we can't critically interpret numbers and statistics, we lose one of our most basic and reliable guides to reality.
60 Jahre DVMLG
This volume celebrates the 60th anniversary of the Deutsche Vereinigung f羹r Mathematische Logik und f羹r Grundlagenforschung der exakten Wissenschaften (DVMLG) which was founded on 28 July 1962. The DVMLG is the learned society representing logic and foundations within the German-speaking world. The volume contains historical papers, personal reflections, descriptions of logic groups in Germany, and descriptions of relevant research areas.
Topics of Thought
This is an open access title available under the terms of a CC BY-NC-ND 4.0 International licence. It is free to read at Oxford Scholarship Online and offered as a free PDF download from OUP and selected open access locations. When one thinks--knows, believes, imagines--that something is the case, one's thought has a topic: it is about something, towards which one's mind is directed. What is the logic of thought, so understood? This book begins to explore the idea that, to answer the question, we should take topics seriously. It proposes a hyperintensional account of the propositional contents of thought, arguing that these are individuated not only by the set of possible worlds at which they are true, but also by their topic: what they are about. The book then builds epistemic, doxastic, probabilistic, and conditional logics based on this view. It applies them to issues ranging from dogmatism, scepticism, and epistemic fallibilism, to imagination and suppositional reasoning, belief revision, framing effects, and the acceptability of indicative conditionals.
The Hasse - Noether Correspondence 1925 -1935
Providing the first comprehensive account of the widely unknown cooperation and friendship between Emmy Noether and Helmut Hasse, this book contains English translations of all available letters which were exchanged between them in the years 1925-1935. It features a special chapter on class field theory, a subject which was completely renewed in those years, Noether and Hasse being among its main proponents. These historical items give evidence that Emmy Noether's impact on the development of mathematics is not confined to abstract algebra but also extends to important ideas in modern class field theory as part of algebraic number theory. In her letters, details of proofs appear alongside conjectures and speculations, offering a rich source for those who are interested in the rise and development of mathematical notions and ideas. The letters are supplemented by extensive comments, helping the reader to understand their content within the mathematical environment of the 1920s and 1930s.
Code Recognition and Set Selection with Neural Networks
In mathematics there are limits, speed limits of a sort, on how many computational steps are required to solve certain problems. The theory of computational complexity deals with such limits, in particular whether solving an n-dimensional version of a particular problem can be accomplished with, say, 2 n n steps or will inevitably require 2 steps. Such a bound, together with a physical limit on computational speed in a machine, could be used to establish a speed limit for a particular problem. But there is nothing in the theory of computational complexity which precludes the possibility of constructing analog devices that solve such problems faster. It is a general goal of neural network researchers to circumvent the inherent limits of serial computation. As an example of an n-dimensional problem, one might wish to order n distinct numbers between 0 and 1. One could simply write all n! ways to list the numbers and test each list for the increasing property. There are much more efficient ways to solve this problem; in fact, the number of steps required by the best sorting algorithm applied to this problem is proportional to n In n .
Introduction to Mathematical Philosophy
First published in 1919, Introduction to Mathematical Philosophy shows Russell drawing on his formidable knowledge of philosophy and mathematics to write a brilliant introduction to the subject. This Routledge Classics edition includes a new Foreword by Michael Potter.
Unravelling the Persistence of Dollarization
A political economic analysis of dollarization in Georgia, structured around three themes: the genesis of dollarization (1991-2003), the persistence of dollarization (2003-12) and the politicization of dollarization (2012-19).
The Joy of Abstraction
Mathematician and popular science author Eugenia Cheng is on a mission to show you that mathematics can be flexible, creative, and visual. This joyful journey through the world of abstract mathematics into category theory will demystify mathematical thought processes and help you develop your own thinking, with no formal mathematical background needed. The book brings abstract mathematical ideas down to earth using examples of social justice, current events, and everyday life - from privilege to COVID-19 to driving routes. The journey begins with the ideas and workings of abstract mathematics, after which you will gently climb toward more technical material, learning everything needed to understand category theory, and then key concepts in category theory like natural transformations, duality, and even a glimpse of ongoing research in higher-dimensional category theory. For fans of How to Bake Pi, this will help you dig deeper into mathematical concepts and build your mathematical background.
The Didactics of Mathematics: Approaches and Issues
Didactics of mathematics is a scientific discipline between math and teaching which deals with various matters of teaching math at individual as well as various school levels. This includes the content and methods on how to teach and learn math. It defines aims and content of a math curriculum, and recommends appropriate methods, procedures and organizational forms of teaching. Didactics of mathematics takes into account psychological relations of learning and provides the necessary teaching technology. At present, there are questions as to the role of a pupil and teacher in an educational process. Didactics of mathematics studies the processes which take place in the minds of the student and teacher when teaching math. This book discusses the fundamentals as well as modern approaches and issues of this field. Different approaches, evaluations, methodologies and advanced studies on the didactics of math have been included herein. This book will help the readers in keeping pace with the rapid changes in this field.
Partial Differential Equations: Spectral and High Order Methods
Partial differential equations (PDEs) are mathematical equations that relate a function of several variables to its partial derivatives. A partial derivative of a function of many variables describes how fast the function changes when one variable is altered, keeping the others constant. The task of discovering a partial derivative can be applied to a function that is itself a partial derivative of another function to obtain a second-order partial derivative. The order and degree of partial differential equations are determined in the same way as ordinary differential equations. Different approaches, evaluations and methodologies on partial differential equations have been included in this book. It strives to provide a fair idea about this discipline and to help develop a better understanding of the latest advances within this field. Coherent flow of topics, student-friendly language and extensive use of examples make this book an invaluable source of knowledge.
Studies in Inductive Logic and Probability, Volume I
Rudolf Carnap and Richard C. Jeffrey's Studies in Inductive Logic and Probability, Volume I gathers seminal work charting Carnap's late-career re-architecture of inductive logic. Moving beyond the single preferred c\*-method of Logical Foundations of Probability (1950), this volume documents the shift to a "continuum" of c-functions and then to a still broader framework sensitive to analogy and similarity, informed by de Finetti's representation theorem and contemporary probability/statistics. Anchored by Carnap's "Basic System" (presented here in detail) and complemented by foundational essays--including the programmatic "Inductive Logic and Rational Decisions"--the collection recasts logical probability in the idiom of events, models, and conditionalization, aligning formal inductive methods with Bayesian decision theory while distinguishing logical from statistical notions of information and entropy. Framed by Jeffrey's editorial introduction and Carnap's own historical notes, the book doubles as an intellectual roadmap through the 1950s-60s renaissance in formal epistemology: collaborations with John Kemeny, dialogue with Savage and Putnam, and the systematic adoption of mathematical tools that were absent from Carnap's earlier work. For philosophers of science, statisticians, and decision theorists, Volume I offers both a definitive statement of Carnap's mature foundations and a launch pad for the unfinished upper stories--issues of confirmation, learning from analogy, and representation--that Volume II continues. It's essential reading for anyone who wants to see how inductive logic became conversant with modern probability while retaining a distinctly logical--normative--conception of rational belief. This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1971.
Studies in Inductive Logic and Probability, Volume I
Rudolf Carnap and Richard C. Jeffrey's Studies in Inductive Logic and Probability, Volume I gathers seminal work charting Carnap's late-career re-architecture of inductive logic. Moving beyond the single preferred c\*-method of Logical Foundations of Probability (1950), this volume documents the shift to a "continuum" of c-functions and then to a still broader framework sensitive to analogy and similarity, informed by de Finetti's representation theorem and contemporary probability/statistics. Anchored by Carnap's "Basic System" (presented here in detail) and complemented by foundational essays--including the programmatic "Inductive Logic and Rational Decisions"--the collection recasts logical probability in the idiom of events, models, and conditionalization, aligning formal inductive methods with Bayesian decision theory while distinguishing logical from statistical notions of information and entropy. Framed by Jeffrey's editorial introduction and Carnap's own historical notes, the book doubles as an intellectual roadmap through the 1950s-60s renaissance in formal epistemology: collaborations with John Kemeny, dialogue with Savage and Putnam, and the systematic adoption of mathematical tools that were absent from Carnap's earlier work. For philosophers of science, statisticians, and decision theorists, Volume I offers both a definitive statement of Carnap's mature foundations and a launch pad for the unfinished upper stories--issues of confirmation, learning from analogy, and representation--that Volume II continues. It's essential reading for anyone who wants to see how inductive logic became conversant with modern probability while retaining a distinctly logical--normative--conception of rational belief. This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1971.
Partial Differential Equations
This book is an attempt to make available to the student a coherent modern view of the theory of partial differential equations. Here equations of the first order and linear second order equations are treated with the tensor calculus, which combines generality and insight, in mind. Since the book is self-contained, much of the material is classical, but an effort has been made to achieve a modern outlook on these topics. A number of significant recent developments are introduced, and treated in relation to the natural background formed by geometry and physics.Special features of the exposition are: (a) the simplified general treatment of first order equations; (b) the geometrical foundations of the theory of linear second order equations (c) unified treatment of boundary value problems and related topics by integral equations; (d) the theory of generalized hyperbolic potentials.
Make: Calculus
When Isaac Newton developed calculus in the 1600s, he was trying to tie together math and physics in an intuitive, geometrical way. But over time math and physics teaching became heavily weighted toward algebra, and less toward geometrical problem solving. However, many practicing mathematicians and physicists will get their intuition geometrically first and do the algebra later. Make: Calculus imagines how Newton might have used 3D printed models, construction toys, programming, craft materials, and an Arduino or two to teach calculus concepts in an intuitive way. The book uses as little reliance on algebra as possible while still retaining enough to allow comparison with a traditional curriculum. This book is not a traditional Calculus I textbook. Rather, it will take the reader on a tour of key concepts in calculus that lend themselves to hands-on projects. This book also defines terms and common symbols for them so that self-learners can learn more on their own.
Mathematics in Ancient Jaina Literature
The volume contains selected articles presented in the ZOOM conference on History of Mathematics in Jain Literature, December 2020, and also contains articles invited by the editors on specific topics.The main objective for the conference was to bring to the attention of historians in mathematics that there is a plenty of literature written by monks and scholars in Jaina literature that contains elements of arithmetic, algebra and geometry, independent of discoveries by other cultures in the past. The talks and the discussions at the conference highlighted a need for a volume that can be recommended as a reference book for a course on History of Mathematics in the Departments of Mathematics and Education in colleges and universities. This is our hope that the present volume would fill up the gap on the lack of knowledge of past Jaina contributions.
Advances in Modal Logic 14
Ever since antiquity, philosophers have recognized that truth comes in many "modes", so that a proposition may not only be true or false, but also e.g. "necessary" or "possible". These ideas led to the modern field of modal logic, a lively are of research at the intersection of philosophy, mathematics, and computer science. Nowadays, the term "modal logic" is understood in a broad sense, which allows it to be used for reasoning about seemingly unrelated phenomena such as knowledge, obligations, time, space, and proofs, among many others. Actual research in modal logic draws on techniques from many disciplines including complexity theory, combinatorics, universal algebra, category theory, topology, and proof theory. These proceedings record the papers presented at the 2022 Advances in Modal Logic, a biennial conference series with an aim to report on important new developments in pure and applied modal logic. The topics in this edition include constructive and substructural modal logic, unification, algebraic and neighbourhood semantics, proof theory and complexity of modal logics, and verification in modal logic.
The Logica Yearbook 2021
This volume of the Logica Yearbook series brings together articles based on selected abstracts accepted for presentation at the annual international symposium Logica 2021, in Hejnice, the Czech Republic. The articles range over mathematical and philosophical logic, history and philosophy of logic, and the analysis of natural language.
Math in Society
Math in Society is a survey of contemporary mathematical topics, appropriate for a college-level topics course for liberal arts major, or as a general quantitative reasoning course.This book is an open textbook; it can be read free online at http: //www.opentextbookstore.com/mathinsociety/
