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A Problem Solving Approach to Mathematics for Elementary School Teachers
by Rick Billstein Shlomo Libeskind Johnny LottMore than 350,000 students have prepared for teaching mathematics with A Problem Solving Approach to Mathematics for Elementary School Teachers since its first edition, and it remains the gold standard today. This text not only helps students learn the material by promoting active learning and developing skills and concepts--it also provides an invaluable reference to future teachers by including professional development features and discussions of today's standards. The Eleventh Edition is streamlined to keep students focused on what is most important. The Common Core State Standards (CCSS) have been integrated into the book to keep current with educational developments.
A Prodigy of Universal Genius: Robert Leslie Ellis, 1817-1859 (Studies in History and Philosophy of Science #55)
by Lukas M. VerburgtThis open access book brings together for the first time all aspects of the tragic life and fascinating work of the polymath Robert Leslie Ellis (1817–1859), placing him at the heart of early-Victorian intellectual culture. Written by a diverse team of experts, the chapters in the book’s first part contain in-depth examinations of, among other things, Ellis’s family, education, Bacon scholarship and mathematical contributions. The second part consists of annotated transcriptions of a selection of Ellis’s diaries and correspondence. Taken together, A Prodigy of Universal Genius: Robert Leslie Ellis, 1817–1859 is a rich resource for historians of science, historians of mathematics and Victorian scholars alike. Robert Leslie Ellis was one of the most intriguing and wide-ranging intellectual figures of early Victorian Britain, his contributions ranging from advanced mathematical analysis to profound commentaries on philosophy and classics and a decisive role in the orientation of mid-nineteenth century scholarship. This very welcome collection offers both new and authoritative commentaries on the work, setting it in the context of the mathematical, philosophical and cultural milieux of the period, together with fascinating passages from the wealth of unpublished papers Ellis composed during his brief and brilliant career.- Simon Schaffer, Department of History and Philosophy of Science, University of Cambridge
A Profile of Mathematical Logic
by Howard DelongAnyone seeking a readable and relatively brief guide to logic can do no better than this classic introduction. A treat for both the intellect and the imagination, it profiles the development of logic from ancient to modern times and compellingly examines the nature of logic and its philosophical implications. No prior knowledge of logic is necessary; readers need only an acquaintance with high school mathematics. The author emphasizes understanding, rather than technique, and focuses on such topics as the historical reasons for the formation of Aristotelian logic, the rise of mathematical logic after more than 2,000 years of traditional logic, the nature of the formal axiomatic method and the reasons for its use, and the main results of metatheory and their philosophic import. The treatment of the Gödel metatheorems is especially detailed and clear, and answers to the problems appear at the end.
A Project-Based Guide to Undergraduate Research in Mathematics: Starting and Sustaining Accessible Undergraduate Research (Foundations for Undergraduate Research in Mathematics)
by Aaron Wootton Pamela E. Harris Erik InskoThis volume provides accessible and self-contained research problems designed for undergraduate student projects, and simultaneously promotes the development of sustainable undergraduate research programs. The chapters in this work span a variety of topical areas of pure and applied mathematics and mathematics education. Each chapter gives a self-contained introduction on a research topic with an emphasis on the specific tools and knowledge needed to create and maintain fruitful research programs for undergraduates. Some of the topics discussed include:• Disease modeling• Tropical curves and surfaces• Numerical semigroups• Mathematics EducationThis volume will primarily appeal to undergraduate students interested in pursuing research projects and faculty members seeking to mentor them. It may also aid students and faculty participating in independent studies and capstone projects.
A Proof Theory for Description Logics (SpringerBriefs in Computer Science)
by Alexandre RademakerDescription Logics (DLs) is a family of formalisms used to represent knowledge of a domain. They are equipped with a formal logic-based semantics. Knowledge representation systems based on description logics provide various inference capabilities that deduce implicit knowledge from the explicitly represented knowledge. A Proof Theory for Description Logics introduces Sequent Calculi and Natural Deduction for some DLs (ALC, ALCQ). Cut-elimination and Normalization are proved for the calculi. The author argues that such systems can improve the extraction of computational content from DLs proofs for explanation purposes.
A Pure Soul: Ennio De Giorgi, A Mathematical Genius
by Andrea ParlangeliThis biography illuminates the life of Ennio De Giorgi, a mathematical genius in parallel with John Nash, the Nobel Prize Winner and protagonist of A Beautiful Mind. Beginning with his childhood and early years of research, into his solution of the 19th problem of Hilbert and his professorship, this book pushes beyond De Giorgi’s rich contributions to the mathematics community, to present his work in human rights, including involvement in the fight for Leonid Plyushch’s freedom and the defense of dissident Uruguayan mathematician José Luis Massera. Considered by many to be the greatest Italian analyst of the twentieth century, De Giorgi is described in this volume in full through documents and direct interviews with friends, family, colleagues, and former students.
A Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic (Undergraduate Texts in Mathematics)
by Ramin Takloo-BighashRight triangles are at the heart of this textbook’s vibrant new approach to elementary number theory. Inspired by the familiar Pythagorean theorem, the author invites the reader to ask natural arithmetic questions about right triangles, then proceeds to develop the theory needed to respond. Throughout, students are encouraged to engage with the material by posing questions, working through exercises, using technology, and learning about the broader context in which ideas developed. Progressing from the fundamentals of number theory through to Gauss sums and quadratic reciprocity, the first part of this text presents an innovative first course in elementary number theory. The advanced topics that follow, such as counting lattice points and the four squares theorem, offer a variety of options for extension, or a higher-level course; the breadth and modularity of the later material is ideal for creating a senior capstone course. Numerous exercises are included throughout, many of which are designed for SageMath. By involving students in the active process of inquiry and investigation, this textbook imbues the foundations of number theory with insights into the lively mathematical process that continues to advance the field today. Experience writing proofs is the only formal prerequisite for the book, while a background in basic real analysis will enrich the reader’s appreciation of the final chapters.
A Python Data Analyst’s Toolkit: Learn Python and Python-based Libraries with Applications in Data Analysis and Statistics
by Gayathri RajagopalanExplore the fundamentals of data analysis, and statistics with case studies using Python. This book will show you how to confidently write code in Python, and use various Python libraries and functions for analyzing any dataset. The code is presented in Jupyter notebooks that can further be adapted and extended.This book is divided into three parts – programming with Python, data analysis and visualization, and statistics. You'll start with an introduction to Python – the syntax, functions, conditional statements, data types, and different types of containers. You'll then review more advanced concepts like regular expressions, handling of files, and solving mathematical problems with Python. The second part of the book, will cover Python libraries used for data analysis. There will be an introductory chapter covering basic concepts and terminology, and one chapter each on NumPy(the scientific computation library), Pandas (the data wrangling library) and visualization libraries like Matplotlib and Seaborn. Case studies will be included as examples to help readers understand some real-world applications of data analysis. The final chapters of book focus on statistics, elucidating important principles in statistics that are relevant to data science. These topics include probability, Bayes theorem, permutations and combinations, and hypothesis testing (ANOVA, Chi-squared test, z-test, and t-test), and how the Scipy library enables simplification of tedious calculations involved in statistics.What You'll LearnFurther your programming and analytical skills with PythonSolve mathematical problems in calculus, and set theory and algebra with PythonWork with various libraries in Python to structure, analyze, and visualize dataTackle real-life case studies using PythonReview essential statistical concepts and use the Scipy library to solve problems in statistics Who This Book Is ForProfessionals working in the field of data science interested in enhancing skills in Python, data analysis and statistics.
A Qualitative Approach to Inverse Scattering Theory (Applied Mathematical Sciences #188)
by David Colton Fioralba CakoniInverse scattering theory is an important area of applied mathematics due to its central role in such areas as medical imaging , nondestructive testing and geophysical exploration. Until recently all existing algorithms for solving inverse scattering problems were based on using either a weak scattering assumption or on the use of nonlinear optimization techniques. The limitations of these methods have led in recent years to an alternative approach to the inverse scattering problem which avoids the incorrect model assumptions inherent in the use of weak scattering approximations as well as the strong a priori information needed in order to implement nonlinear optimization techniques. These new methods come under the general title of qualitative methods in inverse scattering theory and seek to determine an approximation to the shape of the scattering object as well as estimates on its material properties without making any weak scattering assumption and using essentially no a priori information on the nature of the scattering object. This book is designed to be an introduction to this new approach in inverse scattering theory focusing on the use of sampling methods and transmission eigenvalues. In order to aid the reader coming from a discipline outside of mathematics we have included background material on functional analysis, Sobolev spaces, the theory of ill posed problems and certain topics in in the theory of entire functions of a complex variable. This book is an updated and expanded version of an earlier book by the authors published by Springer titled Qualitative Methods in Inverse Scattering Theory Review of Qualitative Methods in Inverse Scattering Theory All in all, the authors do exceptionally well in combining such a wide variety of mathematical material and in presenting it in a well-organized and easy-to-follow fashion. This text certainly complements the growing body of work in inverse scattering and should well suit both new researchers to the field as well as those who could benefit from such a nice codified collection of profitable results combined in one bound volume. SIAM Review, 2006
A Quantitative Portrait of Analytic Philosophy: Looking Through the Margins (Quantitative Methods in the Humanities and Social Sciences)
by Eugenio PetrovichThis book offers an unprecedented quantitative portrait of analytic philosophy focusing on two seemingly marginal features of philosophical texts: citations and acknowledgements in academic publications. Originating from a little network of philosophers based in Oxford, Cambridge, and Vienna, analytic philosophy has become during the Twentieth century a thriving philosophical community with thousands of members worldwide. Leveraging the most advanced techniques from bibliometrics, citations and acknowledgments are used in this book to shed light on both the epistemology and the sociology of this philosophical field, illuminating the intellectual trajectory of analytic philosophy as well as the social characteristics of the analytic community. Special attention is dedicated to the last forty years, providing insights into a phase of analytic philosophy which is still understudied by historians of philosophy. In the eight chapters of the book, readers will find not only numerous quantitative investigations and technical explanations, but also a robust theoretical framework and epistemological reflections on the strengths and limitations of quantitative methods for the study of philosophy. With its strong interdisciplinary appeal, this book will engage a wide range of scholars, including historians of philosophy seeking new methodologies, analytic philosophers interested in a new look at their discipline, and scholars in digital humanities, bibliometrics, and quantitative studies of science, who will find many innovative techniques for investigating disciplinary fields.
A Quantum Mechanics Primer with Solved Exercises (UNITEXT for Physics)
by Daniel Baye Marianne Dufour Benjamin FuksThis book provides a comprehensive introduction to quantum mechanics, supported by numerous solved exercises. Aiming to be both exhaustive and educational, it minimises overly formal aspects by presenting the wave mechanical approach to quantum mechanics. The book simplifies and rigorously covers a large set of fundamental topics such as potential wells and barriers, wave packets, harmonic oscillators, and the hydrogen atom. It also addresses spin and, in simple terms, the conceptual difficulties of quantum physics and Bell’s inequalities. The discussion extends to relativistic quantum mechanics. Each chapter includes exercises designed to test comprehension and facilitate optimal assimilation of the material, and are followed by detailed solutions. Intended for both personal study and course support, this book is valuable for anyone curious about the subject. However, it is specifically targeted at undergraduate and master’s students in physics, chemistry, and mathematics, as well as engineering students.
A Quest Towards a Mathematical Theory of Living Systems (Modeling and Simulation in Science, Engineering and Technology)
by Nicola Bellomo Abdelghani Bellouquid Livio Gibelli Nisrine OutadaThis monograph aims to lay the groundwork for the design of a unified mathematical approach to the modeling and analysis of large, complex systems composed of interacting living things. Drawing on twenty years of research in various scientific fields, it explores how mathematical kinetic theory and evolutionary game theory can be used to understand the complex interplay between mathematical sciences and the dynamics of living systems. The authors hope this will contribute to the development of new tools and strategies, if not a new mathematical theory. The first chapter discusses the main features of living systems and outlines a strategy for their modeling. The following chapters then explore some of the methods needed to potentially achieve this in practice. Chapter Two provides a brief introduction to the mathematical kinetic theory of classical particles, with special emphasis on the Boltzmann equation; the Enskog equation, mean field models, and Monte Carlo methods are also briefly covered. Chapter Three uses concepts from evolutionary game theory to derive mathematical structures that are able to capture the complexity features of interactions within living systems. The book then shifts to exploring the relevant applications of these methods that can potentially be used to derive specific, usable models. The modeling of social systems in various contexts is the subject of Chapter Five, and an overview of modeling crowd dynamics is given in Chapter Six, demonstrating how this approach can be used to model the dynamics of multicellular systems. The final chapter considers some additional applications before presenting an overview of open problems. The authors then offer their own speculations on the conceptual paths that may lead to a mathematical theory of living systems hoping to motivate future research activity in the field. A truly unique contribution to the existing literature, A Quest Toward a Mathematical Theory of Living Systems is an important book that will no doubt have a significant influence on the future directions of the field. It will be of interest to mathematical biologists, systems biologists, biophysicists, and other researchers working on understanding the complexities of living systems.
A Quest for Projects with Scarce Resources: Seeking Schedule Intelligence Through Project Data Discovery (Business Guides on the Go)
by Mario VanhouckeBased on the shared journey of two researchers, this book explores enhancing algorithms for the resource-constrained project scheduling problem. It examines the search for and significance of project data from multiple, distinct perspectives. In the first part, the quest for project data is presented as a continuous exploration of the complexity of the resource-constrained project scheduling problem. This quest is pursued by solving this challenging problem with the aid of state-of-the-art algorithms from the literature, each time gaining a deeper understanding of its challenging nature. To provide insights into the problem’s complexity, project data is created, manipulated, and analyzed in depth to make current projects easier or harder to schedule. This challenging quest for project data has resulted in new project databases for academic research, new ways of testing future algorithms, and insights into how to improve future algorithms to solve this project scheduling problem with limited resources. In turn, the second part discusses the relevance of project data, demonstrating to the reader the importance of the academic research presented in the first part for the professional world. It shows how project data can be used to calibrate real project data, leading to improved decision-making, e.g. for project scheduling, forecasting, and risk analysis. The book extends a warm invitation to academics and practitioners alike, as fellow seekers of knowledge, to enhance their project management skills.
A Random Walk Down Wall Street: The Time-Tested Strategy
by Burton G. MalkielA book meant to provide a comprehensive investment guide for individual investors.
A Random Walk in Physics: Beyond Black Holes and Time-Travels
by Angelo Vulpiani Andrea Puglisi Massimo Cencini Davide VergniThis book offers an informal, easy-to-understand account of topics in modern physics and mathematics. The focus is, in particular, on statistical mechanics, soft matter, probability, chaos, complexity, and models, as well as their interplay. The book features 28 key entries and it is carefully structured so as to allow readers to pursue different paths that reflect their interests and priorities, thereby avoiding an excessively systematic presentation that might stifle interest. While the majority of the entries concern specific topics and arguments, some relate to important protagonists of science, highlighting and explaining their contributions. Advanced mathematics is avoided, and formulas are introduced in only a few cases. The book is a user-friendly tool that nevertheless avoids scientific compromise. It is of interest to all who seek a better grasp of the world that surrounds us and of the ideas that have changed our perceptions.
A Readable Introduction to Real Mathematics (Undergraduate Texts in Mathematics)
by David Rosenthal Daniel Rosenthal Peter RosenthalDesigned for an undergraduate course or for independent study, this text presents sophisticated mathematical ideas in an elementary and friendly fashion. The fundamental purpose of this book is to engage the reader and to teach a real understanding of mathematical thinking while conveying the beauty and elegance of mathematics. The text focuses on teaching the understanding of mathematical proofs. The material covered has applications both to mathematics and to other subjects. The book contains a large number of exercises of varying difficulty, designed to help reinforce basic concepts and to motivate and challenge the reader. The sole prerequisite for understanding the text is basic high school algebra; some trigonometry is needed for Chapters 9 and 12. Topics covered include: mathematical induction - modular arithmetic - the fundamental theorem of arithmetic - Fermat's little theorem - RSA encryption - the Euclidean algorithm -rational and irrational numbers - complex numbers - cardinality - Euclidean plane geometry - constructability (including a proof that an angle of 60 degrees cannot be trisected with a straightedge and compass). This textbook is suitable for a wide variety of courses and for a broad range of students in the fields of education, liberal arts, physical sciences and mathematics. Students at the senior high school level who like mathematics will also be able to further their understanding of mathematical thinking by reading this book.
A Readable Introduction to Real Mathematics (Undergraduate Texts in Mathematics)
by David Rosenthal Daniel Rosenthal Peter RosenthalDesigned for an undergraduate course or for independent study, this text presents sophisticated mathematical ideas in an elementary and friendly fashion. The fundamental purpose of this book is to teach mathematical thinking while conveying the beauty and elegance of mathematics. The book contains a large number of exercises of varying difficulty, some of which are designed to help reinforce basic concepts and others of which will challenge virtually all readers. The sole prerequisite for reading this text is high school algebra. Topics covered include: * mathematical induction * modular arithmetic * the Fundamental Theorem of Arithmetic * Fermat's Little Theorem * RSA encryption * the Euclidean algorithm * rational and irrational numbers * complex numbers * cardinality * Euclidean plane geometry * constructibility (including a proof that an angle of 60 degrees cannot be trisected with a straightedge and compass)* infinite series * higher dimensional spaces. <P><P> This textbook is suitable for a wide variety of courses and for a broad range of students of mathematics and other subjects. Mathematically inclined senior high school students will also be able to read this book.
A Realist Theory of Science (Radical Thinkers)
by Roy Bhaskar<i>A Realist Theory of Science</i> is one of the few books that have changed our understanding of the philosophy of science. In this analysis of the natural sciences, with a particular focus on the experimental process itself, Roy Bhaskar provides a definitive critique of the traditional, positivist conception of science and stakes out an alternative, realist position. Since it original publication in 1975, a movement known as ‘Critical Realism’, which is both intellectually diverse and international in scope, has developed on the basis of key concepts outlined in the text. The book has been hailed in many quarters as a ‘Copernican Revolution’ in the study of the nature of science, and the implications of its account have been far-reaching for many fields of the humanities and social sciences.
A Refresher Course in Mathematics
by F. J. CammReaders wishing to renew and extend their acquaintance with a variety of branches of mathematics will find this volume a practical companion. Geared toward those who already possess some familiarity with its subjects, the easy-to-follow explanations and straightforward tone make this book highly accessible. The contents are arranged logically and in order of difficulty: fractions, decimals, square and cube root, the metric system, algebra, quadratic and cubic equations, graphs, and the calculus are among the topics. Explanations of mathematical principles are followed by worked examples, and the book includes a convenient selection of tables that cover the trigonometrical functions and logarithms necessary for completing some of the examples.
A Relatively Painless Guide to Special Relativity
by Dave GoldbergSerious and accessible—finally the special relativity course book that both physics majors and lifelong learners deserve. Special relativity challenges one’s physical intuition of space, time, matter, and energy in a way that few other topics in physics do. Yet the subject is often treated as an extra in undergraduate courses—something to be picked up in a few random lectures and presented as a combination of geometric and logical puzzles (seemingly with the premise of getting the novice student to concede that Einstein was a genius and that the universe is weird). But special relativity is absolutely fundamental to modern physics. It is the canvas on which electromagnetism, particle physics, field theory, and ultimately general relativity are based. For physics students, developing a relativistic intuition isn’t just a luxury: it’s a requirement. Physicist and popular author Dave Goldberg provides a rigorous but conversational introduction to fill this void in spacetime education. Employing the standard calculus a sophomore or junior university student in science, engineering, or computer science will have encountered, Goldberg connects relativity to a student’s work ahead, acquainting them with topics like tensors, the development of new physical theories, and how relativity directly relates to other disciplines. But more than this, Goldberg welcomes lifelong learners who may have encountered special relativity in popular accounts, but are seeking a mathematical challenge to understand an elegant physical theory.
A Researcher's Guide to Using Electronic Health Records: From Planning to Presentation
by Neal D. GoldsteinIn an age when electronic health records (EHRs) are an increasingly important source of data, this essential textbook provides both practical and theoretical guidance to researchers conducting epidemiological or clinical analysis through EHRs. Split into three parts, the book covers the research journey from start to finish. Part 1 focuses on the challenges inherent when working with EHRs, from access to data management, and raising issues such as completeness and accuracy which impact the validity of any research project. Part 2 examines the core research process itself, with chapters on research design, sampling, and analysis, as well as emerging methodological techniques. Part 3 demonstrates how EHR research can be made meaningful, from presentation to publication, and includes how findings can be applied to real-world issues of public health. Supported by case studies throughout, and applicable across a range of research software programs (including R, SPSS, and SAS), this is the ideal text for students and researchers engaging with EHRs across epidemiological and clinical research.
A Richer Picture of Mathematics: The Göttingen Tradition and Beyond
by David E. RoweHistorian David E. Rowe captures the rich tapestry of mathematical creativity in this collection of essays from the “Years Ago” column of The Mathematical Intelligencer. With topics ranging from ancient Greek mathematics to modern relativistic cosmology, this collection conveys the impetus and spirit of Rowe’s various and many-faceted contributions to the history of mathematics. Centered on the Göttingen mathematical tradition, these stories illuminate important facets of mathematical activity often overlooked in other accounts. Six sections place the essays in chronological and thematic order, beginning with new introductions that contextualize each section. The essays that follow recount episodes relating to the section’s overall theme. All of the essays in this collection, with the exception of two, appeared over the course of more than 30 years in The Mathematical Intelligencer. Based largely on archival and primary sources, these vignettes offer unusual insights into behind-the-scenes events. Taken together, they aim to show how Göttingen managed to attract an extraordinary array of talented individuals, several of whom contributed to the development of a new mathematical culture during the first decades of the twentieth century.
A Risky Business: An Actuary’s Guide to Quantifying and Managing Risk in Society
by Catrin TownsendIntangible, invisible and worth trillions, risk is everywhere. Its quantification and management are key to the success and failure of individuals, businesses and governments. Whether you’re an interested observer or pursuing a career in risk, this book delves into the complex and multi-faceted work that actuaries undertake to quantify, manage and commodify risk—supporting our society and servicing a range of multi-billion-dollar industries. Starting at the most basic level, this book introduces key concepts in actuarial science, insurance and pensions. Through case studies, explanations and mathematical examples, it fosters an understanding of current industry practice. This book celebrates the long history of actuarial science and poses the problems facing actuaries in the future, exploring complex global risks including climate change, aging populations, healthcare models and pandemic epidemiology from an actuarial perspective. It gives practical advice for new and potential actuaries on how to identify an area of work to go into, how best to navigate (and pass!) actuarial exams and how to develop your skills post-qualification. A Risky Business illuminates how actuaries are central to society as we know it, revealing what they do and how they do it. It is the essential primer on actuarial science.
A Royal Road to Algebraic Geometry
by Audun HolmeThis book is about modern algebraic geometry. The title A Royal Road to Algebraic Geometry is inspired by the famous anecdote about the king asking Euclid if there really existed no simpler way for learning geometry, than to read all of his work Elements. Euclid is said to have answered: "There is no royal road to geometry!" The book starts by explaining this enigmatic answer, the aim of the book being to argue that indeed, in some sense there is a royal road to algebraic geometry. From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck's theory of schemes. Contemporary homological tools are explained. The reader will follow a directed path leading up to the main elements of modern algebraic geometry. When the road is completed, the reader is empowered to start navigating in this immense field, and to open up the door to a wonderful field of research. The greatest scientific experience of a lifetime!
A SAS/IML Companion for Linear Models (Statistics and Computing)
by Jamis J. PerrettLinear models courses are often presented as either theoretical or applied. Consequently, students may find themselves either proving theorems or using high-level procedures like PROC GLM to analyze data. There exists a gap between the derivation of formulas and analyses that hide these formulas behind attractive user interfaces. This book bridges that gap, demonstrating theory put into practice. Concepts presented in a theoretical linear models course are often trivialized in applied linear models courses by the facility of high-level SAS procedures like PROC MIXED and PROC REG that require the user to provide a few options and statements and in return produce vast amounts of output. This book uses PROC IML to show how analytic linear models formulas can be typed directly into PROC IML, as they were presented in the linear models course, and solved using data. This helps students see the link between theory and application. This also assists researchers in developing new methodologies in the area of linear models. The book contains complete examples of SAS code for many of the computations relevant to a linear models course. However, the SAS code in these examples automates the analytic formulas. The code for high-level procedures like PROC MIXED is also included for side-by-side comparison. The book computes basic descriptive statistics, matrix algebra, matrix decomposition, likelihood maximization, non-linear optimization, etc. in a format conducive to a linear models or a special topics course. Also included in the book is an example of a basic analysis of a linear mixed model using restricted maximum likelihood estimation (REML). The example demonstrates tests for fixed effects, estimates of linear functions, and contrasts. The example starts by showing the steps for analyzing the data using PROC IML and then provides the analysis using PROC MIXED. This allows students to follow the process that lead to the output.