Understanding Analysis

Book Description :

This elementary presentation exposes readers to both the process of rigor and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim is to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples and concludes with a series of questions.

Book Description :

This elementary presentation exposes readers to both the process of rigor and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim is to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples and concludes with a series of questions.

Book Description :

This elementary presentation exposes readers to both the process of rigor and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim is to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples and concludes with a series of questions.

Book Description :

Understanding Real Analysis, Second Edition offers substantial coverage of foundational material and expands on the ideas of elementary calculus to develop a better understanding of crucial mathematical ideas. The text meets students at their current level and helps them develop a foundation in real analysis. The author brings definitions, proofs, examples and other mathematical tools together to show how they work to create unified theory. These helps students grasp the linguistic conventions of mathematics early in the text. The text allows the instructor to pace the course for students of different mathematical backgrounds. Key Features: Meets and aligns with various student backgrounds Pays explicit attention to basic formalities and technical language Contains varied problems and exercises Drives the narrative through questions

Book Description :

Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition.

Book Description :

This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and includes interesting historical notes, periodic attempts at humor, and occasional diversions into other interesting areas of mathematics. The text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis. The author believes most textbooks are extremely overpriced and endeavors to help change this.Hints and solutions to select exercises can be found at LongFormMath.com.

Book Description :

Written for students as well as professionals who work with and support geophysicists, this book presents a simple and informal discussion of fundamental concepts which underlie the quantitative part of geophysical analysis and interpretation. These general concepts are applicable for an analytical approach to any phenomena that can be measured and recorded. With examples and figures created using Microsoft Excel®, this book is accessible and insightful. Topics covered include: the concept of signals based on the sine function; the summation of sine waves as a more complicated signal; the notion of Fourier series and the spectral representation of signals; digital sampling and discrete representation of signals; the discrete Fourier transform and inverse transform; the concept of filtering in the spectral domain; and the idea of filtering outside of the spectral domain, by convolution, and the relationship between the measurement and spectral domains. This book will be valuable for geologists, junior seismic interpreters, software developers, high school and university students, and geophysical professionals seeking a refresher of the basic concepts.

Book Description :

Getting certified to teach high school mathematics typically requires completing a course in real analysis. Yet most teachers point out real analysis content bears little resemblance to secondary mathematics and report it does not influence their teaching in any significant way. This textbook is our attempt to change the narrative. It is our belief that analysis can be a meaningful part of a teacher's mathematical education and preparation for teaching. This book is a companion text. It is intended to be a supplemental resource, used in conjunction with a more traditional real analysis book. The textbook is based on our efforts to identify ways that studying real analysis can provide future teachers with genuine opportunities to think about teaching secondary mathematics. It focuses on how mathematical ideas are connected to the practice of teaching secondary mathematics–and not just the content of secondary mathematics itself. Discussions around pedagogy are premised on the belief that the way mathematicians do mathematics can be useful for how we think about teaching mathematics. The book uses particular situations in teaching to make explicit ways that the content of real analysis might be important for teaching secondary mathematics, and how mathematical practices prevalent in the study of real analysis can be incorporated as practices for teaching. This textbook will be of particular interest to mathematics instructors–and mathematics teacher educators–thinking about how the mathematics of real analysis might be applicable to secondary teaching, as well as to any prospective (or current) teacher who has wondered about what the purpose of taking such courses could be.

Book Description :

In the second edition of this MAA classic, exploration continues to be an essential component. More than 60 new exercises have been added, and the chapters on infinite summations, differentiability and continuity, and convergence of infinite series have been reorganized to make it easier to identify the key ideas. A Radical Approach to Real Analysis is an introduction to real analysis, rooted in and informed by the historical issues that shaped its development. It can be used as a textbook, or as a resource for the instructor who prefers to teach a traditional course, or as a resource for the student who has been through a traditional course yet still does not understand what real analysis is about and why it was created.

Book Description :

Getting certified to teach high school mathematics typically requires completing a course in real analysis. Yet most teachers point out real analysis content bears little resemblance to secondary mathematics and report it does not influence their teaching in any significant way. This textbook is our attempt to change the narrative. It is our belief that analysis can be a meaningful part of a teacher's mathematical education and preparation for teaching. This book is a companion text. It is intended to be a supplemental resource, used in conjunction with a more traditional real analysis book. The textbook is based on our efforts to identify ways that studying real analysis can provide future teachers with genuine opportunities to think about teaching secondary mathematics. It focuses on how mathematical ideas are connected to the practice of teaching secondary mathematics–and not just the content of secondary mathematics itself. Discussions around pedagogy are premised on the belief that the way mathematicians do mathematics can be useful for how we think about teaching mathematics. The book uses particular situations in teaching to make explicit ways that the content of real analysis might be important for teaching secondary mathematics, and how mathematical practices prevalent in the study of real analysis can be incorporated as practices for teaching. This textbook will be of particular interest to mathematics instructors–and mathematics teacher educators–thinking about how the mathematics of real analysis might be applicable to secondary teaching, as well as to any prospective (or current) teacher who has wondered about what the purpose of taking such courses could be.

Book Description :

Exam Board: SQA Level: National 5 Subject: English First teaching: September 2017 First exam: Summer 2018 Offer the best possible preparation for National 5 English RUAE. Building students' confidence in reading unseen texts, this book focuses on the skills required for the exam before progressing to worked examples and full-length practice papers. - Teaches students how to answer every question type effectively and demonstrate their ability to understand ideas, analyse language and structure, and evaluate writers' techniques - Enables students to put their RUAE skills into practice as they tackle a range of short extracts and questions accompanied by active learning approaches, group work and individual tasks - Tests the skills that students have developed through six full-length passages and practice assessments, with answers and marking guidelines available online at hoddergibson.co.uk/answers-N5-English-RUAE - Provides stretch and challenge opportunities, including extension activities and further reading that will broaden students' Reading for Understanding, Analysis and Evaluation skills

Book Description :

Exam Board: SQA Level: Higher Subject: English First Teaching: September 2014 First Exam: June 2015 A brand new edition of the former Higher English Close Reading, completely rewritten for the new Higher element - worth 30% of marks in the final exam. Written by two highly experienced authors, this book shows you how to practise for the Reading for Understanding, Analysis and Evaluation section of the CfE Higher English exam. It introduces the terms and concepts that lie behind success and offers guidance on the interpretation of questions and targeting answers. - Acquire and improve the skills needed for success in this area of CfE Higher - Includes questions and passages to reflect the style of assessments in this section of the paper - Revise for the exam using the practice papers and accompanying book of suggested answers

Book Description :

Syllabus: CfE (Curriculum for Excellence, from Education Scotland) and SQA Level: BGE S1-3: Third & Fourth Levels Subject: English Capture pupils' imagination and attention with a rich collection of quality texts, handpicked by Jane Cooper to ensure that pupils want to start reading - and keep reading. Tailored to the interests, motivations and abilities of pupils working towards Second and Third Levels in S1 to S3, this book will help you to close the attainment gap. b” Unlock a world of meaning, connection and questioning. b” Apply and develop RUAE skills and monitor progress. b” Save time sourcing non-fiction text extracts. /bDrawing on her long career as an English teacher and author, Jane Cooper has a talent for knowing which topics pupils will find curious or fun - from the world's cheesiest pizza to a mysterious metal column that appeared in a desert.brbrb” Build the foundations for success in National qualifications. /bWith Course Reports showing that N5 and Higher students most commonly lose marks in the RUAE papers, this book offers a proactive solution to building confidence and skills during BGE.brbrb” Support the individual needs of every pupil. /bProviding differentiated, age-appropriate content and tasks, this is the perfect resource for mixed-ability classrooms - especially when used alongside its companion volume,

Book Description :

Changing the temperature of a substance can stimulate dramatic changes of its state. These changes can be intermolecular (physical) and intramolecular (chemical) in nature. Physical changes occur without breaking intramolecular bonds, and lead to transitions between the four major phases: gas, liquid, crystal, and glass. Chemical changes are associated with chemical reactions that originate from breaking intramolecular bonds. Phase transitions as well as chemical reactions occur at finite rates. Measuring the rates of processes is the realm of kinetics. The kinetics of thermally stimulated processes is routinely measured using thermal analysis techniques such as differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA). Knowing the process rates and their dependence on temperature is of vital importance for understanding the behavior of materials exposed to variations in temperature. In recent years, thermal analysis kinetics has made significant progress by developing computational tools for reliable kinetic analysis. It has also expanded its traditional application area to newly developed nano- and biomaterials. This Special Issue is a series of papers that reflect recent developments in the field and highlight the essential role of thermal analysis kinetics in understanding the processes responsible for the thermal behavior of various materials.

Book Description :

Exam Board: SQA Level: National 5 Subject: English First Teaching: September 2013 First Exam: June 2014 This book is ideally suited for use by students working towards National 5 English. Reading for Understanding, Analysis and Evaluation is a key part of this course, and the book provides a selection of practice exercises with questions modelled on those used in the examination, as well as many extracts from various genres, which allow the development of necessary language skills. Some of the material is based on spoken as well as written sources, with follow-up activities suitable for group discussion, and all activities closely related to the principles of Curriculum for Excellence. - Develops skills quickly and effectively using tried and trusted methods from an experienced author team. - Prepares students for assessment with specific advice and plenty of opportunities to test the theory in practice papers. - Teaches - and tests - other language skills essential to success in National 5 English. Answers to and mark schemes for the practice paper questions in this book are available in a separate title: National 5 English: Reading for Understanding, Analysis and Evaluation Answer Book and Marking Schemes (978 1 4441 9211 7). NB: the Answer Book for this title is intended mainly as a Teacher Resource and grants permission to the purchasing institution to make photocopies for the use of staff and students, hence its higher price. Students or parents wishing to view these materials are encouraged to contact their school/college/learning institution to access them free of charge.

Book Description :

This book brings together the essential close reading skills needed by students taking part in the Broad General Education, Levels 3-4 (in S1 to S3). Split into two parts, the first section uses examples, models and active-learning tasks to teach key concepts of reading for understanding, analysis and evaluation. The second section provides 15 practice assessments, based on a variety of fiction and non-fiction texts, which become progressively more challenging. As well as allowing learners to demonstrate BGE reading skills, this section serves as a useful precursor to the style of assessment encountered later on at National levels. This book will help students to: - develop their close reading abilities - understand the distinction between key ideas and supporting details - analyse writers' language and style via a broad range of sample texts.

Book Description :

This book shows that it is possible to provide a fully rigorous treatment of calculus for those planning a career in an area that uses mathematics regularly (e.g., statistics, mathematics, economics, finance, engineering, etc.). It reveals to students on the ways to approach and understand mathematics. It covers efficiently and rigorously the differential and integral calculus, and its foundations in mathematical analysis. It also aims at a comprehensive, efficient, and rigorous treatment by introducing all the concepts succinctly. Experience has shown that this approach, which treats understanding on par with technical ability, has long term benefits for students.

Book Description :

This book (which is photocopiable within the purchasing institution) provides detailed suggested marking instructions to use alongside the student book of the same name, which offers practice in the Higher element of Reading for Understanding, Analysis and Evaluation - worth 30% of marks in the final exam. Written by two highly experienced authors, the student book introduces the terms and concepts that lie behind success and offers guidance on the interpretation of questions and targeting answers. - Acquire and improve the skills needed for success in this area of CfE Higher - Detailed marking instructions to reflect the style of assessments in this section of the paper - Revise for the exam using the practice papers in the accompanying student book

Book Description :

By assuming it is possible to understand regression analysis without fully comprehending all its underlying proofs and theories, this introduction to the widely used statistical technique is accessible to readers who may have only a rudimentary knowledge of mathematics. Chapters discuss: -descriptive statistics using vector notation and the components of a simple regression model; -the logic of sampling distributions and simple hypothesis testing; -the basic operations of matrix algebra and the properties of the multiple regression model; -testing compound hypotheses and the application of the regression model to the analyses of variance and covariance, and -structural equation models and influence statistics.

Book Description :

Understanding Regression Analysis unifies diverse regression applications including the classical model, ANOVA models, generalized models including Poisson, Negative binomial, logistic, and survival, neural networks, and decision trees under a common umbrella -- namely, the conditional distribution model. It explains why the conditional distribution model is the correct model, and it also explains (proves) why the assumptions of the classical regression model are wrong. Unlike other regression books, this one from the outset takes a realistic approach that all models are just approximations. Hence, the emphasis is to model Nature’s processes realistically, rather than to assume (incorrectly) that Nature works in particular, constrained ways. Key features of the book include: Numerous worked examples using the R software Key points and self-study questions displayed "just-in-time" within chapters Simple mathematical explanations ("baby proofs") of key concepts Clear explanations and applications of statistical significance (p-values), incorporating the American Statistical Association guidelines Use of "data-generating process" terminology rather than "population" Random-X framework is assumed throughout (the fixed-X case is presented as a special case of the random-X case) Clear explanations of probabilistic modelling, including likelihood-based methods Use of simulations throughout to explain concepts and to perform data analyses This book has a strong orientation towards science in general, as well as chapter-review and self-study questions, so it can be used as a textbook for research-oriented students in the social, biological and medical, and physical and engineering sciences. As well, its mathematical emphasis makes it ideal for a text in mathematics and statistics courses. With its numerous worked examples, it is also ideally suited to be a reference book for all scientists.