Let G be a reductive group over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne–Lusztig varieties Xx(b), which are indexed by elements b∈G(F) and x∈W, were introduced by Rapoport. Basic questions about the varieties Xx(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. The authors use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of a conjecture of Görtz, Haines, Kottwitz, and Reuman. The authors' approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since they work only in the standard apartment of the building for G(F), their results also hold in the p-adic context, where they formulate a definition of the dimension of a p-adic Deligne–Lusztig set. The authors present two immediate applications of their main results, to class polynomials of affine Hecke algebras and to affine reflection length.
Let G be a reductive group over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne-Lusztig varieties X_x(b), which are indexed by elements b \in G(F) and x \in W, were introduced by Rapoport. Basic questions about the varieties X_x(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. The authors use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of a conjecture of Görtz, Haines, Kottwitz, and Reuman. The authors' approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since they work only in the standard apartment of the building for G(F), their results also hold in the p-adic context, where they formulate a definition of the dimension of a p-adic Deligne-Lusztig set. The authors present two immediate applications of their main results, to class polynomials of affine Hecke algebras and to affine reflection length.
This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry. This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with this fascinating new field.
Winner of the 1983 National Book Award! "...a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." - The New Yorker (1983 National Book Award edition) Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications. The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto (elena.marchisotto@csun.edu) upon request.
Dyslexia, Dyscalculia and Mathematics will be an essential resource for teachers, classroom assistants, and SENCOs who help dyslexic and dyscalculic children with their understanding of mathematics. Written in an accessible style with helpful illustrations, this practical book reveals helpful ways in which to tackle both simple and complex concepts with students of all ages. This second edition has been updated to include references to using technology that will help children with dyslexia and dyscalculia reinforce their mathematical skills and also contains a number of photocopiable resources that can be used in the classroom. Written by Anne Henderson, who is experienced in teaching language and mathematics to pupils with dyslexia and dyscalculia, this book outlines current thinking in the field and shows how the research methods that have been proven as successful can be used with whole classes of children. This book encourages flexible methods and gives teachers the confidence to discuss alternative solutions with their pupils and help them achieve success. It is an ideal handbook for parent-teacher programmes and is also suitable for in-service training.
International research is used to inform teachers and others about how students learn key ideas in higher school mathematics, what the common problems are, and the strengths and pitfalls of different teaching approaches. An associated website, hosted by the Nuffield Foundation, gives summaries of main ideas and access to sample classroom tasks.
You are invited to join a fascinating journey of discovery, as Marcia Birken and Anne C. Coon explore the intersecting patterns of mathematics and poetry — bringing the two fields together in a new way. Setting the tone with humor and illustrating each chapter with countless examples, Birken and Coon begin with patterns we can see, hear, and feel and then move to more complex patterns. Number systems and nursery rhymes lead to the Golden Mean and sestinas. Simple patterns of shape introduce tessellations and concrete poetry. Fractal geometry makes fractal poetry possible. Ultimately, patterns for the mind lead to questions: How do mathematicians and poets conceive of proof, paradox, and infinity? What role does analogy play in mathematical discovery and poetic expression? The book will be of special interest to readers who enjoy looking for connections across traditional disciplinary boundaries. Discovering Patterns in Mathematics and Poetry features centuries of creative work by mathematicians, poets, and artists, including Fibonacci, Albrecht Dürer, M. C. Escher, David Hilbert, Benoit Mandelbrot, William Shakespeare, Edna St. Vincent Millay, Langston Hughes, E.E. Cummings, and many contemporary experimental poets. Original illustrations include digital photographs, mathematical and poetic models, and fractal imagery.
A guide to teaching lower attaining students in secondary mathematics offers an alternative view of attainment and capability, arguing that teaching should be based on a learner's proficiency, rather than on knowledge deficits.
In order to understand the universe you must know the language in which it is written. And that language is mathematics. - Galileo (1564-1642) People have always sought order in the apparent chaos of the universe. Mathematics has been our most valuable tool in that search, uncovering the patterns and rules that govern our world and beyond. This book traces humankind's greatest achievements, plotting a journey through the mathematical intellects of the last 4,000 years to where we stand today. It features the giants of mathematics, from Euclid and Pythagoras, through Napier and Newton, to Leibniz, Riemann, Russell, and many more. Topics include: • Counting and measuring from the earliest times • The Ancient Egyptians and geometry • The movements of planets • Measuring and mapping the world • Fuzzy logic and set theory • The death of numbers ABOUT THE SERIES: Arcturus Fundamentals Series explains fascinating and far-reaching topics in simple terms. Designed with rustic, tactile covers and filled with dynamic illustrations and fact boxes, these books will help you quickly get to grips with complex topics that affect our day-to-day living.
Explores ways to teach math principles using children`s books, shows how to connect children with real-world math, and encourages linking text with relevant manipulatives in a hands-on, minds-on, problem-solving environment. Book lists, suggested activities, assessment strategies. and reproducible graphic organizers are included. Primary level.
This book explains and demonstrates the teaching strategy of asking learners to construct their own examples of mathematical objects. The authors show that the creation of examples can involve transforming and reorganizing knowledge and that, although this is usually done by authors and teachers, if the responsibility for making examples is transferred to learners, their knowledge structures can be developed and extended. A multitude of examples to illustrate this is provided, spanning primary, secondary, and college levels. Readers are invited to learn from their own past experience augmented by tasks provided in the book, and are given direct experience of constructing examples through a collection of many tasks at many levels. Classroom stories show the practicalities of introducing such shifts in mathematics education. The authors examine how their approach relates to improving the learning of mathematics and raise future research questions. *Based on the authors' and others' theoretical and practical experience, the book includes a combination of exercises for the reader, practical applications for teaching, and solid scholarly grounding. *The ideas presented are generic in nature and thus applicable across every phase of mathematics teaching and learning. *Although the teaching methods offered are ones that engage learners imaginatively, these are also applied to traditional approaches to mathematics education; all tasks offered in the book are within conventional mathematics curriculum content. Mathematics as a Constructive Activity: Learners Generating Examples is intended for mathematics teacher educators, mathematics teachers, curriculum developers, task and test designers, and classroom researchers, and for use as a text in graduate-level mathematics education courses.
This is an introductory textbook designed for undergraduate mathematics majors with an emphasis on abstraction and in particular, the concept of proofs in the setting of linear algebra. Typically such a student would have taken calculus, though the only prerequisite is suitable mathematical grounding. The purpose of this book is to bridge the gap between the more conceptual and computational oriented undergraduate classes to the more abstract oriented classes. The book begins with systems of linear equations and complex numbers, then relates these to the abstract notion of linear maps on finite-dimensional vector spaces, and covers diagonalization, eigenspaces, determinants, and the Spectral Theorem. Each chapter concludes with both proof-writing and computational exercises.
Exploring Mathematical Modeling in Biology through Case Studies and Experimental Activities provides supporting materials for courses taken by students majoring in mathematics, computer science or in the life sciences. The book's cases and lab exercises focus on hypothesis testing and model development in the context of real data. The supporting mathematical, coding and biological background permit readers to explore a problem, understand assumptions, and the meaning of their results. The experiential components provide hands-on learning both in the lab and on the computer. As a beginning text in modeling, readers will learn to value the approach and apply competencies in other settings. Included case studies focus on building a model to solve a particular biological problem from concept and translation into a mathematical form, to validating the parameters, testing the quality of the model and finally interpreting the outcome in biological terms. The book also shows how particular mathematical approaches are adapted to a variety of problems at multiple biological scales. Finally, the labs bring the biological problems and the practical issues of collecting data to actually test the model and/or adapting the mathematics to the data that can be collected. Presents a single volume on mathematics and biological examples, with data and wet lab experiences suitable for non-experts Contains three real-world biological case studies and one wet lab for application of the mathematical models Includes R code templates throughout the text, which are also available through an online repository, along with the necessary data files to complete all projects and labs
Focusing on Thomas Burke's bestselling collection of short stories, Limehouse Nights (1916), this book contextualises the burgeoning cult of Chinatown in turn-of-the-century London. London's 'Chinese Quarter' owed its notoriety to the Yellow Perilism that circulated in Britain at the fin-de-siècle, a demonology of race and vice masked by outward concerns about degenerative metropolitan blight and imperial decline. Anne Witchard's interdisciplinary approach enables her to displace the boundaries that have marked Chinese studies, literary studies, critiques of Orientalism and empire, gender studies, and diasporic research, as she reassesses this critical moment in London's history. In doing so, she brings attention to Burke's hold on popular and critical audiences on both sides of the Atlantic. A much-admired and successful author in his time, Burke in his Chinatown stories destabilizes social orthodoxies in highly complex and contradictory ways. For example, his writing was formative in establishing the 'queer spell' that the very mention of Limehouse would exert on the public imagination, and circulating libraries responded to Burke's portrayal of a hybrid East End where young Cockney girls eat Chow Mein with chopsticks in the local cafés and blithely gamble their housekeeping money at Fan Tan by banning Limehouse Nights. Witchard's book forces us to rethink Burke's influence and shows that China and chinoiserie served as mirrors that reveal the cultural disquietudes of western art and culture.
The Initial Teacher Training National Curriculum says that student teachers should be trained to analyse pupil's errors in maths and act accordingly. This is the only book that supports teachers' analysis of mathematical errors and helps them predict potential problems and propose solutions for themselves. Written in an accessible style, Teaching Mathematics with Insight guides the primary and early years teacher, and the student teacher through a series of processes that will enable them to become more effective and enlightened teachers of early mathematics. The processes include: unravelling the complexities of a concept, for example subtraction, by considering its component parts and the knowledge required to acquire understanding; watching children work to observe common mistakes and analyse the underlying misconceptions; discussing the concepts with other adults.
Let G be a reductive group over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne–Lusztig varieties Xx(b), which are indexed by elements b∈G(F) and x∈W, were introduced by Rapoport. Basic questions about the varieties Xx(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. The authors use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of a conjecture of Görtz, Haines, Kottwitz, and Reuman. The authors' approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since they work only in the standard apartment of the building for G(F), their results also hold in the p-adic context, where they formulate a definition of the dimension of a p-adic Deligne–Lusztig set. The authors present two immediate applications of their main results, to class polynomials of affine Hecke algebras and to affine reflection length.
The historiography of English Catholicism has grown enormously in the last generation, led by scholars such as Peter Lake, Michael Questier, Stefania Tutino, and others. In Suspicious Moderate, Anne Ashley Davenport makes a significant contribution to that literature by presenting a long overdue intellectual biography of the influential English Catholic theologian Francis à Sancta Clara (1598–1680). Born into a Protestant family in Coventry at the end of the sixteenth century, Sancta Clara joined the Franciscan order in 1617. He played key roles in reviving the English Franciscan province and in the efforts that were sponsored by Charles I to reunite the Church of England with Rome. In his voluminous Latin writings, he defended moderate Anglican doctrines, championed the separation of church and state, and called for state protection of freedom of conscience. Suspicious Moderate offers the first detailed analysis of Sancta Clara's works. In addition to his notorious Deus, natura, gratia (1634), Sancta Clara wrote a comprehensive defense of episcopacy (1640), a monumental treatise on ecumenical councils (1649), and a treatise on natural philosophy and miracles (1662). By carefully examining the context of Sancta Clara's ideas, Davenport argues that he aimed at educating English Roman Catholics into a depoliticized and capacious Catholicism suited to personal moral reasoning in a pluralistic world. In the course of her research, Davenport also discovered that "Philip Scot," the author of the earliest English discussions of Hobbes (a treatise published in 1650), was none other than Sancta Clara. Davenport demonstrates how Sancta Clara joined the effort to fight Hobbes's Erastianism by carefully reflecting on Hobbes's pioneering ideas and by attempting to find common ground with him, no matter how slight.
Anyone who regularly tackles challenging crossword puzzles will be familiar with the frustration of unanswered clues blocking the road to completion. Together in one bumper volume, Crossword Lists and Crossword Solver provide the ultimate aid for tracking down those final solutions. The Lists section contains more than 100,000 words and phrases, listed both alphabetically and by number of letters, under category headings such as Volcanoes, Fungi, Gilbert & Sullivan, Clouds, Cheeses, Mottoes, and Archbishops of Canterbury. As intersecting solutions provide letters of the unanswered clue, locating the correct word or phrase becomes quick and easy. The lists are backed up with a comprehensive index, which also guides the puzzler to associated tables - e.g. Film Stars; try Stage and Screen Personalities. The Solver section contains more than 100,000 potential solutions, including plurals, comparative and superlative adjectives, and inflections of verbs. The list extends to first names, place names, technical terms, compound expressions, abbreviations, and euphemisms.Grouped according to number of letters - up to fifteen - this section is easy to use and suitable for all levels of crossword puzzle. At the end a further 3,000 words are listed by category, along with an index of unusual words.
Enjoy the “exemplary Victorian company” of this London sleuthing couple with books seven through ten in the long-running New York Times–bestselling series (The New York Times). “Few mystery writers this side of Arthur Conan Doyle can evoke Victorian London with such relish for detail and mood” (San Francisco Chronicle). Now, in a single volume, readers can enjoy more of Anne Perry’s “unfailingly rewarding” series (The New York Times Book Review). Death in the Devil’s Acre: A vicious and depraved serial killer is loose in the slums of Devil’s Acre. When Pitt recognizes one of the victims as a blackmailing footman from a case on Callander Square, his investigation reveals a shocking connection between the city’s brothels and Victorian high society. Now Charlotte and her sister Emily, Lady Ashworth, must unveil the dirty secrets of the aristocracy. Cardington Crescent: When Thomas Pitt’s womanizing brother-in-law is poisoned by his morning coffee, the inspector must exonerate the prime suspect: Lady Ashworth, Charlotte’s sister Emily. With the help of Great-Aunt Vespasia, the couple chip away at a wall of deceit and silence to find the real killer, even after Lord Ashworth’s suspected paramour is strangled—and found by Emily. Silence in Hanover Close: At the behest of his superior, Pitt reopens a case gone cold. Three years prior, amidst whispered rumors of treason, Robert York, an important member of the British Foreign Office, was murdered in his home in London’s exclusive Hanover Close. When a York family housemaid is found dead shortly after Pitt begins his investigation, he is accused and thrown into prison. Now only Charlotte and her recently widowed sister stand between Thomas Pitt and the gallows. Bethlehem Road: When members of Parliament are murdered one-by-one crossing Westminster Bridge, Thomas and Charlotte must sift through a wide range of suspects, including anarchists and suffragettes. As more seats open up in Parliament and fear grips London, the couple wonders: Are the killings political or somehow personal?
This volume has a single goal: to argue that Descartes’s most fundamental discovery is not the epistemological subject, but rather the underlying free agent without whom no epistemological subject is possible. This fresh interpretation of the Cartesian “cogito” is defended through a close reading of Descartes’s masterpiece, the Meditations. Special attention is paid to the historical roots of Descartes’s interest in free agency, particularly his close ties to the French School of spirituality. Three aspects of Descartes’s personal evolution are considered: his aesthetic evolution from Baroque concealment to Classicism, his political evolution from feudal nostalgia to modern secularism, and his spiritual evolution from Stoic wisdom to active engagement in the world through the scientific project.
For more than a decade, there has been growing interest in the role of emotions in academic settings. Written by leading experts on learning and instruction, Emotions at School focuses on the connections between educational research and emotion science, bringing the subject to a wider audience. With chapters on how emotions develop and work, evidence-based recommendations about how to foster adaptive emotions, and clear explanations of key concepts and ideas, this concise volume is designed for any education course that includes emotions in the curriculum. It will be indispensable for student researchers and both pre- and in-service teachers alike.
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