Inverse Problems

Activities for Undergraduates

Author: C. W. Groetsch

Publisher: Cambridge University Press

ISBN: 9780883857168

Category: Mathematics

Page: 222

View: 9019

Problem solving in mathematics is often thought of as a one way process. For example: take two numbers and multiply them together. However for each problem there is also an inverse problem which runs in the opposite direction: now take a number and find a pair of factors. Such problems are considerably more important, in mathematics and throughout science, than they might first appear. This book concentrates on these inverse problems and how they can be usefully introduced to undergraduate students. A historical introduction sets the scene and gives a cultural context for what the rest of the book. Chapters dealing with inverse problems in calculus, differential equations and linear algebra then follow and the book concludes with suggestions for further reading. Whatever their own field of expertise, this will be an essential purchase for anyone interested in the teaching of mathematics.
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The Calculus Collection

A Resource for AP and Beyond

Author: Caren L. Diefenderfer,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883857618

Category: Juvenile Nonfiction

Page: 507

View: 3500

The Calculus Collection is a useful resource for everyone who teaches calculus, in high school or in a 2- or 4-year college or university. It consists of 123 articles, selected by a panel of six veteran high school teachers, each of which was originally published in Math Horizons, MAA Focus, The American Mathematical Monthly, The College Mathematics Journal, or Mathematics Magazine. The articles focus on engaging students who are meeting the core ideas of calculus for the first time. The Calculus Collection is filled with insights, alternate explanations of difficult ideas, and suggestions for how to take a standard problem and open it up to the rich mathematical explorations available when you encourage students to dig a little deeper. Some of the articles reflect an enthusiasm for bringing calculators and computers into the classroom, while others consciously address themes from the calculus reform movement. But most of the articles are simply interesting and timeless explorations of the mathematics encountered in a first course in calculus.The MAA has twice previously issued a calculus reader, collecting articles on calculus from its journals: Selected Papers in Calculus, published in 1969 and reprinted as Part I of A Century of Calculus, and Part II published in 1992. In a sense The Calculus Collection is the third volume in that series, but different in that it is a collection chosen for its usefulness to those who teach first-year calculus in high schools as well as colleges and universities.
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Which Numbers Are Real?

Author: Michael Henle

Publisher: MAA

ISBN: 0883857774

Category: Mathematics

Page: 219

View: 3521

Everyone knows the real numbers, those fundamental quantities that make possible all of mathematics from high school algebra and Euclidean geometry through the Calculus and beyond; and also serve as the basis for measurement in science, industry, and ordinary life. This book surveys alternative real number systems: systems that generalize and extend the real numbers yet stay close to these properties that make the reals central to mathematics.Alternative real numbers include many different kinds of numbers, for example multidimensional numbers (the complex numbers, the quaternions and others), infinitely small and infinitely large numbers (the hyperreal numbers and the surreal numbers), and numbers that represent positions in games (the surreal numbers). Each system has a well-developed theory, including applications to other areas of mathematics and science, such as physics, the theory of games, multi-dimensional geometry, and formal logic. They are all active areas of current mathematical research and each has unique features, in particular, characteristic methods of proof and implications for the philosophy of mathematics, both highlighted in this book.Alternative real number systems illuminate the central, unifying role of the real numbers and include some exciting and eccentric parts of mathematics. Which Numbers Are Real? Will be of interest to anyone with an interest in numbers, but specifically to upper-level undergraduates, graduate students, and professional mathematicians, particularly college mathematics teachers.
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Excursions in Classical Analysis

Pathways to Advanced Problem Solving and Undergraduate Research

Author: Hongwei Chen

Publisher: MAA

ISBN: 0883857685

Category: Mathematics

Page: 301

View: 2813

Excursions in Classical Analysis will introduce students to advanced problem solving and undergraduate research in two ways: it will provide a tour of classical analysis, showcasing a wide variety of problems that are placed in historical context, and it will help students gain mastery of mathematical discovery and proof.The author presents a variety of solutions for the problems in the book. Some solutions reach back to the work of mathematicians like Leonhard Euler while others connect to other beautiful parts of mathematics. Readers will frequently see problems solved by using an idea that might at first glance, might not even seem to apply to that problem. Other solutions employ a specific technique that can be used to solve many different kinds of problems. Excursions emphasizes the rich and elegant interplay between continuous and discrete mathematics by applying induction, recursion, and combinatorics to traditional problems in classical analysis.The book will be useful in students' preparations for mathematics competitions, in undergraduate reading courses and seminars, and in analysis courses as a supplement. The book is also ideal for self study, since the chapters are independent of one another and may be read in any order.
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Introduction to Inverse Problems in Imaging

Author: M. Bertero,P. Boccacci

Publisher: CRC Press

ISBN: 9781439822067

Category: Technology & Engineering

Page: 352

View: 6885

This is a graduate textbook on the principles of linear inverse problems, methods of their approximate solution, and practical application in imaging. The level of mathematical treatment is kept as low as possible to make the book suitable for a wide range of readers from different backgrounds in science and engineering. Mathematical prerequisites are first courses in analysis, geometry, linear algebra, probability theory, and Fourier analysis. The authors concentrate on presenting easily implementable and fast solution algorithms. With examples and exercised throughout, the book will provide the reader with the appropriate background for a clear understanding of the essence of inverse problems (ill-posedness and its cure) and, consequently, for an intelligent assessment of the rapidly growing literature on these problems.
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An Introduction to Inverse Problems with Applications

Author: Francisco Duarte Moura Neto,Antônio José da Silva Neto

Publisher: Springer Science & Business Media

ISBN: 3642325564

Category: Mathematics

Page: 246

View: 4339

Computational engineering/science uses a blend of applications, mathematical models and computations. Mathematical models require accurate approximations of their parameters, which are often viewed as solutions to inverse problems. Thus, the study of inverse problems is an integral part of computational engineering/science. This book presents several aspects of inverse problems along with needed prerequisite topics in numerical analysis and matrix algebra. If the reader has previously studied these prerequisites, then one can rapidly move to the inverse problems in chapters 4-8 on image restoration, thermal radiation, thermal characterization and heat transfer. “This text does provide a comprehensive introduction to inverse problems and fills a void in the literature”. Robert E White, Professor of Mathematics, North Carolina State University
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Methods for Euclidean Geometry

Author: Owen Byer,Felix Lazebnik,Deirdre L. Smeltzer

Publisher: MAA

ISBN: 0883857634

Category: Mathematics

Page: 461

View: 4651

Methods for Euclidean Geometry explores one of the oldest and most beautiful of mathematical subjects. The book begins with a thorough presentation of classical solution methods for plane geometry problems, but its distinguishing feature is the subsequent collection of methods which have appeared since 1600. For example, the coordinate method, which is a central part of the book, has been part of mathematics for four centuries. However, it has rarely served as a tool that students consider using when faced with geometry problems. The same holds true regarding the use of trigonometry, vectors, complex numbers, and transformations. The book presents each of these as self-contained topics, providing examples of their applications to geometry problems. Both strengths and weaknesses of various methods, as well as the ranges of their effective applications, are discussed. Importance is placed on the problems and their solutions. The book contains numerous problems of varying difficulty; over a third of its contents are devoted to problem statements, hints, and complete solutions. The book can be used as a textbook for geometry courses; as a source book for geometry and other mathematics courses; for capstone, problem-solving, and enrichment courses; and for independent study courses.
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Calculus Mysteries and Thrillers

Author: R. Grant Woods

Publisher: MAA

ISBN: 9780883857113

Category: Mathematics

Page: 131

View: 2852

The author presents eleven mathematic problems and their solutions in story form for the reader. The calculus concepts on which the problems are based include; tangent and normal lines, optimization by use of criticla points, inverse trig functions, volumes of solids, surface area integrals, and modeling economic concepts using definite integrals". -- Back cover.
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Topology Now!

Author: Robert Messer,Philip Straffin

Publisher: MAA

ISBN: 9780883857441

Category: Mathematics

Page: 240

View: 8929

An undergraduate textbook on topology designed to have very few prerequisites.
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Books in Print

Author: R.R. Bowker Company

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 5977

Books in print is the major source of information on books currently published and in print in the United States. The database provides the record of forthcoming books, books in-print, and books out-of-print.
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Inverse Problems in the Mathematical Sciences

Author: Charles W. Groetsch

Publisher: Springer Science & Business Media

ISBN: 3322992020

Category: Technology & Engineering

Page: 154

View: 3917

Inverse problems are immensely important in modern science and technology. However, the broad mathematical issues raised by inverse problems receive scant attention in the university curriculum. This book aims to remedy this state of affairs by supplying an accessible introduction, at a modest mathematical level, to the alluring field of inverse problems. Many models of inverse problems from science and engineering are dealt with and nearly a hundred exercises, of varying difficulty, involving mathematical analysis, numerical treatment, or modelling of inverse problems, are provided. The main themes of the book are: causation problem modeled as integral equations; model identification problems, posed as coefficient determination problems in differential equations; the functional analytic framework for inverse problems; and a survey of the principal numerical methods for inverse problems. An extensive annotated bibliography furnishes leads on the history of inverse problems and a guide to the frontiers of current research.
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Resources for Teaching Discrete Mathematics

Classroom Projects, History Modules, and Articles

Author: Brian Hopkins

Publisher: MAA

ISBN: 9780883851845

Category: Mathematics

Page: 323

View: 3432

Resources for Teaching Discrete Mathematics presents nineteen classroom tested projects complete with student handouts, solutions, and notes to the instructor. Topics range from a first day activity that motivates proofs to applications of discrete mathematics to chemistry, biology, and data storage. Other projects provide: supplementary material on classic topics such as the towers of Hanoi and the Josephus problem, how to use a calculator to explore various course topics, how to employ Cuisenaire rods to examine the Fibonacci numbers and other sequences, and how you can use plastic pipes to create a geodesic dome. The book contains eleven history modules that allow students to explore topics in their original context. Sources range from eleventh century Chinese figures that prompted Leibniz to write on binary arithmetic, to a 1959 article on automata theory. Excerpts include: Pascal's "Treatise on the Arithmetical Triangle," Hamilton's "Account of the Icosian Game," and Cantor's (translated) "Contributions to the Founding of the Theory of Transfinite Numbers." Five articles complete the book. Three address extensions of standard discrete mathematics content: an exploration of historical counting problems with attention to discovering formulas, a discussion of how computers store graphs, and a survey connecting the principle of inclusion-exclusion to Möbius inversion. Finally, there are two articles on pedagogy specifically related to discrete mathematics courses: a summary of adapting a group discovery method to larger classes, and a discussion of using logic in encouraging students to construct proofs.
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How People Learn

Brain, Mind, Experience, and School: Expanded Edition

Author: National Research Council,Division of Behavioral and Social Sciences and Education,Board on Behavioral, Cognitive, and Sensory Sciences,Committee on Developments in the Science of Learning with additional material from the Committee on Learning Research and Educational Practice

Publisher: National Academies Press

ISBN: 0309131979

Category: Education

Page: 384

View: 3809

First released in the Spring of 1999, How People Learn has been expanded to show how the theories and insights from the original book can translate into actions and practice, now making a real connection between classroom activities and learning behavior. This edition includes far-reaching suggestions for research that could increase the impact that classroom teaching has on actual learning. Like the original edition, this book offers exciting new research about the mind and the brain that provides answers to a number of compelling questions. When do infants begin to learn? How do experts learn and how is this different from non-experts? What can teachers and schools do-with curricula, classroom settings, and teaching methods--to help children learn most effectively? New evidence from many branches of science has significantly added to our understanding of what it means to know, from the neural processes that occur during learning to the influence of culture on what people see and absorb. How People Learn examines these findings and their implications for what we teach, how we teach it, and how we assess what our children learn. The book uses exemplary teaching to illustrate how approaches based on what we now know result in in-depth learning. This new knowledge calls into question concepts and practices firmly entrenched in our current education system. Topics include: How learning actually changes the physical structure of the brain. How existing knowledge affects what people notice and how they learn. What the thought processes of experts tell us about how to teach. The amazing learning potential of infants. The relationship of classroom learning and everyday settings of community and workplace. Learning needs and opportunities for teachers. A realistic look at the role of technology in education.
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The Heart of Calculus

Explorations and Applications

Author: Philip M. Anselone,John W. Lee

Publisher: The Mathematical Association of America

ISBN: 0883857871

Category: Mathematics

Page: 245

View: 7802

This book contains enrichment material for courses in first and second year calculus, differential equations, modeling, and introductory real analysis. It targets talented students who seek a deeper understanding of calculus and its applications. The book can be used in honors courses, undergraduate seminars, independent study, capstone courses taking a fresh look at calculus, and summer enrichment programs. The book develops topics from novel and/or unifying perspectives. Hence, it is also a valuable resource for graduate teaching assistants developing their academic and pedagogical skills and for seasoned veterans who appreciate fresh perspectives. The explorations, problems, and projects in the book impart a deeper understanding of and facility with the mathematical reasoning that lies at the heart of calculus and conveys something of its beauty and depth. A high level of rigor is maintained. However, with few exceptions, proofs depend only on tools from calculus and earlier. Analytical arguments are carefully structured to avoid epsilons and deltas. Geometric and/or physical reasoning motivates challenging analytical discussions. Consequently, the presentation is friendly and accessible to students at various levels of mathematical maturity. Logical reasoning skills at the level of proof in Euclidean geometry suffice for a productive use of the book.
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Mathematics for Business Decisions: Calculus and optimization

Author: Richard B. Thompson

Publisher: Mathematical Assn of Amer

ISBN: 9780883857311

Category: Business & Economics

Page: N.A

View: 3013

Students are allowed to learn mathematics in a setting that mirrors the professional environment they will encounter after college.
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"Multiplication is for White People"

Raising Expectations for Other People's Children

Author: Lisa D. Delpit

Publisher: The New Press

ISBN: 1595580468

Category: Education

Page: 224

View: 7740

Presents a striking picture of the elements of contemporary public education that conspire against the prospects for poor children of color, creating a persistent gap in achievement during the school years that has eluded several decades of reform. By the best-selling author of Other People's Children.
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Collaborative Learning Techniques

A Handbook for College Faculty

Author: Elizabeth F. Barkley,Claire H. Major,K. Patricia Cross

Publisher: John Wiley & Sons

ISBN: 1118761677

Category: Education

Page: 456

View: 3081

A guide to thirty-five creative assignments for pairs and groups Collaborative Learning Techniques is the bestseller that college and university faculty around the world have used to help them make the most of small group learning. A mountain of evidence shows that students who learn in small groups together exhibit higher academic achievement, motivation, and satisfaction than those who don't. Collaborative learning puts into practice the major conclusion from learning theory: that students must be actively engaged in building their own minds. In this book, the authors synthesize the relevant research and theory to support thirty-five collaborative learning activities for use in both traditional and online classrooms. This second edition reflects the changed world of higher education. New technologies have opened up endless possibilities for college teaching, but it's not always easy to use these technologies effectively. Updated to address the challenges of today's new teaching environments, including online, "flipped," and large lectures, Collaborative Learning Techniques is a wonderful reference for educators who want to make the most of any course environment. This revised and expanded edition includes: Additional techniques, with an all-new chapter on using games to provide exciting, current, technologically-sophisticated curricula A section on effective online implementation for each of the thirty-five techniques Significantly expanded pedagogical rationale and updates on the latest research showing how and why collaborative learning works Examples for implementing collaborative learning techniques in a variety of learning environments, including large lecture classes and "flipped" classes Expanded guidance on how to solve common problems associated with group work The authors guide instructors through all aspects of group work, providing a solid grounding in what to do, how to do it, and why it is important for student learning. The detailed procedures in Collaborative Learning Techniques will help teachers make sure group activities go smoothly, no matter the size or delivery method of their classes. With practical advice on how to form student groups, assign roles, build team spirit, address unexpected problems, and evaluate and grade student participation, this new edition of the international classic makes incorporating effective group work easy.
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