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About
Overview
Reflecting Cengage Learning's commitment to offering flexible teaching solutions and value for students and instructors, this new hybrid edition features the instructional presentation found in the printed text while delivering end-of-section exercises online in Enhanced WebAssign. The result—a briefer printed text that engages students online!
Instructors are loyal to Howard Rolf's text because teaching from it is easy. Why? Exercises at a variety of levels, flexible technology integration, and many interesting business applications make concepts relevant, clear, and easier to understand for students. The new edition of FINITE MATHEMATICS continues to rely on a rich array of examples and exercises and a student-friendly approach to illustrate the applications of mathematical concepts in analyzing and solving problems. Optional graphing calculator problems and Microsoft® Excel® applications are available for instructors who wish to bring technology into the course.
Features and Benefits
Detailed computations enable students to easily follow successive steps in the problem-solving process.
More than 3,200 exercises and 500 examples offer instructors and students with an abundance of homework and practice problems. Examples and exercises are cross-referenced, identifying for students a corresponding exercise to check their understanding after they read through an example and solution.
Exercises are graded by level of difficulty: level 1 for routine problems, level 2 for elementary word problems, and level 3 for more thought-provoking problems.
Discussion questions provide the option of using the graphing calculator or spreadsheet technology to solve the problem. "Explorations" encourage students to think more deeply about mathematical concepts, often providing an opportunity to use the graphing calculator. Many of these exercises may be used for group projects or writing assignments.
References to technology keep the text current with new applications. "Using Your TI Graphing Calculator" and "Using Excel" boxes walk students through the steps necessary to apply the technology to selected exercises. Additional guidance on how to use the graphing calculator is found in an appendix.
What's New
The end-of-chapter review has been expanded to include Important Concepts. This new feature summarizes the key theorems, formulas, and equations from the chapter. Important Concepts together with Important Terms provides students with a comprehensive and useful tool for study and review.
Updated problem sets include many new exercises that use real data with modern business applications, further reinforcing the relevance of this material to applied math students.
The book's technology program has been expanded, offering more options to meet the expectations and needs of today's instructors and students. Enhanced WebAssign®, available with the text, includes an increased number of problems and supporting assets. In addition, the Rolf package now includes YouBook (an interactive eBook version of the text that instructors can customize), CourseMate, and a hybrid version of the text.
Discussions and keystroke instructions in Using Excel have been updated to Excel 2007 and Excel 2010Lecture
(ISBN-10: 1285085094 | ISBN-13: 9781285085098)
This CD-ROM provides the instructor with dynamic media tools for teaching, including Microsoft® PowerPoint® lecture slides, figures from the book, and the Test Bank. You can create, deliver, and customize tests (both print and online) in minutes with ExamView® Computerized Testing, which includes Test Bank items in electronic format. In addition, you can easily build solution sets for homework or exams by linking to Solution Builder's online solutions manual. | 677.169 | 1 |
I'd suggest statistics. It's probably the course that's most useful for the daily life of non-mathematicians. I found it pretty straightforward compared to other classes, but I think the difficulty will vary greatly between schools.
"Finite mathematics" will (presumably) be very unlike what you've encountered so far in your math classes. If you don't really like fractions and graphing and functions and that sort of thing, this might be worth considering. If there's a syllabus for this course somewhere online people can probably give you a better idea what it will be like. | 677.169 | 1 |
books.google.com - Provides a world view of mathematics, balancing ancient, early modern and modern history. Problems are taken from their original sources, enabling students to understand how mathematicians in various times and places solved mathematical problems. In this new edition a more global perspective is taken,... History of Mathematics
A History of Mathematics: An Introduction
Provides a world view of mathematics, balancing ancient, early modern and modern history. Problems are taken from their original sources, enabling students to understand how mathematicians in various times and places solved mathematical problems. In this new edition a more global perspective is taken, integrating more non-Western coverage including contributions from Chinese/Indian, and Islamic mathematics and mathematicians. An additional chapter covers mathematical techniques from other cultures. *Up to date, uses the results of very recent scholarship in the history of mathematics. *Provides summaries of the arguments of all important ideas in the field.
About the author (1998)
Victor J. Katz is professor emeritus of mathematics at the University of the District of Columbia. His many books include the textbook, "A History of Mathematics: An Introduction, 2nd ed." (Addison-Wesley). He is the coeditor of "Historical Modules for the Teaching and Learning of Mathematics. | 677.169 | 1 |
...Level then AS or A2 Level courses can be purchased from the AS/A2 Level section of our website. Each lesson begins with a set of clearly stated objectives... Learn about: GCSE Mathematics, Basic Mathematics, Mathematics Series...
...and can lead to careers in engineering, accounting, teaching, computing and many other areas. The course can be taken as part of a full-time programme... Learn about: GCSE Physics, Full Time, GCSE Mathematics...
...and David Beckham is about to take a free kick that could take England to the World Cup Finals. The crowd are crying out for him to score… and he does... Learn about: Engineering Mathematics, Mathematics Series, Basic Mathematics...
...study a degree in a mathematically related discipline. The Further Maths course is more difficult than the Maths one and is therefore suited to those students... Learn about: Mathematics understanding, Maths skills...
...The modules studied in the first year are described under Mathematics except for Further Pure 1. This last module, and the ones in the second year explore topics such as Matrices... Learn about: GCSE Mathematics...
...programme designed to provide appropriate extension material or timely support for students. In addition we aim to seek ways to enrich and extend the mathematics... Learn about: GCSE Mathematics, Team Training...
...Mechanics and Statistics. An A Level is made up of six units, four of which are Pure while the other two are Mechanics or Statistics. Students are advised... Learn about: Part Time, GCSE Mathematics...
...One of those exams is in January and the other two are in June. Why choose this course? This course gives a solid background in Mathematics and its applications... Learn about: Mathematical Economics, Mathematics Series... | 677.169 | 1 |
Excel HSC General Maths QS
Excel HSC General Mathematics Quick Study is the perfect tool for studying and revising on the go! This app is designed specifically for the HSC General Mathematics course. There are two parts to the app: 1. HSC study cards • There are 134 study cards to revise. • All the Core topics (Measurement; Algebraic Modelling; Probability; Financial Mathematics; Data Analysis) are covered. • Special revision features include: - bookmarks—you can bookmark each card with a green, yellow or red bookmark depending on how well you know each card - revision notes—you can type in your own revision notes to customise your revision for each card or topic. 2. Quick Quiz • There are 553 questions in total. • You can take a randomly generated quick quiz of 10 questions from the topic of your choice, or from all topics combined. • Each question in the quiz is marked instantly for you, with the correct answer highlighted so you can learn from your mistakes. • You are given a score out of ten and your percentage mark at the end of each quiz. • You are also given a comprehensive summary of all your results for each topic including your percentage improvement, which helps you keep track of your progress. • You can take the quizzes as many times as you like until you get the consistently high percentage score you want. Excel Quick Study apps are a convenient and efficient way to revise—any time, anywhere!TechnicalPlease note: due to the large number of different devices supporting Android 2.2 and up, not all brands/models have been tested. Your feedback is appreciated: please email us to report any errors or bugs and we will do our utmost to fix them and to support other | 677.169 | 1 |
Mastering Mathematics: How to Be a Great Math Student / Edition 2
Paperback
Temporarily out of stock online.
Overview perceived "low math ability," be successful on math tests, get back on track when feeling "lost," and much more!
Related Subjects
Table of Contents
Preface to the Instructor. Introduction to the Student. How to Evaluate How Well You Study in Math Courses. PART I: MAKE A FRESH START. 1. How to Have the Right Attitude in Math Courses. PART II: MAKE SUCCESSFUL COURSE PREPARATION YOUR CONSISTENT ROUTINE. 2. How to Begin Before Your Math Course Starts. 3. How to Master the Course Using Four Major Steps. 4. How to Use Class Time Effectively. 5. How to Use Your Time Between Classes—Notes and Textbook. 6. How to Use Your Time Between Classes—Homework and Beyond. PART III: MAKE PREPARING FOR TESTS A SURE THING. 7. How to Aim for Perfection in Your Test Preparation. 8. How to Make a List of Topics That Might Be Covered on the Test. 9. How to Master Each Topic. 10. How to Be a Perfectionist When Preparing for a Test. 11. How to Take a Math Test. PART IV: MAKE USE OF THESE ADDITIONAL STUDY TIPS TO IMPROVE YOUR GRADE. 12. How to Cope with a "Difficult" Teacher. 13. How to Go the Extra Mile In Your Course Preparation. Conclusion. Appendix A: Study Habits Improvement Check. Appendix B: Recommended Books for Students. | 677.169 | 1 |
MAT636 Algebra and Number Theory
Course Description
Provides K-9 advanced study in teaching and learning algebra and number theory content. Presents effective strategies and models of instruction to improve pedagogy and assessment in mathematics classrooms.
Learning Outcomes
Analyze issues, challenges and problems involved in teaching algebra and number theory.
Apply research based solutions and approaches to teaching algebra and number theory.
Create instrument for measuring effectiveness of teaching and assessment of student learning related to algebra and number theory.
Develop learning activity for teaching algebra and number theory, illustrating and justifying the use of varied instructional strategies.
Request Information
Please complete the form below, and an admissions advisor will contact you shortly. | 677.169 | 1 |
Introduction To College Algebra – mth219 (3 credits)
This course introduces algebraic concepts providing a solid foundation for college algebra. Topics range from properties of real numbers, the order of operations, and algebraic expressions to solving equations and inequalities. Additional topics include polynomials, factoring methods, rational and radical expressions as well as graphing and functions.
Linear Equations in Two Variables, Polynomials, and Introduction to Factoring Polynomials
Graph linear equations in two variables.
Determine the slope of a linear equation.
Factor trinomials when the leading coefficient is one.
Factor polynomials using the greatest common factor.
Factor polynomials by grouping.
Perform arithmetic operations on polynomials | 677.169 | 1 |
Mathematics Student Experience
Putnam Exam
Every year in December, we administer the Putnam exam, a national exam that measures mathematical creativity. The national median score on the exam is usually 0 or 1 point out of a possible 120, but students who do well on the exam can open up very nice opportunities for themselves. A Cedarville student who scored in the top 500 students taking the exam (there are usually between 3,000 and 4,000 nationally who take the exam), was awarded a place in a program called Math in Moscow, which allowed him to spend a semester in Moscow studying math in English with Russian mathematicians. The Math in Moscow program is very prestigious and led to this student gaining some very lucrative offers for graduate school.
Computer Programming Contest
Every spring, the computer science department holds a computer programming contest in which many math majors participate in teams of two. Math majors who participate show very good problem solving skills in this contest.
Math Department Colloquium
Each semester, the math department invites a speaker to discuss a mathematical topic with undergraduate students. Recent topics have included The Actuarial Profession, Completeness Properties and Topological Games, and Preemptive Rerouting of Airline Passengers Under Uncertain Delays.
On Campus Tutoring
There are several opportunities for students to tutor in mathematics on campus. The Cove (Cedarville University's Academic Enrichment Center) offers tutoring for all students, and some of our majors work for The Cove in tutoring for calculus and other mathematics courses. Several of our students are also private tutors for students at Cedarville.
Off-Campus Tutoring
Every year there are opportunities for Cedarville students to tutor. Some students are part of tutoring ministries through Cedarville's Christian Ministries. Other students tutor local students for pay.
Events that Involve Travel
The Nebraska Undergraduate Conference for Women in Mathematics: Every January, a group of female math majors travels to Nebraska to attend this conference.
Ohio Space Grant Consortium Student Research Project Symposium: Every spring, a group of students is nominated to attend this symposium. | 677.169 | 1 |
Lesson content has been expanded and includes more video instruction for each lesson. In addition to the video examples each lesson includes interactive practice problems, challenge questions, and worksheets.
Description
Need more than videos to learn math? YourTeacher's Introductory Algebra app is like having a personal math tutor in your pocket.
**
"It's like a private school math classroom, but you are the only student."
John
"I just love YourTeacher and the way you explain things. I felt like I was in a classroom instead of just looking at examples."
Diane
"I was desperately seeking Algebra help. I am working on an Associates Degree at a local community college. Pre-Algebra and Introductory Algebra were required classes for me. I was completely lost because I have absolutely no Algebra background. I was really struggling in my class. Then I found YourTeacher. The lessons were a tremendous help. I ended up making an A in the class and passed the exit test. Thank you for your help."
Catherine
"I can't thank you enough for your patient, easy way of teaching. I am taking a college introductory algebra class this semester and my math teacher runs through the material so fast it's been very hard to keep up with her. By listening to you explain what she ran through during class, I've been able to learn what she was trying to teach. Thanks again for all your help!"
Janiceakers Cyst
This application contains animated exercises for Bakers cyst which can be done in the convenience and privacy of your own home. A Baker's cyst is a pocket of fluid that forms a lump behind the knee. It is also called a popliteal cyst. Exercises: Stretching Quadriceps Stretch,Stretching : Seated Hamstring Stretch,Heel Slides,Strengthening Wall S…
Anterior Cruciate Ligament
ACL injuries are most common during sports that require a sudden change of direction, sudden, abrupt stops and starts and lots of jumping. This application contains more than 30 user friendly animated exercise program. These exercises include strengthening exercises, plyometrics, balance exercise etc which helps to strengthen ligament with less str…
Patellar Tendonitis
It is most common condition in athletes whose does jumping. Other repetitive activities such as running, walking, or cycling may also lead to patellar tendonitis. These activities put repeated stress on the patellar tendon. This causes the tendon to become inflamed. This app consists of causes, symptoms, treatment for patellar tendinitis. It consis… | 677.169 | 1 |
I don't think you will find that a big problem. None of the main ideas of the book require any mathematical knowledge.
Some of the examples are mathematical in nature. For instance, there is about twelve pages in chapter 6 that concerns using an infinite list object to represent a sequence of increasingly accurate approximations to the solution of a certain financial problem. People who have studied calculus will immediately understand the ideas here; people who haven't might understand them anyway, and even if you don't you can always skip those examples; most of them have nothing in particular to do with math.
My own tendency is to write a lot of math stuff, because I find it very interesting, but while I was writing HOP I tried really hard to get rid of the mathematics, because I knew that a lot of people don't like | 677.169 | 1 |
Schaums Easy Outline of Linear Algebra Revised you are looking for a quick nuts-and-bolts overview, turn to Schaum's Easy Outlines! Schaum's Easy Outline of Linear Algebrais a pared-down, simplified, and tightly focused review of the topic. With an emphasis on clarity and brevity, it features a streamlined and updated format and the absolute essence of the subject, presented in a concise and readily understandable form. Graphic elements such as sidebars, reader-alert icons, and boxed highlights stress selected points from the text, illuminate keys to learning, and give you quick pointers to the essentials. Expert tips for mastering linear algebra Last-minute essentials to pass the course Appropriate for the following courses: Beginning Linear Algebra, Linear Algebra, Advanced Linear Algebra, Advanced Physics, Advanced Engineering, Quantitative Analysis Easily understood review of linear algebra Supports all the major textbooks for the linear algebra course | 677.169 | 1 |
Rated:
DVD Features:
Rated: G
Run Time: 20 minutes
Released: June 9, 2009
Originally Released: 2008
Label: am productions, llc
Encoding: Region 1 (USA & Canada)
Audio:
Dolby Digital 2.0 Stereo - English
Product Description:
This easy-to-follow teaching aid for algebra teachers explores the fundamental concepts of functions and relations with the use of a graphic calculator. The program leads viewers through a series of lessons, demonstrating the keystrokes involved in each example, and uses animations to illustrate ideas. | 677.169 | 1 |
News & Events
MA120 - Calculus and Analytical Geometry I
The study of the calculus begins with an
examination of the real number system and
the Cartesian plane. Additional topics to be
considered include: functions and their graphs,
limits and differentiation techniques, the mean
value theorem, application of the derivative,
indefinite integration, the trigonometric
functions. | 677.169 | 1 |
'll love that all of the saved plots are considered public domain; they can be used to create worksheets, presentations, or assessments without copyright concerns. You can also use FooPlot to provide inquiry opportunities for kids. Prior to learning about how a particular function looks and responds, give kids a chance to manipulate it using the Fooplot tools. Kids'll be able to figure out for themselves how changing a function changes the graph. Give them prompts like, "What happens to the shape of the graph when you plot an even exponent as compared to an odd exponent?"
Read MoreRead Less
What's It Like?
FooPlot is a free online tool that lets kids plot functions, polar equations, parametric equations, and points. It's pretty simple to use; you can layer different graphs on top of each other, and a tool bar lets you find intersection points and roots. Other tools let students trace points on a graph, move it, and zoom in or out. Once a plot is created, it can be exported as a PDF or other various file formats for later review. FooPlot could be used in high school math classes ranging from algebra to pre-calculus.
Read MoreRead Less
Is It Good For Learning?
One of the best things about FooPlot: It allows kids to quickly see multiple representations of the same mathematical idea. They immediately see that if they change the equation, the plot changes too. There certainly aren't any bells and whistles, but it's a serviceable tool that can be used in lieu of a graphing calculator. Most of the tools are easy to use, with the exception of the Zoom Box button, which doesn't respond consistently.
As with any tool, there are limitations. FooPlot only finds the roots or intersections for some functions; it won't find them for polar plot types. Since FooPlot uses Newton's method, it won't be able to find roots for certain functions, like those that exhibit fractal behavior or those you can't differentiate.
Read MoreRead Less
Key Standards Supported
Interpreting Functions
HSF.IF: Analyze Functions Using Different Representations
HSF.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★
Reasoning With Equations And Inequalities
HSA.REI: Represent And Solve Equations And Inequalities Graphically
HSA.REI.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
Trigonometric Functions
HSF.TF: Model Periodic Phenomena With Trigonometric Functions
HSF.TF.7
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.★ | 677.169 | 1 |
Algebra 1 - Solving and Graphing Compound Inequalities Practice
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PRODUCT DESCRIPTION
Algebra 1 - Solving and Graphing Compound Inequalities Practice
This is a basic activity that allows students to self check and work at their own pace. There are 20 problems total (10 graphing and 10 solving) and they are divided into 4 parts. Students start with part 1 and then check answers which are posted in the classroom. Once they understand that part they move on to the next. Teacher prep for this activity is printing each part on different colored paper and cutting into quarters. One class set of copies could be used for multiple classes | 677.169 | 1 |
consists of the elementary aspects of linear algebra which depend mainly on row operations involving elementary manipulations of matrices. The field of scalars is typically the field of complex numbers.
Content
Preface
Preliminaries
Sets And Set Notation
Functions
The Number Line And Algebra Of The Real Numbers
Ordered fields
The Complex Numbers
Exercises
Completeness of R
Well Ordering And Archimedean Property
Division And Numbers
Systems Of Equations
Exercises
Fn
Algebra in Fn
Exercises
The Inner Product In Fn
What Is Linear Algebra?
Exercises
Matrices And Linear Transformations
Matrices
Exercises
Linear Transformations
Subspaces And Spans
An Application To Matrices
Matrices And Calculus
Exercises
Determinants
Basic Techniques And Properties
Exercises
The Mathematical Theory Of Determinants
The Cayley Hamilton Theorem
Block Multiplication Of Matrices
Exercises
Row Operations
Elementary Matrices
The Rank Of A Matrix
The Row Reduced Echelon Form
Rank And Existence Of Solutions To Linear Systems
Fredholm Alternative
Exercises
Some Factorizations
LU Factorization
Finding An LU Factorization
Solving Linear Systems Using An LU Factorization
The PLU Factorization
Justification For The Multiplier Method
Existence For The PLU Factorization
The QR Factorization
Exercises
Linear Programming
Simple Geometric Considerations
The Simplex Tableau
The Simplex Algorithm
Finding A Basic Feasible Solution
Duality
Exercises
About the Author
Kenneth Kuttler received his Ph.D. in mathematics from The University of Texas at Austin in 1981. From there, he went to Michigan Tech. University where he was employed for most of the next 17 years. He joined the faculty of Brigham Young University in 1998 and has been there since this time. Kuttler's research interests are mainly in the mathematical theory for nonlinear initial boundary value problems, especially those which come from physical models that include damage, contact, and friction. Recently he has become interested in stochastic integration and the related problems involving nonlinear stochastic evolution equations.
This website uses cookies to improve user experience. By using our website you consent to all cookies in accordance with EU regulation. | 677.169 | 1 |
Popular Textbooks
Details about Discrete Mathematics:
This textbook provides an accessible introduction to discrete mathematics, using an algorithmic approach that focuses on problem-solving techniques. Each chapter has a special section dedicated to showing students how to attack and solve problems.
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Rent Discrete Mathematics 6th edition today, or search our site for other textbooks by Richard Johnsonbaugh | 677.169 | 1 |
Our A-level Maths (Further)
I am a teacher at a GCSE and Sixth Form college in Ealing.
I am a Cambridge Chemical Engineering graduate.
I have been tutoring since 2006.
I am the host of Education Bookcast, a podcast about books in education.
I also develop educational software.
All of my students improve by AT LEAST two grades. We always aim for A and A*. Book your free no obligation trial lesson before deciding to book any paid lessons. I can organise a lesson in my online classroom very quicklyA-level Maths (Further) Articles
Problem solving is a critical skill of our life whether in the examination, workplace or personal situations. A problem solving method is a method where pupils learn by working on problems. Since problems happen endlessly in our lives and need to be addressed, problem solving skills provide...
What should aspiring mathematicians be reading for inspiration before the start of their A Level course? I have been asked this question in recent years and would take along a selection of my favourite mathematical books to the taster sessions laid on for students in between completing their...
Although schools and tutors recommend certain books, there is no one particular book for whole of the GCSE and A level syllabus. Maths can and should be learnt from different places and a range of textbooks, as not all books are the same. I sometimes even ask students do questions from textbooks...
Pierre de Fermat, born 17th August 1601, laid the foundations of Probability Theory, did a lot of work on Number Theory, and is famous today for his "Fermat's Last Theorem" because he died before giving a proof.
In the margin of his copy of Diophantus' Arithmetica, Fermat wrote:
"To divide a...
Further Mathematics has always been of an exceptional standard at A-levels. The key to success in this subject is to try hard, and most importantly to do this effectively. I sat my Edexcel A-level Further Mathematics examinations in 2006 and encountered the same difficulties that most of you out...
Solving quadratic equations has been an important aspect of mathematics since Babylonian times, 3000 years ago. However, as you're no doubt aware, many quadratic equations can't be solved using real numbers – if the curve of the quadratic doesn't pass through (or at least touch) the... | 677.169 | 1 |
This section of a broader work, gives students a series of tutorial exercises in matrix multiplication. Topics include...
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This section of a broader work, gives students a series of tutorial exercises in matrix multiplication. Topics include matrix multiplication, vector multiplication, and the identity matrix. This page presents students with question that the enter answers to. Students are given feedback based on there entry Tutorial:Matrix Multiplication to your Bookmark Collection or Course ePortfolio
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'Brown University has two introductory linear algebra courses. This text is used in the honors course that emphasizes...
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'Brown University has two introductory linear algebra courses. This text is used in the honors course that emphasizes proofs. The book's title suggests that it is not the typical approach to linear algebra even among those books that are more theoretical.For example, the concept of a basis is treated as more fundamental than the concept of linear independence, and linear transformations are introduced before solving systems of linear equations. Especially noteworthy is the motivation and development of determinants. As the author states in the preface:I spent a lot of time presenting a motivation for the determinant, and only much later give formal definitions. Determinants are introduced as a way to compute volumes. It is shown that if we allow signed volumes, make the determinant linear in each column,… and assume some very natural properties, then we do not have any choice and arrive at the classical definition of the determinant Algebra Done Wrong to your Bookmark Collection or Course ePortfolio
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This course is divided into Part I and Part II. The first part of the course provides an introduction to statistical theory....
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This course is divided into Part I and Part II. The first part of the course provides an introduction to statistical theory. A brief review of probability will be given mainly as background material, however, it is assumed to be known. The second part of the course provides an introduction to regression analysis, focusing on specification (approximation) analysis and estimation and inference in large samples. Part II is not covered here, but you can see the material by looking at the previous version of 14.381 14.381 Statistical Method in Economics (MIT) to your Bookmark Collection or Course ePortfolio
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This course is a continuation of 18.014. It covers the same material as 18.02 (Multivariable Calculus), but at a deeper...
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This course is a continuation of 18.014. It covers the same material as 18.02 (Multivariable Calculus), but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral024 Multivariable Calculus with Theory (MIT) to your Bookmark Collection or Course ePortfolio
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This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical...
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This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics02SC Multivariable Calculus (MIT) to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material 18.02SC Multivariable Calculus (MIT)
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The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in...
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The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering03SC Differential Equations (MIT) to your Bookmark Collection or Course ePortfolio
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This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other...
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This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices Linear Algebra (MIT) to your Bookmark Collection or Course ePortfolio
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This course provides students with the basic analytical and computational tools of linear partial differential equations...
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This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix303 Linear Partial Differential Equations: Analysis and Numerics (MIT) to your Bookmark Collection or Course ePortfolio
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This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on...
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This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the Julia Language software337J Parallel Computing (MIT) to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material 18.337J Parallel Computing (MIT) to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
From angles to functions to identities - solve trig equations with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear - this hands-on-guide focuses on helping you solve the many types of trigonometry equations you encounter in a focused, step-by-step... more...
If you're preparing for The American Institute of Professional Bookkeepers' (AIPB) bookkeeping certification test, you need an easy- to-follow test-preparation guide that gets you up to speed quickly in all of the bookkeeping basics, from setting up a company's books and recording transactions to managing employee payroll, handling government paperwork,... more...
From genetics to ecology ? the easy way to score higher in biology Are you a student baffled by biology? You're not alone. With the help of Biology Workbook For Dummies you'll quickly and painlessly get a grip on complex biology concepts and unlock the mysteries of this fascinating and ever-evolving field of study. Whether used as a complement... more...
Gruber's Complete SAT Math Workbook includes shortcuts and tips, strategies and practice tests. It also features the exclusive Gruber system that has raised actual SAT scores by more than 600 points! more...
Helpful instruction and plenty of practice for your child to master the basics of vocabulary Having a strong grasp of vocabulary is essential for your child to read and write with confidence. Practice Makes Perfect: Mastering Vocabulary gives your child bite-sized explanations of the subject, with engaging exercises that keep her or him motivated... more...
Balance the books -- without all the headaches! Trying to get certified and become an accountant? Own a small business but need a little help balancing your books? Don't worry! This hands-on guide provides the learning and vital practice you need to master important accounting concepts and basics. Perfect as a companion workbook for Accounting For... more...
Helpful instruction and plenty of practice for your child to understand the basics of vocabulary Understanding vocabulary is essential for your child to read and write with confidence. Practice Makes Perfect: Exploring Vocabulary gives your child bite-sized explanations of the subject, with engaging exercises that keep her or him motivated and... more...
Helpful instruction and plenty of practice for your child to understand the basics of multiplication and division Understanding multiplying and dividing is essential for your child to do math problems with confidence. Practice Makes Perfect: Multiplication and Division gives your child bite-sized explanations of the subjects, with engaging exercises... more...
Modern Russian Grammar Workbook is an innovative book of exercises and language tasks for all learners of Russian. The book is divided into two parts. Section 1 provides exercises based on essential grammatical structures. Section 2 practises everyday functions such as making introductions, apologizing and expressing needs. A comprehensive... more... | 677.169 | 1 |
TExES Mathematics-Physics 8-12 143 eBook
From understanding the real number system and attributes of functions to identifying and analyzing forces of motion and magnetic materials' properties, this comprehensive study guide provides the core content found on the TEXES Mathematics-Physics 8-12 certification exam. Available for immediate download and formatted for onscreen reading, it covers the sub-areas of Number Concepts; Patterns and Algebra; Geometry and Measurement; Probability and Statistics; Mathematical Processes and Perspectives; Mathematical Learning, Instruction, and Assessment; Scientific Inquiry and Processes; Physics; and Scientific Learning, Instruction, and Assessment. Once you've mastered the content practice for the actual test with 125 sample questions that include full answer rationales. This digital study guide is available for immediate download and use and is formatted for comfortable onscreen reading.
XAMonline eBooks are viewed exclusively using FREE Adobe Digital Editions software. You can only use this eBook on devices that support Adobe Digital Editions software. | 677.169 | 1 |
The Canadian Open Mathematics Challenge (COMC) is Canada's premier national mathematics competition that is open to any student with an interest in and grasp of high school math. The purpose of the COMC is to encourage students to explore, discover, and learn more about mathematics and problem solving. The competition serves to provide teachers with a unique student enrichment activity during the fall term.
Approximately the top 50 students from the COMC will be invited to write the Canadian Mathematical Olympiad (CMO). Students who excel in the CMO will have the opportunity to be selected as part of Math Team Canada -- a small team of students who travel to compete in the International Mathematical Olympiad (IMO). Depending on grade level and performance, students participating in the COMC can also have opportunities to garner awards, to win prizes, to be considered for scholarships, and to get invited to camps.
Problem of the Week!
During the weeks prior to the COMC competition, the CMS is happy to present our Problem of the Week (POTW). Please visit our POTW page for past problems and solutions. Here is the latest problem:
This week we will look at an inscribed circle problem.
Let $ABC$ be an equilateral triangle and $\Gamma$ its incircle. If $D$ and $E$ are points on the sides $AB$ and $AC$, respectively, such that $DE$ is tangent to $\Gamma$, show that
\[
\frac{AD}{DB}+\frac{AE}{EC}=1
\]
McLean Foundation, CAE Inc., Centre de recherches mathématiques, Pacific Institute for the Mathematical Sciences, Fields Institute, Government of Manitoba, Government of Nova Scotia, Government of Ontario, Government of Prince Edward Island and Government of the Northwest Territories.
With the cooperation of:
University of British Columbia, University of Calgary, Dalhousie University, University of Manitoba, Memorial University, University of New Brunswick, University of Prince Edward Island, Department of Mathematics & Statistics (University of Saskatchewan), University of Toronto and York University.
To report errors or omissions for this page, please contact us at comc@cms.math.ca. | 677.169 | 1 |
The MathsFirst website has been
developed by Dr Tammy Smith and Dr Kee Teo in the
University's Institute of Fundamental Sciences. It is
designed to help students sharpen their maths before
enrolling in first-year papers at university, and for
secondary students seeking additional resources.
Dr Smith
says the success of first-year tertiary maths students
depends on their competency in algebra, and that it is
important that secondary school students are equipped with
the basic skills required for tertiary study.
Free to
access, the site contains a course map outlining first-year
maths papers and information about the level of knowledge
required to enrol in each course. Competency tests specific
to papers are available, allowing students to identify any
problem areas. From there, they can link to remedial work
offered online in the form of video tutorials and
interactive practice.
The tutorials use moving graphical
images, colour, and audio explanations of concepts. An
advantage of MathsFirst, and of online resources, is the
user's ability to navigate through tutorials in their own | 677.169 | 1 |
Geometry by Ron Larson(
Book
) 36
editions published
between
1995
and
2016
in
English
and held by
360 WorldCat member
libraries
worldwide
Essentials of geometry -- Reasoning and proof -- Parallel and perpendicular lines -- Congruent triangles -- Relationships
within triangles -- Similarity -- Right triangles and trigonometry -- Quadrilaterals -- Properties of transformations -- Properties
of circles -- Measuring length and area -- Surface area and volume of solids
Beginning algebra with applications by Richard N Aufmann(
Book
) 2
editions published
in
2004
in
English
and held by
30 WorldCat member
libraries
worldwide
Offers an integrated learning system organized by objectives that will provide your students with the steps and tools necessary
to succeed in your math course | 677.169 | 1 |
28 May 2015
Good material to start with.
Description
"Introduction to Vectors" takes learning to a new level by combining written notes with online video. Each lesson is linked with a YouTube video from award-winning teacher and best-selling author Dr Chris Tisdell, where he explains the material in an inspiring and engaging way. Vectors provide a fascinating tool to describe motion and forces in physics and engineering. They also provide a simple framework for understanding geometry. Download the book and subscribe to Dr Chris Tisdell's YouTube channel!
Preface
This workbook is designed to be used in conjunction with the author's free online video tutorials. Inside this workbook each chapter is divided into learning modules (subsections), each having its own dedicated video tutorial.
View the online video via the hyperlink located at the top of the page of each learning module, with workbook and paper or tablet at the ready. Or click on the Introduction to Vectors playlist where all the videos for the workbook are located in chronological order:
There has been an explosion in books that connect text with video since the author's pioneering work Engineering Mathematics: YouTube Workbook [31]. The current text takes innovation in learning to a new level, with all of the video presentations herein streamed live online, giving the classes a live, dynamic and fun feeling.
Content
The basics of vectors
Geometry of vectors
But, what is a vector?
How big are vectors?
Determine the vector from one point to another point
Vectors in Three Dimensions
Parallel vectors and collinear points example
Vectors and collinear points example
Determine the point that lies on vector: an example
Lines and vectors
Lines and vectors
Lines in R3
Lines: Cartesian to parametric form
Lines: Parametric and Cartesian forms given two points
Lines: Convert Parametric to Cartesian
Cartesian to parametric form of line
Planes and vectors
The span of a vector
Equation of plane: Parametric vector form
Planes: Cartesian to parametric form
Equation of plane from 3 points
Dot and cross product
What is the dot product?
Orthogonal vectors
Scalar Projection of vectors
Distance between a point and a line in R3
Cross product of two vectors
Properties of the cross product
What does the cross product measure?
Scalar triple product
What does the scalar triple product measure?
Equation of plane in R3
Distance between a point and a plane in R3
Bibliography
About the Author
"With more than a million YouTube hits, Dr Chris Tisdell is the equivalent of a best-selling author or chart-topping musician. And the unlikely subject of this mass popularity? University mathematics." [Sydney Morning Herald, 14/6/2012
Chris Tisdell has been inspiring, motivating and engaging large mathematics classes at UNSW, Sydney for over a decade. His lectures are performance-like, with emphasis on contextualisation, clarity in presentation and a strong connection between student and teacher.
He blends the live experience with out-of-class learning, underpinned by flexibility, sharing and openness. Enabling this has been his creation, freely sharing and management of future-oriented online learning resources, known as Open Educational Resources (OER). They are designed to empower learners by granting them unlimited access to knowledge at a time, location and pace that suits their needs. This includes: hundreds of YouTube educational videos of his lectures and tutorials; an etextbook with each section strategically linked with his online videos; and live interactive classes streamed over the internet.
His approach has changed the way students learn mathematics, moving from a traditional closed classroom environment to an open, flexible and forward-looking learning model.
Indicators of esteem include: a prestigious educational partnership with Google; an etextbook with over 500,000 unique downloads; mathematics videos enjoying millions of hits from over 200 countries; a UNSW Vice-Chancellor's Award for Teaching Excellence; and 100% student satisfaction rating in teaching surveys across 15 different courses at UNSW over eight years.
Chris has been an educational consultant to The Australian Broadcasting Corporation and has advised the Chief Scientist of Australia on educational policy. | 677.169 | 1 |
The Fourier Transform and its Applications is one of the ten free courses being offered to the public through Stanford...
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The Fourier Transform and its Applications is one of the ten free courses being offered to the public through Stanford Engineering Everywhere. The course belongs to the Linear Systems and Optimization series and is taught by Professor Brad Osgood, of Stanford University's Electrical Engineering Department. The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems Systems and Optimization: The Fourier Transform and its Applications to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Linear Systems and Optimization: The Fourier Transform and its Applications
Select this link to open drop down to add material Linear Systems and Optimization: The Fourier Transform and its Applications Logic Design (邏輯設計) to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Logic Design (邏輯設計)
Select this link to open drop down to add material Logic Design (邏輯設計 Mesh Analysis for DC Circuits to your Bookmark Collection or Course ePortfolio
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Micro Informatics focuses on initiating students to the basic of hardware and software.this course contains 4 labs:Lab1: In...
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Micro Informatics focuses on initiating students to the basic of hardware and software.this course contains 4 labs:Lab1: In this session, we learn the functioning principles of various computer peripherals, we unmount a desktop PC and have an overview of its internal componentsLab2: In this session we build a desktop PC from scratch and make it work, the we move on to another part where we discover the difference between a desktop PC and a laptop by dismantling a laptop and then rebuiding it againLab3: In this session we focus on software in 3 parts: we learn how to configure and access BIOS, in the way we learned the notion of virtualisation and how to create virtual machines, we used that in the second part to try installing Windows OS, in the final part, we saw some diagnosis tools that can be used to monitor the PC health and detect HW & SW problemsLab4: This session is an overview of embedded systems where we create electrical systems and control them using the arduino board that we program it in the session Informatics practical sessions to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Micro Informatics practical sessions
Select this link to open drop down to add material Micro Informatics practical sessions to your Bookmark Collection or Course ePortfolio
MIT's Open CourseWare Initiative is a "large-scale, Web-based electronic publishing initiative" funded by The William and...
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MIT's Open CourseWare Initiative is a "large-scale, Web-based electronic publishing initiative" funded by The William and Flora Hewlett Foundation, the Andrew W. Mellon Foundation, and MIT itself. It has been created in order to "Provide free, searchable,
Material Type:
Reference Material
Author:
MIT has made a long term commitment to the project though initial funds were granted by the Andrew M. Mellon and the William and Flora Hewlett Foundations
Date Added:
Dec 16, 2007
Date Modified:
Jun 29, 2016 your to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material MIT OpenCourseWare
Select this link to open drop down to add material MIT OpenCourseWare to your Bookmark Collection or Course ePortfolio
Ohm's Law describes the fundamental relationship between voltage, resistance, and current. This Formula Solver! Series...
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Ohm's Law describes the fundamental relationship between voltage, resistance, and current. This Formula Solver! Series program shows the step-by-step solution for using any two values (which you can enter yourself) to find the third Ohm's Law to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Ohm's Law
Select this link to open drop down to add material Ohm's Law to your Bookmark Collection or Course ePortfolio
In parallel circuits, power is a function of current and voltage. This Formula Solver! Series program shows the step-by-step...
see more
In parallel circuits, power is a function of current and voltage. This Formula Solver! Series program shows the step-by-step solution for using voltage and resistance to find current and power (all with your own values Parallel Circuit to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Power in a Parallel Circuit
Select this link to open drop down to add material Power in a Parallel Circuit to your Bookmark Collection or Course ePortfolio
In the world of electricity, power is the product of current and voltage. This Formula Solver! Series program shows the...
see more
In the world of electricity, power is the product of current and voltage. This Formula Solver! Series program shows the step-by-step solution for finding power, current, or voltage from the other two values (which you can enter yourself Simple Circuit to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Power in a Simple Circuit
Select this link to open drop down to add material Power in a Simple Circuit Print Free Graph Paper to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Print Free Graph Paper
Select this link to open drop down to add material Print Free Graph Paper to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
Details about Complex Variables:
Complex variables provide powerful methods for attacking problems that can be very difficult to solve in any other way, and it is the aim of this book to provide a thorough grounding in these methods and their application. Part I of this text provides an introduction to the subject, including analytic functions, integration, series, and residue calculus and also includes transform methods, ODEs in the complex plane, and numerical methods. Part II contains conformal mappings, asymptotic expansions, and the study of Riemann–Hilbert problems. The authors provide an extensive array of applications, illustrative examples and homework exercises. This 2003 edition was improved throughout and is ideal for use in undergraduate and introductory graduate level courses in complex variables.
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Rent Complex Variables 2nd edition today, or search our site for other textbooks by Mark J. Ablowitz. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cambridge University Press.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
Who knew how fun and useful exploring graphs could be? Graphs in Action explores different types of graphs and their practical applications and explains how graphs are used to record business growth and productivity. more...
In Hadamard Matrices and Their Applications , K. J. Horadam provides the first unified account of cocyclic Hadamard matrices and their applications in signal and data processing. This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago.... more...
As the school year comes to an end, the students in this book recall their favorite things and activities from the year. They use graphs and charts to record and show their favorites. Join them as they relive their experiences through the year. 32pp. more...
A big, summer storm left the school in this book a mess! The students, parents, and other volunteers worked to clean it up. They make teams to pick up aluminum cans, plastic bottles, and glass. After everything is cleaned up, they use graphs to show how much of each item they collected. They take these items to be recycled. They use the money they... more...
Tiling theory is an elegant branch of mathematics that has applications in several areas of computer science. The most immediate application area is graphics, where tiling theory has been used in the contexts of texture generation, sampling theory, remeshing, and of course the generation of decorative patterns. The combination of a solid theoretical... more...
The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the ?cannonball" packing. In a landmark result, this was proved... more...
This book begins with the basics of graph spectra for ordinary and Laplace and Seidel spectra, and adds material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and more. Includes exercises, tables and references. more...
Based on material from the 2006 NSF workshop on Large-scale Random Graphs, this book describes recent advances made in large scale networks. It includes detailed descriptions of various applications and explores areas for future research. more... | 677.169 | 1 |
Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
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Rent Student Solutions Manual for Van Dyke/Rogers/Adams' Fundamentals of Mathematics 10th edition today, or search our site for other textbooks by James Van Dyke. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cengage.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
Welcome to 2nd Semester Algebra and Geometry!
This course syllabus is meant to inform you of the upcoming and exciting units we will be learning about together. The tentative course outline for the 2nd semester is below. Please take a look!
For February, we will be continuing our work with Linear Quadratic systems of equations, with a focus on what the difference between linear functions and quadratic functions are.
New Course Systems: Homework Buddy System You will be given a partner for whom you will collect homework or get in touch with regarding missed quizzes/work. MVAmath outside course expectations MVAmath is an important part of your mathematics success this year. You are expected to complete all assignments! 2nd semester = no late credit. You do the assignment on time, OR you receive a zero. All Do Nows will now be graded and part of your overall 2nd semester grade. They will be counted as mini-quizzes and also contribute to your participation grade.
Ms. Sara promises to do a better job ..
I.
Giving graded work back in a timely manner.
In order for Ms. Sara to fulfill her promise I must _________________________________________________________________
II.
Giving you more of an indication of how you are doing in class (goal is biweekly)
In order for Ms. Sara to fulfill her promise I must _________________________________________________________________ | 677.169 | 1 |
two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderon-Zygmund and Littlewood-Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman-Meyer theory; Carleson's resolution of the Lusin conjecture; Calderon's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form. | 677.169 | 1 |
Technical Shop Mathematics by Thomas AchatzBasic Technical Mathematics With Calculus by Allyn J Washington is available now for quick shipment to any US location and this book can be substituted for ISBN 0133116530 or ISBN 9780133116533 or the more current 10th edition. You will save lots of cash by using this 9th edition which is nearly identical to the n
The Principles of Mathematics Book 2 focuses on the essential principles of algebra, coordinate graphing, probability, statistics, functions, and other important areas of mathematics; throughout, a biblical worldview focus is emphasized. Students will discover that all of math boils down to a way of describing God's world and is a useful means we can use to serve and worship Him. The Teacher Guide includes student worksheets, a weekly lesson schedule, quizzes, and an answer key. Worksheets include a variety of exercises including review, challenge questions, drill questions, fill-in-the-blanks , word problems, and more. Pages are organized by book, with answer keys in the back. The book page numbers and day number are on the top of the worksheets for easy assignment tracking. This course...
Less
This book focuses exclusively on K-8 mathematics, developing elementary mathematics at the level of teacher knowledge. Themes focus on how the nature of a mathematics topic suggests an order for developing it in the classroom, how topics are developed through 'teaching sequences', and how math builds on itself. Originally designed as a textbook for teachers, this book is divided into short sections, each with a single topic and homework set. The homework sets were designed with the intention that all or most of the exercises will be assigned; many of the questions involve solving problems in actual elementary school textbooks. Others involve studying the textbook - carefully reading a section of the book and answering questions about the mathematics being presented, with attention | 677.169 | 1 |
Most Discussed
Equator: Mac tools for physics students
updated 12:00 am EDT, Mon August 27, 2007
by MacNN Staff
Equator physics tool
Equator Software, a new company formed by the co-founder of ON Technology, has debuted its first software title dubbed Equator. The software allows high school and college-level physics students to easily navigate mathematical problems with its integrated word processor-like math editor, drawing palette, searchable formula reference library, algebra generator and drag-and-drop calculator. Equator records each step of students' work, along with figures and comments, collects all problems of an assignment into a single file and then generates homework-quality documents that can be printed.
With the "Math Symbols" feature, students can format mathematical expressions directly from the keyboard. For example, to type the letter pi, simply type "P - I - spacebar." The reference library offers a bank of standard equations and constants that can be inserted into the current document. The tool can also perform most algebra transformations using drag-and-drop. For example, to divide both sides of Area = πr2 by π, students double-click π to select it, then drag it to the opposite side of the equation.
Equator is available immediately for download, and can be used freely during a seven day trial period. The company offers a number of pricing options that include: $15.00 for a three-month student license, $40.00 for a nine-month student license, $140 for an unlimited personal-use license and $300.00 for a nine-month classroom license.
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TOTAL_COMMENTS Comments
It's not a bad tool from the looks of it, although this is built into Word which most students will have. Mac Office for students is under $200, and this is $139. The price seems to be a bit high for what this does | 677.169 | 1 |
Igcse Revision Guide for Mathematics
Book Description confide... More confidence to succeed in their examinations. Numerical answers are given at the back of the book, whilse full model answers to the exam questions are available free on our website. There is a handy checklist at the front of the book to aid revision planning, and a main vocabulary list has been included.
About Jean Matthews (Author) : Jean Matthews is a published author. Published credits of Jean Matthews include Igcse Revision Guide for Mathematics (Modular Maths for Edexcel), Gcse Mathematics a for Ocr Homework Book Higher (Gcse ... more View Jean Matthews' profile
About John Jeskins (Author) : John Jeskins is a published author. Published credits of John Jeskins include Igcse Revision Guide for Mathematics (Modular Maths for Edexcel), Gcse Mathematics a for Ocr Homework Book Higher (Gcse Ma... more View John Jeskins' profile
Videos
You must be a member of JacketFlap to add a video to this page. Please Log In or Register. | 677.169 | 1 |
Math lab gives options for homework help
The Math Lab at UVU has all the answers you need to pass your math class, but better than that they will give you tools to answer the questions yourself. Christina Ruth/UVU Review
For many students, math is difficult.
General education mathematic requirements notoriously stunt would-be graduates during their last year at UVU.
It can tarnish otherwise flawless grade point averages, and some students will even spend thousands of dollars to take courses off-campus that guarantee the fulfillment of these requirements.
The Math Lab, located in LA 201, is designed to help students who struggle with the subject. Through student tutoring, educational resources and other programs offered in a relaxed atmosphere, this free resource helps students understand mathematical principles at their own pace and apply them to course work.
"Students would be hard pressed to find a school that does as much as UVU does to help students be successful in mathematics," said Kathy Van Wagoner, the manager of the university's Math Lab. "There is plenty of help available for students willing to look for it."
The lab offers tutoring for students in any level of math, from basic through upper division courses.
There is even help for the calculus and statistics courses required by most departments for graduation.
"Here's how the Math Lab works," said Van Wagoner. "Come to the Math Lab and do your homework. If you are really struggling and need more intensive, one-on-one help, then there is free private tutoring, called personal tutoring, that you can sign up for at the front desk."
Students who miss class and are playing catch-up in their course can also check out video lectures at the Math Lab, which are available at the front desk.
While the student tutors are willing to help with questions, the tutors' job is not to teach. Tutors are there to guide and help a student process and learn information.
The Math Lab is not the only resource available to students. In a real-time "math-chat" called MathLabOnline, students who have questions about the subject can chat with a Math Lab tutor using an online whiteboard.
There are also several other locations where students can seek help. There are smaller math labs located in the Tutoring Commons in the Losee and Woodbury buildings, as well as at Wolverine Crossing Apartments on the fourth floor of building G. Math Tutors are also available in the Study Zone in the student center and even at the Wasatch Campus.
For more information on the Math Lab, online resources, hours and locations, visit their website at or visit the UVU Math Lab Blog at | 677.169 | 1 |
study of the geometric properties of solutions to polynomial equations, including solutions in dimensions beyond three. (Solutions in two and three dimensions are first covered in plane and solid analytic geometry, respectively.) Algebraic geometry emerged from analytic geometry after 1850 when topology, complex analysis, and algebra were used to study algebraic curves. An algebraic curve C is the graph of an equation f (x, y) = 0, with points at infinity added, where f (x, y) is a polynomial, in two complex variables, that cannot be factored. Curves are classified by a nonnegative integer—known as their genus, g —that can be calculated from their polynomial. The equation f (x, y) = 0 determines y as a function of x at all but a finite number of points of C. Since x takes values in the complex numbers, which are two-dimensional over the real numbers, the curve C is two-dimensional over the real numbers near most of its points. C looks like a hollow sphere with g hollow handles... | 677.169 | 1 |
Classroom Tips & Techniques: Dot and Cross Products
The standard course in multivariate calculus generally starts with a discussion of vectors, then moves to lessons on lines and planes in three dimensions. This article describes some explanatory calculations that can be used to shed additional light on the dot and cross products, important properties of vectors | 677.169 | 1 |
Kaseberg/Cripe/Wildman's respected INTRODUCTORY ALGEBRA is known for an informal, interactive style that makes algebra more accessible to students while maintaining a high level of mathematical accuracy. This new edition introduces two new co-authors, Greg Cripe and Peter Wildman. The three authors have created a new textbook that introduces new pedagogy to teach students how to be better prepared to succeed in math and then life by strengthening their ability to solve critical-thinking problems. This text's popularity is attributable to the author's use of guided discovery, explorations, and problem solving, all of which help students learn new concepts and strengthen their skill retention.
Additional Product Information
Features and Benefits
A list of Objectives is included at the beginning of the section. They serve as a summary for both students and instructors and coordinate with the titles of the Examples.
The Warm-Up at the beginning of each section is designed to serve as a class opener, reviewing important concepts, beginning exploration, and linking prior and upcoming topics. Warm-Ups tend to be skill-oriented; they generally connect to the algebra needed to solve text examples. The answers to the Warm-Up appear in the Answer Box at the end of the section.
Each Example begins with a title, which states the purpose of the Example. Usually these titles relate back to the objectives for the section. Several Examples link with others elsewhere in the book.
Included throughout the text, Explorations are another type of Example that is intended to be used in class for individual or group exploration.
Answers to the Warm-Up and some Examples, as well as the Practice questions, are placed in the Answer Box at the end of each section. By providing answers as feedback, the Answer Box permits the text to be used in class or as a laboratory manual for group work or independent study.
Exercises include questions on problem solving and Projects that relate to topics from prior chapters.
Instructor Extras are available in the Annotated Instructor's Edition in the margin next to examples. These additional examples can be written on the board for student groups, presented in class, or used as quiz or test questions.
A graphing calculator continues to be incorporated where appropriate, including the addition of more screenshots, and an appendix has been crafted to further support the natural integration of the calculator in the text.
Chapter Projects, included the Chapter Review Exercises at the end of each chapter, synthesize multiple concepts and teach problem-solving skills.
Every chapter ends with a Chapter Summary, Review Exercises, and a Chapter Test in order to review the Objectives presented in each chapter.
What's New
In order to preserve alignment of the content with the AMATYC standards but also to reflect the evolution of these standards over the years, content has been rethought and rearranged to focus on linear functions.
A chapter on Statistics, Chapter 10, has been added and is available online.
In Chapter 2, the more familiar exponent and order of operations topics are now placed before number properties.
In Chapter 3, solving equations with tables and graphs is now before solving with algebraic notation.
Solving formulas has been moved from Chapter 4 to Chapter 3
Chapters 4 and 5 have been exchanged, placing Ratios, Rates, and Proportional Reasoning before Functions, Linear Equations, and Linear Inequalities.
Percents and Right Triangle Trigonometry have been given individual sections in Chapter 4.
Averages, formerly with ratios and rates material, are now in Chapter 10, "Statistics."
Practice problems have been added throughout the text, usually following key Examples, giving students an opportunity to practice the concept demonstrated in the preceding Example. Clear explanations and accompanying practice problems are integrated throughout and serve to reinforce concept comprehension. Answers to the Practice problems appear in the Answer Boxes at the end of the section.
Reading Questions, which appear after the Answer Boxes, encourage students to examine the nuances of the vocabulary and to recognize that the language of algebra and its vocabulary is an essential part of comprehension. As a result, students are encouraged to discuss their results with classmates when they return to the next class. Answers to the Reading Questions can be found in the Annotated Instructor's Edition only.
Activities, located at the end of each section, provide opportunities for students to do meaningful and interactive mathematics during class. The activities are paired or group exercises meant for exploration, summation, or extension of basic ideas. In many sections, the Activity reflects how the mathematics applies to things of interest to the student.
Projects, located at the end of the Exercises, can serve as another opportunity for in-class activities or can be assigned as homework. All have been thoroughly evaluated, some revised, and others added as brand new.
The number of word problems and applications has been increased to replace some of the skill problems.
Exclusively from Cengage Learning, Enhanced WebAssign® combines the exceptional Mathematics content that you know and love with the most powerful online homework solution, WebAssign®. Enhanced WebAssign® engages students with immediate feedback, rich tutorial content, and interactive eBooks helping students to develop a deeper conceptual understanding of their subject matter. Online assignments can be built by selecting from thousands of text-specific problems or supplemented with problems from any Cengage Learning textbook.
Alternate Formats
Choose the format that best fits your student's budget and course goals
Instructor Supplements
This manual contains complete solutions to all the problems in the text.
Solutions Builder
(ISBN-10: 1133725732 | ISBN-13: 9781133725732)
This online instructor database offers complete worked solutions to all exercises in the text, allowing you to create customized, secure solutions printouts (in PDF format) matched exactly to the problems you assign in class. For more information, visit Student Supplements
Go beyond the answers--see what it takes to get there and improve your grade! This manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives you the information you need to truly understand how these problems are solved.Alice KasebergGreg CripePeter Wildman | 677.169 | 1 |
This kit of student materials is designed to be used alongside the sold-separately Stewardship Instruction Pack and CANNOT be used independently. The Student Text provides four worksheets for each of the lessons in the instruction manual. Questions are not multiple-choice and must be supplied by the student. They include practical, math-based problems; questions that ensure mastery of basic financial terms, forms, and practices; and questions that encourage students to think about biblical principles and applications. The Test Booklet provides 30 end-of-chapter tests, one for each lesson. Comprehensive tests quiz students on the practical, mathematical, and biblical portions of the lesson. Answers are not included in this pack, but can be found in the manual sold with the Instruction Kit...
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Click here to take a guided tour of the Second Edition Classroom Assessment for Student Learning, 2/e is a combination textbook and workbook grounded in research shown to increase student motivation and learning through improved classroom assessment. This user-friendly, practical book is full of real-world examples of what assessment for learning looks like in today's classrooms. Presented in a format appropriate for use by individuals or collaborative learning teams, the book teaches two central concepts: How to create accurate classroom assessments of all types and how to integrate assessment with instruction day to day, with a focus on student involvement. The companion CD provides all necessary learning team resources: suggestions for forming and facilitating learning teams,...Guide your students to mastering advanced algebraic concepts with BJU Algebra 2, 3rd Edition! Students will learn concepts such as linear, quadratic, polynomial, radical, and rational functions, exponential and logarithmic functions, and probability and statistics. Complex numbers are also covered throughout the text. Each lesson develops key concepts and detailed examples with practical applications integrated throughout. Exercise sets are graded by difficulty level and each lesson's cumulative review exercises will help students retain previously learned information. This third edition features two new chapters that cover matrix features and sequencing and series. New textbook features include selected Internet keyword searches for helping students locate online tools and enrichment,...This curriculum for high school students is designed to provide students with a thorough understanding of all important physical, mental and social health issues. Written directly to teens from a caring and friendly perspective, Total Health presents a moral basis for a healthy lifestyle based upon scriptural principles. Anatomy, Body systems, food choices, exercise, diseases, stress, emergencies, responsibility, and more. Illustrations and photographs add visual interest to the text; at the end of each chapter, a review contains terms to define, questionsbased on the text, biblical application questions and suggested activities. 464 pages, indexed. Softcover perfect privacy shield for students, these durable cardboard Student Mini Offices include clear vinyl pockets for storing homework, note cards, word of the day strips and more. An excellent teacher resource for creating a private environment during test taking, these sturdy mini desk offices can be used year after year for any subject. (3 pcs. per set) Unfolded, 40 x 12 OTC Not only do these help give my students privacy, but they help me differientiate for each individual student. These mini-offices help me aid struggling students. While many mini-offices are homemade and laminated, these offer me the ability to change the information as the needs of my students changeAmy, Educational Product Development Specialist
Designed for high school students who already have an understanding of how to write a general or persuasive essay, this course will instruct students in conducting research, citing research, evaluating sources, outlining, drafting, arrangement, writing with an objective style, addressing counterarguments, and more. Student Book lessons are primarily divided between beginning research papers and intermediate research papers, with one major paper prompt assigned for each. The chapter-based exercises include rubrics, checklists, fill-in-the-blank note outlines, and short writing exercises which are designed to teach skills students will need to apply in their papers. A final advanced essay allows students to tie together their newly learned skills, which completes the course. The teacher's...
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Structured to build student skills by progressively introducing more detail and more difficult concepts, this workbook provides a detailed introduction to the basics of how to write a research paper. Divided into Beginning, Intermediate, and Advanced sections, individual chapters focus on the prompt, note- taking/research, sources, construction, drafting, and style. This guided approach helps to break down the many steps involved in writing research papers into a manageable, understandable progression of steps to take. Detailed lessons integrate multiple examples and writing samples, which are followed by workbook exercises that help students to apply newly-learned skills. At the end of the three levels, students write a paper, eventually integrating all concepts. Designed for students...
LessThis student workbook set includes everything your student needs to get the consistent practice needed for understanding math. The two workbooks include over 130 worksheets as well as assessment forms. Twenty-six pages of color-coded fact cards provide plenty of addition, subtraction, multiplication, and division review.Get everything you need to teach writing to students in grades 3-5 with this IEW value pack that includes both the teacher's Teaching Writing: Structure and Style course and the Student Writing Intensive Level A course stories, producing reports & critiques, developing essays, translating thoughts and...
LessThis customizable Student Development Transit System Map Print is designed on the poster and would interest those who like student development theory, student affairs, student development, higher education, psychology, development, and theory stuff.
It's one thing to know the story of the Bible; it's another thing to live it. In each of the 30 three-minute teachings from author Randy Frazee, teens will learn how the story of the Bible helps them think, act, and become more like Jesus by examining the key concepts of Christianity. The DVD contains 30 video sessions, reproducible Teacher's Guide with Reading Scripts, and reproducible student handouts.BJ Culture is taught as well. 352 pages, hardcover. This resource is also known as Bob Jones French 2 Student Text.
Prepare your student for responsible citizenship with a discussion of the principles and mechanics of a constitutional republic. Material discussed includes the Constitution, the three branches of government, political parties, elections, foreign policy, and more, all from a biblical perspective. The material can be taught in one or two semesters. 434 pages, softcover. Non-consumable; answers should be written outside of the book. This 3rd edition features a new chapter on state and local governments as well as Christian worldview boxes that are designed to help students think scripturally and critically about governmental issues.This resource is also known as Bob Jones American Government Student Text, Grade 12, 3rd Edition.Designed for additional students using the Phonics Museum Grade 1 Program, this package includes an individual first grade student kit with the primers. This kit includes a workbook, an iron-on, and the primers.
Exceptional Students by Ronald Taylor is available now for quick shipment to any U.S. location. This edition can easily be substituted for ISBN 0078110505 or ISBN 9780078110504 the 2nd edition or 2014 edition or even more recent edition. You will save lots of cash by using this 2008 edition which is nearly identic
(Grade 2 Student Edition) The hardcover Student Editions include age-appropriate songs, music, artwork, and activities that will engage students in active learning. Different styles of music are introduced through creative performing, listening, and movement activities designed to stimulate all types of learners. Table of Contents: Publisher: Alfred Composer/Author: Artist: Arrange r/Editor: Catalog Number: EMC2002 Pages: 208
Winner of the 2012 ECPA Book of the Year Award The ESV Student Study Bible is ideally suited for students who are serious about God's Word--who want to learn more about what the Bible teaches and how the Bible applies to all of life. With 12,000 clear, concise study notes, the ESV Student Study Bible provides numerous features-- including nearly 900 Did You Know? facts, 120 Bible character profiles, and 15 new topical articles. It also features a glossary of key terms, more than 80 maps and illustrations, an extensive concordance, and 80,000 cross- references. These and many other features make it the most comprehensive and content-rich student Bible available today. Suited to high school and college students, the ESV Student Study Bible is also a versatile resource for anyone engaged in...
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The Student Writing Intensive course from the Institute of Excellence in Writing is based on the IEW teacher's seminar Teaching Writing: Structure and Style, and provides a solid foundation of writing skills through direct writing instruction. This Level B course is designed for middle school students in Grades 6-8 and presumes no previous experience in the IEW writing method; it differs from the highersuggested schedule is 30 weeks; doubling lessons allows course to be completed in half the time), and was recorded live with Andrew...
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Picking up where WriteShop I left off, WriteShop II will expand your students' writing abilities from individual descriptive and informative compositions to an engaging narrative made up of multiple writing styles, as well as new styles such as persuasive writing. Lessons teach students to write multiple types of essays, helping them to learn how to expression an opinion, develop, and support it with facts, details, and examples. Students write one composition over a two-week period, revising it twice. Incremental in approach, exercises are designed to be approached consecutively, building upon previously learned material. Working through assignments by brainstorming, sloppy copy writing, first revision, a hand-in, and a final draft, students learn to edit their papers methodologically, ...Flexible and creative, WriteShop is an ideal homeschool writing program! WriteShop lessons give students a chance to practice various kinds of writing, including creative writing, expository writing, and narrative writing. Assignments teach descriptive and informative paragraphs as well as first-and third-person narratives, important techniques of style, and self-editing. Students learn to choose vivid, descriptive words and vary their sentences in order to build strong paragraphs. With each lesson, they learn and practice new skills, apply them to their current writing project, and edit and revise the composition several times. By the end of the lesson--after brainstorming, writing, editing, and revising-- they'll have a polished paper to be proud of! Students write one composition over...The Student Writing Intensive course from the Institute of Excellence in Writing is based on the IEW teacher's seminar Teaching Writing: Structure and Style, and provides a solid foundation of writing skills through direct writing instruction. This Level C course is designed for high school students and presumes no previous experience in the IEW writing method; it differs from the lowerat 1-2 weeks per lesson), and was recorded live with Andrew Pudewa. It teaches students how to use a wide range of structural models and stylistic | 677.169 | 1 |
Intermediate Algebra offers a practical approach to the study of intermediate algebra concepts, consistent with the needs of today's student. The authors help students to develop a solid understanding of functions by revisiting key topics related to functions throughout the text. They put special emphasis on the worked examples in each section, treating them as the primary means of instruction, since students rely so heavily on examples to complete assignments. Many of the applications (both within the examples and exercises) are also uniquely designed so that students have an experience that is more true to life--students must read information as it appears in headline news sources and extract only the relevant information needed to solve a stated problem. This promotes the text's focus on developing better study habits, problem solving and critical thinking skills along with orienting students to think and reason mathematically. Through Intermediate Algebra, students will not only be better prepared for future math courses, they will be better prepared to solve problems and answer questions they encounter in their own lives.
Available with InfoTrac® Student Collections
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Features and Benefits
Applications with data in "as is" form. Many of the application exercises and examples reference information taken directly from headline news, research studies, or reports. The excerpts contain information that is relevant to solve the problems as well as extraneous information that requires students to read and think critically, rather than randomly "plug and chug." No bigger than "texts" or "tweets," these authentic excerpts help students develop the skill of using information found in every day media to solve problems and make decisions not only in Intermediate Algebra, but in their own lives.
Five Steps for Problem-Solving. Based on the work of George Polya, the Five Steps provide a framework for jump starting and organizing student problem solving. Step-by step worked examples throughout the textbook support students as they learn to solve a wide range of applications.
Success in College Mathematics. To help students transition to college-level mathematics, Success in College Mathematics appear at the beginning of each chapter and address topics such as personal responsibility, study skills, and time management. Follow-up exercises appear at the end of each section to help students reflect on their own attitudes and habits and how they can improve their performance.
Examples with Step-by-step Explanation. Each step of the worked examples is accompanied by an annotation that explains how the solution progresses, from the first line to the final answer. Color-coding and boldface are also used to help students easily identify the operation that occurs in each step, enabling the text to act as tutor when students are not in the classroom.
Side-by-Side Examples. To help students apply their knowledge of arithmetic to algebra, some of the worked examples are set up side-by -side with shared annotations. In this format, students can more readily see how certain principles and procedures are used repeatedly in both arithmetic and algebra.
Practice Problems. Following a set of worked examples in each section are a set of Practice Problems that mirror the examples. Students can use the Practice Problems to check their understanding of the concepts or skills presented.
Problem Solving: Practice and Review. Found in the majority of the section-ending exercise sets are a set of four applications-based exercises called Problem-Solving: Practice and Review. Because these problems do not necessarily involve the concepts or skills taught in the section, students need to think critically about the information and relationships in the problem.
Find the Mistake. The section-ending exercise sets include Find the Mistake exercises. Each exercise is a problem and a step-by-step solution which includes one mistake. Students are asked to identify the error and then rework the problem correctly. These exercises help students learn how to find errors in their own work which improves student persistence and self-sufficiency.
Learning the Language of Math. Vocabulary matching exercises that appear before the section-ending exercises help students improve their knowledge of vocabulary and notation.
Checking for Accuracy and Reasonableness. To be successful in mathematics and other disciplines, an important skill for students to develop is checking their work for reasonability and accuracy. The explanations of reasonability in examples of the Five Steps and the checking of solutions in many of the worked examples provide a model for students learning to reflect on their own work. Being able to justify an answer helps students develop confidence and self-sufficiency.
Using Technology. For instructors who wish to integrate calculator technology, optional scientific and graphing calculator instruction is provided at the end of selected sections, where appropriate. The instruction that appears within the Using Technology boxes includes examples with keystrokes, screen shots, and Practice Problems. Follow-up Technology exercises appear at the end of section-ending exercises for continued practice.
End-of-Chapter Study Plan and Review Exercises. This tool offers students an effective and efficient way to prepare for quizzes or exams. The Study Plan, appearing at the end of each chapter, includes study tables for each section that will help students get organized as they prepare for a test or quiz. Each table contains a set of Can I? questions, based on the section objectives, for self-reflection. Can I? questions are tied to one or more Review Exercises that students can use to test themselves. Examples and/or Practice Problems are also linked to each question should students need reference for review.
Real Sources. Among the application problems appearing in this textbook are specially developed exercises and examples that reference information taken directly from news articles, research studies, and other fact-based sources. Many of these applications are set up in two parts with a problem statement given first followed by an excerpt. Since the question appears first and then the information needed to answer the question follows, students experience problem solving in a more true-to-life way. Though similar in length to texts or tweets, the authentic excerpts often contain more information than is needed to solve the applications. Students will need to think critically to select the relevant information and in doing so, practice the skills needed to solve problems outside of the classroom.
Unique Online Homework Problems. Consistent with the authors' approach, unique features and problem types from the text have been integrated into the Enhanced WebAssign® online homework course that accompanies the text including the 5-Steps framework, Success in College Math, Find the Mistake and Solving Equation Exercises with Checks. Online assignments can be built by selecting from thousands of text-specific problems or can be supplemented with Extra Problems have been carefully selected from other Cengage Learning developmental mathematics series. Enhanced WebAssign engages students with immediate feedback, rich tutorial content and interactive, fully customizable eBooks (YouBook) helping students to develop a deeper conceptual understanding of their subject matter and access at:
What's New
Functions early and often. The authors introduce several key concepts related to functions in Chapter 3 and then revisited in more depth in later chapters, where appropriate. The concepts include domain, range, maximum, minimum, zeros, evaluating functions, and translating functions. These concepts appear as soon as students complete their review of elementary algebra and their initial work with polynomials.
Efficacy and Outcomes
Reviews
"I really like [Bracken and Miller's Intermediate Algebra]…. The book's composition is well-thought of and it is evident that a lot of expertise and experience has been interjected in its composition. The book delivers on what it promises in every aspect."
— Shahrokh Parvini, San Diego Mesa University
"The exposition is very good, with nice examples"
— Alex Kolesnik, College of the Desert
"[Bracken and Miller's Intermediate Algebra] deserves high marks for clarity, explicitness and readability."
— Haile Haile, Minneapolis Community andSample Chapters
(ISBN-10: 113394325X | ISBN-13: 9781133943259)
Solutions Manual
(ISBN-10: 1285092163 | ISBN-13: 9781285092164)
The Complete Solutions Manual provides worked-out solutions to all of the problems in the text (or
DVD) along with the PowerPoint versions of the parallel Classroom Examples available in the Annotated Instructor's Edition. A graphing calculator appendix detailing specific calculator commands also appears along with additional support materials for use specifically with the textbook youLaura BrackenEd Miller | 677.169 | 1 |
05216197val Track and Other Permutation Puzzles: And Just Enough Group Theory to Solve Them (Classroom Resource Materials)
Popular puzzles such as the Rubik's cube and so-called oval track puzzles give a concrete representation to the theory of permutation groups. They are relatively simple to describe in group theoretic terms, yet present a challenge to anyone trying to solve them. John Kiltinen shows how the theory of permutation groups can be used to solve a range of puzzles. There is also an accompanying CD that can be used to reduce the need for carrying out long calculations and memorising difficult sequences of moves. This book will prove useful as supplemental material for students taking abstract algebra courses. It provides a real application of the theory and methods of permutation groups, one of the standard topics. It will also be of interest to anyone with an interest in puzzles and a basic grounding in mathematics. The author has provided plenty of exercises and examples to aid study | 677.169 | 1 |
In this number we have interesting studies regarding mathematics teacher training, using algebra and proof to discover "mathematics", analyzing the impact of using graphical artifacts solving statistical problems, and how to use APOS theory to analyze students' understanding of integration. | 677.169 | 1 |
Applying Exponential Functions
Videos, worksheets, games and activities to help PreCalculus students learn how to apply exponential functions.
Applications of Exponential Functions (Part 1)
A number of examples of how exponential functions apply to exponential growth of bacteria, half-life of elements (exponential decay), the exponential decay of the value of new cars, or the growth rate of classic cars. The value of compound interest is also discussed.
Applications of Exponential Functions (Part 2)
Errata: the video shows the amount of Berillium-11 left one hour later instead of one hour ago.
Applications of Exponential Functions (Part 3 | 677.169 | 1 |
A First Course in Mathematics Concepts for Elementary School Teachers
A First Course in Mathematics Concepts for Elementary School Teachers by Marcel B. Finan
Publisher: Arkansas Tech University 2006 Number of pages: 452
Description: Problem-solving is the cornerstone of school mathematics. The techniques discussed in this book should help you to become a better problem solver and should show you how to help others develop their problem-solving skills. | 677.169 | 1 |
Synopsis
Essentials of Elementary School Mathematics is an introductory text on the essentials of mathematics taught in elementary schools. It presents a systematic development of the mathematics of arithmetic. A primary objective is to give students a background sufficient to understand and answer at an appropriate level the various quite penetrating questions asked by young students. Some examples and exercises are concerned primarily with pedagogical aspects of arithmetic.
Comprised of 14 chapters, this book begins with an overview of the language of mathematics, focusing on concepts such as the conjunction (and); negation (not); disjunction (or); and conditional (if...then...). The discussion then turns to the theory of sets; the concept of binary operations; and recognition and identification of properties of various relations. The next section deals with the number systems of arithmetic: whole numbers, integers, rational numbers, and real numbers. Number theory and clock arithmetic are also examined, along with counting techniques and probability. The final section is devoted to motion geometry and analytic geometry.
This monograph should be of interest to students and teachers of mathematicians at the elementaryentials of Elementary School Mathematics
by Larsen, Max | 677.169 | 1 |
Resource Added!
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Lesson Plan
Description:
Subjects:
Mathematics > General
Mathematics > Algebra
Education Levels:
Higher Education
Graduate
Undergraduate-Upper Division
Undergraduate-Lower Division
Keywords:
intermediate algebra
Language:
English
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Elementary Algebra
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Builds on the author's tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students' proficiency and conceptual understanding of algebra. This text continues coverage and integration of geometry in examples and exercises.
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Builds on the author's tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students' proficiency and conceptual understanding of algebra. This text continues coverage and integration of geometry in examples and exercises | 677.169 | 1 |
Make studying statistics simple with this easy-to-read resource Wouldn't it be wonderful if studying statistics were easier? With U Can: Statistics I For Dummies, it is! This one-stop resource combines lessons, practical examples, study questions, and online practice problems to provide you with the ultimate guide to help you score higher in your... more...
About the book? In honor of Edgar Enochs and his venerable contributions to a broad range of topics in Algebra, top researchers from around the world gathered at Auburn University to report on their latest work and exchange ideas on some of today's foremost research topics. This carefully edited volume presents the refereed papers of the participants... more...
Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounterWhat could be better than the bestselling Schaum's Outline series? For students looking for a quick nuts-and-bolts overview, it would have to be Schaum's Easy Outline series. Every book in this series is a pared-down, simplified, and tightly focused version of its predecessor. With an emphasis on clarity and brevity, each new title features a streamlined... more...
This book contains nine refereed research papers in various areas, from combinatorics to dynamical systems, with computer algebra as an underlying and unifying theme. Topics covered are irregular connections, summability of solutions and rank reduction of differential systems, asymptotic behaviour of divergent series, integrability... more...
This insightful book combines the history, pedagogy, and popularization of algebra to present a unified discussion of the subject. Classical Algebra provides a complete and contemporary perspective on classical polynomial algebra through the exploration of how it was developed and how it exists today. With a focus on prominent areas such as the numerical... more... | 677.169 | 1 |
Mathematics - Course Goals, Objectives, and Outcomes
MA 3345 Real Analysis I
Course Goals: The course will develop a deeper and more rigorous understanding of Calculus including defining terms and proving theorems about functions, sequences, series, limits, continuity, derivatives, the Riemann integrals, and sequences of functions. The course will develop specialized techniques in problem solving.
Course Objectives:
prove basic set theoretic statements and emphasize the proofs' development
prove various statements by induction and emphasize the proofs' development
define the limit of a function at a value, a limit of a sequence, and the Cauchy criterion
prove various theorems about limits of sequences and functions and emphasize the proofs' development
define continuity of a function and uniform continuity of a function
prove various theorems about continuous functions and emphasize the proofs' development
define the derivative of a function
prove various theorems about the derivatives of functions and emphasize the proofs' development
define a cluster point and an accumulation point
prove the Bolzano-Weierstrass theorem, Rolle's theorem, extreme value theorem, and the Mean Value theorem and emphasize the proofs' development
define Riemann integrable and Riemann sums
prove various theorems about Riemann sums and Riemann integrals and emphasize the proofs' development
Course Outcomes: Students will be able to
prove a basic set theoretic statement
prove an appropriate statement by induction
define the limit of a function at a value, a limit of a sequence, and the Cauchy criterion | 677.169 | 1 |
Details about Elementary Technical Mathematics:
Elementary Technical Mathematics Tenth Edition was written to help students with minimal math background prepare for technical, trade, allied health, or Tech Prep programs. The authors have included countless examples and applications surrounding such fields as industrial and construction trades, electronics, agriculture, allied health, CAD/drafting, HVAC, welding, auto diesel mechanic, aviation, natural resources, and others. This edition covers basic arithmetic including the metric system and measurement, algebra, geometry, trigonometry, and statistics, all as they are related to technical and trade fields. The goal of this text is to engage students and provide them with the math background they need to succeed in future courses and careers.
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Rent Elementary Technical Mathematics 10th edition today, or search our site for other textbooks by Dale Ewen. Every textbook comes with a 21-day "Any Reason" guarantee. Published by CENGAGE Learning. | 677.169 | 1 |
IntermediateIntermediate Algebra is 1-semester gateway course to other college-level mathematics courses. The goal of the Intermediate Algebra course is to provide students with the mathematical skills that are prerequisites for courses such as College Algebra, Elementary Statistics, Liberal-Arts Math and Mathematics for Teachers.
Preface Chapter R
Real Numbers and Algebraic Expressions
All the Arithmetic Yoursquo;ll Need
Success in Mathematics
What to Do the First Week of the Semester
What to Do Before, During, and After Class
How to Use The Text Effectively
How to Prepare for an Exam
Sets and Classification of Numbers
Use Set Notation
Know the Classification of Numbers
Approximate Decimals by Rounding or Truncating
Plot Points on the Real Number Line
Use Inequalities to Order Real Numbers
Operations on Signed Numbers; Properties of Real Numbers
Compute the Absolute Value of a Real Number
Add and Subtract Signed Numbers
Multiply and Divide Signed Numbers
Perform Operations on Fractions
Know the Associative and Distributive Properties of Real Numbers
Order of Operations
Evaluate Real Numbers with Exponents
Use the Order of Operations to Evaluate Expressions Algebraic Expressions | 677.169 | 1 |
This title provides numerous exercises, worked examples and clear explanations with questions and diagrams. Colour is used to highlight key mathematical elements and enhance learning. Margin notes provide extra support for key topics and formulas (a key formulas page is also included). Review and Technique exercises; Contextual questions; Consolidation 'A' and 'B' exercises and Applications and Activities provide a complete range of challenges and exam practice for complete success. Chapter overviews and summaries consolidate understanding | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Prologue: Literacy for the Modern World. Logic and Problem Solving: Thinking Critically; Approaches to Problem Solving. Quantitative Information in Everyday Life: Numbers in the Real World; Managing Money. Probability and Statistics: Statistical Reasoning; Putting Statistics to Work; Probability: Living with the Odds. Modeling: Exponential Astonishment; Modeling Our World; Modeling with Geometry. Further Applications: Mathematics and the Arts; Mathematics and Politics; Mathematics and Business.
For all readers interested in a quantitative reasoning approach to using and understanding mathematics.
Synopsis
Books à la Carte are unbound, three-hole-punch versions of the textbook. This lower cost option is easy to transport and comes with same access code or media that would be packaged with the bound book. | 677.169 | 1 |
Geometry from Africa
(Classroom Resource Materials94
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About the Book
This book draws on geometric ideas from cultural activities from Subsaharan Africa, and demonstrates how they may be explored to develop mathematical reasoning from school level through to university standard. Paulus Gerdes provides a thoroughly illustrated and researched exploration of mathematical ideas, motifs and patterns. Many important mathematical points are brought to the fore, not via the formal 'theorem-proof' method, but in a more schematic and diagrammatic manner. African artifacts, oral traditions, sand drawing and other forms of artwork with a geometric basis, all provide mathematical ideas for discussion in this unique book. Mathematicians and teachers of mathematics at all levels will be fascinated, as will anybody with an interest in African cultures. | 677.169 | 1 |
This section contains free e-books and guides on Basic Algebra, some of the resources in this section can be viewed online and some of them can be downloaded.
This book is written in such a way as to maintain
maximum flexibility and usability. This textbook will effectively enable
traditional or nontraditional approaches to elementary algebra. Topics covered
includes: Real Numbers and Their Operations, Linear Equations and Inequalities,
Graphing Lines, Solving Linear Systems, Polynomials and Their Operations,
Factoring and Solving by Factoring, Rational Expressions and Equations, Radical
Expressions and Equations, Solving Quadratic Equations and Graphing Parabolas.
This book covers the following topics: Subnormal
and Normal series, Invariant Series and Chief Series, Commutator Subgroup,
Central series and Field extensions, Field Extensions and constructions,
Algebraic Extension and Transcendental Extensions, Roots Of Polynomials, Simple
Extensions, Construction By Straight Edge and Compass, Symmetric Rational
Functions, Normal Extension, Galois Theory and Finite Fields.
First chapter explains the basic arithmetic and algebraic properties of
the familiar number systems the integers, rational numbers, real numbers, and
the possibly less familiar complex numbers. Second chapter introduces some basic
ideas from Number Theory, the study of properties of the natural numbers and
integers.3rd chapter, describes algebra of polynomials in a variable x with
coefficients in a commutative ring S.
This book covers the following
topics: Multiplication of monomial expressions, Factors found by rearrangement
and grouping of terms, Bemainder Theorem, Discussion of roots of a quadratic
equation, Irrational equations, nationalizing factors, Radix fractions, General
convergent, Reduction of quadratic surds to continued fractions, Theory op
Numbers, Congruences, Multiplication of determinants.
This book is written for pupils in the upper grades of grammar
schools and the lower grades of high schools. The introduction of the simple
elements of Algebra into these grades will, it is thought, so stimulate the
mental activity of the pupils, that they will make considerable progress in
Algebra without detriment to their progress in Arithmetic, even if no more time
is allowed for the two studies than is usually given to Arithmetic alone.
Beginning and Intermediate
Algebra is an open source book written by Tyler Wallace. This book covers the
following topics: Pre-Algebra, Solving Linear Equations, Inequalities, Systems
of Equations, Graphing, Polynomials, Factoring, Quadratics, Rational
Expressions, Functions and Radicals.
This note covers the following topics:
From Arithmetic To Algebra, The Making of Algebras, Simultaneous Problems,
Partial Solution: Elements of Complexity, Mathematical Certainty, The First
Hebrew Algebra, The Limits of the Teacherís Function, The Use of Sewing Cards,
The Story of a Working Hypothesis, Macbethís Mistake, Jacobís Ladder. | 677.169 | 1 |
Mathematics
Hardcover
Overview also explored. Written in a tone and at a level appropriate for students (primarily those in upper middle school or high school, as well as undergraduates who are not mathematics majors), Mathematics contains approximately 300 entries that cover the basic disciplines of algebra, geometry, and trigonometry, with the goal of making these topics more accessible and interesting to students and the general reader. Thirty percent of the content (in the form of new and revised essays) highlight the advances that have taken place in mathematics since the beginning of the 21st century. Mathematics is illustrated throughout with images, equations, tables, and figures, and includes sidebars. | 677.169 | 1 |
What covers in a math workshop? •Review past/current contents each week. •Improve study and time management skills as well as reduce math anxiety. •Develop academic support network linked to specific classes with other students. •Work collaboratively to critically analyze course contents to improve understanding of complex material with workshop facilitators. •Provide opportunity to become actively involved in the course material. •Discover study and test preparation strategies. Do I need to sign up? No. Students just show up during the time scheduled.
Assisting
Faculty instructors, instructional assistants, student assistant tutors, and volunteers are available to help SAC math students increase their understanding of various math topics, assist with math assignments and guide students in the use of technology. It is vital that students log in to and out of the appropriate attendance computer in order to use the Math Center
Computers
The Math Center has 49 computers that students may use to complete their math assignments, watch math videos, practice their skills with math tutorials or access software related to their coursework such as STATDISK. To see if your class has related materials on our computers, please CLICK HERE.
Graphing Calculators
The Math Center has some TI-84 and TI-83 graphing calculators attached to some of the tables. Students may download Calculator apps and programs from most of the MC computers. | 677.169 | 1 |
El Sobrante GeometryTim N.
...The concepts behind Calculus are actually simple and few in number. But they are often introduced in ways that seem complex and confusing. I will help explain the concepts and terms in simple English and with everyday examples | 677.169 | 1 |
Combinatorics on Words: Progress and Perspectives covers the proceedings of an international meeting by the same title, held at the University of Waterloo, Canada on August 16-22, 1982. This meeting highlights the diverse aspects of combinatorics on words, including the Thue systems, topological dynamics, combinatorial group theory, combinatorics,... more...
Combinatorics deals with simple combinatorial problems, recurrence relations, and generating functions, particularly the binomial expansions. The book expounds on the general rules of combinatorics, the rule of sum, the rule of product, samples, permutations, combinations, and arrangements of subjects with various restrictions. The text also explains... more...
Introduction to Asymptotics and Special Functions is a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real variable are discussed, along with contour integrals and differential equations with regular and irregular singularities. The Liouville-Green approximation is also... more...
Applications of Group Theory to Combinatorics contains 11 survey papers from international experts in combinatorics, group theory and combinatorial topology. The contributions cover topics from quite a diverse spectrum, such as design theory, Belyi functions, group theory, transitive graphs, regular maps, and Hurwitz problems, and present the state-of-the-art... more...
This volume, as Andrew M. Odlzyko writes in the foreword, ?commemorates and celebrates the life and achievements of an extraordinary person.? Originally conceived as an 80th birthday tribute to Herbert Wilf, the well-known combinatorialist, the book has evolved beyond the proceeds of the W80 tribute. Professor Wilf was an award-winning teacher, who... more...
Yearly math olympiads include at least one combinatrics problem. This book reviews problem-solving techniques for tackling such problems, with examples from recent contests. Includes problems for each topic, with hints and full solutions for reader practice. more...
Written by an expert in the field, this undergraduate textbook will prepare the next generation of pure and applied mathematicians. Key topics include walks on graphs, cubes and the Radon transform, the Matrix?Tree Theorem, and the Sperner property. more...
A brilliant treatment of a knotty problem in computing. This volume contains chapters written by reputable researchers and provides the state of the art in theory and algorithms for the traveling salesman problem (TSP). The book covers all important areas of study on TSP, including polyhedral theory for symmetric and asymmetric TSP, branch and bound,... more... | 677.169 | 1 |
The third book of the Mathematics in Action series, Algebraic, Graphical, and Trigonometric Problem Solving, Fourth Edition, illustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities and the accompanying practice exercises. Along with the activities and the exercises within the text, MathXL® and MyMathLab® have been enhanced to create a better overall learning experience for the reader. Technology integrated throughout the text helps readers interpret real-life data algebraically, numerically, symbolically, and graphically. The active style of this book develops readers' mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines.
IN THIS BOOK, we look at the power of mathematics, not the drudgery of number crunching. We look back through history and see how the power of mathematics has revolutionized the world. We also take an in-depth look at the utility of mathematics--how it is useful to you now--rather than telling you that it might be useful someday. The main theme throughout the book is problem solving.
This book serves as a review of arithmetic, and an introduction to algebra, geometry, and trigonometry and probability.The exercises and answers in this book provide readers with opportunities to test their mastery of each topic in these common branches of mathematics.
Every chapter in this book consists of a theme, a project and a career story. The chapters covers addition, subtraction, patterns, numbers, money, time, measurement, geometry, fractions and probability by providing problem solving strategies and applications.Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly developing area of mathematics and applications and gives complete proofs of many recently solved problems. With MATLAB codes and more than 200 exercises, the book is ideal for a special topics course for graduate or advanced undergraduate students in mathematics or engineering, and will also be a valuable resource for researchers. Often driven by questions from signal processing, control theory, and quantum information, the subject of this book has inspired mathematicians from many subdisciplines, including linear algebra, operator theory, measure theory, and complex function theory. In turn, the applications are being pursued by researchers in areas such as electrical engineering, computer science, and physics. The book is self-contained, has many examples, and for the most part requires only a basic background in undergraduate mathematics, primarily linear algebra and some complex analysis. The book also includes an extensive discussion of the literature, with close to 600 references from books and journals from a wide variety of disciplines | 677.169 | 1 |
Product Description
The Developmental Mathematics workbook series covers basic mathematics through early algebra. Workbooks aren't grade-specific, but rather focus on individual skills, making it an ideal curriculum for self-paced learners at any ability level. This is a self-teaching curriculum that was specifically designed for students to read, learn, and complete themselves, cultivating independent learning skills.
Lessons begin with an explanation of the concept and example problems that are solved step-by-step. A number of practice problems are provided on the following "Applications" pages.
We've used all levels up to 11 of Developmental Mathematics after trying many other math curriculum. This style of learning is what works for my daughter who is challenged by math to a great degree. We'll likely use it for our 6 year old son. I highly recommend it for those who've "tried it all"; it's inexpensive compared to others and you're not out alot of money if it doesn't work for your child, as you can buy one level at a time. ChristianBook is one of only a few places you can find Developmental Mathematics and these prices are quite fair. We've ordered many things from ChristianBook and will continue to shop here; a very reliable store! | 677.169 | 1 |
97801310106Intermediate Algebra: Functions & Authentic Applications
Designed to prepare students for college algebra and some "service" math courses. Written to develop students' critical thinking and problem solving capabilities and make the math they learn in this course relevant to decision-making in real life. Topics are presented in an interesting and inviting format incorporating real world sourced data and modeling.Unique, enthusiastic approach, this text requires students to take an active role in studying mathematics. Focusing more on the mathematical process, students have abundant opportunities to make intuitive leaps to discover patterns -- helping them develop critical thinking skills as well as mathematical confidence. Placing an emphasis on the usefulness of algebra throughout, students discover mathematical concepts while en route to solving true-to-life | 677.169 | 1 |
Applied Calculus for the Managerial, Life, and Social Sciences A Brief Approach, 10th Edition
APPLIED CALCULUS FOR THE MANAGERIAL, LIFE, AND SOCIAL SCIENCES: A BRIEF APPROACH, Tenth Edition balances modern applications, solid pedagogy, and the latest technology to engage students and keep them motivated in the course. Suitable for majors and non-majors alike, the text uses an intuitive approach that teaches concepts through examples drawn from real-life situations from students' fields of interest. In addition, insightful Portfolios highlight the careers of real people and discuss how they incorporate math into their daily professional activities. Numerous exercises, including a Diagnostic Test, ensure that students have a concrete understanding of concepts before advancing to the next topic. The text's pedagogical features coupled with an exciting array of supplements equip students with the tools they need to make the most of their study time and to succeed in the course. | 677.169 | 1 |
Introduction
There are a few barriers in the way of understanding how a computer
algebra system works. Commercial systems like Mathematica, Maple, and
the engines in the TI-89 and HP-50 calculators are closed source. Some
open source ones such as Axiom and Maxima can be incredibly complex.
Joel S. Cohen, professor emeritus at the University of Denver,
authored two volumes which can help dispel the mystery of CAS
internals: | 677.169 | 1 |
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Computer-Based Math™: Rethinking the Curriculum for the Age of Computers
Jon McLoone
computerbasedmath.org has been engaged in a fundamental rethink of math education. This Wolfram Technology for STEM Education: Virtual Conference for Education talk shares some of the thinking behind the new curriculum that computerbasedmath.org is developing and how Wolfram technologies can assist.
Channels: Virtual Events
An electrical and computer engineering graduate student researcher shares his insights from academics and industry about how he uses Mathematica and the Wolfram Language, and how it compares to Matlab.
This free workshop is designed for educators, technology coaches, and anyone interested in incorporating the latest technology into the classroom. Presentations include how to get started using Programming Lab. Learn ...
Learn how the engineering curriculum can be revitalized by making SystemModeler an integral part of the education experience. See how SystemModeler improves the way engineering courses are taught and demonstrate ...
In this free online workshop we discuss methods for introducing computational thinking and programming activities to middle and high school students. We also discuss specific activities that support and introduce ...
Get the basics of neural networks and applications such as image/speech recognition, image captioning, question answering, and game playing. A case study of the ImageIdentify built-in Wolfram Language symbol ...
Solving partial differential equations both symbolically and numerically in the Wolfram Language. Learn about specification of PDEs, boundary conditions, regions, and new functionality like eigensystem computation. | 677.169 | 1 |
books.google.com - Number Theory is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included. The book is divided into two parts. Part A covers key concepts of... Theory
Number Theory: An Introduction to Mathematics
Number Theory is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included.
The book is divided into two parts. Part A covers key concepts of number theory and could serve as a first course on the subject. Part B delves into more advanced topics and an exploration of related mathematics. Part B contains, for example, complete proofs of the Hasse-Minkowski theorem and the prime number theorem, as well as self-contained accounts of the character theory of finite groups and the theory of elliptic functions.
The prerequisites for this self-contained text are elements from linear algebra. Valuable references for the reader are collected at the end of each chapter. It is suitable as an introduction to higher level mathematics for undergraduates, or for self-study.
From the reviews:
"This is a book which many mathematicians could enjoy browsing, and one which a good undergraduate could be encouraged to read to learn something of the interconnections, which exist between apparently disparate parts of mathematics."
--Canadian Mathematical Society
"As a source for information on the 'reach' of number theory into other areas of mathematics, it is an excellent work." | 677.169 | 1 |
N.S.F. Awards $6 Million for Math Curriculum
A group of five university-based teams has been awarded a $6 million
grant to develop a technology-based high school mathematics curriculum
designed to be equally accessible to both college-bound and
non-college-bound students.
The National Science Foundation recently awarded the researchers,
who are affiliated with the Core-Plus Mathematics Project, more than $1
million to begin the five-year project.
The project is a joint undertaking of researchers at Western
Michigan University, the University of Michigan, Ohio State University,
the University of Maryland, and the University of Iowa.
Christian R. Hirsch, the project's director and a math and
statistics professor at Western Michigan, said the project is aimed at
helping to abolish the tracking of math students into separate academic
and vocational courses.
"A close look at the two tracks in high school mathematics today
reveals serious deficiencies in each,'' he said. "Neither does justice
to contemporary math or the needs of students in contemporary
society.''
Working with the National Council of Teachers of Mathematics, Mr.
Hirsch edited a proposed core math curriculum for grades 9-12 that
contains similar proposals for technology immersion and flexible
curriculum. The core curriculum was published as an addendum to the
N.C.T.M.'s teaching standards. (See Education Week, April 8, 1992.)
The Core-Plus project seeks to weave together four major strands
into a simplified curriculum that meets the needs of all students, Mr.
Hirsch said.
Under the curriculum, students would study concepts of algebra and
functions, statistics and probability, geometry and trigonometry, and
discrete math, including graphs and algorithm design, in each of their
three years of instruction.
Representatives of the textbook publisher Glencoe, a division of
MacMillan/McGraw-Hill, and Texas Instruments, which manufactures
graphing calculators, will also participate in the project.
Glencoe hopes to publish the new curriculum materials, while Texas
Instruments may develop a new machine that would be compatible with the
curriculum.--ðŸ÷Ÿ
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Finish your assignments faster, improve your understanding of even the most difficult subjects, and explore on your own. Student products give you exactly the same software as the professional versions, but at a student price! | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
A STUDY GUIDE FOR EXAM 1
This document is intended to help you study for the first midterm. There are
essentially 5 topics that will be on the exam, induction, sets, functions (including
sequences and strings), relations (including equivalence relations)
MATH 61: SAMPLE SOLUTIONS
This document is supposed to give you a template for complete and correct
solutions to different problem types.
Expectations. Recall from the first day of class that a good argument is both
clear and correct. For the moment lets
Lecture 1: Course Overview and Machine Numbers
1
Introduction
In this course, we will learn
1. How to represent numbers and do arithmetic on a computer? Concepts of error, and how to
track and control it? (Chap 1)
2. Algorithms to solve one-dimensional no
The extreme right
in Western Europe
The extreme right
in Western Europe
Success or failure?
Elisabeth Carter
Manchester University Press
Manchester and New York
distributed exclusively in the USA by Palgrave
Copyright Elisabeth Carter 2005
The right of El
170A Final Solutions1
1. Question 1
(a) There exists a continuous random variable X such that P(X = 2) = 1/2. FALSE. If
X is continuous, then P(X = a) = 0 for any a R.
(b) There is some random variable X such that var(X) = 1.
FALSE. The variance must be n | 677.169 | 1 |
Scientific calculators – but not graphing calculators – are permitted in this
Course Description: This course covers topics including special products, factoring, and other operations on polynomials, rational and radical expressions, integral and rational exponents, and scientific notation. In addition, analytic and graphical methods of solving linear equations, linear systems, literal equations, and elementary polynomial equations are covered. Students are also introduced to the analytic geometry of functions, including lines, circles, and parabolas. Diverse applications are emphasized throughout the course.
General Education Goals: MTH 100 is affirmed in the following General Education Foundation Category: Quantitative Knowledge and Skills. The corresponding General Education Goal is as follows: Students will use appropriate mathematical and statistical concepts and operations to interpret data and to solve problems.
Course Goals: Upon successful completion of this course, students should be able to do the following:
demonstrate knowledge of the fundamental concepts and theories from algebra and geometry;
utilize various problem-solving and critical-thinking techniques to set up and solve real-world applications;
communicate accurate mathematical terminology and notation in written and/or oral form in order to explain strategies to solve problems as well as to interpret found solutions; and
use calculators effectively as a tool to solve such problems as those described above. | 677.169 | 1 |
Abstract
Conventional treatments of functions start by building a rule of correspondence betweenx-values andy-values, typically by creating an equation of the formy=f(x). We call this acorrespondence approach to functions. However, in our work with students we have found that acovariational appraoch is often more powerful, where students working in a problem situation first fill down a table column withx-values, typically by adding 1, then fill down ay-column through an operation they construct within the problem context. Such an approach has the benefit of emphasizing rate-of-change. It also raises the question of what it is that we want to cal 'rate' across different functional situations. We make two initial conjectures, first that a rate can be initially understood as aunit per unit comparison and second that a unit is theinvariant relationship between a successor and its predecessor. Based on these conjectures we describe a variety of multiplicative units, then propose three ways of understanding rate of change in relation to exponential functions. Finally we argue that rate is different than ratio and that an integrated understanding of rate is built from multiple concepts.
Confrey, J.: 1990, 'Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions', a paper presented at the annual meeting of theAmerican Educational Research Association, Boston.
Confrey, J.: in press b, 'Splitting, similarity, and rate of change: New approaches to multiplication and exponential functions', in G. Harel and J. Confrey (eds.),The Development of Multiplicative Reasoning in the Learning of Mathematics, Albany
Confrey, J. and Smith, E.: 1989, 'Alternative representations of ratio: The Greek concept of anthyphairesis and modern decimal notation', a paper presented at theFirst Annual Conference of The History and Philosophy of Science in Science Teaching, Tallahassee, pp. 71–82.
Confrey, J., Smith, E., Piliero, S. and Rizzuti, J.: 1991, 'The use of contextual problems and multi-representational software to teach the concept of functions',Final Project Report to the National Science Foundation (MDR-8652160) and Apple Computer, Inc.
Rizzuti, J. and Confrey, J.: 1988, 'A construction of the concept of exponential functions', in M. Behr, C. LaCompagne and M. Wheeler (eds.),Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Dekalb, pp. 259–268.
Smith, E. and Confrey, J.: 1989, 'Ratio as construction: ratio and proportion in the mathematics of ancient Greece', a paper presented at the annual meeting of theAmerican Educational Research Association, San Francisco.
Smith, E. and Confrey, J.: in press, 'Multiplicative structures and the development of logarithms: What was lost by the invention of function?', in G. Harel and J. Confrey (eds.),The Development of Multiplicative Reasoning in the Leanring of Mathematics. Albany.
Steffe, L.: in press, 'Children's multiplying and dividing schemes', in G. Harel and J. Confrey (eds.),The Development of Multiplicative Reasoning in the Learning of Mathematics, Albany.
Thompson, P: 1990, 'The development of the concept of speed and its relationship to concepts of rate', a paper presented at the annual meeting of theAmerican Educational Research Association, Boston.
Thompson, P.: in press, 'The development of the concept of speed and its relationship to concepts of rate', in G. Harel and J. Confrey (eds.),The Development of Multiplicative Reasoning in the Learning of Mathematics, Albany.
Tierney, C. and Nemirovsky, R.: 1991, Young childrens spontaneou representations of changes in population and speed, in R. Underhill (ed.),Proceedings of the Thirteenth Annual Meeting of the NA-PME, Blacksburg, pp. 182–188. | 677.169 | 1 |
Maths for Economics
Geoff Renshaw
The flexible design of the contents means that this book is suitable if your starting point is maths A level, GCSE, or even if you have forgotten most of the maths you learned in school.
A gradual learning gradient underpins the text; each concept is explored comprehensively, including numerous examples and exercises at each stage to ensure that you get a really good understanding of it.
Builds self-confidence in maths through the use of step-by-step examples, progress exercises, hints and rules, encouraging you to develop a 'maths tool-kit' for analysing economics and business problems.
Any unnecessarily long explanations have been rewritten to make the exposition clearer in the appropriate places.
Material on matrix algebra, limits, and necessary and sufficient functions has been revised in line with market feedback
Answers to the progress exercises now appear at the end of the book, rather than online, for ease of use.
Maths for Economics
Third Edition
Geoff Renshaw
Description
Many years of teaching led Geoff Renshaw to develop Maths for Economics as a resource which builds your self-confidence in maths by using a gradual learning gradient and constantly reinforcing learning with examples and exercises. Some students embarking on this module feel that they have lost their confidence in maths, or perhaps never had any in the first place. The author has designed the book so that whether you have a maths A level, GCSE, or perhaps feel that you need to go back over the very basics, knowledge is built up in small steps, not big jumps. Once you are confident that you have firmly grasped the foundations, this book will help you to make the progression beyond the mechanical exercises and into the development of a maths tool-kit for the analysis of economic and business problems. This is a skill which will prove valuable for your degree and for your future employers. | 677.169 | 1 |
Below I have made a copy of the current syllabus. The assignment guide is also included at the end of the syllabus, just in case your printed syllabus is "misplaced".
Each week I will have information regarding when assignments are due and when we will have quizzes or tests. You should check this website to find out what we are doing in class if you absolutely have to miss a class session.
COURSE DISCRIPTION: This is a course in basic algebra concepts. Language of, symbols, notation, and concepts, as well as a basic computational skills will be emphasized. A passing grade in this course will prepare you well for your next mathematics class; additionally, it will enhance your problem-solving strategies for other college courses. You will pass this class only with an A, B, or C. Grades of D, F, or Withdrawal mean that you will be required to re-take this course.
MATERIAL TO BE COVERED: We will attempt to cover all the following chapters this semester, in the following order:
ATTENDANCE REQUIREMENTS: It is important for your success in this class to attend every class session and to be on time. This means you must attend every single day, Monday thru Friday. If you must miss a class, let me know as soon as possible. It is your responsibility to get any missed notes or assignments. No late assignments will be accepted. Each assignment must be turned in at the beginning of class the day it is due, therefore, if you miss a class I must receive your assignment before class.
GRADING POLICY: Algebra, like other subjects, is learned best by doing. It is very important that you spend time outside of class practicing the concepts and studying. In order to assist you in your successful completion of this course, exercises from each section, quizzes, and four or five exams will be given. You are responsible for completing all assignments and keeping them in an organized notebook. No late homework will be accepted. All assignments must be turned in at the beginning of class the day it is due. The top 30 homework assignments will be calculated. Quizzes and exams will be announced in advance. All quizzes and exams will be given at the beginning of class and the top 15 quiz grades will be calculated.
Inform the instructor, in advance, if you will be unable to attend an exam session. Only one make-up exam may be taken and it will be different from the original exam. No make-up exams will be given if you do not contact the instructor before the original exam is given.
A comprehensive final will be given on July 27, 2001 from 12:45-2:00 pm.
GRADING CRITERIA: Grades will be assigned according to the following criteria:
1. Place in the right hand corner, at the top of the page: Your Name Name of Class Name of Instructor Class Time Section # Date
2. Begin each section assignment on a clean sheet of paper.
3. Create an answer column in the right margin of your paper.
4. Homework must be turned in at the beginning of class the day the assignment is due. NO LATE WORK WILL BE ACCEPTED! The top 30 grades will be calculated into your final grade.
ALWAYS show your work! Papers turned in with only answers given and no work shown will not receive credit. You are not allowed to use calculators in this class.
If you have any questions about your assignment or grading please feel free to come during office hours or make an appointment. Additional help can also be found with the math department tutors or at tutoring in the SAS building. | 677.169 | 1 |
On Monday, September 2, 2013 10:51:43 PM UTC-4, Ken Pledger wrote: > In article <6dc55c4a-0f72-4533-b3eb-39daadc61306@googlegroups.com>, > > lite.on.beta@gmail.com wrote: > > > > > On Sunday, September 1, 2013 5:05:55 PM UTC-4, Ken Pledger wrote: > > > > .... I just call it "vector geometry". Some linear algebra textbooks > > > > have a chapter on it. > > > > > > Can you name one? Thanks > > > > > > There are two excellent out-of-print books which concentrate on it: > > A. M. Macbeath, "Elementary Vector Algebra", and Seymour Schuster, > > "Elementary Vector Geometry". A very through and well-written treatment > > that may still be in print is Walter Prenowitz, "Geometric Vector > > Analysis and the Concept of Vector Space", which is Chapter VI in the > > 23rd Yearbook of the National Council of Teachers of Mathematics. Any > > of those three would be well worth a bit of effort to find. > > > > A modern linear algebra book which starts with the geometry is David > > Easdown, "A First Course in Linear Algebra". He introduces geometrical > > vectors in Chapter 1, then develops the theory and geometrical > > applications in the next few chapters. (My students use this text.) > > The well-known book by Howard Anton, "Elementary Linear Algebra" has a > > briefer treatment in Chapter 3. > > > Thank you. The one by Seymour Schuster looks available from amazon.com | 677.169 | 1 |
UPG 031 Math 3
Course Description
This course is designed to prepare you for further university calculus courses. As a Math 31 equivalency, this course covers topics of calculus as well as essential precalculus topics, including: review of functions, review of algebra, limits and continuity, differentiation, applications of differentiation, integration, and applications of integration. | 677.169 | 1 |
About
Overview
Building on the success of its first five editions, the Sixth Edition of the market-leading text explores the important principles and real-world applications of plane, coordinate, and solid geometry. Strongly influenced by both NCTM and AMATYC standards, the text includes intuitive, inductive, and deductive experiences in its explorations. Goals of the authors for the students include a comprehensive development of the vocabulary of geometry, an intuitive and inductive approach to development of principles, and the strengthening of deductive skills that leads to both verification of geometric theories and the solution of geometry-based real world applications.
Updates in this edition include the addition of 150 new problems, new applications, new Discover! activities and examples and additional material on select topics such as parabolas and a Three-Dimensional Coordinate System.
Features and Benefits
Proofs include written and visual proofs, as well as comparisons, to support students with different learning styles.
Technology-related margin features encourage the use of the Geometer's Sketchpad, graphing calculators, and further explorations.
Reminders found in the margin serve as a convenient recall mechanism.
Discover! activities emphasize induction in the development of geometry.
Geometry in Nature and Geometry in the Real World emphasize examples of geometry found in everyday life.
Tables found in chapter endmatter that emphasize chapter properties such as those for triangles, quadrilaterals, circles provide visual connections between figures and concepts and help students better assess their level of mastery and test readiness.
Index of Applications calls attention to the practical applications of geometry.
Warnings are provided so that students might avoid common pitfalls studying geometry.
Perspective on History provides a glimpse of the history of geometry.
Perspective on Application provides illustrations of geometry in practice.
Chapter Summaries review the chapter, preview the chapter to follow, and provide a list of important concepts found in the current chapter.
Chapter Reviews and Tests provide numerous practice problems and tests for each chapter.
Exercise sets in the Student Study Guide, with cross-references to the text, offer additional practice and review.
What's New
Approximately 150 new exercises have been added, many of a challenging nature.
New Discover! activities
New examples to provide a smoother transition within the revelant section as well as to better prepare students for the exercise set that follows
New and updated applications including: -checking for the squareness of the opening of a patio door -the ideal viewer-distance to TV sets of various screen sizes -checking for accuracy in the construction of a garden gazebo -the application of drywall in new construction or remodeling
A new section (Section 10.6)on a Three-Dimensional Coordinate SystemThis guide provides worked-out solutions to select odd-numbered problems from the text as well as new Interactive Exercise sets for additional review. Solutions for the additional Interactive Exercise sets are provided on the Instructor Website.
Student Supplements
This guide provides worked-out solutions to select odd-numbered problems from the text as well as new Interactive Exercise sets for additional review. Solutions for the additional Interactive Exercise sets are provided on the Instructor Website.
Meet the Author
Author Bio
Daniel C. AlexanderGeralyn M. Koeberlein | 677.169 | 1 |
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This book is designed to help parents support their teenagers through school maths. But with its focus on the big picture, the connections within mathematics and the references to history and applications it is also a welcome resource for teachers and even students themselves.
Current physical theories suggest that beautiful symmetries underly the fundamental particles and forces of nature. We describe those symmetries using an analogy from economics, and even rediscover the famous Higgs boson. | 677.169 | 1 |
"Linear algebra is a fantastic subject. On the one hand it is clean and beautiful." – Gilbert Strang This wonderful branch of mathematics is both beautiful and useful. It is the cornerstone upon which signal and image processing is built. This short chapter can not be a comprehensive survey of linear algebra; it is meant only as a brief introduction and review. The ideas and presentation order are modeled after Strang's highly recommended Linear Algebra and its Applications. At the heart of linear algebra is machinery for solving linear equations. In the simplest case, the number of unknowns equals the number of equations. For example, here are a two equations in two unknowns: 2x − y = 1
0.4 Basis 0.5 Inner Products and Projections 0.6 Linear Transforms
y
2x−y=1 (x,y)=(2,3)
x + y = 5.
(1)
x+y=5
x
There are at least two ways in which we can think of solving these equations for x and y . The first is to consider each equation as describing a line, with the solution being at the intersection of the lines: in this case the point (2, 3), Figure 0.1. This solution is termed a "row" solution because the equations are considered in isolation of one another. This is in contrast to a "column" solution in which the equations are rewritten in vector form: 2 1 x+ −1 1 y = 1 5 . (2)
Ax = b. The inverse of a 1 × 1 matrix A = ( a ) is A−1 = ( 1/a ). . In order to multiply two matrices A and B (or a matrix and a vector). Note that this commutative property limits the discussion of matrix inverses to square matrices. In the context of solving linear equations we have introduced the notion of a vector and a matrix. . Ix = x. so that B must equal C . The inverse of a 2 × 2
1 The inverse of a matrix is unique: assume that B and C are both the inverse of matrix A. the column dimension of A must equal the row dimension of B .. j component of the product C is computed from an inner product of the ith row of matrix A and the j th column of matrix B . In other words if A is of size m × n. and for any suitably sized matrix. 0 1
(22)
Given the definition of matrix multiplication. However matrix multiplication is both associative (AB )C = A(BC ) and distributive A(B + C ) = AB + AC . This constraint immediately suggests that matrix multiplication is not commutative: usually AB = BA. . it is easily seen that for any vector x. . but the inverse does not exist when a = 0.. Not all matrices have inverses. IA = A and BI = B . The identity matrix I is a special matrix with 1 on the diagonal and zero elsewhere: 1 0 0 1 = . 0 0
I
.. Let's consider some simple examples. then by definition B = B (AC ) = (BA)C = C . . Multiplying both sides by the matrix inverse A−1 yields the desired solution to the linear equations: A−1 Ax = A−1 b Ix = A−1 b x = A−1 b (23)
A matrix A is invertible if there exists 1 a matrix B such that BA = I and AB = I .. where I is the identity matrix. Notice that this definition is completely consistent with the product of a matrix and vector.. .
0 0 0 0 . then B must be of size n × p (the product is of size m × p).. .. The result is a compact notation for representing linear equations. . . The matrix B is the inverse of A and is denoted as A−1 .
7
.That is. the i.
The transpose of a matrix A. or simultaneously using the Gauss-Jordan method. an
and A−1 =
1/a1 . can itself be calculated by solving a collection of linear equations. . The transpose of a sum of two matrices is the sum of
In order to prove (AB )−1 = B −1 A−1 . (25)
as long as all the diagonal components are non-zero. These can be solved independently for the columns of the inverse matrix. if it exists.
2
8
. the transpose of a m × n matrix is a n × m matrix with (At )ij = Aji . The answer to this question is explored in the next section. and Ax3 = e3 .
. 2 The inverse of an arbitrary matrix. Any diagonal matrix is invertible: A=
a1
. is constructed by placing the ith row of A into the ith column of At . 1/an
. We must naturally wonder the fate of our solution if the matrix is not invertible. Ax2 = e2 ..
(24)
but does not exist when ad − bc = 0.matrix can be calculated as: a c b d
−1
=
1 ad − bc
d −b −c a
. But before moving forward we need one last definition. For example: A= 1 2 1 4 −6 0 1 and At = 2 1
4 −6 0
(27)
In general. and that (B −1 A−1 )(AB ) = I : (B −1 A−1 )(AB ) = B −1 (A−1 A)B = B −1 IB = B −1 B = I. (26) 0 0 1
This matrix equation can be considered "a column at a time" yielding a system of three equations Ax1 = e1 . denoted as At . A system of linear equations Ax = b can be solved by simply left multiplying with the matrix inverse A−1 (if it exists). we must show (AB )(B −1 A−1 ) = I : (AB )(B −1 A−1 ) = A(BB −1 )A−1 = AIA−1 = AA−1 = I .. Consider for example a 3 × 3 matrix A whose inverse we know must satisfy the constraint that AA−1 = I : 2 4 −2
1 1 − 6 0 x1 7 2
x2
x3 = e1
e2
1 0 0 e3 = 0 1 0 . The inverse of a product of matrices AB is (AB )−1 = B −1 A−1 . This is easily proved using the associativity of matrix multiplication.
The sum of any two
9
. That is. A subspace of a vector space is a non-empty subset of vectors that is closed under vector addition and scalar multiplication. With the closure property verified.the transposes: (A + B )t = At + B t. aij = aji . 5. z . 0. Of particular interest will be the class of symmetric matrices that are equal to their own transpose At = A. the eight properties of a vector space automatically hold for the subspace. 2. a vector space must satisfy the following properties. Symmetric matrices are necessarily square. 6. the following constraints are satisfied: (1) the sum of any two vectors in the subspace remains in the subspace. Matrices can also make up a vector space. The transpose of a product of two matrices has the familiar form (AB )t = B t At . by definition. for any vectors x.
Example 0. And the transpose of the inverse is the inverse of the transpose: (A−1 )t = (At )−1 . (2) multiplication of any vector by a scalar yields a vector in the subspace. For example the space of 3 × 3 matrices can be thought of as R9 (imagine stringing out the nine components of the matrix into a column vector). here is a 3 × 3 symmetric matrix: 2 1 4 A = 1 −6 0 . 8.3 Vector Spaces
The most common vector space is that defined over the reals. 3. 7. with rules for vector addition and scalar multiplication.1 Consider the set of all vectors in R2 whose com-
ponents are greater than or equal to zero. x+y =y+x (x + y) + z = x + (y + z ) there exists a unique "zero" vector 0 such that x + 0 = x there exists a unique "inverse" vector −x such that
x + (−x) = 0 1x = x (c1 c2 )x = c1 (c2 x) c(x + y ) = cx + cy (c1 + c2 )x = c1 x + c2 x
Vector spaces need not be finite dimensional. 4 0 3
(28)
notice that. y. This space consists of all column vectors with n real-valued components. R∞ is a vector space. 4. A vector space has the property that the addition and multiplication of vectors always produces vectors that lie within the vector space. and scalar c: 1. denoted as Rn . In addition.
2 Consider the following over-constrained system: Ax 1 5 2 0 4 4 u v = = b b1 b2 b3
The column space of A is the plane spanned by the vectors ( 1 5 2 )t and ( 0 4 4 )t .vectors in this space remains in the space. = u3 v3 b3
v1 v2 v3
x1 x2
b1 = b2 b3
(29)
(30)
In this form. we rewrite the above system according to the rules of matrix multiplication yielding an equivalent form: u1 v1 b1 x1 u2 + x2 v2 b2 . the vector by −1 yields the vector 2 −2 which is no longer in the space. To see why this is so. where the matrix A is either: 1. there are more equations (three) than unknowns (two). we see that a solution exists when the scaled columns of the matrix sum to the vector b.
At this point we have seen three seemingly different classes of linear equations of the form Ax = b. but multiplication of. This is simply the closure property necessary for a vector subspace. It is said that a solution to Ax = b exists if and only if the vector b lies in the column space of A. Therefore. Therefore.
Vector subspaces play a critical role in understanding systems of linear equations of the form Ax = b. 1 −1 for example.
Example 0. this system is over-constrained. the solution b can not be an arbitrary vector in R3 . but is constrained to lie in the plane spanned by these two vectors. A solution to this system exists if the vector b lies in the subspace of the columns of matrix A. The vector subspace spanned by the columns of the matrix A is called the column space of A. Consider for example the following system: u1 u2 u3
Unlike the earlier system of equations. 10
. this collection of vectors does not form a vector space. square and invertible (non-singular).
where the rank of a matrix is the number of linearly independent rows in the matrix.4.2.0)
is true only when c1 = c2 = .. In general it is said that a n × n matrix is invertible if it has rank n or is full rank.2)
(−2.3 Consider the following system: 1 5 2 0 4 4 1 9 6 Ax u v w = = 0 0 0 0 v w )t = ( 0 0 0 )t .
. Figure 0. That is.. a set of vectors u1 . The vectors that lie in the nullspace of a matrix consist of all solutions to the system Ax = 0.... The zero vector is always in the nullspace. two vectors are linearly dependent if they lie along a line.e. + cn un = 0 (31)
(2. In 2-D. . the vectors are linearly dependent. u2 .2)
(−2. a set of vectors are linearly dependent if at least one of the vectors can be expressed as a sum of scaled copies of the remaining vectors. all vectors lying on a
one-dimensional line in R ). therefore the nullspace of A also contains all vectors of the form ( u v w )t = ( c
3
c
−c )t (i. over-constrained..2)
(−1. there is a non-trivial combination of the 11
(−1.4 Linearly dependent (top/bottom) and independent (middle).2) (2.−1)
(2. and hence the only possible solution. In other words. At one extreme is the invertible n×n square matrix whose solutions may be any vector in the whole of Rn . where solutions lie in a subspace of the full vector space. = cn = 0.0)
Figure 0.4 Basis Recall that if the matrix A in the system Ax = b is invertible. The second important vector space is the nullspace of a matrix. At the other extreme is the zero matrix A = 0 with only the zero vector in it's column space. then left multiplying with A−1 yields the desired solution: x = A−1 b. Otherwise. square but not invertible (singular). 3..
Example 0. un are linearly independent if: c1 u1 + c2 u2 + . Linear independence is easy to visualize in lower-dimensional subspaces. In between are the singular and over-constrained cases..
0. Formally. In each case solutions to the system exist if the vector b lies in the column space of the matrix A.
The nullspace of A contains the zero vector ( u
Notice also that the third column of A is the sum of the first two columns.
Linear independence is directly related to the nullspace of a matrix. If the column space of a n × n matrix is all of Rn . the matrix is full rank) if the matrix nullspace contains only the zero vector. Now. Also in 3-D.4. any vector u in the space can be written as a linear combination of these n vectors: u = a1 v1 + . the columns of a matrix are linearly independent (i. consider the following system of linear equations: u1 u2 u3
v1 v2 v3
w1 x1 b1 w2 x2 = b2 . Notice that this is also the condition for linear independence. Linear independence is also related to the column space of a matrix. the vectors u. If there is another combination such that: u = b1 v1 + . consider the following system of linear equations: u1 u2 u3
v1 v2 v3
w1 x1 0 w2 x2 = 0. w3 x3 0
(32)
Recall that the nullspace contains all vectors x such that x1 u + x2 v + x3 w = 0. Given an n dimensional vector space with n basis vectors v1 .vectors that yields the zero vector. In 3-D. Both conditions are important. a linear basis of a vector space is a set of linearly independent vectors that span the space... vn . For example. any three vectors are guaranteed to be linearly dependent. v . For example. (34)
In addition. In 2-D. . w are said to span the space.. If the only solution is the zero vector then the vectors are linearly independent and the matrix is full rank and invertible. the vector ( −1 2 ) can be expressed as a sum of the remaining 3 ( −2 0 ) + ( 2 2 ). in Figure 0.. 12 (35)
.. three linearly independent vectors: 2 vectors are linearly dependent if they lie in the same plane. For example. any four vectors are guaranteed to be linearly dependent. + bn vn . In such cases.. w3 x3 b3
(33)
If the the matrix is full rank. then the solution b can be any vector in R3 .e.. + an vn . the linear independence guarantees that this linear combination is unique. then the matrix is full rank.. Specifically.
0.
Example 0. For example in R2 any two vectors that do not lie on a line form a basis.6 and ( 1 −1
The vectors ( 1
0
0 ). and in R3 any three vectors that do not lie in a common plane or line form a basis.. While the representation is unique. ( 0
0
2 ).5
The vectors ( 1
0
0 ).
Example 0. Although these vectors
span the vector space. + (an − bn )vn = c1 v1 + . ( 0
1
0 ) and ( −1
0
0)
do not form a basis for R3 ..
Example 0.then the difference of these two representations yields 0 = (a1 − b1 )v1 + . + cn vn
(36)
which would violate the linear independence condition. the fourth vector is linearly dependent on the first two.. ( 0
−1
0 ).
0 ) do not form a basis.5 Inner Products and Projections 0.. the basis is not. A vector space has infinitely many different bases.6 Linear Transforms
13
. These vectors are both linearly independent and span the entire vector space. Removing the fourth vector leaves a basis for R3 . These vectors lie in a 2-D plane and do not span the entire vector space.4 The vectors ( 1 0 ) and ( 0 1 ) form the canonical
basis for R2 .
3 Moving erage
Av-
Although in the above example. where the xth number in the sequence is denoted as f [x]: f = {f [x]}. f . 1.1. −∞ < x < ∞.1 Consider the following system: g [x] = 1 2N + 1
N
f[x]
T
g[x]
Figure 1..1 Discrete-Time Signals A discrete-time signal is represented as a sequence of numbers. f [x]. and the output may be a function of all input values. a discrete-time system is a transformation that maps a discrete-time signal. Subsequent output values are computed by sliding these lines to the right. a process that we will cover fully in later chapters.2 Discrete-Time Systems In its most general form. and refer to the entire sequence simply as f [x].
14
.1 Discrete-Time Signals 1. with N = 2.2 Discrete-time system
(1. the output at each position depended on only a small number of input values. Note that from this definition. For example. the ith number in the output sequence is comf[x]
puted as the average of 2N + 1 elements centered around the ith input element. (1. this may not be the case.3 Linear TimeInvariant Systems
f[x]
1.
Figure 1.2 Discrete-Time Systems 1.. 2 4 8 1 f [12] } 7 6 5 4 3 2 1 }. the output value at x = 5 is computed as 1/5 times the sum of the five input
x 3 5 7
elements between the dotted lines.1). and is denoted as: g[x] = T {f [x]}
Example 1. we will often drop the cumbersome notation of Equation (1. the finite-length sequence shown in Figure 1.1 Discrete-time signal
For notational convenience.1)
where x is an integer. As shown in Figure 1. onto a unique g[x].3. Discrete-time signals often arise from the periodic sampling of continuous-time (analog) signals. in general. Discrete-Time Signals and Systems
1. (1.2)
x
Figure 1.1 is represented by the following sequence of numbers f = {0 = { f [1] f [2] .3)
f [x + k].
k = −N
In this system. a discretetime signal is defined only for integer values of x.
Intuitively, this should make perfect sense. This system simply takes an input signal and outputs a delayed copy, therefore, there is no change in the magnitude of each sinusoid, while there is a phase shift proportional to the delay, x0 .
So, why the interest in sinusoids and complex exponentials? As we will show next, a broad class of signals can be expressed as a linear combination of complex exponentials, and analogous to the impulse response, the frequency response completely characterizes the system. Let's begin by taking a step back to the more familiar sinusoids, and then work our way to the complex exponentials. Any periodic discrete-time signal, f [x], can be expressed as a sum of scaled, phase-shifted sinusoids of varying frequencies: f [x] = 1 2π
π
In the language of linear algebra, the sinusoids are said to form a basis for the set of periodic signals, that is, any periodic signal can be written as a linear combination of the sinusoids. Recall 23
that in deriving the convolution sum, the basis consisted of shifted copies of the unit-impulse. But note now that this new basis is not fixed because of the phase term, φk . It is, however, possible to rewrite the Fourier series with respect to a fixed basis of zerophase sinusoids. With the trigonometric identity cos(A + B ) = cos(A) cos(B ) − sin(A) sin(B ), the Fourier series of Equation (2.18) may be rewritten as: f [x] = = = 1 2π 1 2π 1 2π
π
ck cos[kx + φk ]
k =−π π
ck cos[φk ] cos[kx] + ck sin[φk ] sin[kx]
k =−π π
ak cos[kx] + bk sin[kx]
k =−π
(2.19)
In this expression, the constants ak and bk are the Fourier coefficients and are determined by the Fourier transform. In other words, the Fourier transform simply provides a means for expressing a signal in terms of the sinusoids. The Fourier coefficients are given by:
∞ ∞
ak =
j =−∞
f [j ] cos[kj ]
and
bk =
j =−∞
f [j ] sin[kj ]
(2.20)
Notice that the Fourier coefficients are determined by projecting the signal onto each of the sinusoidal basis. That is, consider both the signal f [x] and each of the sinusoids as T -dimensional vectors, f and b, respectively. Then, the projection of f onto b is: f0 b0 + f1 b1 + ... =
j
f j bj ,
(2.21)
where the subscript denotes the j th entry of the vector. Often, a more compact notation is used to represent the Fourier series and Fourier transform which exploits the complex exponential and its relationship to the sinusoids: eiωx = cos(ωx) + i sin(ωx), (2.22)
√ where i is the complex value −1. Under the complex exponential notation, the Fourier series and transform take the form: f [x] = 1 2π
π ∞
Comparing the Fourier transform (Equation (2.24)) with the frequency response (Equation (2.16)) we see now that the frequency response of a linear time-invariant system is simply the Fourier transform of the unit-impulse response:
∞
Linear time-invariant systems can be fully characterized by their impulse. both of which may be used to determine the output of the system to any input signal. This discrepancy makes the system only an approximate.26)
where the output signal g[x] can be determined from its Fourier transform G[ω ].
g[k]e−iωk =
k =−∞ ∞
(f [k] ⋆ h[k])e−iωk
k =−∞
G[ω ] =
k =−∞
∞ j =−∞
f [j ]h[k − j ] e−iωk
27
. or frequency responses. differentiator. h[x]. ε
where. 2
Consider now the derivative of a fixed fre-
quency sinusoid sin(ωx). f [x]: g[x] = f [x] ⋆ h[x] and G[ω ] = F [ω ]H [ω ]. by simply applying the inverse Fourier transform. H [ω ]. Notice that this is roughly in-line with the Fourier transform.5
Frequency (|H [ω ]|)
1
0
−0. Note though that for small ω . differentiating with respect to x gives ω cos(ωx) = ω sin(ωx − π/2). Let's see now if we can better understand the frequency response of this system. ε is given by the distance between samples of the discrete-time signal f [x]. (2.Space (h[x])
0. Namely. This is fairly straight-forward to prove: g[x] = f [x] ⋆ h[x]
∞ ∞
Fourier transforming. not a perfect. the difference being that the amplitude is given by | sin(ω )| instead of ω . in the case of the system T {·}.5 −1 0 1
0 −pi
0
pi
This system is an (approximate) differentiator. convolution in the space domain is equivalent to multiplication in the frequency domain. | sin(ω )| ≈ ω . Note that differentiation causes a phase shift of π/2 and a scaling by the frequency of the sinusoid. This equivalence illustrates an important relationship between the space and frequency domains. and can be seen from the definition of differentiation: df (x) dx =
ε→0
lim
f (x + ε ) − f (x − ε ) . recall that the magnitude was given by | sin(ω )| and the phase by
π .
1)
T
g[x]
for integer values x. f [·]). by the following relationship: f [x] = f (xT ) − ∞ < x < ∞. This sampling operation may be considered as a multiplication of the continuous time signal with an impulse train. It will prove to be mathematically convenient to work with this intermediate sampled signal. fs (x). the quantity T is the sampling period. fs (x).1 Continuous to Discrete: Space
f[x]
3. f [x].note that the impulse train is a continuous-time signal.2 Sampling: space
Note that the sampled signal. In general. In this expression.g.3)
x f[x]
Figure 3. Sampling: Continuous to Discrete (and back)
It is often more convenient to process a continuous-time signal with a discrete-time system.2.1 Processing block diagram
s(x) =
k =−∞
δ(x − kT ).
29
. and now we will concentrate on the conversions between discrete. and (3) the conversion of the output discretetime signal back to a continuous-time signal (D/C converter).. is indexed on the continuous variable x.2 C/D: Frequency 3.. The impulse train is defined as:
∞
D/C
g(x)
Figure 3.and continuoustime signals. Such a system may consist of three distinct stages: (1) the conversion of a continuous-time signal to a discrete-time signal (C/D converter).
(3. Of particular interest is the somewhat remarkable fact that under certain conditions.3.g.2)
f(x)
where δ(·) is the unit-impulse. and T is the sampling period . f (·)). a continuous-time signal can be fully represented by a discrete-time signal! 3. (2) the processing through a discrete-time system. f (x). Earlier we focused on the discrete-time processing. Figure 3. continuous-time signals will be denoted with rounded parenthesis (e. f [x] is indexed on the integer variable x.3 D/C
f(x)
C/D
A discrete-time signal. and discrete-time signals with square parenthesis (e. while the final discrete-time signal. (3. Multiplying the impulse train with a continuous-time signal gives a sampled signal: fs (x) = f (x)s(x).1 C/D: Space 3. is formed from a continuous-time signal. (3.
shown in Figure 3.. In the former case.3. denser sampling). a signal that changes slowly and smoothly can be sampled fairly coarsely. the copies of the frequency response will not overlap if: ωn < ωs − ωn or (3. preventing overlap of the frequency responses.2 Continuous to Discrete: Frequency
F(w)
w −w n S(w) wn
In the space domain. sampling was described as a product between the impulse train and the continuous-time signal (Equation (3.3)). Note however the conditions under which an exact replica is preserved depends on the maximum frequency response ωn of the original continuous-time signal.3 Discrete to Continuous If the Nyquist rate is met. is related to the sampling period T as ωs = 2π/T .3 (from top to bottom) are the Fourier transforms of the continuous-time function. while a signal that changes quickly requires more dense sampling. but this time. and the discrete-time signal does not accurately represent its continuous-time counterpart. Shown in Figure 3. which in turn increases ωs . In other words. ωn . not surprisingly. and the sampling rate. if the Nyquist rate is not met. On the other hand.4 Sampling: aliasing
. F (ω ). the impulse train. the Nyquist rate is not met.5)
F(w)
w −w n S(w) wn
ωs > 2ωn . Fs (ω ).4)
w −w s 0 Fs(w) ws
w w s− w n
Figure 3. Notice that the Fourier transform of the sampled signal contains multiple (yet exact) copies of the Fourier transform of the original continuous signal. then the sampling leads to aliasing. and the results of convolving these two signals. 3. In such a case. In the frequency domain.4 is another example of this sampling process in the frequency domain. ωs which. More precisely. this operation amounts to a convolution between the Fourier transform of these two signals: Fs (ω ) = F (ω ) ⋆ S (ω ) (3. then a discrete-time signal fully characterizes the continuous-time signal from which it was sampled. S (ω ). T must be made smaller (i. Not surprisingly. of the continuous-time signal increases.e.3 Sampling: no aliasing
For example. More precisely. it 30
w
Figure 3. itself an impulse train. as the maximum frequency. the signal is said to be aliased. the Nyquist rate depends on both the characteristics of the continuous-time signal. the sampling period. and the copies of the frequency response overlap. and the sampling interval of the impulse train.
w −w s 0 Fs(w) ws
The frequency ωn is called the Nyquist frequency and 2ωn is called the Nyquist rate.
a sinusoid with frequency ω0 .e.
31
. the Fourier transform of the sampled signal is multiplied by an ideal reconstruction filter as shown in Figure 3. so that π/T > ωn ). this ideal reconstruction filter has the form: sin(πx/T ) .
0 0
Figure 3. So.6 Ideal sync Example 3.5 Reconstruction
h(x) =
(3. we could equivalently reconstruct the signal by convolving with the ideal sync in the space domain. This frequency band is guaranteed to be greater than the Nyquist frequency. In the space domain. the reconstructed signal will be appear as a lower frequency sinusoid . and is equal to π when |A| = |B |. that is. It is easy to show from basic trigonometric identities that cos(A) cos(B ) = 0 when A = B . then the reconstructed signal will be of the form cos((ωs − ω0 )x). from the discrete-time signal.1 Consider the following continuous-time signal: f (x ) = cos(ω0 x). If the sampling rate is greater than 2ω0 . This filter has unit value between the frequencies −π/T to π/T and is zero elsewhere. we need only extract one of these copies. ωn (i. and is 0 otherwise (see below). ωs = 2π/T > 2ωn . the Fourier transform of cos(ω0 x) = π for |ω | = ω0 . The result will be identical to the original signal. let's compute the Fourier transform of this signal:
∞
F (ω )
=
k=−∞ ∞
f (k)e−iωk
=
k=−∞ ∞
cos(ω0 k)(cos(ωk) − i sin(ωk))
=
k=−∞
cos(ω0 k) cos(ωk) − i cos(ω0 k) sin(ωk)
First let's consider the product of two cosines. In particular since the frequency response of the discrete-time signal contains exact copies of the original continuous-time signals frequency response.6)
1
and is often referred to as the ideal sync function.5.is possible to reconstruct the original continuous-time signal. Since reconstruction in the frequency domain is accomplished by multiplication with the ideal reconstruction filter. In order to extract a single copy. but if the sampling rate is less than 2ω0 . and inverse transform the result.it will be aliased.. one can show that cos(A) sin(B ) = 0 for all A and B . We will eventually be interested in sampling this function and seeing how the effects of aliasing are manifested. then there will be no aliasing. Similarly. πx/T
Fs(w)
w pi/T
Figure 3. But first.
. and in this example n = m/2.in this case the first n sinusoids. . . .. . that is.e. . Substituting into the above sampling equation gives: gn = Sn×m Bm×n wn = Mn×n wn .8)
where the columns of the matrix B contains the basis set of sinusoids . (3.
f1 0 0 f2 . from the sub-sampled signal g .9)
If the matrix M is invertible. (3. then the original weights (i.7)
where the subscripts denote the vector and matrix dimensions. .e..F(w)
w −w 0 w0
Sampling
No Aliasing
Aliasing Fs(w)
Fs(w)
w −w 0 w0 −w 0 w0
w
We will close this chapter by drawing on the linear algebraic framework for additional intuition on the sampling and reconstruction process. We may express this operation in matrix form as follows: 1 0 g1 . First we will need to restrict ourselves to the sampling of an already sampled signal.. . We can express this constraint in matrix notation: fm = Bm×n wn .
0 0 0 0 0 0 0 1 0 0 0 0
gn = Sn×m fm . . In other 32
. ... . Our goal now is to determine when it is possible to reconstruct the signal f . can be written as a sum of a finite number of sinusoids). = 0 gn 1
0 0 0 0 1 0 0 0 0 0 0 0
.. . .. consider a m-dimensional signal sub-sampled to a n-dimensional signal.. the representation of the original signal) can be determined by simply left-multiplying the sub-sampled signal g by M −1 . then we can sample it without loss of information. . 0 fm−1 1 fm
(3. . The Nyquist sampling theory tells us that if a signal is band-limited (i. ...
33
.words. Nyquist sampling theory can be thought of as simply a matrix inversion problem. the trick to sampling and perfect reconstruction is to simply limit the dimensionality of the signal to at most twice the number of samples. This should not be at all surprising.
More specifically. of course. The design of such filters consists of four basic steps: 1. 2.
4
In multi-dimensional filter design. and its selection is left to their discretion.W.4. We will however provide some general guidelines for choosing a frequency response that is amenable to a successful design.
34
. In the frequency domain. a lowpass. bandpass. Parks and C. The designer should experiment with varying size filters until an equitable balance is found. Burrus for a full coverage of many other approaches).1)
4. With a frequency response and filter size in hand.. a large filter allows for a more accurate match to the desired frequency response.S.g. Digital Filter Design
Recall that the class of linear time-invariant systems are fully characterized by their impulse response. choose the desired frequency response choose the length of the filter define an error function to be minimized choose a minimization technique and solve
The choice of frequency response depends. a filter modifies the frequencies of the input signal. the output of a linear time-invariant system to any input f [x] can be determined via a convolution with the impulse response h[x]: g[x] = f [x] ⋆ h[x]. (4. 3. 4. Although there are numerous techniques for the design of digital filters we will cover only two such techniques chosen for their simplicity and generally good performance (see Digital Filter Design by T. (4. separability is also a desirable property. or highpass filters). however a small filter is desirable in order to minimize computational demands 4 .2)
In other words.1 Choosing a Frequency Response 4. this expression takes on the form: G[ω ] = F [ω ]H [ω ]. In choosing a filter size there are two conflicting goals.4 Weighted Least-Squares
Therefore the filter h[x] and the linear-time invariant system are synonymous. this chapter will provide the computational framework for realizing a finite length filter that "best" approximates the specified frequency response.3 Least-Squares 4.2 Frequency Sampling 4. It is often the case that such filters pass certain frequencies and attenuate others (e. on the designers particular application.
In general this problem is hard because by restricting ourselves to a small finite-length filter we are in effect asking to fit an arbitrarily complex function (the desired frequency response) with the sum of a small number of sinusoids. Our goal is to design a filter h whose Fourier transform best approximates the specified response H .
(4.4. that is: F (h) = H (4. this design is equivalent to solving for n unknowns (the filter taps) from n linearly independent equations.3)) in matrix notation: Mh = H (4. (4. and n is the chosen filter size.frequency sampling.
36
. the design simply involves inverse Fourier transforming the specified frequency response. This series of steps can be made more explicit and practical for computer implementation by expressing the initial constraint (Equation (4.4)
In other words. M is the n × n Fourier matrix (Equation (2.6)
Since the matrix M is square. This fact illustrates the shortcomings of this approach. The filter h can be solved for by left multiplying both sides of Equation (4. but places no constraints on the response between sampled points.5) by the inverse of the matrix F : h = M −1 H.3)
where F is the Fourier transform operator. the more difficult task of designing a finite-length filter that closely approximates this response begins.5)
where H is the n-dimensional sampled frequency response. We begin with the most straight-forward design method . By applying the inverse Fourier transform we obtain a solution: F −1 (F (h)) = F −1 (H )
h = F −1 (H ).2 Frequency Sampling Once the desired frequency response has been chosen. This restriction can often lead to poor results that are partially alleviated by the least-squares design presented next. namely. that this method produces a filter with a frequency response that exactly matches the sampled response.28)).
H is the N sampled frequency response. this equation is over constrained.
37
.8)
In order to minimize 5 .8)) is uniform across all frequencies. We can solve this system of equations in a least-squares sense by first writing a squared error function to be minimized: E (h) = | M h − H |2 (4. π ]. As before this constraint can be expressed as: M h = H. note that the previous frequency sampling design is equivalent to the least-squares design when the sampling of the Fourier basis is the same as the filter size (i. the frequency response was of the form given in Equation (4. This is easily rectified by introducing a weighting on the least-squares error function: E (h) = W | M h − H |
5
0.7)
where M is the N × n Fourier matrix (Equation (2. 4. then set equal to zero. Note that unlike before. N = n). π/2]. and highpass
Because of Parseval's theorem (which amounts to the orthonormality of the Fourier transform).28)).3 Least-Squares Our goal is to design a filter h that "best" approximates a specified frequency response.e.11)
Figure 4. The frequency response was sampled at a rate of N = 512. ω1 ] = [0. and highpass 16-tap filters designed using this technique. In this design. Finally.5
0 0 1
pi/2
pi
0. bandpass. (4. we differentiate with respect to the unknown h: dE (h) dh = 2M t | M h − H | = 2M t M h − 2M t H.5
0 0
pi/2
pi
2
(4.10)
1
(4. with start and stop bands of [ω0 . bandpass.4 Weighted Least-Squares One drawback of the least-squares method is that the error function (Equation (4.. and the filter size is n. [π/4.9)
Shown in Figure 4.4 Leastsquares: lowpass. and solve: h = (M t M )−1 M t H (4. and a transition width of ∆ω0 = ∆ω1 = π/4. the minimal error in the frequency domain equates to a minimal error in the space domain.4. 3π/4]. having n unknowns in N > n equations.4 are a set of lowpass.3). and [π/2.5
0 0 1
pi/2
pi
0.
e.
pi/2 pi
0 0 1
0.
(4. the errors in the low frequencies are smaller.5
0 0
Shown in Figure 4. As before.5 is a comparison of an 8-tap lowpass filter designed with a uniform weighting.5
Note that this solution will be equivalent to the original leastsquares solution (Equation (4. W (ω ) = (|ω|+1) 8.
pi/2 pi
Figure 4. while the errors in the high frequencies have increased. we minimize by differentiating with respect to h: dE (h) dh = 2M t W | M h − H | = 2M t W M h − 2W M t H. Note that in the case of the later.5 Least-squares and weighted least squares
38
. and with a weighting that 1 emphasizes the errors in the low frequency range. and solve:
1
(4. then set equal to zero.where W is a diagonal weighting matrix. the diagonal of the matrix contains the desired weighting of the error across frequency. That is.10)) when W is the identity matrix (i.. uniform weighting).13)
0.12)
h = (M t W M )−1 M t W H.
the image coordinate system is chosen to be orthogonal to the Z-axis. It is straight-forward to show from a similar triangles argument that the relationship between the world and image point is: dX x=− Z and dY y=− .2 Perspective projection
39
. and continues to be popular today among students. the origins of the two systems are related by a one-dimensional translation along the Z −axis or optical axis. the pinhole camera is convenient because it affords a simple model of more complex imaging systems..3 CCD
Figure 5. this simple device is capable of producing a photograph. artists. Photons to Pixels
5. the pinhole camera is not a particularly efficient imaging system (often requiring exposure times as long as several hours) and is more popular for its artistic value than for its practical value.
Figure 5.C. Z (5. Note that the world and image points are expressed with respect to their own coordinate systems. Denote a point in the three-dimensional world as a column vector. and for convenience.1) is often referred to as the focal length. We do not adopt this convention primarily because it is a misnomer. The Chinese philosopher Mo Ti is believed to be the first to notice that objects reflect light in all directions and that the light rays that pass through a small hole produce an inverted image. That is. i.2 Lenses 5.1)
5. under the pinhole model all points are imaged in perfect focus.1 Pinhole Camera The history of the pinhole camera (or camera obscura) dates back as early as the fifth century B. the projection of points from the three-dimensional world onto the two-dimensional sensor takes on a particularly simple form. In its simplest form a pinhole camera is a light-tight box with a tiny hole in one end and a photo-sensitive material on the other. and scientists..1 Pinhole image formation
Y P Z
y
p x X
where d is the displacement of the image plane along the Z-axis 6 These equations are frequently referred to as the perspective projection equations.1 Pinhole Camera 5. Although non-linear in their nature. Nevertheless. However. P = ( X Y Z )t and the projection of this point onto the two dimensional image plane as p = ( x y )t . with a pinhole camera model. the perspective projection equations may be expressed in matrix form
6 The value d in Equation (5. Remarkably.e.5.
3 Orthographic projection
1 0 0 1
X 0 Y 0 Z
(5. In the special case when all the points lie on a single frontal-parallel d is constant in Equasurface relative to the image plane (i. That is.e. where light rays are assumed to travel from a point in the world parallel to the optical axis until they intersect the image plane.2 Lenses It is important to remember that both the perspective and orthographic projection equations are only approximations of more complex imaging systems. whereas a pinhole camera captures a single light ray.using the homogeneous equations:
X xs −d 0 0 0 Y ys = 0 −d 0 0 Z . Z tion (5. If the point P 40
Y P Z
y
Figure 5. Under the thin-lens model the projection of the central or principal ray obeys the rules of perspective projection. light emanates from a point in the world in all directions and. Such complex imaging systems are often described with the simpler thin-lens model. Unlike the pinhole camera and perspective projection equations.4)
Orthographic projection is a reasonable approximation to perspective projection when the difference in depth between points in the world is small relative to their distance to the image plane.4 Thin lens
. 5.
(5.2)
where the final image coordinates are given by ( x
s y )t = ( x s
ys t s ) . Equation (5. a lens collects a multitude of light rays and focuses the light to a small region on the image plane. this model is not physically realizable and is used primarily because the projection equations take on a particularly simple linear form:
Y P p Z x X y
x=X And in matrix form: x y =
and
y = Y. the difference between perspective and orthographic is only a scale factor.1): the point P = ( X Y Z )t is projected onto the image plane cen−dY t dX tered about the point ( x y )t = ( −Z Z ) .
An approximation to the above perspective projection equations is orthographic projection.1)).3)
Figure 5. Commercial cameras are constructed with a variety of lenses that collect and focus light onto the image plane. s 0 0 1 0 1
(5..
In addition to this geometric argument for the non-invertibility of image formation. Note also that as the lens radius R approaches 0 (i. Inherent to such a transformation is a loss of information. Points at a depth of Zo = Z are imaged onto a small region on the image plane. d
(5. independent of the particular model. the projection of each light ray can be described in the following more compact matrix notation: l2 α2 = 1
1 −R n2 −n1 n2
0
n1 n2
l1 α1
. then the remaining light rays captured by the lens also strike the image plane at the point p.5 Non-invertible projection
.
(5. all points of the form Pc = ( cX cY cZ )t . and α2 is the angle between the exiting light ray and the optical axis.
Y
Image formation. in this case depth information.7)
where R is the radius of the lens. is a threedimensional to two-dimensional transformation..6)
where R is the radius of the lens. the blur radius is zero.is in perfect focus. f (5. Alternatively.the projection is not one-to-one and thus not invertible. the blur radius also approaches zero for all points independent of its depth (referred to as an infinite depth of field). The focal length is defined to be the distance from the lens to the image plane such that the image of an object that is infinitely far away is imaged in perfect focus. l1 and l2 are the height at which a light ray enters and exits the lens (the thin lens idealization ensures that l1 = l2 ). a pinhole camera). respectively.e. Specifically. often modeled as a blurred circle with radius r : r =
1 f
R 1 −Z o
1 1 − f Zo
−
1 . 41
y
P 1 Z
P 2
P 3 p x X
Figure 5. α1 is the angle between the entering light ray and the optical axis. and f is the focal length of the lens. n1 and n2 are the index of refraction for air and the lens material.5). are projected to the same point ( x y )t . for any c ∈ R. A point is imaged in perfect focus if its distance from the lens satisfies the following thin-lens equation: 1 1 + Z d = 1 . Note that when the depth of a point satisfies Equation (5.5)
where d is the distance between the lens and image plane along the optical axis. a similarly straight-forward linear algebraic argument holds. This formulation is particularly convenient because a variety of lenses can be described in matrix form so that a complex lens train can then be modeled as a simple product of matrices.
6 MOS capacitor
42
. 5.2)). An analog-to-digital converter then translates this voltage into a number (i. where n < m (e. When photons (i. the matrix M is not invertible and thus the projection is not invertible. light) enter the depletion region. p = Mn×m P . is applied to the surface of a P-type MOS capacitor. we have seen that the image formation equations may be written in matrix form as.g. The stored charge is transferred across a series of MOS capacitors (e. The value of the stored charge is proportional to the intensity of the light striking the capacitor. when a positive voltage.e. positive charge migrates toward ground. the intensity of an image pixel). first introduced in 1969. V . each one corresponding to a single image pixel. More specifically.In particular. As charge passes through the last capacitor in the series. Since the projection is from a higher dimensional space to a lower dimensional space. In its most basic form a CCD is a charge storage and transport device: charge is stored on the MOS capacitors and then transported across these capacitors for readout and subsequent transformation to a digital image. A digital image is subsequently formed by transferring the stored charge from one depletion region to the next.. a row or column of the CCD array) by sequentially applying voltage to each MOS capacitor. To complete the image formation process we need to discuss how the light that strikes the image plane is recorded and converted into a digital image. the electrons released are stored in this region. The core technology used by most digital cameras is the chargecoupled device (CCD). an amplifier converts the charge into a voltage. The region depleted of positive charge is called the depletion region..3 CCD To this point we have described the geometry of image formation.. how light travels through an imaging system.
Light
V
Depletion region
MOS Ground
Figure 5..g.e. A basic CCD consists of a series of closely spaced metal-oxide-semiconductor capacitors (MOS). Equation (5.
For example. that is white is mapped to black. where α is a real number. the underlying image is left untouched. gamma correction). 43
6. it is only the mapping from pixel value to display brightness that is being effected. with the corresponding LUTs shown in the same figure.1 Lookup Table 6. A variety of interesting visual effects can be achieved by simple manipulations of the functional form of the LUT.3 Gamma Correction 6. An image is first passed through a lookup table (LUT) that maps the image intensity values to brightness values. Point-Wise Operations
6.4 Quantize /Threshold 6. and D and I are the displayed and image values. an image can be both contrast enhanced and brightened or darkened by simply passing the image through two (or more) LUTs. The image contrast is increased with a large slope and negative intercept (in the limit. Figure 6. Of course.1 Lookup Table The internal representation of a digital image is simply a matrix of numbers representing grayscale or color values.2 Brightness /Contrast 6. The bright and dark images of Einstein were created by passing the middle image through the LUTs shown in the same figure.2 Brightness/Contrast Perhaps the most common and familiar example of a LUT manipulation is to control the brightness or darkness of an image as shown in Figure 6.. and the contrast is reduced with a small slope and positive intercept (m → 0 and b → 1/2). with the image intensity values u ∈ [0. where the relationship between the slope and intercept is b = 1/2(1 − m). If the lookup table is linear with unit slope and zero intercept then the image is directly mapped to the display. Keep in mind though that in manipulating the lookup table. otherwise. the displayed image will not be an exact representation of the underlying image.e.2 Autoscale
.4. The top image is said to be high contrast and the bottom image low contrast. The functional form of these LUTs is linear: g(u) = mu + b.1. 6.3. The functional form of the LUT is a unit-slope line with varying intercepts: g(u) = u + b. and black is mapped to white. most computer monitors intentionally impose a non-linear LUT of the general form D = I α (i. Another common manipulation is that of controlling the contrast of an image as shown in Figure 6.6.1 Lookup table
white
black min max
Figure 6. A value of b > 0 results in a brightening of the image and b < 0 a darkening of the image.5 Histogram Equalize
white
black 0 255
Figure 6. 1]. m → ∞ and b → −∞). The image is inverted with m = −1 and b = 1. But when an image is displayed on a computer monitor we typically do not see a direct mapping of the image.
The problem with this sort of autoscaling is that a few stray pixels can dictate the contrast resulting in a low-contrast image. This may be avoided by employing a non-linear contrast adjustment scheme. This may be alleviated by employing a sigmoidal non-linearity of the form g(u) = 1 .4 Contrast
44
. Shown in Figure 6.Autoscaling is a special case of contrast enhancement where the minimum image intensity value is mapped to black and the maximum value is mapped to white.
Low
Figure 6. This non-linearity. The most standard approach is gamma correction.
Dark
Bright
(6.6 is a contrast enhanced image with α = 12 and β = 6. where the LUT takes the form: g(u) = uα . there is no saturation at the low or high end. respectively.2)
High
In order that g(u) be bound by the interval [0. and α < 1 reduces contrast.5). it must be scaled as follows: (g(u) − c1 )/c2 . Shown in Figure 6. black (0) and white (1) are mapped to themselves. 1 + e−αu+β (6. high contrast images are visually more appealing. 1].2. Although this will lead to a small amount of saturation. with its two degrees of freedom. Note that with the intensity values u scaled into the range [0.3 Gamma Correction Typically. That is. 6. A less sensitive approach is to sort the intensity values in increasing order and map the 1% and 99% intensity values to black and white.5 are contrast enhanced (top: α = 2) and contrast reduced (bottom: α = 1/2) images.3 Brightness
where α > 1 increases contrast. is more versatile and can produce a more balanced contrast enhancement. where c1 = 1/(1 + eβ ) and c2 = 1/(1+ e−α+β ) − c1. it rarely fails to produce a high-contrast image. also realizable as a lookup table (LUT) manipulation.1)
Figure 6. One drawback to this scheme is that the gray values are mapped in an asymmetric fashion with respect to midlevel gray (0. Gamma correction is widely used in a number of devices because it yields reasonable results and is easily parameterized. Autoscaling maximizes the contrast without saturating at black or white. Figure 6. 1]. However a drawback of linear contrast enhancement described above is that it leads to saturation at both the low and high end of the intensity range.
Notice that the effect is similar to contrast enhancement.e. Histogram equalization then amounts to simply inserting this function into the LUT. a twostep function. More specifically.. the image is said to be thresholded. 255]. typically in the range [0.9 is Einstein before and after histogram equalization.9.
(6. which intuitively should make sense since we increased the number of black and white pixels. by its very nature. 6. that is.
Why anyone would want to histogram equalize an image is a mystery to me. An image can be further quantized to a lower number of bits (b) or intensity values (2b ). with most the mass near mid-gray (128) and falling off on either side.6 Contrast: Sigmoid
C (y ) =
i=0
N (i).4 Quantize/Threshold A digital image.8 is a thresholded image of Einstein and the corresponding LUT. The point at which the step function transitions from zero to one is called the threshold and can of course be made to be any value (i. Quantization can be accomplished by passing the image through a LUT containing a step function. C (u) is the number of pixels with intensity value less than or equal to u. but here it is in case you do. Figure 6. This process is known has histogram equalization 7 .5 Contrast: Gamma
Figure 6. the cumulative distribution function is defined as:
u
Figure 6.
7
45
. when an image is quantized to one bit or two intensity values.6. Shown in Figure 6. For example an image quantized to 8-bits contains 28 = 256 possible intensity values. define N (u) as the number of pixels with intensity value u.7. An image can be transformed so that the distribution of intensity values is flat.7 is an image of Einstein quantized to five intensity values. where the number of steps governs the number of intensity values. Figure 6. Shown in Figure 6. notice that all the subtle variations in the curtain and in his face and jacket have been eliminated. In the limit. is quantized to a discrete number of intensity values. Although it may not be immediately obvious an image is histogram equalized by passing it through a LUT with the functional form of the cumulative distribution function.3)
that is. this is the image histogram and a discrete approximation to the probability distribution function. Shown in Figure 6. slid left or right).5 Histogram Equalize The intensity values of a typical image are often distributed unevenly across the full range of 0 to 255 (for an 8-bit image). each intensity value is equally represented in the image. Then.
all pixels that exceed the value 255 will be truncated to 255. Since such an operation is destructive it may not be inveritble.9 Histogram equalize
. for example when brightening an 8-bit image. and then displaying with a linear LUT.7 Quantize
Figure 6.In all of these examples the appearance of an image was altered by simply manipulating the LUT.8 Threshold
15000 10000 5000 0 0
100
200
15000 10000 5000 0 0
100
200
46
Figure 6. Such a manipulation leaves the image content intact. These operations can be made destructive by applying the LUT operation directly to the image. the mapping from image intensity value to display brightness value.
Figure 6. it is a non-destructive operation and thus completely invertible. For example an image can be brightened by adding a constant to each pixel.
5 Wiener Filter
ωy
g[x. y − l]. π ].2 Low-.2 Derivative Filters 7.0625 0.1250 0. Band-.2500 0.1 is a simplified decomposition of the 2-D Fourier domain parameterized by ωx and ωy ∈ [−π. The same intuition holds in 2-D. formed the basis for much of our discussion on discrete-time signals and systems.7. The inner disc corresponds to the lowest frequencies. Shown in Figure 7. Linear Filtering
7. a 2-D convolution with the blur filter: 0. Both of these operations can be accomplished with a 2-D filter and 2-D convolution.3 Steerable Filters 7. Similarly the two-dimensional convolution sum will form the basis from which we begin our discussion on image processing and computer vision. For example. now 2-D. High-pass
.1 Convolution The one-dimensional convolution sum.1 2-D Frequency
F [ωx . or more efficiently with a 1-D filter and a pair of 1-D horizontal and vertical convolutions. l]h[x − k.1 Convolution 7. The 1-D convolution sum extends naturally to higher dimensions.1250 0. The 2-D convolution sum is then given by:
∞ ∞
7. and high-pass filtering extends naturally to two-dimensional images. Equation (2. l]e−i(ωx k+ωy l) . y ]. the intuition for a convolution is that of computing inner products between the filter and signal as the filter "slides" across the signal.4 Edge Detection 7. In the Fourier domain.0625 0.1250 0.1)
In 1-D.
(7. where the 2-D Fourier transform is given by:
∞ ∞
ωx
Figure 7. y ] =
k =−∞ l=−∞
f [k. band-pass.5). Fourier transforms of the filter and image. the center annulus to the middle (band) frequencies.2)
The notion of low-pass. this convolution is equivalent to multiplying the.
(7.0625
Figure 7. Consider an image f [x. Two of the most common (and opposing) linear filtering operations are blurring and sharpening.1250 47
0. Inner products are computed between the 2-D filter and underlying image as the filter slides from left-to-right/top-to-bottom. and the outer dark area to the highest frequencies. y ] and a two-dimensional filter h[x. ωy ] =
k =−∞ l=−∞
f [k.0625
0.
so why not just differentiate the continuous signal f (x) and then sample the result? Surely this is what we have in mind when we 48
. But our intuition may be that this is not such an unreasonable request.3 Blur Sharpen and
Discrete differentiation forms the foundation for many applications in image processing and computer vision. A generic separable sharpening filter is of the form: sharp = ( 0. We are all familiar with the definition of the derivative of a continuous signal f (x): D {f (x)} = lim f (x + ε) − f (x) . ε (7. D {f [x]}. The amount of blur is then directly proportional to the size of the filter.25 0.08 −1. Shown in Figure 7. A generic blur filter may be constructed from any row of the binomial coefficients: 1 1 1 3 2 3 1 1
1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 where each row (filter) should be normalized by it's sum (i. is meant to enhance the high-frequencies.08 ) . 7.4)
This filter leaves the low-frequencies intact while enhancing the contribution of the high-frequencies. Blurring simply reduces the high-frequency content in an image. sharpening.25 ) . The opposing operation.e. which is only defined over an integer sampling lattice? Strictly speaking. no. does it make sense to differentiate a discretely sampled signal.2 Derivative Filters
Figure 7. from which the sampled signal f [x] was derived.5)
ε→0
This definition requires that the signal f (x) be well defined for all x ∈ R.50 0. (7. After all.the filters are xy-separable. an outer-product of the 1-D filter with itself yields the 2-D filter . we know how to differentiate f (x).can be realized by convolving in the horizontal and vertical directions with the 1-D filter: blur = ( 0. blur filters should always be unit-sum so as not to increase or decrease the mean image intensity).3 are results from blurring and sharpening. (7. The separability of 2-D filters is attractive for two reasons: (1) it is computationally more efficient and (2) it simplifies the filter design.00 0.3)
That is.. So.
Recall from earlier chapters that under certain conditions (Nyquist theory). the relationship between the continuous and sampled signals can be expressed precisely as: f (x) = f [x] ⋆ h(x). if the signal f (x) is sampled above the Nyquist rate and if it is in 49
Figure 7. (πx/T )2 (7.10)
0
0
where T is the sampling period at which f (x) was sampled.8)
Notice that the derivative operator D {·} is being applied only to continuous entities. the sampled derivative of the sync and the original sampled signal. Having computed the desired derivatives.
(7. But one should not be fooled by the seeming simplicity of our intuition.9)
0 0
On the left-hand side of the above equation is the desired quantity. On the right-hand side is a discrete convolution between two known quantities. (7.6)
where h(x) = sin(πx/T )/(πx/T ) is the ideal sync.ask for the derivative of a sampled signal. Now. denoting S {·} as the sampling operator: S { D {f (x)} } = f [x] ⋆ S { D {h(x)} } S {f (x)} = f [x] ⋆ S {h (x)} f ′ [x] = f [x] ⋆ h′ [x]. and ⋆ is the convolution operator. So.4 Ideal and its derivative
sync
. (7.7)
and expressing the right-hand side in terms of the convolution sum: D {f (x)} = D =
∞ k =−∞
f [k]h(x − k)
∞
k =−∞
f [k]D {h(x − k)}
= f [x] ⋆ D {h(x)}. The derivative of the sync can be expressed analytically by simply differentiating the sync function: h′ (x) = π 2 x/T 2 cos(πx/T ) − π/T sin(πx/T ) . the derivative of the sampled signal. we need only sample the results.
′ ′
1
(7. as we will soon discover the design of an accurate and efficient discrete derivative operator will prove to be sufficiently challenging. applying the continuous differential operator to both sides yields: D {f (x)} = D {f [x] ⋆ h(x)}.
Trying to directly design a finite length filter to this response is futile because of the discontinuity at −π /π . at least in the low-frequency range. To begin we need to compute the frequency response of the ideal derivative filter. is spatially infinite in extent. (7. We can compute the response indirectly by first expressing f [x] in terms of its Fourier series: f [x] = 1 2π
π
F [ω ]eiωx . an imaginary ramp is the frequency response of the ideal derivative filter.11)
and then differentiating both sides with respect to x: D {f [x]} = = 1 2π 1 2π
π ω =−π π ω =−π
F [ω ]D {eiωx } iωF [ω ]eiωx . So we are going to have to part with mathematical perfection and design a finite-length filter. If you are feeling a bit uneasy it is for a good reason. And since multiplication in the frequency domain is equivalent to convolution in the space domain. which of course accounts for the spatially infinite extent of h′ (x).8)) we formulate a quadratic error function to be minimized: E (h) = | M h − H |2 . meaning that it cannot be implemented on a finite machine. In particular the derivative of the sync. (7. The simplest such approximation is that of a sinusoid where. h′ (x) falls off slowly from the origin so that truncation will cause significant inaccuracies. Although mathematically correct.5 Ideal and approximate derivative frequency response
Differentiation in the space domain is then seen to be equivalent to multiplying the Fourier transform F [ω ] by an imaginary ramp iω . the match is reasonably good (i. Employing the least-squares filter design technique (Equation (4. H is the N sampled desired frequency response. And even worse. for small ω ).12)
pi
0
−pi −pi
0
pi
Figure 7. then Equation (7.fact differentiable.e..9) tells us that we can exactly compute the derivative of the sampled signal f [x].
ω =−π
(7. we have a solution for differentiating a discretely sampled signal that is physically unrealizable. and n the filter size is. sin(ω ) = ω . 50
.13)
where M is the N × n Fourier matrix (Equation (2. an altogether happy ending. h′ (x).28)). So we are resigned to designing a filter with a periodic frequency response that "best" approximates a ramp.
y ) = h(x) ⋆ h(y )). y ] ⋆ h′ (x) ⋆ h(y ). y ) = f [x. frequency domain differentiation).. these functions are sampled to produce discrete filters.. and. y ] ⋆ Dx {h(x) ⋆ h(y )} = f [x. y ] ⋆ Dx {h(x)} ⋆ h(y ) (7.e.
(7. but in the final step. h[y ].20)
f'[x]
Notice that calculating the partial derivative requires a pair of one-dimensional convolutions: a derivative filter.7 are the magnitudes of the Fourier transform of the derivative filter (solid line) and the interpolation filter times iω (i. the above equation can be rewritten as: fx (x.e. in the dimension of differentiation. So how is this different from the one-dimensional case? In the one-dimensional case only the derivative filter is employed. For example √ consider the often √ used Sobel derivative filters with h[x] = ( 1 2 1 ) /(2 + 2) and h′ [x] = ( 1 0 −1 ) /3. which they clearly 52
f[y]
Figure 7. In some ways the choice of filters seems trivial: chose an interpolation function h(x). Since the two-dimensional sync is separable (i. This sampling step typically destroys the required derivative relationship.19)
And finally. although a seemingly subtle point. Since two-dimensional differentiation reduces to a pair of onedimensional convolutions it is tempting to simply employ the same differentiation filter used in the one-dimensional case.7 Sobel frequency response
. has dramatic effects on the accuracy of the resulting derivative operator. And in fact this constraint is automatically enforced by the very nature in which the continuous functions are chosen. y ] ⋆ S {h′ (x)} ⋆ S {h(y )} fx [x. and an interpolation filter. sampling both sides gives an expression for the partial derivative of the discretely sampled two-dimensional signal: S {fx (x. h(x.6 Horizontal partial differentiation
−pi
0
pi
Figure 7. h′ [x]. If the filters obeyed the required derivative relationship than these curves would be exactly matched.noting again that the differential operator is only applied to continuous entities. And by our formulation we know that the pair of filters should satisfy the relationship that one is the derivative of the other h′ (x) = D (h(x)). all remaining dimensions would be convolved with the interpolation filter). Shown in Figure 7. But since a pair of filters are now required perhaps we should give this some additional thought. in the other dimension (for multi-dimensional signals. and sample these functions to get the final digital filters h[x] and h′ [x]. differentiate it to get the derivative function h′ (x). y )} = f [x. y ] = f [x. y ] ⋆ h′ [x] ⋆ h[y ]. whereas in the two-dimensional case we require the pair of filters. = f [x.
(7. This error function can be expressed more concisely as: E (u) = |M u|2 .are not. (7. we write a discrete approximation of this continuous error functional over the n-vectors h and h′ containing the sampled derivative and interpolation filters. H ′ ) = dω W (ω )(iωH (ω ) − H ′ (ω )) . We begin by writing down the desired relationship between the derivative and interpolation filters.21)
from which we can write a weighted least-squares error functional to be minimized: E (H. Let's see then if we can design a better set of filters. respectively: E (h. h′ ) = |W (F ′ h − F h′ )|2 .22)
2
where W (ω ) is a frequency weighting function.
53
. the derivative and interpolation filters can be "unpacked" and normalized so that the interpolation filter is unit sum. Note that the dimension n is determined by the filter size and the dimension m is the sampling rate of the continuous Fourier basis functions.e. the matrix ′ Fm ×n = iωFm×n . which should be chosen to be sufficiently large to avoid sampling artifacts. After solving for u. (7.23)
where the columns of the matrix Fm×n contain the first n Fourier basis functions (i..24)
where the matrix M and the vector u are constructed by "packing together" matrices and vectors: M = (WF′ | −W F ) and u= h h′ . The mismatching of the filters results in gross inaccuracies in derivative measurements.25)
The minimal unit vector u is then simply the minimal-eigenvalue eigenvector of the matrix M t M . Next. most conveniently expressed in the frequency domain: H ′ (ω ) = iωH (ω ). and Wm×m is a diagonal frequency weighting matrix. (7. (7. a discrete-time Fourier transform). Below are the resulting filter values for a 3-tap and 5-tap filter pair.
9 Differential motion estimation
7.036420 0. t)vx (x.0 -0. y. The standard formulation for motion estimation is: fx (x.8 Frequency response of matched derivative filters
where the motion vector is v = ( vx vy )t . t). Quite remarkably it turns out that we need not design a new set of filters for each possible direction because the derivative in any direction can be synthesized from a linear combination of the horizontal and vertical derivatives. t) = 0.280353 0. Notice that the 5-tap filters are nearly perfectly matched.453014 0. t)vy (x.26)
−pi
0
pi
Figure 7.8 are the matching of these filters in the frequency domain. h h′ h h′
−pi 0 pi
0.Shown in Figure 7. Rather. y.223755 0. Shown in Figure 7. 54
.3 Steerable Filters In the previous section we showed how to compute horizontal and vertical partial derivatives of images. y.108415
Higher-order derivative filters can be designed by replacing the initial constraint in Equation (7.21) with H ′ (ω ) = (iω )k H (ω ) for a kth order derivative. and in this way is quite distinct from the onedimensional case.0 -0. y ) + ft (x. The proper choice of derivative filters can have a dramatic impact on the applications which utilize them.9 are the resulting motion fields for a simple translational motion with the Sobel (top panel) and matched (bottom) derivative filters used to compute the various derivatives.280353 -0.248972 0.248972 0. and ft (·) are the partial derivatives with respect to space and time. and fx (·). fy (·). For example.453014
0. (7. we chose to work in the frequency domain where differentiation takes on a particularly simple form. the difference in accuracy is significant. A peculiar aspect of this filter design is that nowhere did we explicitly try to model a specified frequency response. This property of derivatives has been termed steerability.
Figure 7.429217 0.036420 0. One may naturally wonder how to compute a derivative in an arbitrary direction. Although these filters are the same size.108415 0. a common application of differential measurements is in measuring motion from a movie sequence f (x.552490 0. y. There are several formulations of this property.223755 0. the design fell naturally from the relationship between the continuousand discrete-time signals and the application of the continuous derivative operator. y ) + fy (x.
More specifically. Expressing this oriented ramp in terms of the horizontal and vertical ramps provides the basis and coefficients needed to steer derivatives of arbitrary order: (ωα )N = (cos(α)ωx + sin(α)ωy )N .Notice that we obtain the horizontal and vertical derivatives when α = 0 and α = 90. y ) and fy (x.33)
where for notational simplicity the spatial arguments (x. 56 (7.32). the N th -order derivative is steerable with a basis set of size N + 1. Steerability is not limited to first-order derivatives. For example. its vertical derivative fy (x. Note that three partial derivatives are now needed to steer the second-order derivative. (−jωα )N . y ). From the previous section we know how to compute the horizontal and vertical derivatives via convolutions with an interpolation and derivative filter. as in Equation(7. (7. Pause to appreciate how remarkable this is.. its horizontal derivative fx (x.30) the N th -order derivative in the frequency domain is computed by multiplying the Fourier transform by an imaginary oriented ramp raised to the N th power.10 Steerability
. (7. y ). To compute any other directional derivative no more convolutions are required.10 from top to bottom is a disc f (x. the second-order derivative in an arbitrary direction can be synthesized as follows: fαα = cos2 (α)fxx + 2 cos(α) sin(α)fxy + sin2 (α)fyy . Higher-order derivatives are also steerable. a pair of directional derivatives is sufficient to represent an infinite number of other directional derivatives.34) You may have noticed that the coefficients needed to steer the basis set look familiar.e. y ). Similarly.35)
Figure 7. y ) have been dropped. where the steered derivative was synthesized from the appropriate linear combinations of the horizontal and vertical derivatives. Shown in Figure 7. α can take on any real-valued number. y ). This equation embodies the principle of steerability . and the multiple subscripts denote higher-order differentiation.the derivative in any direction α can be synthesized from a linear combination of the partial horizontal and vertical derivatives. i. they are the binomial coefficients that come from a polynomial expansion. The obvious benefit of steerability is that the derivative in any direction can be synthesized with minimal computational costs. and its steered derivative f45 (x. simply take the appropriate linear combinations of the horizontal and vertical derivatives as specified in Equation (7. the third-order derivative can be steered with a basis of size four: fααα = cos3 (α)fxxx + 3 cos2 (α) sin(α)fxxy +3 cos(α) sin2 (α)fxyy + sin3 (α)fyyy . fx (x. y ).
y ]. but is only briefly touched upon here.Although presented in the context of derivatives. y ]: fx [x. where h(θ ) is band-limited. The goal of de-noising is to recover the original signal s from the corrupted signal s ˆ. More specifically.11 Edges
| ▽ [x. y ] = (f [x. One such example is edge detection .12 Additive noise
. The introduction of noise into a signal is often modeled as an additive process.36)
then the filter is steerable with a basis size of 2N .a topic that has received an excessive amount of attention. In the most general case.
′
(7. the principle of steerability is not limited to derivatives. A basic edge detector begins by computing first-order spatial derivatives of an image f [x. y ) is steerable in orientation if it can be expressed as a polar-separable function.38)
where h′ [·] and h[·] are the derivative and prefilter defined in Section 7. for an arbitrary radial component g(r ).4 Edge Detection Discrete differentiation forms the foundation for many applications in computer vision.5 Wiener Filter For any of a number of reasons a digital signal may become corrupted with noise.37) (7. y ] ⋆ h′ [x]) ⋆ h[y ] fy [x. An edge is loosely defined as an extended region in the image that undergoes a rapid directional change in intensity. The "strength" of an edge at each spatial location is defined to be the magnitude of the gradient vector ▽[x. y ] ).39)
As shown in Figure 7.11. s ˆ = s + n. Differential techniques are the obvious choice for measuring such changes. y ] ⋆ h[x]) ⋆ h [y ]. 7. y ] = ( fx [x. defined as:
Figure 7. y ] fx
+
2 [x. y ] = (f [x. g(r )h(θ ). y ] fy [x.2. the gradient magnitude is only the beginning of a more involved process (not discussed here) of extracting and localizing the salient and relevant edges. Given a single constraint in two unknowns this problem is equivalent to my asking you "37 is the sum of two numbers. 7. a two-dimensional filter f (x. But by 57
n
s
+
s
Figure 7. y ]| =
2 [x. what are they?" Lacking clairvoyant powers or knowledge of how the individual numbers were selected we have little hope of a solution. fy
(7. and for h(θ ) expressed as:
N
h(θ ) =
n=1
an cos(nθ ) + bn sin(nθ )
(7.
for example a common choice is to assume white noise. Shown in Figure 7. But we can instantiate them by making assumptions about the general statistical nature of the signal and noise. to assume that S (ω ) = 1/ω p . Note that the noise levels are reduced but that much of the sharp image structure has also been lost. which is an unfortunate but expected side effect given that the Wiener filter is low-pass in nature. The frequency response in the top panel of Figure 7.13 was constructed under these assumptions. This one-dimensional formulation can easily be extended to two or more dimensions.frequency response in terms of the signal and noise characteristics. and. but of course we don't typically know what those are. and the results of applying a 7 × 7 Wiener filter. Einstein plus noise. is Einstein. for natural images.14 from top to bottom. Shown in the bottom panel is a 7-tap filter derived from a least-squares design. N (ω ) is constant for all ω .
59
.
one where a small number of pixels are corrupted due to. Here we look at a different noise model. Depending on the density of the noise the median filter may need to be computed over a larger neighborhood. Shown in the middle panel of Figure 8. a faulty transmission line. for example. Because of the non-linear sorting step of a median filter it cannot be implemented via a simple convolution and is thus often more costly to implement.1 is the disastrous result of applying the solution from the additive noise model (Wiener filter) to the salt and pepper noise image in the top panel. however this loss of detail is quite distinct from that of averaging. this is often referred to as posterization. hence the name salt and pepper used to describe such noise patterns (Figure 8. More specifically. This effect could never be achieved with an averaging filter which would indiscriminately smooth over all image structures.1 Median Filter Noise may be introduced into an image in a number of different ways.2 is the result of applying a 15 × 15 median filter.
Salt & Pepper Noise
Wiener
Median
Figure 8. Outside the scope of this presentation there are a number of tricks for reducing the computational demands of a median filter.1). the center pixel of each 3 × 3 neighborhood is replaced with the median of the nine pixel values in that neighborhood. Non-Linear Filtering
8-1 Median Filter 8-2 Dithering
8. the mean is best replaced with the median. As the income of Gates will skew the average salary so does each noise pixel when its value is so disparate from its neighbors. For example. computed by sorting the set of numbers and reporting on the value that lies midway. shown in Figure 8. In the previous section we talked about how to remove noise that has been introduced in an additive fashion. The corrupted pixels randomly take on a value of white or black. The tradeoff being that a larger neighborhood leads to a loss of detail.1 Median filter
Figure 8.2 15 × 15 median filter
60
. In such cases. Notice that although many of the internal details have been lost the boundary contours (edges) have been retained. Shown in the bottom panel of Figure 8. Trying to average out the noise in this fashion is equivalent to asking for the average salary of a group of eight graduate students and Bill Gates.8.1 is the much improved result of applying a 3 × 3 median filter to the salt and pepper noise image.
Although in both images each pixel takes on only one of two gray values (black or white). (3/16)120 is added to the pixel at 61
Figure 8. For example. The error. Feynman. A standard and reasonably effective algorithm is a stochastic error diffusion algorithm based on the Floyd/Steinberg algorithm. is distributed to four neighbors as follows: (7/16)120 is added to the pixel at position (x + 1. and all values greater than 128 are thresholded to 255 (white).. Since this algorithm makes only a single pass through the image. 2 gray values) images. y ). As an example. and the position of the weights represent spatial position on a rectangular sampling lattice. Shown below are two 1 bit (i. so all pixels less than 128 (mid-level gray) are thresholded to 0 (black). The first was produced by thresholding. the algorithm is casual). y ) with intensity value 120 is thresholded to 0.3 Thresholding and Dithering
. it is clear that the final image quality is critically dependent on the way in which pixels are quantized. the algorithm extends naturally to an arbitrary number of gray levels. 256 gray values) grayscale image of Richard P.3 is an 8-bit (i. Typically. For simplicity.e. Note also that since the weights have unit sum. The basic Floyd/Steinberg error diffusion dithering algorithm tries to exploit local image structure to reduce the effects of quantization.. the error is distributed to four neighbors with the following weighting:
1 16
×
• 7 3 5 1
. the gray value is first thresholded into "black" or "white". left to right and top to bottom. A pixel at position (x. consider the quantization of an 8-bit image to a 1-bit image.2 Dithering Dithering is a process by which a digital image with a finite number of gray levels is made to appear as a continuous-tone image. shown at the top of Figure 8. This algorithm operates by scanning through the image. the difference between the new pixel value and the original value is then computed and distributed in a weighted fashion to its neighbors. a 1-bit version of this algorithm is described here.
where the • represents the thresholded pixel. There are numerous dithering algorithms (and even more variants within each algorithm). 120-0 = 120.e. the neighbors receiving a portion of the error must consist only of those pixels not already visited (i.. the error is neither amplified nor reduced. At each pixel.e. An 8-bit image has 255 gray values. Sadly there are few quantitative metrics for measuring the performance of these algorithms. and the second by dithering.8.
y + 1) and (1/16)120 to pixel (x + 1. (5/16)120 to pixel (x.1)... randomly scaling the error in the range 0. and alternating the scanning direction (e.9 to 1. odd lines are scanned left to right. y + 1).g. the error diffusion algorithm reduces the effects of quantization via simple thresholding. y + 1). By propagating this error the surrounding pixels are brightened making it more likely that they will be thresholded to something brighter.
62
. These problems may be partially alleviated by introducing a stochastic process in a variety of places. However this algorithm does introduce correlated artifacts due to the deterministic nature of the algorithm and scanning order. Qualitatively.position (x − 1.g. Two possibilities include randomizing the error before distributing it to its neighbors (e. As a result the local neighborhood maintains it's average gray value. and even lines from right to left). The intuition behind this algorithm is that when the pixel value of 120 is thresholded to 0 that pixel is made darker.
9. Multi-Scale Transforms
63
.
y (t). N − 1]. Shown in Figure 10. where vx and vy denote the horizontal and vertical components of the velocity vector v . we require an estimate of motion (velocity) v = ( vx vy ). People. walking across the room. or smaller motions. With respect to notation. t) ∂t = 0. In order to estimate motion.3)
. etc. y ) and an image sequence is denoted as f (x(t). an image is denoted as f (x. Also shown is the corresponding estimate of motion often referred to as a flow field. Our task is to estimate such image motions from two or more images taken at different instances in time. dogs. (10. The flow field consists of a velocity vector at each point in the image (shown of course are only a subset of these vectors). Expanding this constraint according to the chain rule yields: ∂f ∂x ∂f ∂y ∂f + + ∂x ∂t ∂y ∂t ∂t = 0. our task is to estimate the amount of motion at each point in the image. a sequence of N images taken in rapid succession may be represented as f (x(t). it is assumed that as a small surface patch is moving. Given such an image sequence. t + i∆t) with i ∈ [0. For example. scratching behind your ear. This constraint can be expressed with the following partial differential equation: ∂f (x(t). 64 (10. y (t).10. an assumption of brightness constancy is made.1 Differential Motion Our visual world is inherently dynamic. These may be gross motions. and ∆t representing the amount of time between image capture (typically on the order of 1/30th of a second). are (usually) moving. t). where x(t) and y (t) are the spatial parameters and t is the temporal parameter. Motion Estimation
10-1 Differential Motion 10-2 Differential Stereo
10. That is. For a given instant in space and time.2)
where the partials of the spatial parameters x and y with respect to time correspond to the velocity components: fx vx + fy vy + ft = 0. its brightness value remains unchanged.1)
Figure 10. y (t). (10. cars.1 are a pair of images taken at two moments in time as a textured square is translating uniformly across the image.1 Flow field
This constraint holds for each point in space and time.
65
.y
ft
. fx = 0 or fy = 0) or zero-dimensionally (fx = 0 and fy = 0). A least-squares error function takes the form:
2
E (v ) =
x. The constraint can be solved by assuming that the motion is locally similar. when is the matrix M invertible? Generally speaking the matrix is rank deficient. and integrating this constraint over a local image neighborhood. when the intensity variation in a local image neighborhood varies only one-dimensionally (e. leaving us with a single constraint in two unknowns (the two components of the velocity vector. This transformed brightness constancy constraint is rewritten by packing together the partial derivatives and velocity components into row and column vectors. then the velocity can be estimated by simply left multiplying by the inverse matrix: v = − M −1 b (10.5)
To solve for the motion this error function is first differentiated ∂E (v ) ∂v = 2 = 2 fs
t fs v+
ft fs ft .
If the matrix M is invertible (full rank). These singularities are sometimes referred to as the aperture and blank wall problem.g.y
t v fs
+
x.6)
t fs fs v+2
Setting equal to zero and recombining the terms into matrix form yields: fx fy (
2 fx fx fy
fx
fy ) fx fy 2 fy
vx vy vx vy
= − = −
fx ft fy ft fx ft fy ft (10.8)
The critical question then is.
(10.
The space/time derivatives fs and ft are measured quantities. The motion at such points simply can not be estimated. v )..7)
M v = −b.4)
t fs v + ft = 0.The subscripts on the function f denote partial derivatives. (10. ( fx fy ) vx vy + ft = 0 (10. Note again that this constraint holds for each point in space and time but that for notational simplicity the spatial/temporal parameters are dropped. and hence not invertible.
Given a temporal sequence of N images. y ). The spatial derivative in the horizontal direction fx is estimated by prefiltering this image in the vertical y direction and differentiating in x.2 Motion and Stereo
where the disparity ∆ = 2δ. Of course the temporal derivative requires a minimum of two images. Similarly. the image that is formed by translating the camera in a purely horizontal direction is given by f (x + ∆(x. Stereo estimation typically involves a pair of cameras spatially offset in the horizontal direction such that their optical axis remain parallel (Figure 10. and ft . Motion is thought of in a continuous (differential) framework. relative to distant objects).2.Motion estimation then reduces to computing. Z ) is imaged to the image position (x. the spatial derivative in the vertical direction fy is estimated by prefiltering in the horizontal x direction and differentiating in y . Y. and is typically estimated from between two and seven images. these problems are virtually identical: velocity (v ) ≡ disparity (∆). the spatial/temporal derivatives fx . fy . the disparity (δ) between the left and right images. to find the shift that brings the stereo pair back 66
. As illustrated in Figure 10. This dichotomy is unnecessary: stereo estimation can be cast within a differential framework. Stereo estimation involves determining the displacement disparity of a stationary object as it is imaged onto a pair of spatially offset cameras. (10. 10. Motion and stereo estimation are often considered as separate problems. Denoting an image as f (x. If a point in the world (X. y ). Given this. The choice of filters depends on the image sequence length: an N tap pre/derivative filter pair is used for an image sequence of length N (See Section 7).e. for each point in space and time. y ).2). That is. with its discrete pair of images. a stereo pair of images is denoted as: fL (x + δ(x.2 Differential Stereo Motion estimation involves determining. Our task is to determine. then the shift ∆(x. while stereo. y ). The spatial/temporal derivatives are computed as follows. from a single stationary camera. is thought of in terms of a discrete matching problem. Finally. y ).. y ). for each point in the image. the temporal derivative is estimated by temporally differentiating the original N images. and prefiltering the result in both the x and y directions. the spatial derivatives are computed by first creating a temporally prefiltered image. nearby objects have large disparities. y ). how much an object moves over time (its velocity).9)
V
D
Figure 10. y ) is inversely proportional to the distance Z (i. y ) and fR (x − δ(x.
this leads to a single constraint with a single unknown which can be solved for directly: vx = − ft .where the partial derivative of the spatial parameter x with respect to time correspond to the motion (speed) in the horizontal direction: fx vx + ft = 0. In both the differential motion and stereo formulations there exists singularities when the denominator (spatial derivative) is zero. if the spatial derivative is zero over a large area. then disparities at these points simply can not be estimated. In the stereo case. the denominator is a spatial derivative. However. As with the earlier motion estimation this can be partially alleviated by integrating the disparities over a local image neighborhood. This may seem odd. corresponding to a surface in the world with no texture. but recall that differentiation of a multi-dimensional function requires differentiating along the desired dimension and prefiltering along all other dimensions (in this case the viewpoint dimension).16)
Unlike before.
68
. in one case with respect to time (motion) and in the other with respect to viewpoint (stereo). the denominator consists of the spatial derivative of the sum of the left and right image pair. (10. fx (10.17)
This solution now looks very similar to the solution for differential stereo in Equation 10. Also in both solutions. In both solutions the numerator is a derivative.13.
if we are told which data points were generated by which model. for each model k. . b2 . shown in Figure 11. In so doing the model assignment is made in a "soft" probabilistic fashion. If we are told the model parameters. In either case. and the system is modeled with additive noise n1 (i) and n2 (i). On the other hand. b1 and a2 .1 Expectation/Maximization The Expectation/Maximization (EM) algorithm simultaneously segments and fits data generated from multiple parametric models. The "E-step" of EM assumes that the model parameters are known (initially. . the model k that minimizes the error between the data and the model prediction: rk (i) = |ak x(i) + bk − y (i))|. the model parameters can be assigned random values) and calculates the likelihood of each data point belonging to each model. an over-constrained set of linear equations: xk (1) xk (2) .
(11. instead each data point i is assigned a 69
.(11. 2 in our current example. y ) generated from one of two linear models of the form: y (i) = a1 x(i) + b1 + n1 (i) or y (i) = a2 x(i) + b2 + n2 (i). But. . then estimating the model parameters reduces to solving.
ak bk
yk (1) yk (2) = . For example. The EM algorithm is an iterative two step algorithm that estimates both the model assignment and parameters. . . .1 Data two models
from
xk (n) 1
1 1 . lacking either piece of information makes this a considerably more difficult estimation problem. . (11. for each data point i.11. each data point is not explicitly assigned a single model. That is. then determining which data point was generated by which model would be a simple matter of choosing. Useful Tools
11.1 are a collection of data points (x. knowing one piece of information (the model assignment or parameters) makes determining the other relatively easy.3)
yk (n)
where the xk (i) and yk (i) all belong to model k.2)
for k = 1.1)
11-1 Expectation/ Maximization 11-2 Principal Component Analysis 11-3 Independent Component Analysis
where the model parameters are a1 .
Figure 11.
the algorithm converges to a solution. and after six iterations. At the end of this chapter is a Matlab implementation of EM for fitting multiple linear models to two-dimensional data. Initially.2 Principal Component Analysis 11. The EM algorithm iteratively executes the "E" and "M" step.Rewriting in matrix form:
2 i wk (i)x(i) i wk (i)x(i) i
wk (i)x(i) i wk (i)
ak bk
=
i wk (i)x(i)y (i) i w(i)y (i)
Axk = b xk = A−1 b. repeatedly estimating and refining the model assignments and parameters. (11.2 are several iterations of EM applied to fitting data generated from two linear models.11)
yields a weighted least squares solution for the model parameters.3 Independent Component Analysis
Figure 11. Note that this solution is identical to solving the set of linear equations in Equation (11.2 Six iterations of EM
71
. the model parameters are randomly assigned. Beyond the current scope are proofs on the convergence and rate of convergence of EM. 11. Shown in Figure 11.3) using weighted least-squares. | 677.169 | 1 |
Description
Strictly as per the new CBSE course structure and NCERT guidelines, this thoroughly revised and updated textbook is meant for class XII of senior secondary schools (under the 10 + 2 pattern of education). The subject matter of this book is presented in a very systematic and logical manner. Every effort has been made to make the contents as lucid as possible so that the beginners will grasp the fundamental concepts in an unambiguous manner.
KEY FEATURES
Large number of solved examples to understand the subject.
Categorization of problems under:
Level of Difficulty A (Cover the needs of the students preparing for CBSE exams)
Level of Difficulty B (Guide the students for engineering entrance examinations).
A Smart Table at the beginning of each chapter to decide the relative importance of topics in the CBSE exam.
Problem Solving Trick(s) to enhance the problem solving skills.
A list of Important Formulae at the beginning of the book.
Besides this, each chapter is followed by a Chapter Test and an exercise in which the questions from the CBSE papers of previous years are provided. Working hints to a large number of problems are given at the end of each and every exercise. In a nut shell, this book will help the students score high marks in CBSE, and at the same time build a strong foundation for success in any competitive examination.
About The Auhtor DINESH KHATTAR (Ph.D.), an alumnus of Hindu College, is with the Department of Mathematics at Kirori Mal College, University of Delhi. Besides teaching UG classes at Kirori Mal College for more than 20 years, he is also taking PG classes at the University. He is a topper (Gold Medalist) in his B.Sc. and M.Sc. exams. Dr. Khattar is an author of many books on mathematics including CBSE Mathematics for Class XII (Part 1 and Part 2).
ANITA KHATTAR (M.Sc., B.Ed.), an alumna of Hindu College, University of Delhi, is a mathematics teacher at Sarvodaya Vidyalaya, I.P. Extension, Delhi for more than 17 years. She has co-authored a number of books on mathematics including CBSE Mathematics for Class XII (Part 1 and Part 2 | 677.169 | 1 |
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The History of Modern Mathematics, Volume I: Ideas and their Reception documents the proceedings of the Symposium on the History of Modern Mathematics held at Vassar College in Poughkeepsie, New York on June 20-24, 1989. This book is concerned with the emergence and reception of major ideas in fields that range from foundations and set theory, algebra... more...
Lectures in General Algebra is a translation from the Russian and is based on lectures on specialized courses in general algebra at Moscow University. The book starts with the basics of algebra. The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the... more...
Modern Syllabus Algebra presents topics of traditional and modern algebra found in the Teachers Certificate and B.Ed, part I syllabuses of University Institutes of Education. It also contains additional exercises taken from examination papers of the University of London Institute of Education (the Teachers' Certificate). The book discusses several... more...
Some Modern Mathematics for Physicists and Other Outsiders: An Introduction to Algebra, Topology, and Functional Analysis, Volume 1 focuses on the operations, principles, methodologies, and approaches employed in algebra, topology, and functional analysis. The publication first offers information on sets, maps, and algebraic composition laws and... more...
Intermediate Algebra, Third Edition focuses on operations, principles, and approaches involved in intermediate algebra. The manuscript first ponders on basic properties and definitions, linear equations and inequalities in one variable, and exponents and polynomials. Discussions focus on factoring trinomials, special factoring, solving equations... more...
Intermediate Algebra with Trigonometry focuses on principles, operations, and approaches employed in intermediate algebra with trigonometry. The publication first elaborates on basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on polynomials, sums, and differences, multiplication... more...
Intermediate Algebra: A Text/Workbook, Second Edition focuses on the principles, operations, and approaches involved in intermediate algebra. The publication first takes a look at basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on properties of exponents, polynomials, sums,... more... | 677.169 | 1 |
Description
This edition features the exact same content as the traditional text in a convenient, three-hole- punched, loose-leaf version. Books a la Carte also offer a great value–this format costs significantly less than a new textbook.
Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. David Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.
This title is also sold in the various packages listed below. Before purchasing one of these packages, speak with your professor about which one will help you be successful in your course. | 677.169 | 1 |
Analytic Trigonometry - Your Complete Guide
From simplifying to verifying to solving to the Law of Sines and Cosines you will learn all step by step460 will take you from simplifying basic trigonometric expressions to solving complex trigonometric equations. We will sprinkle in verifying trigonometric identities in there as well. This course starts with an introductory video for each topic and then provides over 76 examples on how to solve different examples. If you need help finding the right answer each example has a step by step tutorial showing you what to do. There is also hints and processes used in the description to give you a better idea.
This course should take you a couple of days to complete while you work through the problems and use the videos as reinforcement. This course is created to be your complete guide for Analytic Trigonometry.
What are the requirements?
Basic Arithmetic Skills
Basic Algebra Skills
What am I going to get from this course?
Simplify, Verify, and Solve Trigonometric Equations
Evaluate and solve equations using the double angle formulas
Evaluate the verify identities using the half angle formulas
Evaluate and solve equations using the sum and difference formulas
Determine the missing parts of a triangle using Law of Sines and Law of Cosines
Remember that the reciprocal of a number such as a is 1/a and the reciprocal of a number a/b is b/a. To prove the reciprocal identities apply the functions to the values on the unit circle to confirm that they are equal to one another.
The cofunction identities can best be understood by looking at the values of your trigonometric functions within the first quadrant. You will notice that the sine of 30 degrees has the same value as the cosine of 60 degrees. This is represented in the notation of the cofunction identity as 90-30 =60. This is the same for the remaining trigonometric functions.
Remember that the quotient identities mean division. Look at a right triangle and understand that the ratio of tangent is opposite over adjacent where if that right triangle is on the unit circle the opposite side is represented by y and the adjacent side is represented by x.
Look to prove the pythagorean identities by looking at a right triangle that is within the Unit circle. The two legs are represented by sine and cosine as the hypotenuse has a length of 1. Applying the Pythagorean theorem you can prove all three Pythagorean Identities.
A basic understanding of even and odd functions as well as the graphs of the three basic trigonometric graphs is helpful in understanding the even and odd identities. Like all even functions the graph of cosine is symmetrical about the y axis so the input value will not change the output value if it is positive or negative. Similar with odd functions and sine and tangent.
We will break down the process and thinking that will be needed when looking at an expression to rewrite the expression in the simplest of terms possible. There is usually many routes to take to simplify an expression so practice and a solid foundation of algebraic processes will be helpful when applying the identities to simplify the expression.
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I am a high school that is on a mission to improve math education. I was that student that sat in the back of class frustrated with the boredom of class and the lack of understanding. I made the decision to become a math teacher to make a difference in others lives. I knew that with the struggles I had I could relate well to students that struggled with math. With a weak math background I set out to get a degree in mathematics. In was a difficult journey and I worked very hard not just to pass my math classes but to have an understanding of what I was doing. I learned a lot about myself, mathematics and what it takes to be successful in class through my time at college. I want to pass along my experience to you the student so that may have your own success with mathematics. | 677.169 | 1 |
This insightful book combines the history, pedagogy, and popularization of algebra to present a unified discussion of the subject. Classical Algebra provides a complete and contemporary perspective on classical polynomial algebra through the exploration of how it was developed and how it exists today. With a focus on prominent areas such as the numerical... more...
Handbook of Algebra defines algebra as consisting of many different ideas, concepts and results. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. Each chapter of the book combines some of the features... more...
This is a memorial volume dedicated to A. L. S. Corner, previously Professor in Oxford, who published important results on algebra, especially on the connections of modules with endomorphism algebras. The volume contains refereed contributions which are related to the work of Corner. It contains also an unpublished extended paper of CornerThis upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing,... more... | 677.169 | 1 |
Sequências Numéricas
Sequências Numéricas
The study of sequences, although seen as an incipient numerical progression, is the foundation of mathematical analysis. This concept in turn is based on set theory and Dedekind cuts, and is studied by mathematical analysis.
The study of sequences, although seen as an incipient numerical progression, is the foundation of mathematical analysis. This concept in turn is based on set theory and Dedekind cuts, and is studied by mathematical Sequências Numéricas to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Sequências Numéricas
Select this link to open drop down to add material Sequências Numéricas to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
Mathematics Curriculum Studies 5
Description
This course introduces students to the key concepts underlying a deep understanding of statistical methods of data analysis and probability leading to an appreciation of variability and related distributions of measure. The course will also examine current related pedagogical models within the field of secondary mathematics including communicating with students and parents.
Availability
WebLearn GradSchool
Semester 2 - 2016
Semester 2 - 2017
Learning Outcomes
On successful completion of the course students will be able to:
1. understand the key concepts related to methods of data collection;
2. understand the key concepts related to methods of data analysis and display;
3. appreciate the mathematical knowledge and beliefs that learners bring to a learning task;
4. understand the relationship between statistical analysis and probability;
5. apply a range of strategies for teaching secondary mathematics; and
6. recognise the common misconceptions that students may have about the mathematical content covered. | 677.169 | 1 |
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Algebra I
Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process. It's also seen as a "gatekeeper" subject. Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it's impossible to move forward. It's used by people with lots of different jobs, like carpentry, engineering, and fashion design. In these tutorials, we'll cover a lot of ground. Some of the topics include linear equations, linear inequalities, linear functions, systems of equations, factoring expressions, quadratic expressions, exponents, functions, and ratios. | 677.169 | 1 |
workbook guides students through basic skills, mathematical review, and successful problem-solving techniques. Practice tests are also included for each chapter. | 677.169 | 1 |
College Algebra with Modeling and VisualizationGary Rockswold focuses on teaching algebra in context, answering the question, "Why am I learning this?" and ultimately motivating the reader to succeed. Introduction to Functions and Graphs. Linear Functions and Equations. Quadratic Functions and Equations. Nonlinear Functions and Equations. Exponential and Logarithmic Functions. Systems of Equations and Inequalities. Conic Sections. Further Topics in Algebra. Basic Concepts From Algebra and Geometry. For all readers interested in college algebra. | 677.169 | 1 |
Product Description
A bridge to upper-level math, BJU Press' Fundamentals of Math Grade 7 will ensure students have a solid foundation in the skills they'll need for 8th grade and beyond! Whole numbers, decimals, number theory, fractions, rational numbers, percents, measurement, geometry, area/volume, probability/statistics, integers, algebra, relations/functions, and logic/set theory are all taught in detail with review to keep concepts fresh. Chapters include clear explanations of new concepts, plenty of practice, "skill check" reviews, example problems, and a cumulative review of the new concept | 677.169 | 1 |
Master Math: AP Statistics / Edition 1
Overview concepts of AP Statistics including exploring data, sampling and experimentation, anticipating patterns, and statistical inference. The example problems in each chapter are written with the AP Statistics Exam in mind to help you understand the concepts and learn how to effectively answer the exam questions. You'll also find useful appendices that help with exam preparation, including all the tables and formulas that are given and needed, as well as a quick-reference summary of assumptions and conditions for inference. A helpful glossary will help you brush up on terminology. Master Math: AP Statistics is an invaluable resource for anyone studying and preparing for the AP Statistics Exam.
Most Helpful Customer Reviews
Gerry McAfee is perhaps the greatest statistician since Sir Ronald Fisher. Master Math really is an incredible studying tool! Master Math is extremely helpful. This year I was enrolled in an AP Statistics coarse. My math teacher was completely incompetent and I struggled to understand even the most basic principles of Statistics. A month ago, when I realized that I would have to take the AP Exam I knew i was completely unprepared. When I told my unpreparedness to a friend he recommended Master Math. I began reading I learned all of the things my stats teacher had failed to teach, and it only took me a couple of days to read it! With the help of Master Math I took the AP Exam with confidence, knowing that I got a great score. Master Math is a quick read, for someone that doesn't want to dedicate a lot of time to studying but still wants to get a great score. Master Math is in depth and teaches all of the concepts needed for the AP Exam. This is an extremely necessary book. Gerry McAfee is to Master Math as Michelangelo is to David. Master Math is really that good! | 677.169 | 1 |
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An Introduction to Mathematics for Economics introduces quantitative methods to students of economics and finance in a succinct and accessible style. The introductory nature of this textbook means a background in economics is not essential, as it aims to help students appreciate that learning mathematics is relevant to their overall understanding of the subject. Economic and financial applications are explained in detail before students learn how mathematics can be used, enabling students to learn how to put mathematics into practice. Starting with a revision of basic mathematical principles the second half of the book introduces calculus, emphasising economic applications throughout. Appendices on matrix algebra and difference/differential equations are included for the benefit of more advanced students. Other features, including worked examples and exercises, help to underpin the readers' knowledge and learning. Akihito Asano has drawn upon his own extensive teaching experience to create an unintimidating yet rigorous textbook | 677.169 | 1 |
Well paying careers demand skills like problem solving, reasoning, decision making, and applying solid strategies etc. and Algebra provides you with a wonderful grounding in those skills - not to mention that it can prepare you for a wide range of opportunities.
This is a COMPLETE Pre-Algebra guide to well over 325 rules, definitions and examples, including number line, integers, rational numbers, scientific notation, median, like terms, equations, Pythagorean theorem and much more!
Our guide will take you step-by-step through the basic building blocks of Algebra giving you a solid foundation for further studies in our easy-to-follow and proven format!
Algebra is a very unique discipline. It is very abstract. The abstractness of algebra causes the brain to think in totally new patterns. That thinking process causes the brain to work, much like a muscle. The more that muscle works out, the better it performs on OTHER tasks. In simple terms, algebra builds a better brain! Believe it or not algebra is much easier to learn than many of us think and this guide helps make it easier!
Like all our 'phoneflips', this lightweight application has NO ads, never needs an internet connection and wontGetting a Job Overseas
The book "Getting a Job Overseas" is just what you need to help you find a job in another country so that you can officially move there. If you've always wanted to live and work in another nation then you need to know exactly what it takes to get that job that is going to allow you to live overseas.
How to Be a Better Lover
Are you in a loving relationship and you want to know how to spice up your love life so you can become a better lover? If you want to become a better lover this book "How to Be a Better Lover" is going to give you some valuable information towards having a more fulfilling sex life.
How To Quit Smoking Today
If you want to quit smoking but aren't sure how, or aren't sure if you have the strength to do it, this book can help you tremendously. How to Quit Smoking Today will teach you exactly what you need to do in order to break your addiction to nicotine and cigarettes.
CMU Mobility Monitor
Interested in Helping with a Mobility Research Study?
We are conducting user studies to understand where people go and what location-based applications they use. Participants will be included in a raffle for Amazon gift cards for up to $50. We require that participants are of age 18 or up and have an Android phone.
Build Muscle Mass Quickly
Have you always wanted to have a bit more muscle mass that gives your body more definition and shape? Or perhaps you've always been that skinny guy or girl who people envy but inside you're secreting wishing that you looked better? If this is you, then the book "Build Muscle Mass Quickly" is just what you need.
How to Get a Flat Stomach
Have you always wanted to have that flat stomach that you had when you were younger and in better shape? Well now you can. By reading the book, "How to Get a Flat Stomach", you'll find out exactly what you need to do to achieve a flat stomach and how you can maintain it. | 677.169 | 1 |
How to Get Mathematica
Mathematica can be installed on:
Faculty/staff school-owned machines
Request Mathematica through
this link
and download through the Wolfram User Portal.
Faculty/staff personally owned machines
Request Mathematica through
this link. Our site license number is L3399-2488.
Students' personally owned machines
Request Mathematica for Students through
this link
and download through the Wolfram User Portal. Students are requested
to enroll using their LCC email address.
Are you interested in putting Mathematica elsewhere? Please let IT or
Roger Graves at Wolfram Research know.
First Steps for New Users
If you are brand new to Mathematica, here are some suggestions on the best
ways to get started.
Step 1: Watch a tutorial screencast
Hands-on Start to Mathematica: Follow along in Mathematica as you watch this
multi-part screencast that teaches you the basics—how to create your first
notebook, calculations, visualizations, interactive examples, and more.
Step 2: Get a quick overview of Mathematica
How To
Topics: Access step-by-step instructions ranging from how to create
animations to basic syntax information.
Step 3: Explore the Learning Center
Learning
Center: Browse through this extensive collection of Mathematica tutorials,
examples, and other resources to learn more about using Mathematica.
Step 4: What's New in Mathematica 10
What's
New in Mathematica 10: Provides examples to help you get started with new
functionality in Mathematica 10, including machine learning, computational
geometry, geographic computation, and device connectivity. | 677.169 | 1 |
Publisher Description
Basic Algebra Shape-Up helps students master specific basic algebra skills, while providing teachers with measurable results. Concepts covered include creating using ratios, proportions, and working with integers, simple and multi-step equations, and variables. Students start with an assessment and receive immediate instructional feedback throughout. Step-by-step tutorials, which introduce each level, can be referred to during practice. Problems are broken down into small, easily understood steps. The program is self-paced and self-monitored. Students advance as they demonstrate readiness. They may track their own improvement through progress-to-date and last session scores. Scores are kept in a record management system that allows teachers to view and print detailed reports. Designed for students in U.S. grades 6 through 9 (age 10 and up), the program can also be used by ESL and adult students interested in improving their algebra skills.
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This is one of a series of units intended for both preservice and inservice elementary school teachers to satisfy a need for materials on "new mathematics" programs which (1) are readable with minimal instruction, (2) show the pedagogical objectives and uses of such mathematical structural ideas as the field axioms, sets, and logic, and (3) relate mathematics to the "real world," its applications, and other areas of the curriculum. This unit contains a collection of references on the application of mathematics to other subject areas in the elementary school curriculum. The major mathematical categories included are applications involving (1) number and operation, (2) measurement and approximation, (3) organization and presentation of data, (4) algebra, (5) ratio, Proportion, or comparison, (6) geometry, (7) probability and statistics, (8) proof and deductive inference, and (9) sets. (RP) | 677.169 | 1 |
Calculus, known in its early history as infinitesimal calculus, is a mathematical
discipline focused on limits, functions, derivatives, integrals, and infinite series.
Isaac Newton and Gottfried Lei...
The antecedents of calculus are many. Eudoxus (408–355) developed the
method exhaustion ... did much more than "draw tangents". Fermat invented the
mathematics of differential and integral calculus:
BVCa. | 677.169 | 1 |
Almost everyoneis acquainted with plane Euclidean geometry as it is usually taught in high school. This book introduces the reader
to a completely different way of looking at familiar geometrical facts.
It is concerned with transformations of the plane that do not alter the
shapes and sizes of geometric figures. Such transformations play a
fundamental role in the group theoretic approach to geometry. The
treatment is direct and simple. The reader is introduced to new ideas
and then is urged to solve problems using these ideas. The problems form
an essential part of this book and the solutions are given in detail in
the second half of the book | 677.169 | 1 |
Eliminate the requirement for Algebra 2 from the Texas student curriculum? Why not also eliminate sudoku and crossword puzzles from the daily newspapers?
As regular physical exercise improves the health of your physical body, so also do mental puzzles improve the condition of your mental health. Guess what? So do the rigorous logical reasoning processes of higher-level algebra improve the development of the still-formative minds of young kids.
No, you may never use the principles of algebra in your adult employment, but your cognitive intelligence will be greatly enhanced to address that employment. | 677.169 | 1 |
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