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This book is a well-organized text with carefully constructed examples, a full quota of exercises with solutions, and an emphasis that is algebraic rather than geometric. The book is Sage-enabled with approximately 90 examples of Sage code spread throughout. The book sections can be loaded into Sage as worksheets so that the code can be evaluated immediately; however, it is not necessary to use Sage in order to make use of this textbook.

The HTML version is a fully featured electronic text with cross references that expand in place. It works well on an iPad or any device with Internet access. The Sage examples are executed by the Sage single cell server over the Internet, and students can experiment with their own examples as they read without needing to install Sage or log into a Sage server. Most students will prefer this version over both the pdf and print versions.

As stated in the preface of the book, the dual aims are “to teach the fundamental concepts and techniques of matrix algebra and abstract vector spaces, and to teach the techniques associated with understanding the definitions and theorems forming a coherent area of mathematics.”

Verion 3.50 (spring 2016) is archived here. Online and Hardcover versions are available via the author's website.

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The formation of coalitions to achieve both collaborative and competitive goals is a phenomenon we see all around us. The list of examples of this phenomenon is long and varied: production cartels, political lobbies, customs unions, environmental coalitions, and ethnic alliances are just a few everyday instances. This book looks at coalition formation from the perspective of game theory. How are agreements determined? Which coalitions will form? And are such agreements invariably efficient from a social perspective? The book brings together developments in both cooperative and non-cooperative game theory to study the analytics of coalition formation and binding agreements. It concentrates on pure theory, but discusses several potential applications, such as oligopoly and the provision of public goods.

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Despite the importance of mathematics in our educational systems little is known about how abstract mathematical thinking emerges. Under the uniting thread of mathematical development, we hope to connect researchers from various backgrounds to provide an integrated view of abstract mathematical cognition. Much progress has been made in the last 20 years on how numeracy is acquired. Experimental psychology has brought to light the fact that numerical cognition stems from spatial cognition. The findings from neuroimaging and single cell recording experiments converge to show that numerical representations take place in the intraparietal sulcus. Further research has demonstrated that supplementary neural networks might be recruited to carry out subtasks; for example, the retrieval of arithmetic facts is done by the angular gyrus. Now that the neural networks in charge of basic mathematical cognition are identified, we can move onto the stage where we seek to understand how these basics skills are used to support the acquisition and use of abstract mathematical concepts.

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This new and expanded edition is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Paper). STEP is an examination used by Cambridge Colleges for conditional offers in mathematics. They are also used by some other UK universities and many mathematics departments recommend that their applicants practice on the past papers even if they do not take the examination.Advanced Problems in Mathematics bridges the gap between school and university mathematics, and prepares students for an undergraduate mathematics course. The questions analysed in this book are all based on past STEP questions and each question is followed by a comment and a full solution. The comments direct the reader’s attention to key points and put the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently.This book is a must read for any student wishing to apply to scientific subjects at university level and for anyone interested in advanced mathematics.

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Large sparse linear systems of equations are ubiquitous in science, engineering and beyond. This open access monograph focuses on factorization algorithms for solving such systems. It presents classical techniques for complete factorizations that are used in sparse direct methods and discusses the computation of approximate direct and inverse factorizations that are key to constructing general-purpose algebraic preconditioners for iterative solvers. A unified framework is used that emphasizes the underlying sparsity structures and highlights the importance of understanding sparse direct methods when developing algebraic preconditioners. Theoretical results are complemented by sparse matrix algorithm outlines. This monograph is aimed at students of applied mathematics and scientific computing, as well as computational scientists and software developers who are interested in understanding the theory and algorithms needed to tackle sparse systems. It is assumed that the reader has completed a basic course in linear algebra and numerical mathematics.

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This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018.

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The present text book contains a collection of six high-quality articles. In particular, this book is devoted to Linear Mathematics by presenting problems in Applied Linear Algebra of general or special interest.

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CK-12 Foundation’s Basic Algebra FlexBook® textbook is an introduction to the algebraic topics of functions, equations, and graphs for middle-school and high-school students.

Teacher's resources at CK-12.

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CK-12’s Algebra I Second Edition is a clear presentation of algebra for the high school student. Topics include: Equations and Functions, Real Numbers, Equations of Lines, Solving Systems of Equations and Quadratic Equations.

You can say thanks to CK-12.

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Algebra II is considered the third year of math in the on-level mathematics high-school sequence. Depending on the level of the students in your class, the option of Trigonometry can be added at the end of the coursework. There is a lot of material to be covered in this course, including: systems of equations, quadratics, higher level polynomials, exponential and logarithmic functions, modeling, rational functions, conics, sequences and series, probability, and trigonometry.

Answer keys and related material are available. You can say thanks at CK-12.

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This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory.

 

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This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory.

 

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Quantum physics; Mathematical physics; Matrix theory; Algebra

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This book reproduces the complete extant correspondence between Emmy Noether and Helmut Hasse. There are 82 such letters, of which 79 are from Noether to Hasse, dating from 1925 until Noether‘s sudden death in 1935. The correspondence refl ects a crucial period in the development of 20th century algebra and number theory, in particular class fi eld theory. Details of proofs appear alongside with conjectures and speculations. Also discussed are questions of textbook presentation, e.g., of Galois theory. Aside from mathematical details, the spontaneity of Noether‘s style allows many glimpses at the image that Emmy Noether and Helmut Hasse had of the topics they were working in. The Hasse – Noether correspondence is a rich source for those who are interested in the rise and the development of mathematical notions and ideas. Each letter is accompanied by a detailed commentary supplied by the editors. For the convenience of the reader, numerous cross-references, extended indexes, and short biographies of all persons mentioned in the correspondence have been added.

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In this thesis we experimentally developed high-resolution groundbreaking imaging techniques and novel methods suitable for nanophotonics materials. The experimental results are carefully supported by theory and numerical calculations. We engineered the propagation of light by exploiting devices that strongly localize electromagnetic fields at the nanoscale. The proposed techniques have a large field of application. We deeply investigated ordered and disordered based single and coupled nano-resonators, called photonicmolecules, and develop a laser-assisted local oxidation of the dielectric environment. These results put the basis for an unprecedented investigation of light behaviour in optical nano-resonators. Therefore, they would pave the way for novel devices that exploit the strong coupling regime between single light emitters and localized optical modes.

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Inverse Problems - Recent Advances and Applications examines some recent advances in inverse problems, new aspects of their mathematical modeling of inverse problems regarding in relation to their applications in physical systems and used the computational methods used. It consists of five chapters divided into two sections. Section 1, “Modeling and Formulations of Inverse Problems”, discusses new approaches to modeling and formulations of inverse problems in some physical systems. Section 2, “Some Computational Aspects”, contains research related to mathematical methods of solving some inverse problems.

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Algebra

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Investigating Human Interaction through Mathematical Analysis offers a new and unique approach to social intragroup interaction by using mathematics and psychophysics to create a mathematical model based on social psychological theories. It draws on the work of Dr. Stanley Milgram, Dr. Bibb Latane, and Dr. Bernd Schmitt to develop an algebraic expression and applies it to quantitatively model and explain various independent social psychology experiments taken from refereed journals involving basic social systems with underlying queue-like structures. It is then argued that the social queue as a resource system, containing common-pool resources, meets the eight design principles necessary to support stability within the queue. Making this link provides a means to advance to more complex social systems. It is envisioned that if basic social systems as presented can be modeled, then, with further development, more complex social systems may eventually be modeled for the purpose of identifying and validating social structures that might eventually support stable governments in our common environment called Earth. This is a fascinating reading for academics and advanced students interested in political theory, detection theory, social psychology, organizational behavior, psychophysics, and applied mathematics in the social and information sciences. The Open Access version of this book, available at www.taylorfrancis.com, has been made available under a Creative Commons Attribution-Non Commercial-No Derivatives 4.0 license.

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Investigating Human Interaction through Mathematical Analysis offers a new and unique approach to social intragroup interaction by using mathematics and psychophysics to create a mathematical model based on social psychological theories. It draws on the work of Dr. Stanley Milgram, Dr. Bibb Latane, and Dr. Bernd Schmitt to develop an algebraic expression and applies it to quantitatively model and explain various independent social psychology experiments taken from refereed journals involving basic social systems with underlying queue-like structures. It is then argued that the social queue as a resource system, containing common-pool resources, meets the eight design principles necessary to support stability within the queue. Making this link provides a means to advance to more complex social systems. It is envisioned that if basic social systems as presented can be modeled, then, with further development, more complex social systems may eventually be modeled for the purpose of identifying and validating social structures that might eventually support stable governments in our common environment called Earth. This is a fascinating reading for academics and advanced students interested in political theory, detection theory, social psychology, organizational behavior, psychophysics, and applied mathematics in the social and information sciences.

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This work is based on the “Notes for the course of dynamic systems” written by Professor Riccardo Ricci in 2005 and subsequently reworked until the year of his death, which occurred in 2013. Starting from the latest version, the scholars have undertaken a review and expansion that led to this manual. The text is aimed at bachelor students of the courses in Mathematics, Physics and Engineering. Moreover, it presents the fundamental topics of Lagrangian mechanics and the dynamics of rigid bodies and variational principles, with a hint at Hamiltonian mechanics.

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Linear algebra occupies a central place in modern mathematics. Also, it is a beautiful and mature field of mathematics, and mathematicians have developed highly effective methods for solving its problems. It is a subject well worth studying for its own sake. This book contains selected topics in linear algebra, which represent the recent contributions in the most famous and widely problems. It includes a wide range of theorems and applications in different branches of linear algebra, such as linear systems, matrices, operators, inequalities, etc. It continues to be a definitive resource for researchers, scientists and graduate students.

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In order not to intimidate students by a too abstract approach, this textbook on linear algebra is written to be easy to digest by non-mathematicians. It is also designed such that no other material is required for an understanding of the topics covered. As the basis for courses on space and atmospheric science, remote sensing, and satellite communications, applications of the formal theory in physics and engineering are discussed as well.

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This book is open access, meaning free unlimited access to the eBook Maintains the classic approach of previous editions Fourth edition gives an expanded treatment of the singular value decomposition and its consequences Includes 250 additional exercises and several new topics

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This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book.

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Matrix Theory - Classics and Advances examines matrix theory and its application in solving a series of problems related to natural phenomena and applied science. It consists of eleven chapters divided into two sections. Section 1, “Theory and Progress”, discusses the classical problems of matrix theory and its contribution to different fields of pure mathematics. Section 2, “Applications”, contains the research related to the application of matrix theory in applied science.

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Leonhard Euler (1707–1783) was born in Basel, Switzerland. Euler's formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. When its variable is the number pi, Euler's formula evaluates to Euler's identity. On the other hand, the Yang–Baxter equation is considered the most beautiful equation by many scholars. In this book, we study connections between Euler’s formulas and the Yang–Baxter equation. Other interesting sections include: non-associative algebras with metagroup relations; branching functions for admissible representations of affine Lie Algebras; super-Virasoro algebras; dual numbers; UJLA structures; etc.

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Operator theory is a fascinating model that includes essential theorems and diverse applications in all science. This book discusses some fundamental aspects of pure and applied operator theory and the use of the theory to solve a large array of problems. As such, it will be of interest to scholars, researchers, and undergraduate and graduate students in pure and applied mathematics, classical and modern physics, engineering, and science in general.

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Norway is a leading country in the world when it comes to not prioritizing algebra in school mathematics, the mathematical language many students will need for further education and professions. Analysis of data from TIMSS, TIMSS Advanced and other international comparative studies have concluded that this result is consistent across 20 years of research. Norwegian students perform well in domains such as statistics/data, and alarmingly low in algebra. This conclusion is based on data from all levels in school: Primary school, lower secondary school, upper secondary school, and teacher education. The international studies indicate that East Asian, Eastern European and Latin European countries prioritize algebra more strongly than Norway or other Nordic and English-speaking countries. Several of the countries performing well in algebra, also seem to have some sort of balance between teaching algebra and teaching statistics, while there is a huge difference between the student scores on these domains in Norway. It almost seems as though “students learn statistics instead of learning algebra”. The importance of taking into account the type of mathematical content which is tested in different studies before drawing conclusions, is also addressed in the book. For instance, PISA mathematics is compared to TIMSS grade 8 mathematics when it comes to mathematical theory involvement. On the other hand, it is emphasized that it is necessary to have information from different studies to make valid conclusions concerning mathematics in schools. The importance of cooperation between countries all over the world in developing school mathematics, with countries learning from each other, is also discussed. The book is a result of cooperation between researchers at the Department of teacher education and school research (ILS) and the Department of mathematics, both at the University of Oslo, and school teachers in an upper secondary school. The results are discussed from a teacher education perspective, a school teacher perspective and from the perspective of the national curriculum for schools in Norway. Some of the chapters in the book are devoted to going through all the test items in TIMSS Advanced 2015 which can be published, and it is described how these items can be used both by teachers in school and in teacher education at universities.

"Denne boka presenterer forskningsresultater om matematikk på alle nivåer i skolen (barneskole, ungdomsskole, videregående skole) over de siste 20 årene. Norge har deltatt i internasjonale komparative studier fra 1995 til 2015 i regi av IEA (TIMSS Advanced, TIMSS, TEDS-M) og OECD (PISA). De fleste av disse studiene gjennomføres nå med jevne mellomrom. Sentrale forskningsspørsmål for boka er:-Hvilke konsistente forskningsresultater for skolematematikk finner vi, basert på data over 20 år fra alle disse studiene?-Hvilke utfordringer har norsk skole hvis elevene skal få den matematiske basiskunnskapen de trenger for videre utdanninger og yrker?Hvordan resultatene fra disse studiene kan bidra til bedre undervisning i skolen, og til å lette overgangen fra videregående skole til høyere utdanning for elevene, står sentralt i boka. Disse problemstillingene er noe av bakgrunnen for samarbeidet om boka mellom Institutt for lærerutdanning og skoleforskning (ILS), Universitetet i Oslo; Lillestrøm videregående skole i Akershus; Matematisk institutt, Universitetet i Oslo; og Matematikksenteret, NTNU i Trondheim."

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The main purpose of the conference is to stimulate the exchange of new ideas in algebra, topology and geometry, and to celebrate 50 years of the Jan Kochanowski University.

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This is the first of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains combinatorial and homological surveys. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics.

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Intensive studies on light–matter interactions and technological breakthroughs, especially conducted in the field of dressed photon research, have led to a growing concern regarding unsettled off-shell quantum field interactions. The Special Issue, entitled “Quantum Fields and Off-Shell Sciences”, was organized to promote the progress of such research activities from a wider perspective, not limited to dressed photon studies. This book contains excellent papers that were published in this Special Issue. It will provide scientific and technical information on the quantum fields and off-shell sciences to scientists, engineers, and students who are and will be engaged in this field.

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The last few years have been characterized by a tremendous development of quantum information and probability and their applications, including quantum computing, quantum cryptography, and quantum random generators. In spite of the successful development of quantum technology, its foundational basis is still not concrete and contains a few sandy and shaky slices. Quantum random generators are one of the most promising outputs of the recent quantum information revolution. Therefore, it is very important to reconsider the foundational basis of this project, starting with the notion of irreducible quantum randomness. Quantum probabilities present a powerful tool to model uncertainty. Interpretations of quantum probability and foundational meaning of its basic tools, starting with the Born rule, are among the topics which will be covered by this issue. Recently, quantum probability has started to play an important role in a few areas of research outside quantum physics—in particular, quantum probabilistic treatment of problems of theory of decision making under uncertainty. Such studies are also among the topics of this issue.

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This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.

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This book provides a broad overview of recent developments in polynomials and their applications. It includes eight chapters that address such topics as characteristic functions of polynomials, permutations, Gon?arov polynomials, irreducible factors, polynomial regression algorithms, and the use of polynomials in fractional calculus, and much more.

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An approachable introduction to elementary sheaf theory and its applications beyond pure math. Sheaves are mathematical constructions concerned with passages from local properties to global ones. They have played a fundamental role in the development of many areas of modern mathematics, yet the broad conceptual power of sheaf theory and its wide applicability to areas beyond pure math have only recently begun to be appreciated. Taking an applied category theory perspective, Sheaf Theory through Examples provides an approachable introduction to elementary sheaf theory and examines applications including n-colorings of graphs, satellite data, chess problems, Bayesian networks, self-similar groups, musical performance, complexes, and much more. With an emphasis on developing the theory via a wealth of well-motivated and vividly illustrated examples, Sheaf Theory through Examples supplements the formal development of concepts with philosophical reflections on topology, category theory, and sheaf theory, alongside a selection of advanced topics and examples that illustrate ideas like cellular sheaf cohomology, toposes, and geometric morphisms. Sheaf Theory through Examples seeks to bridge the powerful results of sheaf theory as used by mathematicians and real-world applications, while also supplementing the technical matters with a unique philosophical perspective attuned to the broader development of ideas.

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The Stacks project is an open source textbook and reference work on algebraic stacks and the algebraic geometry needed to define them. For more general information and the latest revision, see the extensive about page.

Archived Version is # bd7e5af, compiled on Sep 07, 2015.

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"It is increasingly clear that the shapes of reality – whether of the natural world, or of the built environment – are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself ‘tick’, and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside. The Essence of Mathematics consists of a sequence of 270 problems – with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline. The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works."

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"It is increasingly clear that the shapes of reality – whether of the natural world, or of the built environment – are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself ‘tick’, and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside.The Essence of Mathematics consists of a sequence of 270 problems – with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline.The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works."

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