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Cohomology operations are at the center of a major area of activity in algebraic topology. This treatment explores the single most important variety of operations, the Steenrod squares. It constructs these operations, proves their major properties, and provides numerous applications, including several different techniques of homotopy theory useful for computation. 1968 edition.
The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems. To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated. Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book. The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic $K$-theory and the s-cobordism theorem. A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the “big picture”, teaches them how to give mathematical lectures, and prepares them for participating in research seminars. The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.
Every single person in Scotland has some kind of stake in the effectiveness of the nation's schools, so in writing this book my goal was to explain the intricacies and inconsistencies of the system, and to explore its strengths and weaknesses, in a way that would make sense to as many people as possible. How much do we really know about the state of Scottish education? Why do inequalities continue to dictate the school experiences of children across the country? What can be done to address the problems in the school system? James McEnaney does what he claims pundits and politicians cannot or will not do... tell the truth about Scottish schools. Class Rules makes the key issues and information surrounding Scotland's education system accessible to all. McEnaney delves into the successes and failures of the Curriculum for Excellence, interrogates the rhetoric around closing the 'attainment gap' between the richest and poorest pupils, and considers the impact of the global Covid-19 pandemic. Most importantly, this book also looks to the future to ask what changes can be made to improve the system for young people across the country, and is a must-read for anyone interested in the future of schools in Scotland.
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Intermediate 1 Maths offers an attractive and motivating textbook designed to enthuse students. Attractively illustrated with colour coded explanations and worked examples, the book brings Maths into real-life contexts with activities that make the subject relevant to Intermediate 1 students, whilst offering comprehensive coverage of the syllabus in a focused and concise manner. There is an extensive range of relevant exam-style questions and assessment items to prepare for examination. A CD is included with the book, containing answers to all of the student exercises. The disc can be retained for use solely by the teacher, or released for student self-access according to preferred teaching and assessment strategies. The book covers all the outcomes in the four units of the Intermediate 1 Mathematics course, i.e. units 1, 2, 3 and the Applications of Mathematics unit. Each chapter concentrates on the required learning outcomes with questions, exercises, and end of topic summaries. Questions are designed to help students prepare and practise for unit tests, and the final exam features prominently throughout the book, whilst students are directed to the relevant chapters of the course depending on which units they are studying. Attention is given to both calculator and non-calculator examples in line with the two papers which make up the final exam, and each chapter contains student exercises (with answers supplied). A summary appears at the end of each chapter telling students what they should be able to do.
This is a collection of the 2002-2005 official SQA past papers for Advanced Higher mathematics. A comprehensive answer section shows exactly what examiners are looking for and how to aim for the best grade.