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Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The Appendix provides a summary of the basic properties of each Lie algebra of finite and affine type | 677.169 | 1 |
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How to Study Math –Short Version Delano P. Wegener, Ph.D.
Spring 2005 HOW TO STUDY MATHEMATICS The advice contained in this document will encourage you to study mathematics differently than you probably have in the past. Take my word for it, it works. This method has worked for tens of thousands of students. It will work for you, but some effort will be required. HOW TO STUDY MATHEMATICS It is worth observing that this is the approach used by professional mathematicians. This observation is worth while not because you want to become a professional mathematician. It is worth while because professional mathematicians have been successful in their study of mathematics. It worked for them, and with some effort it will work for you. HOW TO STUDY MATHEMATICS You should be honest with yourself. You do not have a long history of success with mathematics. In fact you might have a long history of a lack of success with mathematics. Neither of these situations is a stigma. Simply be honest and acknowledge the fact. HOW TO STUDY MATHEMATICS
First: Change your attitude about how mathematics is learned. Mathematics is not learned by working problems. Mathematics is learned in order to work problems. HOW TO STUDY MATHEMATICS If you work 100 problems of a particular type, there is a good chance that you have learned to mimic a process which leads to a solution for that one problem type. It is unlikely that you will have learned any mathematical concepts. In fact I have witnessed many instances where the student inferred a totally incorrect set of reasons for the process. If you learn the concepts then you are able to solve this particular type of problem as well as many others. HOW TO STUDY MATHEMATICS
Second: Change your frame of mind think of studying mathematics rather than doing mathematics. HOW TO STUDY MATHEMATICS Listen to the words used when students, teachers, parents, and the general public speak about mathematics. Compare those words and that language with the words and language used when other subjects are discussed. HOW TO STUDY MATHEMATICS Students "study" history, social studies, language arts, etc. They read literature. They neither read nor study mathematics. Almost universally when mathematics is the topic of discussion you hear that students "do" mathematics. HOW TO STUDY MATHEMATICS Teachers encourage their students to "do" their math every day to avoid falling behind. Parents check to insure that their children have "studied" spelling, "read" their literature, and "done" their math. HOW TO STUDY MATHEMATICS Students take their queue from this language and routinely skip the reading material in a math book. They seldom read and certainly do not "study" the concepts presented. Students do precisely what is expected of them, they turn to the exercises and attempt to "do" their math. HOW TO STUDY MATHEMATICS
Doing rather than studying leads to: lack of understanding collection of unrelated rules to memorize collection of tricks for solving nonsensical problems found only in mathematics textbooks. HOW TO STUDY MATHEMATICS Third: Make up your mind to concentrate on concepts rather than manipulative activities. HOW TO STUDY MATHEMATICS If you concentrate on manipulation, your focus is so narrow that you will never understand nor be able to use mathematics. HOW TO STUDY MATHEMATICS Fourth: Watch for and make use of concepts that remain the same when working with different mathematical objects. HOW TO STUDY MATHEMATICS This is difficult for the student to do because you cannot see similarities until you have studied quite a bit of mathematics. Your instructor will point out those concepts that remain the same and will be useful in future work. HOW TO STUDY MATHEMATICS YOU ARE NOW READY FOR THE SIX STEPS FOR STUDYING MATHEMATICAL CONCEPTS HOW TO STUDY MATHEMATICS
Step 1: Memorize all definitions and special properties as soon as you encounter them. In the following days and weeks continue to go over these items until the statements that you have memorized are "your own words" for the concepts. HOW TO STUDY MATHEMATICS As with any other language, you must memorize the meanings of words in order to understand what you read, and what you hear. Imagine trying to read, write, speak, or listen if you do not understand any nouns or verbs. You must know the meanings of mathematical terms in order to communicate about mathematics. HOW TO STUDY MATHEMATICS You must be able to: communicate in the discipline HOW TO STUDY MATHEMATICS If you do not memorize definitions, you are trying to make sense of things that look like:
(1) If the same ________ is ________ to (or ________ from) both sides of an ________ the resulting ________ will be ________ to the original ________. HOW TO STUDY MATHEMATICS or: (2) If both sides of an ________are ________ (or ________) by the same ________ ________, the resulting ________ is ________ to the original ________. HOW TO STUDY MATHEMATICS
or: The process to ________ a ________: Start with the original ________ (the one to be ________ ) and use the above two properties to ________ ________ ________, all ________ to the original ________, until we arrive at the ________ ________. HOW TO STUDY MATHEMATICS
Step 2: Read and study the explanations Step 2:
provided in text, class, or website. HOW TO STUDY MATHEMATICS Be sure you understand the concepts being explained. The examples provided can help clarify some items. HOW TO STUDY MATHEMATICS When looking at an example, always ask yourself; What concepts are being illustrated? If you cannot answer that question, the example is useless to you. HOW TO STUDY MATHEMATICS
Step 3: Work a few related exercises to determine if you have learned the concepts. If these exercises give you trouble, go back to the explanations and study some more. Then try some more exercises. HOW TO STUDY MATHEMATICS If you are unable to work a textbook problem, it means either you do not understand a necessary concept or you were unable to recall the concept. Very rarely is your inability to solve a textbook problem due to a lack of manipulative skills. HOW TO STUDY MATHEMATICS The primary purpose of exercises in textbooks is to provide the student with a means of demonstrating to himself that he has learned the concept. HOW TO STUDY MATHEMATICS
Step 4: If a concept simply eludes you, get help. Ask for an explanation in class or Get help during office hours. See a tutor HOW TO STUDY MATHEMATICS Individual Learning Preferences have long been recognized by educators and cognitive scientists. Obtaining assistance from another source may present the concept in a manner which makes it "learnable" by you. HOW TO STUDY MATHEMATICS
Step 5: When your work on a few exercises Step 5:
indicates that indeed you do understand the concept, you may want to work a few more (possibly more difficult) exercises to insure that your assessment of your understanding is valid, that the concept is available for recall, and that you are able to transfer the concept to new situations. ….. HOW TO STUDY MATHEMATICS
Step 5: …. Sometimes you may want to work even more problems simply to improve your speed for dealing with that particular kind of problem. HOW TO STUDY MATHEMATICS
Step 6: Constantly review all memorized material as well as major concepts. At regular intervals of a week or two work a few exercises to enhance retention and ability to transfer. HOW TO STUDY MATHEMATICS The need for Steps 5 and 6 is justified by the last three External Conditions of Learning as specified by Gagne [1] . These three conditions of learning address feedback, assessment, retention, and transfer. HOW TO STUDY MATHEMATICS This concludes the short version of a seminar on How to Study Mathematics. Thank You for Your Attention ...
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This 1:1 This Teach Interactive™ digital activity is designed is designed for AP Calculus AB, AP Calculus BC, Honors Calculus, and College Calculus 1. Practicing derivatives and the chain rule has never been more fun, engaging, and interactive! First students find the derivatives of seven functions using the chain rule, matching the missing parts (the chain). The second part of the activity contains three more rigorous chain rule problems. Students must rearrange the steps in the correct order. This requires higher thinking skills and will engage and challenge even your best students. Note: The problems include transcendentals, but not inverse trig functions. This product is not editable.Please look closely at the previews.
This resource includes:
• Interactive slides with ten Chain Rule problems
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use this book and when to use your calculator --
Mathematics vocabulary --
The rules you should know --
Direction decoding --
Little pieces lead to big problems --
Study strategies --
"When will I use this, anyway?" --
Parent-teacher communication. | 677.169 | 1 |
Synopsis
Complete Mathematics for Cambridge IGCSE Student Book by David Rayner
Updated in line with the Cambridge IGCSE Extended Mathematics syllabus, this engaging Student Book provides complete content coverage in a clear and visual style, suitable for both native and non-native speakers of English. The book is easy-to-use, with an accessible format of worked examples and practice questions. Accompanying digital materials enhance the printed content, presenting hundreds of additional homework questions, self-assessment checklists, revision and exam tips and examiner comments. An Teacher's Guide and Revision Guide are also available.
Reviews
Of the 33 students in one of our IGCSE extended Maths class last year, 25 got A passes (including 8 A*'s with one student on 100%) which is a credit to the comprehensive quality of the Rayner textbook. Mathematics Teacher, New Zealand Best Maths book for Extended IGCSE. I really like this book. My son was struggling with Maths and was getting about 35% in Core Maths tests. For his intended career he needs to have good Maths grades so we decided to get this book and work through it slowly. He is now scoring above 80% for Extended Maths tests. Any type of question you can possibly imagine could be asked in an exam is in this book. Each chapter starts with a few examples or Maths facts/formulas, then the exercises start with simple questions. The exercises get progressively more challenging, sometimes requiring searching other books or Internet for help. The end of each chapter has real exam questions that have appeared in the past. Amazon Parent review | 677.169 | 1 |
Interactive Video Skillbuilder CD-ROM for Hungerford's
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Pages: 507
Publisher: Brooks Cole; 1 edition (July 19, 2005)
ISBN: 0534466435
Academic trigonometry. Plane and spherical
Functions and Change: A Modeling Approach to College Algebra and Trigonometry
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Elements of Geometry and Trigonometry. Revised and Adapted to the Course of Mathematical Instruction in the United States...
Exact Values in Trigonometry: Five New Techniques
International Library of Technology 134B
Spherical trigonometry, for colleges and secondary schools
Modern algebra with trigonometry (Allendoerfer Undergraduate series)
Let's Laugh at Trig!: A Simple Introduction to Trigonometry
Elements of Geometry and Trigonometry from the Works of A. M. Legendre
A Key to Papers in Trigonometry
An Introduction to the Study of Numerical Trigonometry
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Angel and Porter's a Survey of Mathematics with Applications, 5/E with the Trigonometry Appendix Bun
Die Trigonometrie In Tabellen Nach Theorie Und Praxis
Solve linear equations grade 10 ontario word equations, factors calculator for mixed numbers, orleans hanna prognostic test practice questions, egyptians quadratic, how to do foil with a addition and subtraction sign, ti 89 quadratic equation system, free Intergers work sheet Algebra and Trigonometry: Graphs and Models (5th Edition). Trigonometry word has been raised from the Greek word which has meaning of "3-sided measure" and it is the branch of mathematics in which we study of the relationships between sides and angles of a triangle pdf. He also gave the power series of π and the θ, radius, diameter, and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century. [46] [47] The Indian text the Yuktibhāṣā contains proof for the expansion of the sine and cosine functions and the derivation and proof of the power series for inverse tangent, discovered by Madhava Analytic Trigonometry with Applications -- Eighth 8th Edition. Then by the definition in section 20, the angle OQP represents the angle between the plane of the base and the plane of the side OAB Algebra and Trigonometry : Early Functions Approach. We saw a maths teacher using them recently, and we also found pictures and explanations of them on the Internet A Treatise On Plane Trigonometry: Containing An Account Of Hyperbolic Functions, With Numerous Examples (1888). Use this control to limit the display of threads to those newer than the specified time frame. These lecture videos are organized in an order that corresponds with the book we have used recently for our Math1060 courses. (Precalculus by Larsen, Hostetler) We have numbered the videos for quick reference so it's reasonably obvious that each subsequent video presumes knowledge of the previous videos' material Text-Specific DVDs for Larson's College Algebra, 9th. The course concentrates on geometry and trigonometry but includes a few other topics as well. While most of the content is accessible by a fee only, some lessons are available as free samples. Here is a non-intimidating way to prepare students for formal geometry. Key to Geometry workbooks introduce students to a wide range of geometric discoveries as they do step-by-step constructions A treatise on spherical trigonometry Volume 2; with applications to spherical geometry and numerous examples. The directional analysis of quantities such as force and acceleration is also studied in this subject. Questions like sketching of graph fro sin X or Cos x at various degrees of X, finding the real value of various expression and functions, and proving of expression and then finding out expression in that term are few questions that gets repeated almost every year. Apart from that, questions like finding out the value of x from a given expression or finding value of various Sin and Cos degrees using the formula etc gets repeated almost every year online. Check out our CyberExams to prepare for a test, or ask a question on our popular CyberBoard online! In a triangle, there are six basic elements: 3 sides and 3 angles. Not any three line segments may serve as the sides of a triangle. They do iff they satisfy the triangle inequality, or rather three triangle inequalities. Not any three angles may be the angles of a triangle. In Euclidean geometry, the three angles of a triangle add up to a straight angle Bundle: College Physics, 10th + Enhanced WebAssign Multi-Term LOE Printed Access Card for Physics. I've used it several times, always the night before some sort of aptitude test or exam that included pre-calculus math. I used it to pass the math CLEP test, placement exams for college and grad school, and the ASVAB test to get into the Navy download Interactive Video Skillbuilder CD-ROM for Hungerford's Contemporary Trigonometry: A Graphing Approach pdf. | 677.169 | 1 |
Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examplesMore...
Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is self-contained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables. Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material. Supplemented with numerous exercises, Advanced Calculus is a perfect book for undergraduate students of | 677.169 | 1 |
"What does this have to do with real life?" is a question that plagues mathematics teachers across America, as students are confronted with abstract topics in their high school mathematics courses. The National Council of Teachers of Mathematics emphasizes the importance of making real world connections in teaching mathematics so that...
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Here's an innovative shortcut to gaining a more intuitive understanding of both differential and integral calculus. In Calculus Demystified an experienced teacher and author of more than 30 books puts all the math background you need inside and uses practical examples,...
Even as developments in photorealistic computer graphics continue to affect our work and leisure activities, practitioners and researchers are devoting more and more attention to non-photorealistic (NPR) techniques for generating images that appear to have been created by hand. These efforts benefit every field in which illustrationsthanks to...
The huge number and broad range of the existing and potential applications of fuzzy logic have precipitated a veritable avalanche of books published on the subject. Most, however, focus on particular areas of application. Many do no more than scratch the surface of the theory that holds the power and promise of fuzzy logic. Fuzzy Automata and...
MASTER ONE LIFE'S MOST USEFUL SKILLS--EVEN IF YOU'VE NEVER BEEN GOOD AT MATH
Knowing algebra gives you a better choice of jobs, helps you perform better in science, computing, and math courses, ups your score on competitive exams, and improves your ability to do daily computations. And there's no faster or more painless way...
Since the publication of the first edition in 1998, programmers and developers have been waiting eagerly for an update to this expert guide on how to use Java in conjunction with the timesaving design patterns that have surfaced in the past few years. With the new edition of his bestselling Patterns in Java, Volume 1, Mark Grand bringsYou may be a hacker and not even know it. Being a hacker has nothing to do with cyberterrorism, and it doesn't even necessarily relate to the open-source movement. Being a hacker has more to do with your underlying assumptions about stress, time management, work, and play. It's about harmonizing the rhythms of your creative work...
Isometric Game Programming with DirectX takes you from zero to DirectX, literally! Beginning with game programming basics you will explore such topics as WIN32 programming basics, GDI fundamentals, fonts, regions and bitmaps, DirectDraw, and Game Logic. You'll then move on to Isometric fundamentals and methodology as well as advanced topics such as... | 677.169 | 1 |
PDF (Acrobat) Document File
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0.52 MB | 3 pages
PRODUCT DESCRIPTION
This graphing test covers comparing rates from graphs and tables, functions, systems, and creating equations based on scenarios.
You will receive a PDF of the two page test, which has ten multiple choice questions, and the answer key. The questions are modeled after Common Core based tests. The test will take students about 45 minutes.
Common Core Standards:
CCSS.Math.Content.8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
CCSS.Math.Content.8.EE.C.8.b
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
CCSS.Math.Content.8.EE.C.8.c
Solve real-world and mathematical problems leading to two linear equations in two variables.
CCSS.Math.Content.8.F.A.1
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1
CCSS.Math.Content.8.F.A.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
CCSS.Math.Content.8.F.B.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
CCSS.Math.Content.HSF.IF.A.1
UnderstandCCSS.Math.Content.HSF.IF.C.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
CCSS.Math.Content.HSF.BF.A.1
Write a function that describes a relationship between two quantities.
CCSS.Math.Content.HSA.REI.C.6
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
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Work & Power Problems
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Here are a variety of work and power problems for students either in pre-algebra or algebra 1. Some problems have multiple parts in order to tie together the Work (W = F*d) and Power (P = W/t) equations with the occasional time unit conversion. As with my other problem sets, I aimed for a variety of situations both to hold students' interest and to give them different contexts in which to picture the concept. My students often take anywhere from 15 to 40 minutes on this, depending on their math ability and motivation | 677.169 | 1 |
High School
Mathematics
Algebra 1- Rising 9-10 Grade
Algebra 1 is designed to build a foundation of algebraic proficiencies, to improve computational skills, to develop problem-solving ability, and to generate a positive attitude towards mathematics. Problem solving is taught in a strategic fashion to ensure students master fundamental algebra skills and concepts. Linear equations, quadratic equations, and inequalities are graphed and used for solving word problems. UC/CSU approved course NCAA Core Course
Algebra 1 Accelerated- Mitty Only Rising 9th Grade
Algebra 1 Accelerated is a 3 weeks course designed to teach Algebra 1 with extended work and exploration of additional Algebraic topics. Emphasis is placed on application and development of problem solving skills necessary for advancement into higher-level mathematics. UC/CSU approved course NCAA Core Course
Geometry- 9–10 Grade
Geometry teaches the properties and relationships of points, lines, and planes for both two and three-dimensional figures and their applications to area, volume, and proportion. It includes an introduction to analytic geometry, a study of the structure of logic and deductive, inductive and intuitive reasoning, and the development of proof and construction. Understanding the relationship between geometry and algebra, and how geometry assists algebra in developing solutions to problems are expected outcomes as well. UC/CSU approved course NCAA Core Course
Trigonometry- 9 -12 Grade
Trigonometry is a one-semester (3 weeks in Summer) course designed to prepare students to succeed at higher-level mathematics studied in pre-calculus, calculus and physics. Study of problems involving angles, triangles and objects that move in circular motion, or orbits are undertaken. Additional topics include angle measure in degrees and radians, trigonometric and circular functions and their graphs, right triangles, oblique triangles, inverse trigonometric functions and vectors. Trigonmetry Course Syllabus UC/CSU approved course NCAA Core Course
Science
Biology- 9-10 Grade
Biology teaches life science principles as powerful conceptual tools to make sense of the complexity, diversity and interconnectedness of life on Earth. Students engage in laboratory and authentic learning experiences that encourage the application of biological knowledge for the purpose of making personal decisions and solving problems in an evolving global 21st century community. Development of mathematical and analytical skills in this course support and prepare students as knowledge seekers capable of analyzing scientific data collected from hands-on instructional activities. UC/CSU approved course NCAA Core Course
Chemistry- 10-12 Grade
Chemistry teaches the composition of matter and the changes in composition that matter can undergo. In investigating the composition of matter, the structure of the atom and the changes the atom undergoes in the formation of new substances will be explored. As a traditional chemistry class, emphasis is given to topics that form the basis for college-level courses as well as provide general chemistry knowledge in everyday experiences. Problem-solving and critical-thinking skills along with the quantitative investigation of matter and its changes are the basis for this course, and prepares students to study other fundamental sciences such as physics and environmental science. UC/CSU approved course NCAA Core Course
English
Contemporary Authors- 9-12 Grade
Contemporary Authors is designed to review and practice skills taught in English 1 and 2. Emphasis is placed upon close-reading and reading comprehension; writing various types of responses and essays; learning and application of new vocabulary; polishing grammar and usage knowledge; and editing proficiency. This includes review and practice of MLA format in various writing assignments. The student will be reading and writing daily in class and for homework. Goals of the course are for students to improve as a reader and as a writer, and to become knowledgeable about the two novels used as primary texts. UC/CSU approved course NCAA Core Course
Introduction to College Preparatory English- Mitty Only Rising 9th Grade
Introduction to College Preparatory English is intended to benefit students in need of reading and writing skills improvement to ensure academic success in English 1. Course work includes daily close reading, reading quizzes, class discussions, group activities, and short-writing assignments on literary elements and issues within the Summer Reading novel. Writing strategies are taught sequentially beginning with diagnostic interviews, short paragraph writing, and finally a short essay that is assesses the organization and development of written expression.
World Language
Spanish 2- 9-10 Grade
Spanish 2 instructs students with the goal to continue building on oral and written communication studied in Level 1 Spanish. Communication will be enhanced and developed through greater use of Spanish in the classroom, cooperative learning activities, listening, writing and reading exercises, and oral presentations. Additionally, students will appreciate the relevance and influence of music, geography, and culture to the Spanish language. UC/CSU approved course NCAA Core Course | 677.169 | 1 |
Revision Notes For Class 11 Mathematics / Page 2
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As you know that before exam which is your favorite subject it doesn't matter because before exam student have to do study of all subject for find maximum mark in all subject . take math subject many student think it is more hard subject they save himself for study of this subject .the main reason of this problem that doesn't understanding properly any topic . but Scholarslearning is giving a best way where student can prepare himself in this subject . Revision Notes For Class 12 Mathematics by this student can learn this subject very easily .for download it visit .
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PRENTICE HALL HIGH SCHOOL 2009 PREALGEBRA HOME SCHOOL BUNDLE KIT GRADE 9/12
Solid preparation for algebra and geometry. Integers and algebraic concepts are introduced beginning in Chapter 1 to develop algebraic thinking skills. Throughout the text, connections are made to arithmetic skills. Geometry concepts are integrated when appropriate to foster connections. With an emphasis on the mastery of basic skills, the text provides numerous opportunities to assess progress in basic skills along with abundant remediation and intervention activities | 677.169 | 1 |
Reducing Radicals
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0.03 MB | 8 pages
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Students in A2/Trig come in with a basic knowledge of reducing radicals, but it is one of their weakest skills. this guides them through reducing integers, reducing with variables and introduces what to do with a negative underneath the radical. This lesson prepares students to use complete the square method and quadratic formula for solving quadratic equations (day 8,9,10 | 677.169 | 1 |
THE BASICS 2: Evaluate and Simplify Using the Order of Operations
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0.35 MB
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Teach your students how to evaluate both numerical and algebraic expressions. Discuss how to simplify an expression by combining like terms. Tackle real-life examples using expressions as well as the various ways of remembering the order of operations.
Why use my notes and my assignment?
1. The fill-in-the-blank format is excellent for struggling students who otherwise would not complete notes.
2. Each note sheet and assignment has a QR code on top and a web address on bottom. If a student is absent or just needs more help, they can either use the QR Reader on their phone or manually visit the web address and I will personally coach them through my YouTube videos! There is a 15-minute note sheet tutorial and a 5-10 minute assignment tutorial | 677.169 | 1 |
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Students will understand more about the various Math class methods including sqrt, pow, abs, ceil, floor, the trig functions sine, cosine, and tangent, and the constant values PI and E. The Java documentation help file, or API, will be explored specifically in regards to the Math class. Also, the use of static import statements for the System and Math classes will be introduced. Objectives Lesson 5A - Methods – Math class3
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Labs include work with classic math functions including number line distance, coordinate plane distance, area and volume formulas for various geometric 2D and 3D figures, problems involving the use of ceil and floor, the Pythagorean theorem, quadratic formula and various trig functions such as the law of sines and law of cosines. Objectives Lesson 5A - Methods – Math class4
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Previously, several Math class items were introduced, including: pow – returns the power given a base and exponent sqrt – returns the square root of a value PI – a fairly precise constant for PI, the ratio of the circumference of a circle to its diameter E – a fairly precise constant for E, the base of the natural logarithms Math class definition Lesson 5A - Methods – Math class5
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A complete definition of the Math class can be found in the Java help files, or external documentation, which is often referred to as the Java API (application programming interface). On the next screen you can see the beginning of this definition. Math class definition Lesson 5A - Methods – Math class6
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Math class API Lesson 5A - Methods – Math class7 You can find this help file very easily by typing the word Math anywhere in your program, placing your cursor on the word, then doing the key sequence CTRL-F1.
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Math class API Lesson 5A - Methods – Math class8 Lets examine several features of this help file… The Math class belongs to the java.lang package It is a descendant of, or extends the java.lang.Object class, which means it has inherited all of its features, and added a few more of its own (more on inheritance later). A brief description of the class is provided.
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Math class API Lesson 5A - Methods – Math class9 Here you see the beginning of the listing of Math class fields and methods, (over 40 in all) including the constants E and PI, as well as the first few methods.
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Math class API Lesson 5A - Methods – Math class10 The abs method is overloaded, or defined in several ways to be able to handle four different data type parameters: double, float, int, and long.
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Math class API Lesson 5A - Methods – Math class11 Notice that the return type is the same as the parameter type for this method. These are parameters!
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Math class API Lesson 5A - Methods – Math class12 And these are return values! Notice that the return type is the same as the parameter type for this method.
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Math class API Lesson 5A - Methods – Math class13 Eureka! They match!!! Notice that the return type is the same as the parameter type for this method.
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Math class API Lesson 5A - Methods – Math class14 Not all match, however. Most of the return types are doubles, and sometimes there are multiple parameters.
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Math class API Lesson 5A - Methods – Math class15 Just look carefully for the method you want, and see what parameters it requires, and what the return type is.
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Math class API Lesson 5A - Methods – Math class16 In all of the API class listings, there are two main sections. The column on the left indicates the return type, or the type of data for the answer returned by the method. All of the Math class methods are static, which means they are stand-alone methods and work independently of any object (more explanation on that later). The column on the right contains the name of the method, plus any parameter(s) it receives, along with a description of its purpose.
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Math class API Lesson 5A - Methods – Math class17 On the next few slides, the few methods that we will study are highlighted. Study them carefully and become familiar with them. On this slide, check out the ceil and cos methods
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ceil and floor Lesson 5A - Methods – Math class26 Most of the Math functions are familiar to you, or will be as a part of your math-based learning, but the ceil and floor functions are not part of the standard math curriculum and require some explanation.
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ceil and floor Lesson 5A - Methods – Math class27 In a similar way to the rounding function, ceil and floor convert any decimal value to the nearest whole number, but in a specific direction, NOT according to the rounding rule.
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ceil and floor Lesson 5A - Methods – Math class30 They seem opposite, but if you think of the vertical number line, it all makes sense.
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ceil and floor Lesson 5A - Methods – Math class31 Notice carefully that although the resulting value is always a whole number, it is still a double type.
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ceil and floor situations Lesson 5A - Methods – Math class32 These two interesting functions play a critical part of problem- solving, especially in situations that require interpreting the remainder, or fractional part of a division problem.
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ceil and floor situations Lesson 5A - Methods – Math class33 For example, if you need to purchase enough plywood to do a job, and your precise calculations come up to 4.5 sheets of plywood, you would ceil the value and purchase 5 sheets. A lumber company will NOT sell you 4.5 sheets of plywood, sorry!
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ceil and floor situations Lesson 5A - Methods – Math class34 Another situation would be if you have a certain amount of money and calculate that you can purchase exactly 4.5 items of something you need, clearly you would floor the value and only be able to purchase 4 items, with some change leftover.
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Static imports Lesson 5A - Methods – Math class35 The previous program examples used the Math class methods and the output statement System.out.println quite often, which brings up a really nifty feature of Java…the use of static imports.
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Static imports Lesson 5A - Methods – Math class36 Since the two most commonly used classes in many programs are the System and Math classes, it is quite useful to list two static import statements at the top of the program: import static java.lang.Math.*; import static java.lang.System.*;
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Static imports Lesson 5A - Methods – Math class37 import static java.lang.Math.*; import static java.lang.System.*; These import statements enable the use of the System and Math class methods WITHOUT having to say System or Math each time, saving lots of typing and greatly simplifying your code!
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Static import program examples Lesson 5A - Methods – Math class38 Notice that with the two static import statements, the words System and Math have not been included in the commands, but the program still works.
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In this lesson, you learned more about the Math class methods, specifically about the ceil and floor methods, how to examine the class in great detail using the Java API, and about how to use static imports for System and Math to greatly simplify and streamline your code. Lesson Summary Lesson 5A - Methods – Math class39
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The following labs are all intended to help you practice using the Math methods. There may be more than one way to solve the program. Be sure that the output is precise, especially whether or not a value is an integer or a decimal, and how many decimal places are used. Labs Lesson 5A - Methods – Math class40
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Lab 5A- 1 Lesson 5A - Methods – Math class41 WAP to find the distance between two input integers representing values on the number line. The formula is: |a-b| find the distance between two input decimals representing values on the number line. The formula is: |a-b| Input: (data file lab5A1.in) Output: The distance from 3 to 5 is 2 The distance from -3.3 to 5.8 is 9.1 The distance from 65 to -88 is 153 The distance from 24.9 to is
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Lab 5A- 2 Lesson 5A - Methods – Math class42 WAP to input an integer num from the keyboard and: : find the area of a square whose side length is num. : find the volume of a sphere whose radius is num. : find the length of the side of a square whose area is num. Input Data File (lab5A2.in): 5 82 Output: The area of a square whose side length is 5 is 25 The volume of a sphere whose radius is 5 is The side length of a square whose area is 5 is 2.24 The area of a square whose side length is 82 is 6724 The volume of a sphere whose radius is 82 is The side length of a square whose area is 82 is 9.06
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Lab 5A- 3 Lesson 5A - Methods – Math class43 WAP to find the minimum number of 4 X 8 sheets of plywood it would take to cover a roof when the total amount needed in square feet is input from a data file. Input Data File (lab5A3.in): Output: A 200 square-foot roof needs a minimum of 7 sheets of plywood. A 988 square-foot roof needs a minimum of 31 sheets of plywood.
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Lab 5A- 4 Lesson 5A - Methods – Math class44 WAP to find out the maximum number of DVDs Johnny can buy from WalMart if the price for each DVD (including tax) is $8.99 and the amount of money he has in his wallet is a value input from a data file. Input Data File (lab5A4.in): Output: With $26.25, Johnny can buy 2 DVDs. With $53.94, Johnny can buy 6 DVDs.
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Lab 5A- 5 Lesson 5A - Methods – Math class45 WAP to find the distance from two points in a coordinate plane, inputting the x and y coordinate for each. Output the distance formatted to two decimal places. Use the Math.hypot method to accomplish this task. Input coordinates in this order: x1 y1 x2 y2 Input Data File (lab5A5.in): Output: The distance between the points (3,5) and (6,10) is 5.83 units. The distance between the points (0,-8) and (-16,5) is units.
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Lab 5A- 6 Lesson 5A - Methods – Math class46 In the mathematics field of trigonometry, the Law of Cosines states that the third side of an oblique triangle (any triangle that is not a right triangle) can be found by knowing the length of any two sides and the measure of the included angle (SAS). Given the Law of Cosines formula shown below, WAP to input two sides and the included angle (in degrees) of a triangle and find the length of the third side. Remember that the Math.cos method must have a radian measure as the parameter, which means you will have to convert the input angle from degree measure to radians using the Math.toRadians method.
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Lab 5A- 6 Lesson 5A - Methods – Math class48 Output: A triangle with sides of length 5.18 and 6.00 and an included angle measuring 60 degrees has a third side of length A triangle with sides of length and and an included angle measuring 35 degrees has a third side of length A triangle with sides of length and and an included angle measuring 120 degrees has a third side of length
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Lab 5A- 7 Lesson 5A - Methods – Math class49 Another classic problem found in trigonometry deals with the right triangle. When the lengths of any two sides of a triangle are known, it is possible to find the measure of the two acute angles of the triangle, using the inverse of sin, cos, and tan functions (asin, acos, atan). To summarize, when the opposite side (0) to an angle (A) and the hypotenuse (h) are known, you use the sin function in the following equation to solve for A: sin A = o/h Use the cos function when the adjacent side (a) and hypotenuse (h) are known: cos A = a/h o A h a A h
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Lab 5A- 7 Lesson 5A - Methods – Math class50 Use the tan function when the opposite (o) and adjacent (a) sides are known: tan A = o/a To solve for A in the above tangent situation, transform the tan A = o/a equation into A = atan(o/a). Since the measure calculated will be in radians, you must convert it to degrees to find the final answer. The same process is used for cos and sin. Given a data file with two sides given (always the two shorter sides), find and output the two acute angles of the right triangle. o Aa
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Lab 5A- 7 Lesson 5A - Methods – Math class51 Input Data File (lab5A7.in): Output: A right triangle with shorter sides of length 3 and 4 has acute angle measures of 36.9 and 53.1 degrees. A right triangle with shorter sides of length 5 and 12 has acute angle measures of 22.6 and 67.4 degrees. A right triangle with shorter sides of length 10 and 10 has acute angle measures of 45.0 and 45.0 degrees. A right triangle with shorter sides of length 50 and 87 has acute angle measures of 29.9 and 60.1 degrees.
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You now understand how to use many more of the Math class methods, as well as the Java API, and static import statements. The Lesson 5B will elaborate more about the String class methods. CONGRATULATIONS! Lesson 5A - Methods – Math class52
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Please copy, paste, and customize the brief acknowledgement shown below in an to indicating that you used this lesson. Feel free to include any comments you wish to offer, positive or negative. I, (your name), from (educational institution) in (city, state, country) used Lesson 5A on (date), understand, respect and honor all of the rights claimed by the author, Mr. John Owen, and offer the following comments..._______________. THANK YOU! Acknowledgement of use Lesson 5A - Methods – Math class53 | 677.169 | 1 |
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We have a range of print and digital resources specifically for the Australian Curriculum. | 677.169 | 1 |
Sharon Stricklin
MTH100 - Intermediate College Algebra
This course provides a study of algebraic techniques such as linear equations and inequalities, quadratic equations, systems of equations, and operations with exponents and radicals. Functions and relations are introduced and graphed with special emphasis on linear and quadratic functions. This course does not apply toward the general core requirements for mathematics | 677.169 | 1 |
Concise Introduction to Pure Concise Introduction to Pure Mathematics, Second Edition provides a robust bridge between high school and university mathematics, expanding upon basic topics in ways that will interest first-year students in mathematics and related fields andMore...
A Concise Introduction to Pure Mathematics, Second Edition provides a robust bridge between high school and university mathematics, expanding upon basic topics in ways that will interest first-year students in mathematics and related fields and stimulate further study. Divided into 22 short chapters, this textbook offers a selection of exercises ranging from routine calculations to quite challenging problems.The author discusses real and complex numbers and explains how these concepts are applied in solving natural problems. He introduces topics in analysis, geometry, number theory, and combinatorics.What's New in the Second Edition: Contains extra material concerning prime numbers, forming the basis for data encryption Explores "Secret Codes" - one of today's most spectacular applications of pure mathematics Discusses Permutations and their importance in many topics in discrete mathematicsThe textbook allows for the design of courses with various points of emphasis, because it can be divided into four fairly independent sections related to: an introduction to number systems and analysis; theory of the integers; an introduction to discrete mathematics; and functions, relations, and countability | 677.169 | 1 |
Pre-Calculus: Part 1
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Starr Books offers Quantum Scientific Pre-Calculus, Part 1 (of 4) begins in Unit 1 with an exploration of graphing equations and functions as well as performing operations on functions. Topics include transformations of functions, finding inverse functions, and variation. Unit 2 focuses on quadratic functions and division of polynomials, including long division and synthetic division. Complex numbers are introduced in Unit 2 and continued in Unit 3. This last unit also includes lessons on the Fundamental Theorem of Algebra, finding zeros of polynomial functions, and graphing rational functions | 677.169 | 1 |
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Read More while new pedagogy help students 'bridge the concepts.' These 'bridges' guide students and help them make successful connections from concept to concept-and from this course to the next. "Elementary Algebra, 9th edition, International Edition" is one of the most current and reliable texts you will find for the course, and is ideally structured and organized for a lecture-format. Each section can be discussed in a 45- to 50-minute class session, allowing you to easily construct your course to fit your needs | 677.169 | 1 |
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Summary
Hundreds of solved examples, exercises, and applications help students gain a firm understanding of the most important topics in the theory and applications of complex variables. Topics include the complex plane, basic properties of analytic functions, analytic functions as mappings, analytic and harmonic functions in applications, and transform methods. Perfect for undergrads/grad students in science, mathematics, engineering. A three-semester course in calculus is sole prerequisite. 1990 edition. Appendices.
Table of Contents
Read more
The complex plane
The complex numbers and the complex plane
A formal view of the complex numbers
Some geometry
Subsets of the plane
Functions and limits
The exponential, logarithm, and trigonometric functions
Line integrals and Green's theorem
Basic properties of analytic functions
Analytic and harmonic functions; the Cauchy-Riemann equations
Flows, fields, and analytic functions
Power series
Cauchy's theorem and Cauchy's formula
The Cauchy-Goursat theorem
Consequences of Cauchy's formula
Isolated singularities
The residue theorem and its application to the evaluation of definite integrals
Analytic functions as mappings
The zeros of an analytic function
The stability of solutions of a system of linear differential equations | 677.169 | 1 |
This text is based on the idea that teachers of mathematics at any level will find it advantageous to be able to point to interesting, useful, and diverse applications of the mathematics they teach. The first chapter describes several graphing techniques used for displaying data and finding meaningful relationships as a result. Subsequent chapters consider problems of current or historical interest in different fields. After some description of background information in the area of application, the text shows how the essential features are represented mathematically. For each topic, exercises are included that either develop the main ideas further or serve as concrete examples of the mathematics involved. The final chapter briefly considers some aspects of the process of mathematical modeling. An index of mathematical topics refers the reader to an appropriate chapter. The topics listed are: calculus, equations, exponents and logarithms, geometry, graphing, groups, matrices, modular arithmetic, percent, permutations and combinations, ratio and proportion, sequences, and trigonometry. (MP) | 677.169 | 1 |
This survey of both discrete and continuous mathematics focuses on the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics, rather than on rote symbolic manipulation. Coverage begins with the fundamentals of mathematical language and proof techniques (such as induction); then applies them to easily-understood questions in elementary number theory and counting; then develops additional techniques of proofs via fundamental topics in discrete and continuous mathematics. Topics are addressed in the context of familiar objects; easily-understood, engaging examples; and over 700 stimulating exercises and problems, ranging from simple applications to subtle problems requiring ingenuity. ELEMENTARY CONCEPTS. Numbers, Sets and Functions. Language and Proofs. Properties of Functions. Induction. PROPERTIES OF NUMBERS. Counting and Cardinality. Divisibility. Modular Arithmetic. The Rational Numbers. DISCRETE MATHEMATICS. Combinatorial Reasoning. Two Principles of Counting. Graph Theory. Recurrence Relations. CONTINUOUS MATHEMATICS. The Real Numbers. Sequences and Series. Continuity. Differentiation. Integration. The Complex Numbers. For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics.
"synopsis" may belong to another edition of this title.
From the Publisher:
Offering a survey of both discrete and continuous mathematics, Mathematical Thinking begins with the fundamentals of mathematical language and proof techniques such as induction. These are applied to easily-understood questions in elementary number theory and counting. Further techniques of proofs are then developed via fundamental topics in discrete and continuous mathematics. The text can be used for courses emphasizing discrete mathematics, continuous mathematics, or a balance between the two. It contains many engaging examples and stimulating exercises010132449021). Hardback. Book Condition: new. BRAND NEW, Mathematical Thinking: Problem-solving and Proofs (2nd Revised edition), John P. D'Angelo, Douglas B. West, B9780130144126 | 677.169 | 1 |
Queuing Theory
About the
Course
Queuing Theory, also known as- Sequencing Problem; is a subject which is taught in engineering disciplines, management studies and in B.Sc Math Honours.
Did you know Queuing Theory is used all over the world in many industries to solve real-life everyday problem of job sequencing and resource allocation ? It's also used in computer algorithms. It's a very interesting subject and is quite easy too if you can understand it!
He'll tell you the concepts and theories before solving the problem just when you need it. You will really enjoy the tutorial!
Topics
Covered
1. Queuing Theory #1: All Formulae and Applicaitons:
You'll learn all the formulas you may need for solving Queuing Theory problems. You'll learn how to calculate following things and all formulas related to that-
1. Finding Utilization Parameter (Traffic Intensity)
2. Probability that queuing system is idle
3. Average time the system is idle in an 8 hour working day
4. Probability that there is 4 customers in the system
5. Number of customers in the system
6. Expected number of customers in the system
7. Expected number of customers waiting for service
8. Expected length of non-empty queue
9. Expected time a customer spends in the queue
10. Expected time a customer spends in the system
11. Probability that customer will spend more than 10 minutes in the queue waiting for service.
2. Queuing Theory #2: Airlines Industry Problem:
In this video we'll solve a Queuing Theory word problem in Operations Research(OR), related to airplane landing in an airport.
3. Queuing Theory #3: Assistant Employment Problem:
In this video we'll solve a decision making word problem in Queuing Theory in Operations Research. Our question is- There is only one cashier in a movie theatre to handle all the show bookings. We have to decide whether they need employing an assistant to assist the cashier. Based on the given data,you need to do the calculation and make the decision.
4. Queuing Theory #4: Slow-Fast Repairmen Problem:
Today we'll solve a decision making word problem in Queuing Theory in Operations Research(OR). Our question is- A factory has two repairmen for repairing of faulty machines,one is fast-but-expensive and another is slow-but-cheap; both charge Rs 5 for repairing of one machine but their repairing capacity is different. There is breakdown charge or downtime charge associated with every non-productive machine hours. Now you've to decide by appropriate calculations which repairman to employ for minimum operational cost but fast repair service.
5. Queuing Theory #5: Medical Industry Problem:
Today we'll solve a Queuing Theory word problem which includes logical decision making. Our question is- A multispeciality clinic has only one MRI machine and due to increased patient inflow,many patients have to wait in queue to get serviced. Management wants to decrease the queue size and increase the patient handling capacity. But that comes with an increased treatment fees. Now you've to calculate what should be the fees if management wants to decrease the queue size to half or less.
6. Queuing Theory #6: Restaurant Problem:
Today we'll solve a simple problem in Queuing Theory for a restaurant.
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7. Queuing Theory #7: Multiple Channel Problem:
Solving a multiple channel Queuing Theory problem in Operations Research with details explanation in very easy language.
You'll learn to find-
1. Probability that queuing system is idle
2. Expected number of customers in the system
3. Expected number of customers waiting for service
4. Expected time a customer spends in the queue
5. Probability that customer has to wait for the service
All these on Multiple Channel Queuing Model.
8. Queuing Theory #8: Outsourcing Problem:
In this video we'll solve an outsourcing related IT industry problem using Queuing Theory. Our question is- A bank want to outsource maintenance of its servers to an IT company. IT company agreed to provide onsite engineers at cost of Rs 50/hr whereas downtime of each server costs the bank Rs 100/hr. Now the bank want to know how many engineers should they hire for optimal cost saving?
Who
should attend
Anyone who has any of the following subjects in his/her curriculum. Or wants to know about Project Management.
1) Project Management
2) Management for Engineering Students
3) Network Analysis
4) Operations Research (OR)
5) Management Science
6) Business Analytics
Pre-requisites
Nothing
What
you need to bring
Nothing
Key
Takeaways He'll tell you the concepts and theories before solving the problem just when you need it. You will really enjoy the tutorial! | 677.169 | 1 |
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Update Standards?
CCSS.Math.Content.HSF-IF.A.1: Common Core State Standards forCCSS.Math.Content.HSF-IF.A.2: Common Core State Standards for Mathematics
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
CCSS.Math.Content.HSF-IF.A.3: Common Core State Standards for Mathematics
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
CCSS.Math.Content.HSF-IF.C.7a: Common Core State Standards for Mathematics
Graph linear and quadratic functions and show intercepts, maxima, and minima.
CCSS.Math.Content.HSF-IF.C.7b: Common Core State Standards for Mathematics
CCSS.Math.Content.HSF-IF.C.7e: Common Core State Standards for Mathematics
CCSS.Math.Content.HSF-IF.C.9: Common Core State Standards for Mathematics
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
CCSS.Math.Content.HSF-BF.B.3: Common Core State Standards for Include recognizing even and odd functions from their graphs and algebraic expressions for them.
MA.9-12.CCSS.Math.Content.HSF-IF.A.1:MA.9-12.CCSS.Math.Content.HSF-IF.A.2: Mathematics
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MA.9-12.CCSS.Math.Content.HSF-IF.A.3: Mathematics
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
MA.9-12.CCSS.Math.Content.HSF-IF.C.7a: Mathematics
Graph linear and quadratic functions and show intercepts, maxima, and minima.
MA.9-12.CCSS.Math.Content.HSF-IF.C.7e: Mathematics
MA.9-12.CCSS.Math.Content.HSF-IF.C.9: Mathematics
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
MA.9-12.CCSS.Math.Content.HSF-BF.B.3:
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Understanding the Fundamental Theorem of Calculus
In this Fundamental Theorem of Calculus worksheet, students demonstrate their understanding of the theorem by identifying the derivative and anti-derivative of given functions. This two-page worksheet contains ten | 677.169 | 1 |
Discrete Mathematics by Example
"Discrete Mathematics by Example" is designed for an undergraduate course and provides many exercises and examples, enabling the development of students' understanding of the principles of discrete mathematics through appropriate pedagogical methods. The text includes chapters on propositional and predicate logic; typed set theory; boolean algebra; relations; functions; sequences; induction and graph theory.
Book Description McGraw-Hill Inc. PAPERBACK. Book Condition: New. 0077098404 International Edition Paperback OR Softcover Edition with Same Contents. Fast Delivery. Book Cover and ISBN may be different from the US Edition. Customer satisfaction guaranteed. Bookseller Inventory # INDMRK-7098407471333675347098407369 | 677.169 | 1 |
About this product
Description
Description
It's secret that algebra gives students the 'jitters,' and word problems give them the 'shakes.' As a dastardly duo, the word problems of algebra are just about as nerve-wracking as anything in the teenage years. But w there's a new book that rescues students from this combination of confusion: the Algebra Survival Guide, Second Edition. Following on the success of the Algebra Survival Guide's First Edition, written by longtime teacher/tutor Josh Rappaport, the Second Edition takes a hard look at algebra's word problems and offers time-tested advice for cracking them. With a new 62-page chapter devoted to these word problems, the new edition tackles the ultimate math nightmares of the puberty years: problems involving rate, time and distance, work performed, and mixture formulas, among others. With twelve additional content chapters that explain fundamental and advanced areas of algebra, the 352-page Second Edition guides students to success in pre-algebra and Algebra I. The Guide features a unique question/answer format so students hear their own questions echoed in the text. The book's answers, written in the voice of a friendly tutor, provide lucid explanation and answers, along with step-by-step instructions with explanations in English right next to the math steps. At the end of each one-page lesson, the book presents practice problems, giving students instant feedback on their grasp of the math. Additional chapter tests provide comprehensive checks for understanding. The new edition boasts an expanded glossary and index, and contains a navigation tool that helps students kw exactly where they are in the program. In addition, the Guide is adorned with lively illustrations by award-winning artist Sally Blakemore. These cartoons t only provide well-deserved comic relief for math learners, they also offer a visual way to grasp algebra's challenging abstractions. (Example: to illustrate the Reflexive Property of a = a, the cartoon shows a sad 'x' variable looking in the mirror while experiencing a 'bad hair day.') With all of these features, the Second Edition Algebra Survival Guide is ideal for homeschoolers, tutors and students striving for algebra excellence. The Second Edition also aligns with the current Common Core State Standards for Math, so it's ideal for today's teachers, as well. Its content chapters tackle the trickiest topics of algebra: Properties, Sets of Numbers, Order of Operations, Absolute Value, Exponents, Radicals, Factoring, Cancelling, Solving Equations, and the Coordinate Plane. Loaded with thorough explanations, practice problems and answers, the new Algebra Survival Guide gives students the boost they need in a style they'll enjoy to master the skills of algebra.
Author Biography
Josh Rappaport runs the Now I Get It! Tutoring Service in Santa Fe, New Mexico. As a longtime tutor, Josh has heard just about every question about math ever uttered. To help children, Josh relates math's complexities to life situations through playful analogies. Josh put his ideas together in the Algebra Survival Guide, winner of a Parents Choice commendation. The Guide has been used by individuals, schools and school districts across the United States. Sally Blakemore, an award winning paper-engineer and book designer has illustrated for Esquire Magazine, The New Yorker, The Washington Post, Oppenheimer Funds; and books, for Singing Turtle Press, Perseus Press, Western National Parks Press, Gibbs Smith Books, North Carolina Press, Michigan Press, and Texas Press, to name a few. She lives in Santa Fe, NM. | 677.169 | 1 |
Precalculus: Graphing and Data Analysis
This book motivates students by highlighting real people facing real challenges finding real solutions. This series features real workers at Motorola, along with hundreds of applications and real data sets highlighting the relevance and scope of activities a reader may encounter in life. Covers such topics as graphs, functions, polynomial and rational functions, the zeros of a polynomial function, exponential and logarithmic functions, trigonometric functions, analytic trigonometry, applications of trigonometric functions, polar coordinates, vectors, analytic geometry, systems of equations and inequalities, sequence, induction, the binomial theorem, counting and probability, and more. For anyone interested in Precalculus.
"synopsis" may belong to another edition of this title.
From the Inside Flap:
PREFACE TO THE INSTRUCTOR
As professors at respectively an urban public university and a community college, Michael Sullivan and Michael Sullivan III are aware of the varied needs of Precalculus students, ranging from those having little mathematical background and fear of mathematics courses to those who have had a strong mathematical education and are highly motivated. For some of your students, this will be their last course in mathematics, while others might decide to further their mathematical education. This text is written for both groups. As the author of precalculus, engineering calculus, finite math, and business calculus texts, and as a teacher, Michael understands what students must know if they are to be focused and successful in upper level math courses. However, as a father of four, including the co-author, he also understands the realities of college life. Michael Sullivan III believes passionately in the value of technology as a tool for learning that enhances understanding without sacrificing important skills. Both authors have taken great pains to insure that the text contains solid, student-friendly examples and problems, as well as a clear and seamless writing style. We encourage you to share with us your experiences teaching from this text. In the Second Edition
The second edition builds upon a strong foundation by integrating new features and techniques that further enhance student interest and involvement. The elements of the previous edition that have proved successful remain, while many changes, some obvious, others subtle, have been made. The text has been streamlined to increase accessibility. A huge benefit of authoring a successful series is the broad-based feedback upon which improvements and additions are ultimately based. Virtually every change to this edition is the result of thoughtful comments and suggestions made by colleagues and students who have used the previous edition. This feedback has proved invaluable and has been used to make changes that improve the flow and usability of the text. For example, some topics have been moved to better reflect the way teachers approach the course. One significant change is the inclusion of the "Field Trip to Motorola" chapter projects. These projects take the incorporation of real life in mathematics to a higher level. The supplements package has been enhanced through upgrading traditional supplements and adding innovative media components such as MathPak, an Integrated Learning Environment. MathPak combines all of the text's key supplements into one easy-to-navigate software package. Changes to the Second Edition
Chapter 1 • "Data in Ordered Pairs" has been moved to the section where linear curve fitting is introduced. • "Circles" is now covered in a stand-alone section. • The section on inequalties now contains a discussion of interval notation.
Chapter 2 • "More about Functions" has been divided into "Characteristics of Functions" and "Library of Function: Piecewise Defined Functions".
Chapters 3 & 4 • Chapter 3 has been divided into two parts: Polynomial and Rational Functions and the Zeros of a Polynomial Function. • The section on Rational Functions has been split into two sections making the material more manageable.
Chapter 6 • A rewrite of the introduction to trigonometric functions makes the distinction between the functions of a real number and that of an angle clearer. • The section on graphing of trigonometric functions is divided in two sections—one graphing the sine and cosine functions and the other graphing the four remaining trigonometric functions. • The section on Inverse Trigonometric Functions have been moved to Chapter 7. • The section on Right Triangle Trigonometry now appears in Chapter 8.
Chapter 7 • Now includes two sections on the section on the Inverse Trigonometric Functions and two section on Trigonometric Equations.
Chapter 8 • The first section discusses right triangle trigonometry and applications.
Chapter 9 • Static equilibrium problems were added. • This chapter now includes a new section on the cross product.
Chapter 10 • Asymptotes are introduced earlier in the section on hyperbolas.
Chapter 11 • Now includes two introductory sections on systems of linear equations. • The section on determinants includes a discussion on minors and cofactors. • Systems of Inequalities and Linear Programming are combined into one section.
Chapter 12 • The first five sections of Chapter 11 from the first edition. • The section on sequences utilizes the power of recursive functions and the graphing calculator to discuss amortization.
Chapter 13 • The second part of Chapter 11 from the 1st Edition. • Now contains two sections on probability: classical and empirical.
Chapter 14 • A new section on the Area Problem; the Integral has been added.
Appendix • Rewritten to provide a more thorough review of prerequisite material. • Can be used as a "just-in-time" review by students. PREFACE TO THE STUDENT
As you begin your study of Precalculus, you may feel overwhelmed by the number of theorems, definitions, procedures, and equations that confront you. You may even wonder whether or not you can learn all of this material in the time allotted. These concerns are normal. Keep in mind that many elements of Precalculus are all around us as we go through our daily routines. Many of the concepts you will learn to express mathematically, you already know intuitively. For many of you, this may be your last math course, while for others, it is just the first in a series of many. Either way, this text was written with you in mind. One of the coauthors, Michael Sullivan, has taught Precalculus courses for over thirty years. He is also the father of four college graduates, including this text's other co-author, who called home from time to time frustrated and with questions. We both know what you're going through. So we have written a text that doesn't overwhelm, or unnecessarily complicate Precalculus, but at the same time gives you the skills and practice you need to be successful.
This text is designed to help you the student, master the terminology and basic concepts of Precalculus. These aims have helped to shape every aspect of the book. Many learning aids are built into the format of the text to make your study of the material easier and more rewarding. This book is meant to be a "machine for learning," that can help you focus your efforts and get the most from the time and energy you invest.
Please do not hesitate to contact us through Prentice Hall with any suggestions or comments that would improve this text. | 677.169 | 1 |
This section contains free e-books and guides on Differential Calculus, some of the resources in this section can be viewed online and some of them can be downloaded.
Aims of this note is to position you to a level of
competence in the techniques or skills in solving any type of problem on
differentiation. This note will consist of ten units divided into 2 modules. In
the first module you shall be introduced to basic mathematical concepts such as
real numbers functions, units and continuity. Differentiating a function from
first principles is shield in the first module. In the second module you will
acquire techniques of differentiation of functions such Sin x, Cos x, Log x,
Cosh x, Sinh x, arc sinh x, arc cosh x, and arcsinh x.
This book emphasis on
systematic presentation and explanation of basic abstract concepts of
differential Calculus. Topics covered includes: Limits, Continuity and
Differentiation of Real Functions of One Real Variable, Differentiation and
Sketching Graphs Using Analysis.
This
book covers the following topics: Ordinary Differential Equations, First Order PDE, Second Order PDE,
Characteristics and Canonical Forms, Characteristics and Discontinuities, PDE in
N-dimensions The Potential Equation, Harmonic Functions, Green's Function,
Consequences of Poisson's Formula The Diffusion Equation, The Wave Equation. | 677.169 | 1 |
Math 114 Exercises Set 2
1. Let V be the set of all positive real numbers. Define the following operations:
i. u v = uv
ii. v = v
Prove that V is a vector space.
2. Explain why the following are not vector spaces.
a. The set of all integers Z with the fo
1
The Column Space & Column Rank of a Matrix
E. L. Lady
0
Let A = 0
0
1
2
1
2
4
2
3
0
5
4
20 .
0
The column space of A is the subspace of R3 spanned by the columns of A ,
in other words it consists of all linear combinations of the columns of A :
4
3
MATH10212 Linear Algebra Brief lecture notes
48
Similarity and Diagonalization
Similar Matrices
Let A and B be n n matrices. We say that A is similar to B if there is an
invertible n n matrix P such that P 1 AP = B. If A is similar to B, we
write A B.
Rem
These notes closely follow the presentation of the material given in David C. Lays
textbook Linear Algebra and its Applications (3rd edition). These notes are intended
primarily for in-class presentation and should not be regarded as a substitute for
thor
Lesson 4
Definition and Properties of Determinants
One application of determinants is to determine whether a linear system with n equations in n unknowns has a
unique solution. We first define the concept of permutations before defining the determinants.
C+ Programming: From Problem Analysis to
Program Design, Fifth Edition
Chapter 2: Basic Elements of C+
Objectives
In this chapter, you will:
Become familiar with the basic
components of a C+ program, including
functions, special symbols, and identifiers
C+ Programming: From Problem
Analysis to Program Design, Fifth Edition
Chapter 5: Control Structures II
(Repetition)
Objectives
In this chapter, you will:
Learn about repetition (looping) control
structures
Explore how to construct and use countcontroll
Department of Mathematics
University of the Philippines-Diliman
Math 114 - TFV
J. Basilla
Third Long Examination
6 Oct 2007
I. Write the letter(in Upper Case) of the correct answer on the rst page of your blue book.
(1) (1 point) Which of the following is
Cofactor Expansion and
Applications of the Determinant
Definition. If A is a square matrix, then the (i, j)-minor of A denoted by Mij (A) (or simply
Mij ) is defined to be the determinant of the submatrix that remains after the ith row and jth
column are
Definition and Properties of
Determinants
One application of determinants is to determine whether a linear system with n equations
in n unknowns has a unique solution. We first define the concept of permutations before
defining the determinants.
Definitio
Recent Documents
8. Diagonalization
E-mail: hogijung@hanyang.ac.kr
http:/web.yonsei.ac.kr/hgjung
8.1. Matrix Representations of Linear Transformations
Matrix of A Linear Operator with Respect to A Basis
We know that every linear transformation T: RnRm has an associated st | 677.169 | 1 |
Series solutions of second-order linear differential equations.
Power series, solutions about an ordinary point.
Solutions about a regular singular point.
Equal roots of indicial equation and roots differing by an integer.
Introduction to systems of first-order equations.
Two linear first-order equations.
On completion of the course students should be able to:
- use some of the standard methods for solution of first- and second-order ordinary differential equations;
- be aware of the implications of existence and uniqueness theorems;
- solve systems of linear first-order equations in two unknowns with constant coefficients;
- analyse the stability characteristics of non-linear systems in two unknowns.
Learning and Teaching Methods
This course runs at 3 hours per week. There are 5 lectures and one class in every fortnight.
In the Summer term 3 revision lectures are given.
Assessment
20 per cent Coursework Mark, 80 per cent Exam Mark
Coursework
The coursework comprises 2 tests worth 10% each.
Other details
Information about coursework deadlines can be found in the "Coursework Information" section of the Current Students, Useful Information Maths web pages: Coursework and Test Information
Exam Duration and Period
2:00 during Summer Examination period.
Other information
Available to Socrates /IP students spending all relevant terms at Essex.
Bibliography
All reading lists are available here: (Essex login required). | 677.169 | 1 |
Success in the electronics field requires a substantial background in mathematics. This updated book is written to provide beginning students with these needed skills. Practical, easy-to-understand problems help prepare students for the types of problems that professional electronic technicians face everyday. As part of the successful Practical Problems in Mathematics series, this fourth edition features expanded coverage of scientific notation, increased problems to be solved using a calculator, additional information on RLC circuits, and a new unit on simultaneous equations that includes coverage of Kirchoff's Law. | 677.169 | 1 |
Introduction to Numerical Methods and MATLAB Implementations undergraduate engineering students a practical and rigorous introduction to the fundamentals of numerical computation. This book is a thoroughly modern exposition of classic numerical methods using MATLAB. The fundamental theory ofMore...
Designed to give undergraduate engineering students a practical and rigorous introduction to the fundamentals of numerical computation. This book is a thoroughly modern exposition of classic numerical methods using MATLAB. The fundamental theory of each method is briefly developed. Rather than providing a detailed numerical analysis, the behavior of the methods is exposed by carefully designed numerical experiments. The methods are then exercised on several nontrivial example problems from engineering practice. The material in each chapter is organized as a progression from the simple to the complex. This leads the student to an understanding of the sophisticated numerical methods that are part of MATLAB. An integral part of the book is the Numerical Methods with MATLAB (NMM) Toolbox, which provides 150 programs and over forty data sets. The NMM Toolbox is a library of numerical techniques implemented in structured and clearly written | 677.169 | 1 |
A comparative study of Euclidean and non-Euclidean geometries, their respective histories and technologies, and their applications in mathematics, the sciences, and modern life. Prerequisite: MATH 166, 215, 217. | 677.169 | 1 |
Resource Added!
Type:
E-book
Description:
CK-12's Basic Geometry FlexBook is designed to present students with geometric principles in a simpler, more graphics-oriented course. Students will explore geometry at a slower pace with an emphasis placed on visual aids and approachability.
Description: CK-12's Basic Geometry FlexBook is designed to present students with geometric principles in a simpler, more graphics-oriented course. Students will explore geometry at a slower pace with an emphasis placed on visual aids and approachability. | 677.169 | 1 |
CCSS.Math.Practice.MP2: Common Core State Standards for Mathematics
CCSS.Math.Practice.MP4: Common Core State Standards for Mathematics
CCSS.Math.Practice.MP5: Common Core State Standards for Mathematics
Use appropriate tools strategically.
CCSS.Math.Practice.MP6: Common Core State Standards for Mathematics
Attend to precision.
CCSS.Math.Content.HSF-IF.B.4: Common Core State Standards for Mathematics
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.? | 677.169 | 1 |
GED Basic Math
Instructor
Dbasics Educational Services
Our mission is simple, it is to: Help Learners Pass The Classes That Counts. We do this by either searching for, or creating ways to make the learning experience more absorbable to each student. We welcome homeschoolers, Online, Distant and Adult Learners.
Reviews
Course reviews will be shown here
Overview
Measurements and Data Analysis
Measurements and data analysis are important areas of math that we use every day. We use measurements when we make things. Sometimes we measure items such as gallons of paint or cups of chocolate chips. At other times we measure length or space, such as the perimeter or area of a floor, wall, or garden or the volume of a container. Data analysis is becoming an increasingly important activity in our lives. The increasing use of computers has accelerated the collection, display, and analysis of data, especially information in numerical form. Most often, we see such data presented in the forms of tables, charts, and graphs.
Questions related to measurement and data analysis will account for more than 25% of the GED mathematical test. Additionally, you will find that you can use many of the skills in this unit in your personal life and on the job.
In the previous in lesson, you learned about fractions, ratios, decimals, and percents. In this lesson you will learn about the measurement systems, the customary U.S. System, how to convert measurements, operations with the measurements, measuring common figures, area of squares and rectangles, and volume.
@ Copyright Dbasics Educational Services. All rights reserved.
Course content
Request invitation
Content of this course is available by invitation only. You can not access this course if you don't have an invitation from the course instructor. | 677.169 | 1 |
plenty of exercises, examples, questions, and problems in Physics that can be solved without Calculus, but there are many more that require it. | 677.169 | 1 |
2331_syllabus_sp12 - MATH 2331 Spring 2012 Course Syllabus...
MATH 2331 Spring 2012 Course Syllabus Page 1 of 3 Instructor: Dr. Melahat Almus E-mail: almus@math.uh.edu Website: Office: 609 PGH Office hours: Check instructor's website (main page) The information contained in this class syllabus is subject to change without notice. Students are expected to be aware of any additional course policies presented by the instructor during the course . Learning Objectives Students will be expected to learn: 1. How to solve systems of linear equations, when there is a solution. 2. How to determine when a system of linear equations has a solution, and to determine how many solutions exist. 3. When a system of linear equations has no solution, how to find the best approximate solution using least squares. 4. How to work with vectors and matrices: vector addition and scalar multiplication, dot products of vectors, and matrix multiplication. 5. How to find eigenvectors and eigenvalues of a square matrix. 6. How to find the determinant of a matrix, and to understand what it means. 7.
This
preview
has intentionally blurred sections.
Sign up to view the full version. | 677.169 | 1 |
Applied Differential Equations: An Introduction presents a contemporary treatment of ordinary differential equations (ODEs) and an introduction to partial differential equations (PDEs), including their applications in engineering and the sciences. Designed for a two-semester undergraduate course, the text offers a true alternative to books published for past generations of students. It enables students majoring in a range of fields to obtain a solid foundation in differential equations.
The text covers traditional material, along with novel approaches to mathematical modeling that harness the capabilities of numerical algorithms and popular computer software packages. It contains practical techniques for solving the equations as well as corresponding codes for numerical solvers. Many examples and exercises help students master effective solution techniques, including reliable numerical approximations.
This book describes differential equations in the context of applications and presents the main techniques needed for modeling and systems analysis. It teaches students how to formulate a mathematical model, solve differential equations analytically and numerically, analyze them qualitatively, and interpret the | 677.169 | 1 |
Computer graphics is important in many areas including engineering design, architecture, education, and computer art and animation. This book examines a wide array of current methods used in creating real-looking objects in the computer, one of the main aims of computer graphics.
Key features:
- Good foundational mathematical introduction to curves and surfaces; no advanced math required
- Topics organized by different interpolation/approximation techniques, each technique providing useful information about curves and surfaces
- Includes a gallery of color images, Mathematica code listings, and sections on curves and surfaces by refinement and on sweep surfaces
- Web site maintained and updated by the author, providing readers with errata and auxiliary material
This engaging text is geared to a broad and general readership of computer science/architecture engineers using computer graphics to design objects, programmers for computer gamemakers, applied mathematicians, and students majoring in computer graphics and its applications. It may be used in a classroom setting or as a general reference.
"The book gives an introduction to curve and surface representation and modelling techniques that are used in the related fields of geometric modelling, computer aided geometric design (CAGD), and computer graphics. It mainly introduces newcomers to these domains as well as those interested in the mathematical background of graphics applications. It is written in a mathematically very accessible style … . Historical remarks, citations from the literature adapted to the context as well as numerous examples contribute to a pleasant and readable style." (Gudrun Albrecht, Mathematical Reviews, 2006 i)
"This book is concerned with the computation of surfaces. … The book also deals with curves … . after reading and understanding a topic, the reader should be able to design and implement the concepts discussed. … The notions are clearly introduced and are ... illustrated by figures and examples. … The book has many examples, which are important for a better understanding of the presented concepts … . Many exercises are sprinkled throughout the text. " (Marian Ioan Munteanu, Zentralblatt MATH, Vol. 1083, 2006) | 677.169 | 1 |
ISBN 9780792330059
ISBN-10
0792330056
Binding
Hardback
Number of Pages
576 Pages
Subject
Complex analysis, complex variables
These proceedings discuss the interface and interconnections between the fields of complex variables and potential theory. The following topics were covered: real and complex potential theory, complex dynamics and Banach algebras and infinite dimensional holomorphy. | 677.169 | 1 |
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Basic Absolute Value Worksheet Math | 677.169 | 1 |
Amazing Fetal Growth
Eleventh graders investigate the determinant of a matrix. For this Algebra II lesson, 11th graders represent a triangle with a matrix and use the determinant to find the area of three given triangles. Students also verify the formula for area of a triangle. | 677.169 | 1 |
Big Idea:
This lesson will provide students with the opportunity to investigate a conceptual understanding of exponential functions. Students will begin with a mathematical model and then extend their knowledge to providing "mathematical" financial advice based off | 677.169 | 1 |
8th Grade Math Academic Intervention Workbook
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
3.83 MB | 477 pages
PRODUCT DESCRIPTION
A printable consumable workbook for 8th grade aligned to the common core! Using a workbook is much more feasible than bringing a textbook back and forth each day! Teachers use with an AIS course. Students always keep the book on them and correct their work right on the page!
First 152 pages are Do Nows to get the class started, the other 335 pages are all practice problems covering the common core standards.
This can be used to remediate higher levels are used in younger grades to help advance. Great intervention to use | 677.169 | 1 |
Equations Bundle ~ Middle School Math
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
15.26 MB | 25 pages
PRODUCT DESCRIPTION
The resources in this bundle emphasizes solving equations involving rational numbers including positive and negative integers, decimals, and fractions. Problem solving and application is also emphasized in the resources. Students will be asked to write their own equations for real-life situations, and solve geometric application questions involving equations. One-step, two-step, and multi-step equations are all included | 677.169 | 1 |
Fundamentals of Math Online Only (2nd ed.)
Step 1: Choose Format
Step 2: Include textbooks and tests?
Fundamentals of Math for Distance Learning
Fundamentals of Math (2nd edition) focuses on problem solving and real-life uses of math with special features in each chapter while reinforcing computational skills and building a solid math foundation. Dominion through Math problems regularly illustrates how mathematics can be used to manage God's creation to His glory.
Teacher Information: Mr. Harmon teaches this course. Read more about him by clicking on the Instructor tab.
What's Included: This kit does not include textbooks. Click on the Contents tab to learn more about what is included with this product.
Scheduling Information: 164 days; lessons are 30 minutes in length.
Author - BJU Press
Copyright Year - 2010
Edition - 2nd ed.
Format - Distance Learning Online
Grade Level - 7
Publisher - BJU Press
Rated 5 out of 5Â by 1
reviewer. | 677.169 | 1 |
4
Examination Advice General Advice Answer questions to the required degree of accuracy. If a question asks for an exact answer then a decimal approximation is not acceptable. When an exact answer is required, appropriate working must be shown.
5
Examination Advice General Advice When an instruction to use calculus is stated for a question, an appropriate derivative or antiderivative must be shown. Label graphs carefully – coordinates for intercepts and stationary points; equations for asymptotes. Pay attention to detail when sketching graphs.
6
Examination Advice General Advice Marks will not be awarded for questions worth more than one mark if appropriate working is not shown.
7
Examination Advice Notes Pages Well-prepared and organised into topic areas. Prepare general notes for each topic. Prepare specific notes for each section of Examination 2. Include process steps as well as specific examples of questions.
8
Examination Advice Notes Pages Include key steps for using your graphic calculator for specific purposes. Be sure that you know the syntax to use with your calculator (CtlgHelp is a useful APP for the TI-84+)
9
Examination Advice Strategy - Examination 1 Use the reading time to carefully plan an approach for the paper. Momentum can be built early in the exam by completing the questions for which you feel the most confident. Read each question carefully and look for key words and constraints.
10
Examination Advice Strategy - Examination 2 Use the reading time to plan an approach for the paper. Make sure that you answer each question in the Multiple Choice section. There is no penalty for an incorrect answer. It may be sensible to obtain the "working marks" in the extended answer section before tackling the multiple choice questions.
11
Examination Advice Strategy - Examination 2 Some questions require you to work through every multiple-choice option – when this happens don't panic!! Eliminate responses that you think are incorrect and focus on the remaining ones. Multiple Choice questions generally require only one or two steps – however, you should still expect to do some calculations.
12
Examination Advice Strategy - Examination 2 If you find you are spending too much time on a question, leave it and move on to the next. When a question says to "show" that a certain result is true, you can use this information to progress through to the next stage of the question.
23
A function is undefined when: a)The denominator is equal to zero b)The square root of a negative number is present. c)The expression in a logarithm results in a negative number. Maximal (or implied) Domain The largest possible domain for which the function is defined
24
Consider the function So the maximal domain is:
25
Using Transformations NATURE – Reflection, Dilation, Translation MAGNITUDE (or size) DIRECTION When identifying the type of transformation that has been applied to a function it is essential to state each of the following:
26
1.Translations a) Parallel to the x-axis – horizontal translation. b) Parallel to the y-axis – vertical translation. To avoid mistakes, let the bracket containing x equal zero and then solve for x. If the solution for x is positive – move the graph x units to the RIGHT. If the solution for x is negative – move the graph x units to the LEFT.
27
2.Dilations a)Parallel to the y-axis – the dilation factor is the number outside the brackets. This can also be described as a dilation from the x-axis. b)Parallel to the x-axis – the dilation factor is the reciprocal of the coefficient of x. This can also be described as a dilation from the y-axis. Note: A dilation of a parallel to the y-axis is the same as a dilation of parallel to the x-axis.
28
3.Reflections a) Reflection about the x-axis b) Reflection about the y-axis c) Reflection about both axes d) Reflection about the line
44
Inverse Functions Key features: The original function must be one-to-one Reflection about the line y = x Domain and range are interchanged Intersections between the graph of the function and its inverse occur on the line y = x
45
To find the equation of an inverse function Step 1:Complete a Function, Domain, Range (FDR) table. Step 2:Interchange x and y in the given equation. Step 3:Transpose this equation to make y the subject. Step 4:Express the answer clearly stating the rule and the domain.
46
ANSWER: A Question 18
47
ANSWER: C Question 19 Graph of the inverse function
48
Question 21
49
a) exists because the function f is one-to-one b) i)
50
b) ii)
51
Algebra of Functions Sum and Difference of Functions Points of intersection of the two functions. Points where either graph crosses the x-axis The dominant function in different parts of the domain. Key features:
52
53
Solving indicial equations Step 1:Use appropriate index laws to reduce both sides of the equation to one term. Step 2:Manipulate the equation so that either the bases or the powers are the same. Step 3:Equate the bases or powers. If this is not possible then take logarithms of both sides to either base 10 or base e.
54
Question 26 ANSWER: C
55
Step 1:Use the logarithmic laws to reduce the given equation to two terms – one on each side of the equality sign. Step 2:Convert the logarithmic equation to indicial form. Step 3:Manipulate the given equation so that either the bases or the powers are the same. Solving logarithmic equations
56
Step 4:Equate the bases or powers. If this is not possible then take logarithms of both sides to either base 10 or base e. Step 5:Check to make sure that the solution obtained does not cause the initial function to be undefined.
61
Question 32 Dilation by a factor of 2 from the x-axis ANSWER: C Reflection about the x-axis
62
Solving Trigonometric Equations Put the expression in the form sin(ax) = B Check the domain – modify as necessary. Use the CAST diagram to mark the relevant quadrants. Solve the angle as a first quadrant angle. Use symmetry properties to find all solutions in the required domain. Simplify to get x by itself.
63
ANSWER: E Question 34
64
Question 36 Analysis Question a)
65
b)
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c)
67
d)
68
e) i)
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e) ii)
70
Revision Quiz
71
Question 1 The derivative ofis equal to a)b)c) d) e) A
72
Bonus Prize!!
73
Angie notes that 2 out of 10 peaches on her peach tree are spoilt by birds pecking at them. If she randomly picks 30 peaches the probability that exactly 10 of them are spoilt is equal to Question 3 a) d) b) e) c) D
74
Question 4 a) d) e) c) b) The total area of the shaded region shown is given by D
75
Which one of the following sets of statements is true? a) 2121 and b) 2121 and c) 2121 and d) 2121 and e) 2121 and A Question 5
76
DIFFERENTIAL CALCULUS Chain Rule: Product Rule: Quotient Rule:
77
Further Rules of Differentiation Square Root Functions
78
Further Rules of Differentiation Trigonometric Functions
79
Further Rules of Differentiation Logarithmic Functions
80
Further Rules of Differentiation Exponential Functions
81
ANSWER: D Question 37
82
ANSWER: B Question 38
83
Graphs of Derived Functions ANSWER: C Question 40
84
Question 42 a)
85
b)
86
Question 43
87
b)
88
c)
89
Approximations ANSWER: D Question 45
90
Related Rates Question 46
91
92
Question 47 Analysis Question a)
93
b) Find the EXACT COORDINATES of the two stationary points and their nature. Local maximum at: Stationary point of inflexion at:
94
c)i)
95
ii)
96
iii)Select a pointand find the equation of the tangent at this point.
97
This tangent will pass through the origin when x = 0 Therefore the only two tangents that pass through the origin are when
98
d) i) Use CALCULUS to find the exact values of the constants p, q and k.
99
d)ii)
100
Antidifferentiation and Integral Calculus
101
ANSWER: B Question 48
102
Trigonometric Functions Rules of Antidifferentiation
103
Exponential Functions
104
Rules of Antidifferentiation Logarithmic Functions
105
Example
106
Definite Integrals
107
Properties of Definite Integrals ANSWER: E Question 50
108
Integration by recognition
109
ANSWER: A Question 52 On the interval (a, b) the gradient of g(x) is negative.
110
Calculating Area Sketch a graph of the function, labelling all x-intercepts. Shade in the region required. Divide the area into parts above the x-axis and parts below the x-axis. Find the integral of each of the separate sections, using the x-intercepts as the terminals of integration. Subtract the negative areas from the positive areas to obtain the total area.
111
The total area bounded by the curve and the x-axis on the interval [a, c] is given by: ANSWER: D Question 53
112
Question 54 a)
113
b)Hence, find the exact area of the shaded region
114
Area between curves
115
Sketch the curves, locating the points of intersection. Shade in the required region. If the terminals of integration are not given – use the points of intersection. Check to make sure that the upper curve remains as the upper curve throughout the required region. If this is not the case then the area must be divided into separate sections. Evaluate the area. Method
116
Question 55 Find the solution to the equation
117
b) Use CALCULUS to find the area of the shaded region.
118
Numerical techniques for finding area ANSWER: D Question 56
119
Question 57 Analysis Question a) Write down the equation in x, the solutions of which give the x-coordinates of the stationary points of the curve
120
b) i)
121
b) ii) A repeated root at x = -1 indicates that the normal is a tangent to the curve at this point.
122
c) i)
123
124
c) ii)
125
Discrete Random Variables A discrete random variable takes only distinct or discrete values and nothing in between. Discrete variables are treated using either discrete or binomial distributions. These values are usually obtained by counting. A continuous random variable can take any value within a given domain. These values are usually obtained through measurement of a quantity. Continuous variables are often treated using normal distributions.
126
Expected value and expectation theorems
127
Variance and Standard Deviation
128
The number of hours each day, X, spent cycling has the following probability distribution. The proportion of days for at least two hours of cycling is: Question 59 ANSWER: D
129
ANSWER: B x Question 60
130
Markov Chains A Markov chain is a chain of events for which the probabilities of outcomes or states depend on what has happened previously. Tree diagrams are a useful tool for solving problems.
131
Question 61 If it has snowed the day before the probability of snow is 0.6. If it has not snowed on the previous day then the probability of snow is 0.1. If it has snowed on Thursday, what is the probability that it will not snow on the following Saturday?
132
Question 62 A bag contains three white ball and seven yellow balls. Three balls are drawn without replacement. The probability that they are all yellow is: ANSWER: D
139
Continuous Random Variables Mean: Mode:the value for which is a maximum Variance: Median:the value m such that
140
Question 67 ANSWER: E
141
The Normal Distribution The mean, mode and median are the same. The total area under the curve is one unit.
142
Using symmetry properties
143
144
145
Question 71 X is normally distributed with a mean of 72 and a standard deviation of 8. Use the result that to find: a)
146
b)
147
c)
148
Draw a diagram, clearly labelling the mean. Shade the region required. Use either the appropriate symmetry properties or a calculator to find the required probability Remember that: Solving normal distribution problems
149
Volunteers for a weight loss program have weights which are normally distributed with a mean of 100 kg and a standard deviation of 8 kg. One person is selected at random. The probability that this person's weight is over 110 kg is approximately Question 72
150
ANSWER: E
151
Applications of the normal distribution Draw a diagram, clearly shading the region that corresponds to the given probability. Use the symmetry properties of the curve to write down the appropriate z value. Use the inverse normal function to find the required probability and the corresponding z value. Use the relationship to calculate the required x value.
152
Question 73 Black Mountain coffee is sold in packets labeled as being of 250 grams weight. The packing process produces packets whose weight is normally distributed with a standard deviation of 3 grams. In order to guarantee that only 1% of packets are under the labeled weight, the actual mean weight (in grams) would be required to be closest to a) 243b) 247c) 250d) 254e) 257 | 677.169 | 1 |
Course Description
Physics is a foundational science on which many of the concepts of chemistry, geology, biology, and other natural sciences rely. Because physics is a math-intensive subject, it is sometimes intimidating to those who wish to study it.
This course focuses on some of the critical mathematical tools required for studying physics and is geared toward students with a basic knowledge of algebra, geometry, and trigonometry. Starting with an introduction to vectors, the course then begins to develop a set of mathematical tools for studying classical Newtonian mechanics as well as some basic topics in classical electromagnetism. Although the course does not cover calculus, the math that is presented is more than sufficient for analyzing and solving a wide range of problems involving the motion of objects and forces such as gravity, the electric force, and magnetism.
The first half of the course deals with objects in motion, including how to quantify and relate characteristics such as force, velocity, momentum, energy, and power. The second half of the course focuses on waves and electromagnetism, starting with a look at the electric and magnetic forces and their sources, then moving to acoustic and electromagnetic waves, and ending with an overview of geometrical optics. By the end of the course, students should be armed with many of the tools they need to study physics in depth.
Have a question? Check out our FAQ or contact us for more information.
Course Lessons
Average Lesson Rating: (29 votes)
"Extraordinarily Helpful"
1/16/2017 11:23:15 AM
Lesson 1 - Introduction and Mathematical Foundations
This lesson discusses some simple, but fundamental, mathematical tools that will underlie the principles and methods presented in the remainder of the course. 19 Total Points
Lesson 1 Video
Review Article: Understanding Vectors
Complete: Lesson 1 Assignment
Complete Exam: Lesson 1 – Introduction and Mathematical Foundations
Lesson 2 - Force and Newton's Laws I
In the previous lesson, we introduced the idea of vectors as quantities with magnitude (or length) and direction. We now take a deeper look at the properties of vectors in the context of forces and Newton's first law of motion. 18 Total Points
Lesson 2 Video
Complete: Lesson 2 Assignment
Complete Exam: Lesson 2 - Force and Newton's Laws I
Lesson 3 - Force and Newton's Laws II
Building on the material we studied in the previous two lessons, we now apply our understanding of vectors and forces to Newton's second and third laws of motion. 18 Total Points
Lesson 3 Video
Review Article: Newton's third law of motion
Complete: Lesson 3 Assignment
Complete Exam: Lesson 3 – Force and Newton's Laws II
Lesson 4 - Uniform Circular Motion
In this lesson, we look at how to apply both vectors and the geometry of circles and triangles to uniform circular motion. Examples of such motion include the orbits of celestial objects, such as planets and stars. 19 Total Points
Lesson 4 Video
Review Article: Uniform Circular Motion
Complete: Lesson 4 Assignment
Complete Exam: Lesson 4 Uniform Circular Motion
Lesson 5 - Momentum
In this lesson, we define the concept of momentum in the context of physics and use the mathematics of vectors and differences (such as the difference in time, Δt) to derive the law of conservation of linear momentum 18 Total Points
Lesson 5 Video
Complete: Lesson 5 Assignment
Complete Exam: Lesson 5 Momentum
Lesson 6 - Work
In this lesson, we turn from momentum to work, which is the amount of energy required to perform a given task (such as moving an object from one point to another). 17 Total Points
Lesson 6 Video
Review Article: Work: The Physics of Hypertextbook
Complete: Lesson 6 Assignment
Complete Exam: Lesson 6 Work
Lesson 7 - Work and Kinetic Energy
This lesson takes the concept of work developed in the previous lesson and applies it to the concept of kinetic energy, or energy of motion. 18 Total Points
Lesson 7 Video
Review Article: Work and Energy
Complete: Lesson 7 Assignment
Complete Exam: Lesson 7 Work and Kinetic Energy
Lesson 8 - Mechanical Energy and Power
This lesson concludes our look at basic Newtonian mechanics (motion) by considering potential energy and its relationship to kinetic energy through the concept of mechanical energy. 10 Total Points
Lesson 8 Video
Review 2 Articles: Mechanical Energy; Work, Energy and Power
Complete Exam: Lesson 8 Mechanical Energy and Power
Lesson 9 - Electrostatics
The course now turns from an examination of the mechanics of motion resulting from applied forces to an overview of a specific type of force: the electric force. 9 Total Points
Lesson 9 Video
Review 2 Articles: Electrostactics; Superposition Principle
Complete Exam: Lesson 9 Electrostatics
Lesson 10 - Currents and Magnetism
In this lesson, we turn from stationary charges to charges in motion. Charged particles in motion exert a magnetic force on each other. 18 Total Points
Lesson 10 Video
Review 2 Articles: Basics of Magnetism; Vector Products of Vectors
Complete: Lesson 10 Exercises
Complete Exam: Lesson 10 Currents and Magnetism
Lesson 11 - Waves
This lesson considers some of the mathematical aspects of waves that we will need to briefly study sound and light in later lessons. 9 Total Points
Lesson 11 Video
Review Article: Waves Basics Outline
Complete Exam: Lesson 11 Waves
Lesson 12 - Acoustic Waves
In this lesson, we begin to apply our knowledge of waves gained in the previous lesson to the topic of sound (acoustic waves). 14 Total Points | 677.169 | 1 |
Mathematics
Mathematics is a multifaceted subject of great beauty and application. The study of math explores some of the deepest questions and puzzles that have ever been encountered and equips the student with a universal language used to study quantities and relationships in all fields. Through the study of mathematics, the student develops both the ability to think logically and abstractly as well as the problem-solving and computational skills necessary for success in any field of study.
Highlights
A professional and innovative staff committed to providing the best possible mathematics education, including the use of computers and multimedia presentations in the teaching of mathematics.
A comprehensive mathematics curriculum addressing the needs of both the transfer student and the non-transfer student.
A math resource center providing free tutoring and alternative models of instruction and support for students.
A technology-mediated alternative for students in arithmetic and algebra.
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Math Resource Center
The Math Resource Center (MRC), located in SM 5, offers free tutoring in all levels of mathematics. The MRC is staffed by math instructors and peer tutors. Computers are available for use in the Math Resource Center for students to do homework, view lecture notes, and work on projects.
You must be enrolled in a mathematics class at West Valley College to obtain free tutoring for the Math Resource Center. Tutoring is available on a drop-in basis only. Follow the following instructions to register for free tutoring at West Valley College:
Math Placement Exam
You should take your placement test seriously. Preparing for the placement test will save you lots of time and money. Remember, your score on the placement test will determine which math class you are eligible to enroll in. Placing in the right courses will help you achieve your education goals faster.
All students who started on or after Fall 2014 must take a placement test before registering for classes. To schedule a placement test, you must have a valid student ID number. Visit the Assessment Office web site for more information on how to schedule your assessment test.
After scheduling your assessment test, make sure you take some time to review the topics that you will be tested on. Below are some free resources to help you prepare for the math placement test at West Valley College. | 677.169 | 1 |
Introductory Logic - eighth grade and up, these lessons cover logical statements, fallacies, syllogisms, and many other elements. This course is a thorough introduction and serves as both a self-contained course as well as a preparatory course for moreMore...
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Designed for eighth grade and up, these lessons cover logical statements, fallacies, syllogisms, and many other elements. This course is a thorough introduction and serves as both a self-contained course as well as a preparatory course for | 677.169 | 1 |
Publisher Description
Interactive, problem-solving lessons, and customizable real time tests with solutions and evaluations. Topics covered: rectangular coordinate system, functions and graphs, linear and absolute value functions, quadratic functions, polynomial and rational functions, exponential and logarithmic functions, second degree equations, determinants and Cramer''s rule, matrices, complex numbers. The demo version contains selected lessons from the full version, fully functional, all features included.
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Hypothesis: Mathematics Essay
Hypothesis is a preliminary assumption or a tentative explanation that accounts for a set of facts, taken to be true for the purpose of investigation and testing; a theory. In this research paper, we are trying to figure out what are the probable causes of Math anxiety as well as the possible solutions regarding this matter. We came up with the following:
1. Teacher Factor
This simply means that a teacher has an effect on how a student grows or understands Math itself. There are times that if a student doesn't like his/her teacher in Mathematics, he/she will not study well. His/her desire to learn new things is just suddenly fading. But if a student likes his/her teacher in Math, he/she will participate in class and is eager to know more.
2. Past generations/ older years or levels
Students from the previous year/level can affect the mentality of the students younger than them. If this will happen, the students will set their minds or have this certain mentality that the subject Math is difficult causing students to get lazy and be bored.
3. Pressure and Stress
Some parents want their children to be top achievers in school causing them to persuade their children study even more. Because of the mentality that math is hard, other parents pressure their children to study math in order to pass. This causes children what you call stress. Sometimes when the students can take it anymore, they rebel against their parents, teachers and the like.
4. Stereotypes
Stereotype is an oversimplified standardized image of a person or group. This affect the performance of a person in what he's doing because he has this idea that if he belongs to this group of people that are not good or doesn't do in this area, he will suck or fail too.
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The guide that helps students study faster, learn better, and get top grades
About the Author:
Murray Speigel, Ph.D., was Former Professor and Chairman of the Mathematics Department at Rensselaer Polytechnic Institute, Hartford Graduate Center.
Seymour Lipschutz, Ph.D. (Philadelphia, PA), is presently on the Mathematics faculty at Temple University. He has written more than 15 Schaum's Outlines.
John J. Schiller, is an Associate Professor of Mathematics at Temple University. He received his Ph.D. at the University of Pennsylvania and has published research papers in the areas of Riemann surfaces, discrete mathematics biology. He has also coauthored texts in finite mathematics, precalculus, and calculus202
Book Description McGraw-Hill Education, 2009. Paperback. Book Condition: New. All books ship SAME or NEXT business day!! 2nd Day Shipping Available! Absolutely brand NEW! May not include supplements, codes etc. Contact us with any questions!. Bookseller Inventory # 23952, 2009. Book Condition: New. Brand New, Unread Copy in Perfect Condition. A+ Customer Service! Summary: The guide that helps students study faster, learn better, and get top grades6957008538115693 | 677.169 | 1 |
Calculus Quick Reference Calculus Quick Reference lists down all the important formulas and evaluation techniques used in calculus which makes it easier for you to memorize and apply them in solving problemsKid Math A fun way for kids to learn numbers and build basic math skills | 677.169 | 1 |
语言: English
图书介绍
Dr. Carleen Eaton continues on to Algebra 2, and brings with her 10
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Summary and Info
Engineers looking for an accessible approach to calculus will appreciate Precalculus: With Limits, 2nd Edition. The book offers a clear writing style that helps reduce any math anxiety they may have while developing their problem-solving skills. It incorporates Parallel Words and Math boxes that provide detailed annotations which follow a multi-modal approach. Your Turn exercises reinforce concepts by allowing them to see the connection between the exercises and examples. A five-step problem solving method is also used to help engineers gain a stronger understanding of word problems.
More About the Author
Cynthia A. Young is an associate professor of English and the director of the African and African Diaspora Studies program at Boston College, in the U.S.
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★★★★★★★★★★Precalculus0 out of 5 stars based on 0 ratings. Your Rating: ☆☆☆☆☆★★★★★ | 677.169 | 1 |
Introduction to Information Theory
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Wann: Freie Auswahl
Beschreibung
This tutorial introduces fundamental concepts in information theory. Information theory has made considerable impact in complex systems, and has in part co-evolved with complexity science. Research areas ranging from ecology and biology to aerospace and information technology have all seen benefits from the growth of information theoryLogic
Maths
Probability
Information Processing
Information
Computation
Forms of Information
Formula of Information
Mutual Information
Communication C
Themenkreis
In this tutorial, students will follow the development of information theory from bits to modern application in computing and communication. Along the way Seth Lloyd introduces valuable topics in information theory such as mutual information, boolean logic, channel capacity, and the natural relationship between information and entropy.
Lloyd coherently covers a substantial amount of material while limiting discussion of the mathematics involved. When formulas or derivations are considered, Lloyd describes the mathematics such that less advanced math students will find the tutorial accessible. Prerequisites for this tutorial are an understanding of logarithms, and at least a year of high-school algebra. | 677.169 | 1 |
Introduction to Graph Theory98132.61
FREE
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Good TEXT ONLY. NO ACCESS CODE. Used texts with varying degrees of highlighting and wear. Email for details. Could be a former library book.
About the Book
This book fills a need for a thorough introduction to graph theory that features both the understanding and writing of proofs about graphs. Verification that algorithms work is emphasized more than their complexity. An effective use of examples, and huge number of interesting exercises, demonstrate the topics of trees and distance, matchings and factors, connectivity and paths, graph coloring, edges and cycles, and planar graphs. For those who need to learn to make coherent arguments in the fields of mathematics and computer science. | 677.169 | 1 |
About this product
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Description
Workshop problems usually can be solved with mathematics, but as this interesting book shows, there's often a -math solution too. In most situations, the savvy craftsman can get the answers either with math or by applying common-sense rules of thumb. This handbook presents both approaches so readers can choose what works best in their particular situation. With this handbook on the bench, readers will enjoy success in the workshop whether or t they are comfortable with mathematics.Hundreds of illustrations guide readers through the solutions to everyday workshop problems. Straightforward writing and a simplified approach to calculations lead readers step-by-step to the answers they need. Dozens of useful tables put essential reference information into this single, easy-to-follow resource. This practical handbook, packed full of facts, formulas and shortcuts, is essential for woodworkers of all skill levels. | 677.169 | 1 |
Publisher Description
F algebraic, trigonometric, hyperbolic, date, statistical, and financial functions.In addition to Farsight Calculator contains a set of powerful tools,such as "Finance box","date calculations" and "Unit convertor" etc.
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Calculate loan payments, depreciation, cash flows, and interest rate conversions.
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Over 100 units of measure in six categories makes converting easy.
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Introduction To Calculus
Students engage in a lesson that is about an introduction to the course of calculus. They practice some of the basic concepts and solve some problems applying different principals of calculus. The teacher uses the sample problems provided in the lesson for direct instruction | 677.169 | 1 |
In fact this is what I am using this year and both the kids and myself are. Boost your students understanding of Saxon Math with DIVE's easy-to-understand lectures. ISBN# 1565771389 Read more Show
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Students never get enough practice factoring, but now they can! Students will need to factor many problems just to solve the maze, The Factor Maze has 2 Mazes: (ax^2 + bx + c), and Rational Expression Maze
1) Factoring Maze: Start at the top left hand corner, and solve the maze by factoring every polynomial you come across. Write the factorization under the polynomial for easy reference. You can only move one square up, down, left, or right when it SHARES a factor with the current square. You have solved the maze when you exit at the bottom right.
2) Rational Expression Maze: Simplify the following problems. Move through the maze by locating the answer for each problem until you reach "End Here" You must show your work for full | 677.169 | 1 |
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In this activity, students will be analyzing nonproportional relationships in the context of a real world problem. They will be graphing the relationship and answering questions based on the results. This is a good task to use when working with functions/linear equations and touches on the the topic of systems | 677.169 | 1 |
Simplifying Radicals - Lesson and Practice Problems
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This is an interactive lesson on simplifying radicals (square roots and cube roots with integers and variables). You can go through this as a class or give your students a chance to explore it individually or in groups. Students practice steps throughout the lesson, and are also given six problems to try entirely on their own at the end. Can also be used for review and test prep. **This product is also available as part of "Precalculus Summer Prep / Algebra 2 Review Packet," a concept and skill review packet consisting of six main | 677.169 | 1 |
Percentage problems in bridging courses
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Research on teaching high school mathematics shows that the topic of percentages often causes learning difficulties. This article describes a method of teaching percentages that the authors used in university bridging courses. In this method, the information from a word problem about percentages is presented in a two-way table. Such a table gives a logical structure to the problem and provides an algorithm for finding a simple equation for the unknown value of interest. The use of this procedure is illustrated by several examples of different levels of difficulty. The method can be applied to many types of percentage problems, so it is quite universal. | 677.169 | 1 |
Christmas Examination:Common Exam given to all First Years in DecemberTime: 1 hour. Questions: To be agreed amongst all teachersThe result of this exam alone should be put on the Christmas report with a comment
TERM 2: JANUARY TO FEBRUARY MID-TERM
Perimeter & Area
Geometry 1: Points – Angles – Lines
Ratio & Proportion
TERM 2: FEBRUARY TO EASTER
Collecting Data
Coordinates
Solving Equations
Geometry 2 – Triangles
TERM 3: EASTER TO SUMMER
Presenting Data
Measure
Transformation Geometry
Revision of Topics covered in year
Summer Examination:Common Exam given to all First Year students formally in the hall in May.Time:1 hourQuestions: To be agreed amongst all teachersThe result of this exam alone should be put on the Summer report with a comment.
Homework:
Choose appropriate exercises from the textbook relevant to the material covered in examples given in class.
Regular chapter tests in class at the end of every chapter or group of shorter chapters set by the individual teachers themselves
Common Christmas Examination in December set by a selected teacher with an agreed marking scheme. This is based on the work covered in the first term. The result should go on the Christmas report with a comment.
Formal common Summer Examination in May set by a selected teacher with an agreed marking scheme. This is based on the whole year's work and is given to all first years in the Exam hall. Time: 1 hour. The result should go on the summer report with a comment. | 677.169 | 1 |
This text enables students to become active participants in developing their understanding of mathematics, while avoiding the danger of making technology an end in itself. Robust pedagogy including chapter opening applications, graphing explorations, technology tips, calculator investigations, and discovery projects are some of the tools students will use to master the material and begin applying the mathematics to solve real-world problems. Algebra Review. Graphs and Technology. Equations and Inequalities. Functions and Graphs. Polynomial and Rational Functions. Exponential and Logarithmic Functions. Systems of Equations and Inequalities. Discrete Algebra. Analytic Geometry. [Triangle Trigonometry. Trigonometric Functions. Trigonometric Identities and Equations. Applications of Trigonometry.] Geometry Review Appendix. Program Appendix
"synopsis" may belong to another edition of this title.
Product Description:
Thomas Hungerford's CONTEMPORARY COLLEGE ALGEBRA is known and well respected for its thorough integration of the graphing calculator, clear writing, and outstanding applications problems. Many adopters laud the use of real data in examples and exercises, and they appreciate the flexibility of the book's organization. This market-leading text is now accompanied by an outstanding array of innovative supplements that facilitate teaching and enhance learning, such as iLrn--a robust suite of online course management, testing, and tutorial resources for instructor and student, with vMentor live online tutoring. The text also includes a free Interactive Video Skillbuilder CD-ROM with MathCue, and a Book Companion Web Site featuring online graphing calculator resources.
About the Author:
Thomas W. Hungerford received his M.S. and Ph.D. from the University of Chicago. He has taught at the University of Washington (Seattle) and at Cleveland State University, and is now at St. Louis University. His research fields are algebra and mathematics education. He is the author of many notable books for undergraduate and graduate level courses. These include: ALGEBRA, Springer Verlag, Graduate Texts in Mathematics #73, 1974; ABSTRACT ALGEBRA: AN INTRODUCTION, Second Edition, Harcourt, 1997; MATHEMATICS WITH APPLICATIONS, Eighth Edition (with M. Lial), Addison-Wesley, 2003; and CONTEMPORARY PRECALCULUS: A GRAPHING APPROACH, Fourth Edition, Brooks/Cole, 2004. | 677.169 | 1 |
Description of the book "Mathematics for Algorithm and Systems Analysis":
Discrete mathematics is fundamental to computer science, and interesting, challenging problems in discrete mathematics arise in programming languages, computer architecture, networking, distributed systems, database systems, artificial intelligence, theoretical computer science, and elsewhere. This up-to-date text prepares undergraduates with four units of study: counting and listing, functions, decision trees and recursion, and basic concepts of graph theory. Each unit is divided into four sections, and each section contains a selection of problems that vary in difficulty. Solutions to these problems are given at the end of the text.
Multiple-choice questions for review appear at the end of PDF each unit. The text offers students the mathematical language and sophistication to recognize and articulate the ideas behind these questions and to answer questions similar in concept and difficulty. Those who master the necessary ideas and mathematical language to understand these review questions gain the ability to formulate, in the neutral language of mathematics, the problems that arise in various applications of computer science--a skill that enables them to discuss problems in discrete mathematics with other computer scientists and with mathematicians.
Reviews of the Mathematics for Algorithm and Systems Analysis
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Pen-based Proofs in Euclidean Geometry
The concept of digital ink has the potential to dramatically transform and enhance the teaching
and learning process by becoming widely used in classrooms - replacing the use of desktops
or laptops. One of the potential advantages of the new technology is that it allows the
expression and exchange of ideas in an interactive environment using sketch based interfaces.
The project combines the concept of geometrical sketching on tablets with an automated theorem proving systems.
The goal of the work is to promote an intuition for problem solving and a way of algorithmic thinking on examples
from planar geometry. Choosing the pen-based interface, we will expose students to contemporary
software design and development. Integrating elements
of formal proofs by definitions and axioms will
propel students into a higher level of geometrical thinking.
The idea of proving will be developed and refined
as the above cycle progresses (revolves) until students
obtain a strong grasp on a logical construction. | 677.169 | 1 |
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Painless Algebra
Suitable both for students beginning their study of algebra and for those who want to remember what they once knew, the nine chapters of this book progress from basic principles of addition and subtraction to the solution of quadratic equations. But everything is presented as painlessly as possible, with step-by-step guides to solving each kind of problem, advice on common mistakes to avoid and illustrative drawings and diagrams | 677.169 | 1 |
have the palace olympic sweat and machine wash it with no problems. funny you say that cos i was in london yesterday and impressed by the quality of their tees. deffo not gildan and ye i would happily spend £35 to get one
with the livestream channel surely we've never had more sips content than this, i don't see a problem? they are essentially the same as series except more often and less cut out. never understood why people want less of what they like
its on par with warwick, both known to have strong maths depts, warwick maybe slightly better. to be honest a maths degree in the uk is enough to get you a really good job- employers know how hard the subject is, and how it trains someone to be a critical thinker. And if you get a first at any of these stronger unis? You won't even have to look for a job.
Could be that I'm not american. We have loads of homework which was compulsory but only exams count towards your grade. Though I must say even in the pure modules where home works are mostly proofs it still required practice!
As a maths student I can tell you that your pro about workload is completely wrong in my experience. I don't know any other subject at my uni that requires more work. We get a tonne of homework (especially in first year when you have to cover a wide variety of modules) and because the subject is naturally difficult you also have to put in the time to understand the stuff and be able to complete assignments etc
I haven't covered linear operators yet but I think what you're describing is similar to why we need n linearly independent solutions for the homogeneous case of a differential equation of order n? Since it is essentially a linear map that has a dimension 2? Thus a basis consists of the sum of the two terms?
Are you sure? This isn't the product rule at all. I've known that since school and also covered it in several 1st year modules.
Maybe I haven't explained this properly? I'm not trying to find the derivative of a product of functions; I'm trying to find partial derivatives of functions that have substitutions (are compositions) using the chain rule.
edit:
an example would be finding the gradient of a curve that is originally expressed in polar coordinates. | 677.169 | 1 |
Basic Algebra
PRODUCT DESCRIPTION
This is the best book around to teach the essentials of pre-algebra and first-year algebra in an easy-to-follow, low-anxiety way.
Basic Algebra makes complex ideas simple:
• Teaching material is broken down into one- or two-page lessons, each explaining a single principle
• Clear explanations mean a minimum of intervention is required
• Can be used for self-instruction or as a classroom text
• A special section with practice exercises follows each lesson
• Chapter checkout tests reinforce individual lessons | 677.169 | 1 |
Trusted Guide to Discrete Mathematics with Proof?Now in a Newly Revised Edition Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. Discrete Mathematics with Proof, Second Edition continues to facilitate an up-to-date understanding of this important topic, exposing readers to a wide range of modern and techlogical applications. The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics. Subsequent chapters explore additional related topics including counting, finite probability theory, recursion, formal models in computer science, graph theory, trees, the concepts of functions, and relations. Additional features of the Second Edition include:* An intense focus on the formal settings of proofs and their techniques, such as constructive proofs, proof by contradiction, and combinatorial proofs* New sections on applications of elementary number theory, multidimensional induction, counting tulips, and the bimial distribution* Important examples from the field of computer science presented as applications including the Halting problem, Shann's mathematical model of information, regular expressions, XML, and Normal Forms in relational databases* Numerous examples that are t often found in books on discrete mathematics including the deferred acceptance algorithm, the Boyer-Moore algorithm for pattern matching, Sierpinski curves, adaptive quadrature, the Josephus problem, and the five-color theorem* Extensive appendices that outline supplemental material on analyzing claims and writing mathematics, along with solutions to selected chapter exercises Combinatorics receives a full chapter treatment that extends beyond the combinations and permutations material by delving into n-standard topics such as Latin squares, finite projective planes, balanced incomplete block designs, coding theory, partitions, occupancy problems, Stirling numbers, Ramsey numbers, and systems of distinct representatives. A related Web site features animations and visualizations of combinatorial proofs that assist readers with comprehension. In addition, approximately 500 examples and over 2,800 exercises are presented throughout the book to motivate ideas and illustrate the proofs and conclusions of theorems. Assuming only a basic background in calculus, Discrete Mathematics with Proof, Second Edition is an excellent book for mathematics and computer science courses at the undergraduate level. It is also a valuable resource for professionals in various technical fields who would like an introduction to discrete mathematics.
Eric Gossett, PhD , is Professor of Mathematics and Computer Science at Bethel University. Dr. Gossett has thirty years of academic and industry experience in the areas of Web programming, discrete mathematics, data structures, linear algebra, and algebraic structures. He is the recipient of the Bethel Faculty Service Award for his work developing Bethel's first generation of Web services | 677.169 | 1 |
Go
Math - Middle & High School. For a quick
review of the subject, click on Mini-Lessons. They will guide
you through the topics with graphics & examples. For additional
practice, click on Worksheet. All worksheet are in printable
format.
PBS
Teacher - Great detailed lessons for Functions&
Algebra that includes links to resources. | 677.169 | 1 |
ISBN 9788175154636
ISBN-10
8175154632
Binding
Paperback
Number of Pages
448 Pages
Language
(English)
Subject
Engineering & Technology
This book is developed as a self-contained, comprehensiveclassroom test for the third semester B.E. classes in CSE, ISE, MCA branches ofVTU. The book fully covers the syllabus of third semester. It can also be madeuseful for MCA classes of VTU. The matter of the book has been presented in asimple and lucid language. Groups and Coding Theory have also been discussed inthe book. The book covers the foundation of Modern Mathematics which is beingused by almost all branches of Science, Engineering and Economics. Chapters areexplained in detail and presented in a reader-friendly manner. | 677.169 | 1 |
Trigonometry
Customize your search:This course covers mathematical topics in trigonometry. Trigonometry is the study of triangle angles and lengths, but trigonometric functions have far reaching applications beyond simple studies of triangles. This course is designed to help prepare students to enroll for a first semester course in single variable calculus.
Learn basic engineering mathematics and how to apply basic mathematics to solve engineering problems. The goal of this mathematics course is to provide high school students and college freshmen an introduction to basic mathematics and especially show how mathematics is applied to solve fundamental engineering problems.
This course provides a brisk, entertaining treatment of differential and integral calculus, with an emphasis on conceptual understanding and applications to the engineering, physical, and social sciences. | 677.169 | 1 |
Clearly written and comprehensive, the eleventh edition of Gustafson and Hughes' popular book, COLLEGE ALGEBRA, provides in-depth and precise coverage ... more », incorporated into a framework of tested teaching strategy. The authors combine carefully selected pedagogical features and patient explanations to give readers a book that preserves the integrity of mathematics, yet does not discourage them with material that is confusing or too rigorous. Long respected for its ability to help learners quickly master difficult problems, this book also helps them develop the skills they'll need in future courses and in everyday life. Retaining the mathematical precision instructors have come to expect, the authors have focused on making this new edition more modern to better illustrate the importance of math in our world | 677.169 | 1 |
35 Years' Chapterwise Solved Papers (2013-1979) IIT JEE Mathematics, available in paperback, is a guide for aspirants preparing for the IIT JEE examination. But it also caters to the needs of all the students who are appearing for other competitive examinations as well.
The book contains solutions for all the questions from the question papers of the main and advanced level IIT JEE examinations from the past few years. The solution set is accurate and reliable and is aimed at helping students to figure out easier ways to solve the same problems. Different methods are provided that help them to tackle a concept from more than one perspective.
Most of the fundamental concepts in Mathematics that are necessary to crack the IIT JEE and other competitive examinations are explained in 35 Years' Chapterwise Solved Papers (2013-1979) IIT JEE Mathematics. Detailed overview is given and each concept is studied in depth with an aim of helping students to grasp the concepts in an effective manner.
The book deals with the complete syllabus of mathematics from class 11th and 12th. The question papers also give students an idea about the pattern they can expect in their examination. in addition to this, there are model papers and practice sets included to help students in analyzing their level of understanding and preparation. 35 Years' Chapterwise Solved Papers (2013-1979) IIT JEE Mathematics was published in the year 2013.
Product Description
About the Author
About the Author: Amit M. Agarwal is a well-known Indian author. Apart from this book, he has authored many other books like Objective Approach To Mathematics For JEE Main And Advanced And Other Engineering Entrances, A Textbook of Vectors And 3D Geometry For JEE Main And Advanced: Leading Edge Texts and Play With Graphs: A Magical Book Teaching You The Techniques of Problem Solving Through Graphs.
Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. | 677.169 | 1 |
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Developmental Mathematics Courses
While developmental mathematics courses certainly must focus on content and address mathematic deficiencies, faculty work with students to develop other "good-student" skills in order to prepare them for academic success. Attending class regularly, completing assignments thoroughly and on time, being an active learner, taking notes, etc. are all areas of focus. Classes meet two or three times each week and are limited to 20 or 24 students per section but generally, classes are much smaller. Student Academic Success Center faculty use a variety of instructional techniques for students who have had less than positive experiences with math in the past. Students are strongly encouraged to take advantage of the individual tutoring available at the Student Academic Success Center.
SAS 021 - Intro Algebra &Problem Solving
Credits: 3.00
The goal of this developmental level course is to develop proficiency in using algebra to communicate mathematically and to solve applied problems. Topics include solving equations, applying formulas, graphing, and operations with equalities, polynomials, exponents, and roots. Credits count toward full-time enrollment, but do not satisfy core curriculum or graduation requirements. Course grades are computed into students' cumulative grade point averages. SAS 021 is open to matriculated students only.
SAS 022 - Adv Algebra & Problem Solving
Credits: 3.00
The goal of this developmental course is to develop proficiency in using algebra to communicate mathematically, represent mathematical relationships, and solve applied problems. Topics include functions, exponents, solving and graphing linear and quadratic equations, rational and radical equations, inequalities, and systems of equations. Credits count toward full-time enrollment, but do not satisfy core curriculum or graduation requirements. Course grades are computed into students' cumulative grade point average. SAS 022 is open to matriculated students only.
SAS 040 - Chemistry Math Lab
Credits: 1.00
This developmental level course is designed to support General Chemistry I students who might otherwise struggle with the mathematics required for chemistry. This course provides reinforcement of in fall semester only.
College: No College Designated Division: Developmental Course Department: Student Support Services Pre-requisites: Concurrent enrollment in General Chemistry I
SAS 041 - Chemistry Math Prep
Credits: 1.00
This developmental level course is designed to support developmental mathematics students preparing to take General Chemistry I, students who did not successfully complete General Chemistry I because of difficulty with the mathematics required for the course, or students whose majors do not require chemistry until their second year of studies. The course provides an introduction to spring semester only. | 677.169 | 1 |
Comprehensive Linear Equation Review - 50 Problems - Ready to Print
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0.56 MB | 9 pages
PRODUCT DESCRIPTION
Do you need additional resources for your students? After years of teaching Algebra I have created this study guide that covers most of the basic topics covered in a linear equation section. This study guide has 6 sections that build upon each other to ensure that students are able to write equations in slope-intercept form.
This packet is ready to print and comes with complete and detailed answer key that could be shared with your students after they complete the assignment.
If you find this resource helpful I would appreciate any feedback you would be willing to leave. Please check out my other Algebra and Geometry resources if are looking for more printables for your classroom. Thanks for looking and happy teaching | 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 |
Sample records for algebraic approach based
In order to improve the degree of conceptual learning in our algebra-based general physics course, the second semester (of a two-semester sequence) has been taught in a nontraditional format during the past year. The key characteristics of this course were: 1) Intense and continuous use of interactive-engagement methods and cooperative learning; 2) coverage of less than half of the conventional number of topics, 3) heavy emphasis on qualitative questions as opposed to quantitative problems, 4) adjustment of the pacing of the course based on continuous (twice per week) formative assessment. The students enrolled in the course were relatively poorly prepared, with weak mathematical skills. Open-book quizzes stressing qualitative concepts in electricity and magnetism were given twice per week; most were given in "group quiz" format, allowing collaboration. Exams (also open-book) were all done individually. Most of the class time was taken up by quizzes, and by interactive discussion and group work related to quiz questions. New topics were not introduced until a majority of the class demonstrated competence in the topic under discussion. Despite lengthy and intensive focus on qualitative, conceptual questions and simple quantitative problems, only a small minority of the class ultimately demonstrated mastery of the targeted concepts. Frequent testing and re-testing of the students on basic concepts disclosed tenacious persistence of misconceptions.
In this article we describe a partially flipped Introductory Linear Algebra course developed by three faculty members at two different universities. We give motivation for our partially flipped design and describe our implementation in detail. Two main features of our course design are team-developed preview videos and related in-class activities.…
Various algebraicapproaches… controlled analysis of the effects of TBL on the student learning experience in the first year of TBL useThe study proves that a didactical model based in a method to solve word problems of increasing complexity which uses a numerical approach was essential to develop the analytical ability and the competent use of the algebraic language with students from three different performance levels in elementary algebra. It is shown that before using the… algebra algebraLinear algebra has become one of the most useful fields of mathematics since last decade, yet students still have trouble seeing the connection between some of the abstract concepts and real-world applications. In this article, we propose the use of thought-provoking questions in lesson designs to allow two-way communications between instructors…
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraicapproach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.approach in the sense that the earlier results are not used. However, this paper is not really independent because for many proofs, the reader is referred to the originalapproachA novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a… methodology which is explored widely in the literature and exploited in various commercial implementations. PMID:27289536
One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraicapproach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.
The present report introduces a novel module of the QuBiLS-MIDAS software for the distributed computation of the 3D Multi-Linear algebraic molecular indices. The main motivation for developing this module is to deal with the computational complexity experienced during the calculation of the descriptors over large datasets. To accomplish this task, a multi-server computing platform named T-arenal was developed, which is suited for institutions with many workstations interconnected through a local network and without resources particularly destined for computation tasks. This new system was deployed in 337 workstations and it was perfectly integrated with the QuBiLS-MIDAS software. To illustrate the usability of the T-arenal platform, performance tests over a dataset comprised of 15 000 compounds are carried out, yielding a 52 and 60 fold reduction in the sequential processing time for the 2-Linear and 3-Linear indices, respectively. Therefore, it can be stated that the T-arenal based distribution of computation tasks constitutes a suitable strategy for performing high-throughput calculations of 3D Multi-Linear descriptors over thousands of chemical structures for posterior QSAR and/or ADME-Tox studies. PMID:27490863
A simple liquid-liquid equilibrium (LLE) system involving a constant partition coefficient based on solute ratios is used to develop an algebraic understanding of multistage contacting in a first-year separation processes course. This algebraicapproach to the LLE system is shown to be operable for the introduction of graphical techniques…
Computer-Intensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings using computing tools for easy access to numerical, graphical, and symbolic representations of mathematical ideas. (MKR)As one of the most studied post-translational modifications (PTM), protein phosphorylation plays an essential role in almost all cellular processes. Current methods are able to predict and determine thousands of phosphorylation siteslation siteslation sitesThis article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…
We study the algebraic structure of an integrable Hubbard-Shastry type lattice model associated with the centrally extended su (2 | 2Based on the quantized universal enveloping (QUE) algebras, a quantization of the automorphism group of some nonassociative algebras is given in the formulation employing noncommuting matrix entries. A quantum group of G2 type included in this scheme is studied in detail in the Faddeev-Reshetikhin-Takhtajan (FRT) approach. Also in the formulation employing noncommuting matrix entries, the QUE-algebra of G2 type is reconstructed through the pairing induced by the R-matrix between the quantum group and the QUE-algebra.
In this paper, we study reasoning about actions following purpose of this research is to relax this assumption and establish the retailer`s inventory system as a cost minimization problem to determine the retailer`s optimal inventory cycle time. Then, an algebraicapproach is provided and an easy-to-use theorem is derived to efficiently determine the optimal cycle time. From the final numerical examples, result implies that the retailer will order less quantity to take the benefits of the permissible delay in payments more frequently when the larger the differences between the unit selling price per item and the unit purchasing price per item. certain application of algebraic system theory to the design of controllers for multivariable (MV) systems is explored analytically using an approachbased on state-space representations and matrix-fraction descriptions. Chapters are devoted to characteristic lambda matrices and canonical descriptions of MIMO systems; spectral analysis, divisors, and spectral factors of nonsingular lambda matrices; feedback control of MV systems; and structural decomposition theories and their application to MV control systems.
The present paper describes the empirical evaluation of a linear algebraapproach to model-based diagnosis, in case the behavioral model of the device under examination is described through a Petri net model. In particular, we show that algebraic analysis based on P-invariants of the net model, can significantly improve the performance of a model-based diagnostic system, while keeping the integrity of a general framework defined from a formal logical theory. A system called INVADS is described and experimental results, performed on a car fault domain and involving the comparison of different implementations of P-invariant based diagnosis, are then discussed.
An on-line algebraic filtering scheme, based on the recently introduced algebraicapproach to parameter and state estimation, is presented along with successful experimental results. The proposed filtering algorithm is based on the connections between a time derivative estimator and an algebraicallybased signal filtering option. The main advantages of the proposed approach are: (i) there are no appreciable delays in the filtered signal; (ii) the method does not require any statistical assessment of the noises corrupting the signal; (iii) high attenuation of the noise effects is achieved; (iv) the on-line computations are carried out in real time; and (v) high versatility and ease of implementation. Several experiments related to real depth measurements were conducted to show the effectiveness of the proposed algorithm. Comparisons are performed with different filtering alternatives.
We approach the relationship between classical and quantum theories in a new way, which allows both to be expressed in the same mathematical language, in terms of a matrix algebra in a phase space. This makes clear not only the similarities of the two theories, but also certain essential differences, and lays a foundation for understanding their relationship. We use the Wigner-Moyal transformation as a change of representation in phase space, and we avoid the problem of "negative probabilities" by regarding the solutions of our equations as constants of the motion, rather than as statistical weight factors. We show a close relationship of our work to that of Prigogine and his group. We bring in a new nonnegative probability function, and we propose extensions of the theory to cover thermodynamic processes involving entropy changes, as well as the usual reversible processes.
Using a calculus and an algebraicapproach, the Cartesian coordinates of the Fermat-Torricelli point are deduced for triangles with no internal angle greater than 120°. Although in theory, the deduction of these coordinates could be made 'by hand', in practice it is very laborious to obtain them without the aid of mathematical computer software, but with human guidance, since there are mathematical artifices not yet incorporated into the software. It is also shown that these coordinates can be conveniently expressed in terms of the side lengths and the area of the triangle. These coordinates are contrasted with the coordinates of a similar point: one whose sum of the squares of the distances to the vertices of an arbitrary triangle is a minimum.
The maritime environment poses a number of challenges for autonomous operation of surface boats. Among these challenges are the highly dynamic nature of the environment, the onboard sensing and reasoning requirements for obeying the navigational rules of the road, and the need for robust day/night hazard detection and avoidance. Development of full mission level autonomy entails addressing these challenges, coupled with inference of the tactical and strategic intent of possibly adversarial vehicles in the surrounding environment. This paper introduces PACIFIC (Process Algebra Capture of Intent From Information Content), an onboard system based on formal process algebras that is capable of extracting actions/activities from sensory inputs and reasoning within a mission context to ensure proper responses. PACIFIC is part of the Behavior Engine in CARACaS (Cognitive Architecture for Robotic Agent Command and Sensing), a system that is currently running on a number of U.S. Navy unmanned surface and underwater vehicles. Results from a series of experimental studies that demonstrate the effectiveness of the system are also presented.
The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions. PMID:15724067
The Mat-Rix-Toe project utilizes a matrix-based game to deepen students' understanding of linear algebra concepts and strengthen students' ability to express themselves mathematically. The project was administered in three classes using slightly different approaches, each of which included some editing component to encourage the…
The algebraic Riccati equation, which we denote by 'ARE' in the rest of the paper, is one of the most important equations of the post modern control theory. It plays important role for solving H 2 and H ∞ optimal control problems.
In this paper, is presented one software development and implementation of an algebraic method for XML data processing, which accelerates XML parsing process. Therefore, the proposed in this article nontraditional approach for fast XML navigation with algebraic tools contributes to advanced efforts in the making of an easier user-friendly API for XML transformations. Here the proposed software for XML documents processing (parser) is easy to use and can manage files with strictly defined data structure. The purpose of the presented algorithm is to offer a new approach for search and restructuring hierarchical XML data. This approach permits fast XML documents processing, using algebraic model developed in details in previous works of the same authors. So proposed parsing mechanism is easy accessible to the web consumer who is able to control XML file processing, to search different elements (tags) in it, to delete and to add a new XML content as well. The presented various tests show higher rapidity and low consumption of resources in comparison with some existing commercial parsers. algebrabased physics course. The subjects each filled out a survey at the beginning of the semester of their first semester of algebrabasedThis study explores pupils' performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from… digital multiplication by analog convolution algorithm, the bits of two encoded numbers are convolved to form the product of the two numbers in mixed binary representation; this output can be easily converted to binary. Attention is presently given to negative base encoding, treating base -2 initially, and then showing that the negative base system can be readily extended to any radix. In general, negative base encoding in optical linear algebra processors represents a more efficient technique than either sign magnitude or 2's complement encoding, when the additions of digitally encoded products are performed in parallel.
We present a computational approach for fast approximation of nonlinear tomographic reconstruction methods by filtered backprojection (FBP) methods authors have developed an object-based linear algebra package, currently with emphasis on sparse Krylov methods, driven primarily by needs of the Los Alamos National Laboratory parallel unstructured-mesh casting simulation tool Telluride. Support for a number of sparse storage formats, methods, and preconditioners have been implemented, driven primarily by application needs. They describe the object-based Fortran 90 approach, which enhances maintainability, performance, and extensibility, the parallelization approach using a new portable gather/scatter library (PGSLib), current capabilities and future plans, and present preliminary performance results on a variety of platforms.
This paper introduces a lattice algebra procedure that can be used for the multispectral analysis of historical documents and artworks. Assuming the presence of linearly mixed spectral pixels captured in a multispectral scene, the proposed method computes the scaled min- and max-lattice associative memories to determine the purest pixels that best represent the spectra of single pigments. The estimation of fractional proportions of pure spectra at each image pixel is used to build pigment abundance maps that can be used for subsequent restoration of damaged parts. Application examples include multispectral images acquired from the Archimedes Palimpsest and a Mexican pre-Hispanic codex. PMID:23385905
Holyoak and Koh (1987) and Holyoak (1984) propose four critical tasks for analogical transfer to occur in problem solving. A study was conducted to test this hypothesis by comparing a multiple components (MC) approach against worked examples (WE) in helping students to solve algebra word problems in chemistry classes. The MC approach incorporated…It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrödinger-Dirac picture of quantum mechanics on Hilbert spaces. In this picture, states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture, which has evolved in the Csstarf-algebraicapproach to quantum mechanics, starts with the algebra of observables and introduces states as a derived concept. The equivalence between these two pictures amounts, essentially, to the Gelfand-Naimark-Segal construction. In this construction, the abstract Csstarf-algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation that is defined may be reducible or irreducible, but in either case it allows us to identify a unitary group associated with the Csstarf-algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group; it also follows that quantum tomograms are strictly related with appropriate functions (positive-type) on the group. In this paper we present, using very simple examples, a tomographic description emerging from the set of ideas connected with the Csstarf-algebra picture of quantum mechanics. In particular, we introduce the tomographic probability distributions for finite and compact groups, and formulate an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state.This study examined the effects of types of representations, constructivist teaching approaches, and student engagement on middle school algebra students' procedural knowledge and conceptual understanding. Data gathered from 16 video lessons and algebra pretest/posttests were used to run three multilevel structural equation models. Symbolic…
This article reports the results of a study examining the utility of curriculum-based measurement (CBM) in algebra for predicting performance on a state math assessment and informing instructional placement decisions for students in seventh, eighth, and tenth grades. Students completed six Basic Skills algebra probes across different time…
This paper analyses the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is…
This quantitative quasi-experimental study examined the effect of brain-based instruction compared to teacher-centered instruction on student achievement in algebra I. A pre-test and post-test were given to a control group of 30 and experimental group of 42 before and after a unit if study in algebra I, which was taught using the two instructional…
Caselle-Ponzano fusion rules, based on graph theory, are investigated in the framework of Wigner-Racah algebras. While in the graph theory parameters of fusion rules are taken over specific values, the restriction of parameters is removed in the present formalism. Formal extension to q-analogs is also given.
We develop in this paper the principles of an associative algebraicapproach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed {gl(1|1)} spin-chain and its continuum limit—the {c=-2} symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245-288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289-329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL N , goes over in the continuum limit to a bigger algebra than {V}, the product of the left and right Virasoro algebras. This algebra, {S}—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field {S(z,bar{z})≡ S_{αβ} ψ^α(z)bar{ψ}^β(bar{z})}, with a symmetric form {S_{αβ}} and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL N in the {gl(1|1)} spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of {sp_{N-2}}. The semi-simple part of JTL N is represented by {U sp_{N-2}}, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL N image represented in the spin-chain. On the continuum side, simple modules over {S} are identified with "fundamental" representations of {sp_∞}.approach. All known N=4, 5 superconformal Chern-Simons matter theories can be recovered in the present superspace formulation for super-Lie algebra realization of symplectic 3-algebras.We discuss the semimicroscopic algebraic cluster model introduced recently, in which the internal structure of the nuclear clusters is described by the harmonic oscillator shell model, while their relative motion is accounted for by the Vibron model. The algebraic formulation of the model makes extensive use of techniques associated with harmonic oscillComputer algebra systems (CASs) have been used by thousands of teachers and students for teaching and learning algebra. They have the ability to perform efficiently almost all of the algebraic expansions and simplifications. Nevertheless, the traditional approach of using paper and pencil in acquiring procedural knowledge is still widely…
We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gr\\"obner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed).
The spatio-temporal data simulating Ice-Land-Ocean interaction of Antarctic are used to demonstrate the Clifford algebra-based data model construction, spatio-temporal query and data analysis. The results suggest that Clifford algebra provides a powerful mathematical tool for the whole modelling and analysis chains for complex geo-simulation data. It can also help implement spatio-temporal analysis algorithms more clearly and simply.
In this paper we present a new global image orientation approach for a set of multiple overlapping images with given homologous point tuples which is based on a two-step procedure. The approach is independent on initial values, robust with respect to outliers and yields the global minimum solution under relatively mild constraints. The first step of the approach consists of the estimation of global rotation parameters by averaging relative rotation estimates for image pairs (these are determined from the homologous points via the essential matrix in a pre-processing step). For the averaging we make use of algebraic group theory in which rotations, as part of the special orthogonal group SO(3), form a Lie group with a Riemannian manifold structure. This allows for a mapping to the local Euclidean tangent space of SO(3), the Lie algebra. In this space the redundancy of relative orientations is used to compute an average of the absolute rotation for each image and furthermore to detect and eliminate outliers. In the second step translation parameters and the object coordinates of the homologous points are estimated within a convex L∞ optimisation, in which the rotation parameters are kept fixed. As an optional third step the results can be used as initial values for a final bundle adjustment that does not suffer from bad initialisation and quickly converges to a globally optimal solution. We investigate our approach for global image orientation based on synthetic data. The results are compared to a robust least squares bundle adjustment. In this way we show that our approach is independent of initial values and more robust against outliers than a conventional bundle adjustment.
A highly structured and compact algebraic representation of grey-level images is expanded. Addition and multiplication are defined for the set of all grey-level images, which can then be described as polynomials of two variables. Utilizing this new algebraic structure, an innovative, efficient edge-detection scheme is devised. A robust method for linear feature extraction is developed-control system CAS. In the case of real numbers, the Archimedes-Cantor approach satisfies this requirement. The name of Archimedes brings back the exhaustion method. Cantor's name reminds us of the use of Cauchy rational sequences to represent real numbers. The usage of CAS with the Archimedes-Cantor approach enables the discussion of various representations of real numbers such as graphical, decimal, approximate decimal with precision estimates, and representation as points on a straight line. Exercises with numbers such as e, π, the golden ratio ϕ, and algebraic irrational numbers can help students better understand the real numbers. The Archimedes-Cantor approach also reveals a deep and close relationship between real numbers and continuity, in particular the continuity of functionsIn this paper we apply theWe focus on constructing a generic representation of Y( s o(3)) based on the Lie algebraic basis of s o(3) basis, and further developing transition of Yangian operator hat Y. As an application of Y( s o(3)), we calculate all the matrix elements of unit vector operators hat n in angular momentum basis. It is also discovered that the Yangian operator hat Y may work in quantum vector space. In addition, some shift operators hat {O}^{(± )}_{μ } are naturally built on the basis of the representation of Y( s o(3)). As an another application of Y( s o(3)), we can derive the CG cofficients of two coupled angular momenta from the down-shift operator hat {O}^{(-)}_{-1} acting on a s o(3)-coupled tensor basis. This not only explores that Yangian algebras can work in quantum tensor space, but also provides a novel approach to solve CG coefficients for two coupled angular momenta.
Superintendents from districts in the Minority Student Achievement Network (MSAN) challenged the Strategic Education Research Partnership (SERP) to identify an approach to narrowing the minority student achievement gap in Algebra 1 without isolating minority students for intervention. SERP partnered with 8 MSAN districts and researchers from 3…Loop Quantum Cosmology (LQC) is a tentative approach to model the universe down to the Planck era where quantum gravity settings are needed. The quantization of the universe as a dynamical spacetime is inspired by Loop Quantum Gravity (LQG) ideas. In addition, LQC could bridge contact with astronomical observations, and thus potentially investigate quantum cosmology modelings in the light of observations. To do so however, modeling both the background evolution and its perturbations is needed. The latter described cosmic inhomogeneities that are the main cosmological observables. In this context, we present the so-called deformed algebraapproach implementing the quantum corrections to the perturbed universe at an effective level by taking great care of gauge issues. We particularly highlight that in this framework, the algebra of hypersurface deformation receives quantum corrections, and we discuss their meaning. The primordial power spectra of scalar and tensor inhomogeneities are then presented, assuming initial conditions are set in the contracting phase preceding the quantum bounce and the well-known expanding phase of the cosmic history. These spectra are subsequently propagated to angular power spectra of the anisotropies of the cosmic microwave background. It is then shown that regardless of the choice for the initial conditions inside the effective approach for the background evolution (except that they are set in the contracting phase), the predicted angular power spectra of the polarized B-modes exceed the upper bound currently set by observations. The exclusion of this specific version of LQC establishes the falsifiability of the approach, though one shall not conclude here that either LQC or LQG excluded. PMID:22572100This dissertation analyzes differences found in Spanish-speaking middle school and high school students in algebra-based problem solving. It identifies the accuracy differences between word problems presented in English, Spanish and numerically based problems. The study also explores accuracy differences between each subgroup of Spanish-speaking…
Starting from the four DNA bases bases. The new algebraic structure shows strong connections workThe algebra-based physics course is taken by Biology students, Pre-Pharmacy, Pre-Medical, and other health related majors such as medical imaging, physical therapy, and so on. Nearly 500 students take the course each Semester. Student learning is adversely impacted by poor math backgrounds as well as extensive work schedules outside of the classroom. We have been researching the use of an intensive flipped-classroom approach where students spend one to two hours each week preparing for class by reading the book, completing a series of conceptual problems, and viewing videos which describe the material. In class, the new response system Learning Catalytics is used which allows much richer problems to be posed in class and includes sketching figures, numerical or symbolic entries, short answers, highlighting text, etc in addition to the standard multiple choice questions. We make direct comparison of student learning for 1200 sudents who have taken the same tests, 25% of which used the flipped classroom approach, and 75% who took a more standard lecture. There is significant evidence of improvements in student learning for students taking the flipped classroom approach over standard lectures. These benefits appear to impact students at all math backgrounds consider an approximation to the two-body spinless Salpeter equation which is valid for the case of heavy quarks with the Cornell potential. We then replace the square of kinetic term with its nonrelativistic equivalent and obtain an equation which can be alternatively viewed as the generalization of the Schrödinger equation into the relativistic regime. In the calculations, we use the Lie algebraicapproach within the framework of quasi-exact solvability. With the help of the representation theory of sl(2) , the ( n+1 -dimensional matrix equation of the problem is constructed in a quite detailed manner and thereby the quasi-exact expressions for the energy eigenvalues and the corresponding wave functions as well as the allowed values of the potential parameters are obtained.
The purposeThe purposeA new Clifford algebra-based vector field filtering method, which combines amplitude similarity and direction difference synchronously, is proposed. Firstly, a modified correlation product is defined by combining the amplitude similarity and direction difference. Then, a structure filtering algorithm is constructed based on the modified correlation product. With custom template and thresholds applied to the modulus and directional fields independently, our approach can reveal not only the modulus similarities but also the classification of the angular distribution. Experiments on exploring the tempo-spatial evolution of the 2002-2003 El Niño from the global wind data field are used to test the algorithm. The results suggest that both the modulus similarity and directional information given by our approach can reveal the different stages and dominate factors of the process of the El Niño evolution. Additional information such as the directional stability of the El Niño can also be extracted. All the above suggest our method can provide a new powerful and applicable tool for geophysical vector field analysis.
All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraicapproachAlgebraic grid generation methods based on transfinite interpolation called the two-boundary and four-boundary methods are applied for generating grids with highly complex boundaries. These methods yield grid point distributions that allow for accurate application to regions of sharp gradients in the physical domain or time-dependent problems with small length scale phenomena. Algebraic grids are derived using the two-boundary and four-boundary methods for applications in both two- and three-dimensional domains. Grids are developed for distinctly different geometrical problems and the two-boundary and four-boundary methods are demonstrated to be applicable to a wide class of geometries.
Background The inference of gene regulatory networks (GRNs) from experimental observations is at the heart of systems biology. This includes the inference of both the network topology and its dynamics. While there are many algorithms available to infer the network topology from experimental data, less emphasis has been placed on methods that infer network dynamics. Furthermore, since the network inference problem is typically underdetermined, it is essential to have the option of incorporating into the inference process, prior knowledge about the network, along with an effective description of the search space of dynamic models. Finally, it is also important to have an understanding of how a given inference method is affected by experimental and other noise in the data used. Results This paper contains a novel inference algorithm using the algebraic framework of Boolean polynomial dynamical systems (BPDS), meeting all these requirements. The algorithm takes as input time series data, including those from network perturbations, such as knock-out mutant strains and RNAi experiments. It allows for the incorporation of prior biological knowledge while being robust to significant levels of noise in the data used for inference. It uses an evolutionary algorithm for local optimization with an encoding of the mathematical models as BPDS. The BPDS framework allows an effective representation of the search space for algebraic dynamic models that improves computational performance. The algorithm is validated with both simulated and experimental microarray expression profile data. Robustness to noise is tested using a published mathematical model of the segment polarity gene network in Drosophila melanogaster. Benchmarking of the algorithm is done by comparison with a spectrum of state-of-the-art network inference methods on data from the synthetic IRMA network to demonstrate that our method has good precision and recall for the network reconstruction task, while also
Seventy Cleveland, Ohio, inner city adult illiterates, 33 from an experimental group and 37 from a contrast group, were studied to determine the efficiency and effectiveness of Words in Color or the Morphologico-Algebraicapproach to teaching reading. Results indicated that the reading achievement gain of functionally illiterate adults taught by…The m-shifting generators of su(2) algebra together with a pair of l-shifting ladder symmetry operators have been used in the space of all spherical harmonics Ylm(θ,ϕ) in order to introduce a new set of operators, expressing the transitions between them. It is shown that the space of spherical harmonics whose l +2m or l -2m is given presents negative and positive irreducible representations of a deformed su(1,1) algebra, respectively. These internal symmetries also suggest new algebraic methods to construct the spherical harmonics in the framework of the spectrum-generating algebras.
Examined student performance in distance computer-based calculus and linear algebra courses offered by Stanford University to pre-college students as part of their Education Program for Gifted youth (EPGY). Puts special emphasis on modeling student performance over time and on capturing long-term trend effects using stochastic and nonlinear…
Representation bases of unitary parasupersymmetry algebra of arbitrary order p is constructed by some one-dimensional models which are shape invariant with respect to the main quantum number n. Consequently, the isospectral Hamiltonians and their exact solutions are obtained as labelled by the main quantum number n. were collected from the North Carolina…
Lack of effort is a major reason for non-success among students in remedial mathematics, and the presence of a reward or an incentive may increase motivation in this learner group. A survey investigated the attitudes of 421 students toward incentive options in a competency-based remedial algebra course at Iowa State University. Students rated the…
In this paper, we study the Dirac equation with spin and pseudospin symmetry by the quadratic algebraapproachCollege algebra fulfills general education requirements at many colleges in the United States. The study reported here investigated differences in mathematics achievement between undergraduates in college algebra classes using one of two homework methods: "WeBWorK," an open-source system for web-based homework, or traditional…
In this article, we report on a semester-long study of the incorporation into our general chemistry course, of advanced algebraic and computer algebra techniques for solving chemical equilibrium problems. The method presented here is an alternative to the commonly used concentration table method for describing chemical equilibria in general…
Many students with learning disabilities (LD) in mathematics receive their mathematics education in general education inclusive classes; therefore, these students must be capable of learning algebraic concepts, including developing algebraic thinking abilities, that are part of the general education curriculum. To help students develop algebraic…
Transferring existing active classroom educational efforts to new faculty is a challenge that must be met to ensure sustainability of changes. We describe a flipped class approach to teaching algebra-based Physics being transferred to a new faculty member. This flipped class includes extensive video and reading-based preparation materials outside of class, and the use of Learning Catalytics for in-class work is developed and tested by one of the authors. These materials are of course idiosyncratic to the style of the developer. Student results using the new materials are compared with students in more standard classes which suggest significant positive benefit over several years. A faculty member decided to use these materials in his own section of the same course. Our experience shows that it takes some time for the new faculty member to use and adapt the materials in a way which matches his own style, which in the end results in equivalently enhanced results. Lessons learned from this transfer process will be discussed. We acknowledge the financial support of the NSF through DUE 1544001 and 1431350.
Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)
The renewed interest in siting new nuclear power plants in the United States has brought to the center stage, the need to site interim facilities for long-term management of spent nuclear fuel (SNF). In this paper, a two-stage approach for identifying potential areas for siting interim SNF facilities is presented. In the first stage, the land area is discretized into grids of uniform size (e.g., 100m x 100m grids). For the continental United States, this process resulted in a data matrix of about 700 million cells. Each cell of the matrix is then characterized as a binary decision variable to indicate whether an exclusion criterion is satisfied or not. A binary data matrix is created for each of the 25 siting criteria considered in this study. Using map algebraapproach, cells that satisfy all criteria are clustered and regarded as potential siting areas. In the second stage, an optimization problem is formulated as a p-median problem on a rail network such that the sum of the shortest distance between nuclear power plants with SNF and the potential storage sites from the first stage is minimized. The implications of obtained results for energy policies are presented and discussedWe construct a topological sigma model and a current algebrabased on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.
This article presents features of a qualitative research study concerning the teaching and learning of school algebra using a function-basedapproach in a grade 8 class, of 23 students, in 26 lessons, in a state school of Athens, in the school year 2003-2004. In this article, we are interested in the inequality concept and our aim is to…
The development of a multicellular organism is a dynamic process. Starting with one or a few cells, the organism develops into different types of cells with distinct functions. We have constructed a simple model by considering the cell number increase and the cell-type order conservation, and have assessed conditions for cell-type diversity. This model is based on a stochastic Lindenmayer system with cell-to-cell interactions for three types of cells. In the present model, we have successfully derived complex but rigorous algebraic relations between the proliferation and transition rates for cell-type diversity by using a symbolic method: quantifier elimination (QE). Surprisingly, three modes for the proliferation and transition rates have emerged for large ratios of the initial cells to the developed cells. The three modes have revealed that the equality between the development rates for the highest cell-type diversity is reduced during the development process of multicellular organisms. Furthermore, we have found that the highest cell-type diversity originates from order conservation. PMID:17293029 approachesMethod of computing dynamics of multiple-flexible-link robotic manipulators based on spatial-operator algebra, which originally applied to rigid-link manipulators. Aspects of spatial-operator-algebraapproach…Florida Atlantic University implemented a number of changes in the College Algebra course in an attempt to improve student success. We summarize these changes, and how they have affected the course. We also discuss possibilities for future improvements. (Contains 3 figures and 1 footnote.)
The present study explores students' abilities in conversions between geometric and algebraic representations, in problem-solving situations involving the concept of "limit" and the interrelation of these abilities with students' constructed understanding of this concept. An attempt is also made to examine the impact of the "didactic contract" on…
In this paper, we present the results of an experimental classroom activity, history-based with a phylogenetic approach, to achieve algebra properties through geometry. In particular, we used Euclidean propositions, processed them by a dynamic geometry system and translate them into algebraic special products.
We present a symbolic approachbased on matrix methods that allows for the analysis and computation of intraocular lens power following cataract surgery. We extend the basic matrix approach corresponding to paraxial optics to include astigmatism and other aberrations. The symbolic approach allows for a refined analysis of the potential sources of errors ("refractive surprises"). We demonstrate the computation of lens powers including toric lenses that correct for both defocus (myopia, hyperopia) and astigmatism. A specific implementation in Mathematica allows an elegant and powerful method for the design and analysis of these intraocular lenses.
The aim of the present paper is to provide a metalogical systematisation in the area of deontic action logic based on Boolean algebra. Differences among the systems in question lie in two aspects: the level of closedness of a deontic action logic and the possibility of performing no action at all. It is also shown that the existing definitions of obligation in those systems are not acceptable due to their unintuitive interpretation or paradoxical consequences. As a solution we propose an axiomatic characterisation of obligation with an adequate class of models.
A multi-customer finite production rate (FPR) model with quality assurance and discontinuous delivery policy was investigated in a recent paper (Chiu et al. in J Appl Res Technol 12(1):5-13, 2014) using differential calculus approach. This study employs mathematical modeling along with a two-phase algebraic method to resolve such a specific multi-customer FPR model. As a result, the optimal replenishment lot size and number of shipments can be derived without using the differential calculus. Such a straightforward method may assist practitioners who with insufficient knowledge of calculus in learning and managing the real multi-customer FPR systems more effectively. PMID:27186457 three-dimensional universal complex Clifford algebra Cbar3,0 is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.
An algorithm is presented for the symbolic integration of a class of algebraic functions. This class consists of functions made up of rational expressions of an integration variable x and square roots of polynomials, trigonometric and hyperbolic functions of x. The algorithm is shown to consist of the following components:(1) the reduction of input integrands to conical form; (2) intermediate internal representations of integrals; (3) classification of outputs; and (4) reduction and simplification of outputs to well-known functions.
We study a relativistic quantum particle in cosmic string algebrabased successes and failures of research based courses such as the introductory calculus-based physics course for biologists at Cal State San Marcos.
In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. The experimental results show that this method has a large advantage over the classical methods on geometric intuition, computation and real-time, and can be directly extended to all serial robot manipulators and completely automatized, which provides a new tool on the analysis and application of general robot manipulators. PMID:23918347
The refined topological vertex of Iqbal—Kozçaz—Vafa has been investigated from the viewpoint of the quantum algebra of type W1+∞ by Awata, Feigin, and Shiraishi. They introduced the trivalent intertwining operator Φ which is normal ordered along with some prefactors. We manage to establish formulas from the infinite operator product of the vertex operators and the generalized ones to restore this prefactor, and obtain an explicit formula for the vertex realization of the topological vertex as well as the refined topological vertex.Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)
We share results from a quasi-experimental study in which we compared achievement between traditional face-to-face and computer-based sections of Intermediate Algebra on a common multiple choice exam as well as performance on open-response tasks. Students in the computer-based group performed better on the final exam and were also more likely to…
The physics of light interference experiments is well established for nematic liquid proposedSmoking problem is considered as one of the hot topics for many years. In spite of overpowering facts about the dangers, smoking is still a bad habit widely spread and socially accepted. Many people start smoking during their gymnasium period. The discovery of the dangers of smoking gave a warning sign of danger for individuals. There are different statistical methods used to analyze the dangers of smoking. In this study, we apply an algebraic statistical method to analyze and classify real data using Markov basis for the independent model on the contingency table. Results show that the Markov basis based classification is able to distinguish different date elements. Moreover, we check our proposed method via information theory by utilizing the Shannon formula to illustrate which one of these alternative tables is the best in term of independent. PMID:26858555
We report our experience in using the IPAD as a pedagogical tool for enhancing physics learning in an introductory algebra-based physics laboratory course for primarily pre-med students. We used several applications including (1) video analysis for experiments in accelerated motion (2) virtual oscilloscope for studying wave motion and circuit response to low frequency driving voltages; (3) applications for visualization of electric fields and magnetic fields. We compare student responses to this platform versus more traditional experiments. Using student surveys and polls. We also evaluate the IPAD as a new and familiar interface versus traditional interfaces like the standard oscilloscope. We report on the advantages and disadvantages of using this mobile, popular platform in delivering experimental physics content and promoting student engagement.
Many world interpretations (MWIs) of quantum mechanics avoid the measurement problem by considering every term in the quantum superposition as actual. A seemingly opposed solution is proposed by modal interpretations (MIs) which state that quantum mechanics does not provide an account of what 'actually is the case', but rather deals with what 'might be the case', i.e., with possibilities. In this paper we provide an algebraic framework which allows us to analyze in depth the modal aspects of MWI. Within our general formal scheme we also provide a formal comparison between MWI and MI, in particular, we provide a formal understanding of why - even though both interpretations share the same formal structure - MI fall pray of Kochen-Specker-type contradictions while MWI escape them.
The motion of an electric charge in the field of a magnetic monopole is described by means of a Lagrangian model written in terms of the Clifford algebra of the physical space. The equations of motion are written in terms of a radial equation (involving r=| r|, where r( t) is the charge trajectory) and a rotor equation (written in terms of an unitary operator spinor R). The solution corresponding to the charge trajectory in the field of a magnetic monopole is given in parametric form. The model can be generalized in order to describe the motion of a charge in the field of a magnetic monopole and other additional central forces, and as an example, we discuss the classical ones involving linear and inverse square interactions.
During the last decade, the field of quantum computation has attracted a lot of interest and motivated many theoretical and experimental studies of n-qubit quantum systems. But apart from the promise of more efficient quantum algorithms, these investigations also revealed a number of obstacles which still have to be overcome in practice. In this context, the use of simulation programs has proved to be an appropriate method. In order to facilitate the simulation of n-qubit quantum systems, we present the Feynman software program to provide the necessary tools to define and to deal with quantum registers as well as the operators acting on them. Using an interactive design within the framework of the computer algebra system Maple, we hope that the Feynman software program will be useful not only for teaching the basic elements of quantum computing but also for studying their physical realization in the future.
In the last decade we have elaborated a mathematical tool for the description of the interaction of polarized light with polarization devices, alternative to the standard matrix (Jones and Mueller) formalisms, namely a vectorial pure operatorial Pauli algebraic one. After a brief, coherent survey of this formalism, we present some applicative results obtained in this frame, referring to the gain and the modification of the state of polarization at the interaction of the polarized light with deterministic devices. Due to an adequate parameterization of the problem, specific to this method, symmetric expressions of the gain and of the generalized Malus' law are obtained. On the other hand, the equation of the ellipsoid in which a Poincaré sphere of a given degree of polarization is mapped by such a device can be established. PMID:22505099
We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry-Levy-Leblond classification.
An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible representations with naturally defined generalized coherent states. We have selected the Lie algebra \\mathfrak{su}(1,1) , because this algebra possesses the necessary characteristics to realize the hypercomputation and also because such algebra has been identified as the dynamical algebra associated with many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra \\mathfrak{su}(1,1) , we presented an adaptation of KHQA over some concrete physical referents: the infinite square well, the infinite cylindrical well, the perturbed infinite cylindrical well, the Pöschl Teller potentials, the Holstein Primakoff system and the Laguerre oscillator. We conclude that it is possible to have many physical systems within condensed matter and quantum optics in which it is possible to consider an implementation of KHQA.
This note demonstrates multiple representations (numerical and graphical) of spreadsheet-based quadratic sequences together with prospective teachers' pencil-paper transformations of these numerical sequences into a corresponding symbolization as algebraic expressions. With the majority of prospective teachers, the experience of school mathematicsIn the current paper, a nonlinear differential algebraicapproach is proposed for the modeling of hysteretic dynamics of polycrystalline ferromagnetic materials. The model is constructed by employing a phenomenological theory to the magnetization orientation switching. For the modeling of hysteresis in polycrystalline ferromagnetic materials, the single crystal model is applied to each magnetic domain along its own principal axis. The overall dynamics of the polycrystalline materials is obtained by taking a weighted combination of the dynamics of all magnetic domains. The weight function for the combination is taken as the distribution function of the principal axes. Numerical simulations are performed and comparisons with its experimental counterparts are presented. The hysteretic dynamics caused by orientation switching processes is accurately captured by the proposed model. Minor hysteresis loops associated with partial-amplitude loadings are also captured. Rate dependence of the hysteresis loops are inherently incorporated into the model due to its differential nature.
Quantification of magnetization-transfer (MT) experiments are typically based on the assumption of the binary spin-bath model. This model allows for the extraction of up to six parameters (relative pool sizes, relaxation times, and exchange rate constants) for the characterization of macromolecules, which are coupled via exchange processes to the water in tissues. Here, an approach is presented for estimating MT parameters acquired with arbitrary saturation schemes and imaging pulse sequences. It uses matrix algebra to solve the Bloch-McConnell equations without unwarranted simplifications, such as assuming steady-state conditions for pulsed saturation schemes or neglecting imaging pulses. The algorithm achieves sufficient efficiency for voxel-by-voxel MT parameter estimations by using a polynomial interpolation technique. Simulations, as well as experiments in agar gels with continuous-wave and pulsed MT preparation, were performed for validation and for assessing approximations in previous modeling approaches. In vivo experiments in the normal human brain yielded results that were consistent with published data.This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is time-discrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in which variables have only two possible states. The use of a larger number of possible states allows a finer discretization of experimental data and more than one possible mode of action for the variables, depending on threshold values. Furthermore, with a suitable choice of state set, one can employ powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to meso-scale networks rather than networks on a genome-wide scale. An analysis of the complexity of the algorithm is performed. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster. PMID:15246788
Quantitatively delineating multiple source inputs is a common but often difficult task in geological and environmental researches. Binary mixing model works well for simple two-component mixing problems when the two end members can be accurately defined. However, real world mixing problems usually involve more than two source components, and each component may have many composite profiles. Therefore, binary model is inadequate to provide an accurate estimate of contribution of all the end member sources. In such cases, linear combination model by solving linear algebra equations will have to be developed. In this study, we demonstrate the application of the linear algebra method to accurately calculate the percentage inputs of leaf waxes from different plant groups to lake sediments of Blood Pond and Rocky Pond in Massachusetts. We systematically sampled 28 species, including 6 tree species, 6 grass species, 7 emergent plants, 4 floating and 3 submerged aquatic plants around Blood Pond as well as 6 sediments samples of different depth. The distributions of these plants leaf waxes (n-alkanes and n-alkanoic acids) show distinctively different patterns, but also significant overlaps. To accurately quantify the percentage source inputs, we develop the 4-end member linear combination model. As we have an equation for each of the total 6 n-alkane compounds we use (C23 alkane ~C33 alkane), a 'least square fitting' is applied to solved the overdetermined linear system. Our results show that 91.7% of the total C23 alkane in sediments is derived from floating and submerged plants. We also applied our approach to Rocky Pond in southeastern Massachusetts and found >92 % of mid chain n-alkyl lipids are contributed by floating and submerged plants. Our results provide critical basis for using D/H ratios of mid-chain n-alkyl lipids (such as behenic acid) in lake sediments for reconstructing lake water D/H ratios.
This paper discusses student difficulties in grasping concepts from Linear Algebra--in particular, the connection of eigenvalues and eigenvectors to other important topics in linear algebra. Based on our prior observations from student interviews, we propose technology-enhanced instructional approaches that might positively impact studentIt is a bitter reality that the curricula and traditional pedagogy prevailing in public schools of Pakistan in general and Sindh in particular do not incorporate the algebraic concepts properly. Both the content and the presentation therein cannot be considered up to the mark, thereby making "Algebra" a tough and dry subject. This…
I will describe methods used at the University of Cincinnati to enhance student success in an algebra-based physics course. The first method is to use ALEKS, an adaptive online mathematics tutorial engine, before the term begins. Approximately three to four weeks before the beginning of the term, the professor in the course emails all of the students in the course informing them of the possibility of improving their math proficiency by using ALEKS. Using only a minimal reward on homework, we have achieved a 70% response rate with students spending an average of 8 hours working on their math skills before classes start. The second method is to use a flipped classroom approach. The class of 135 meets in a tiered classroom twice per week for two hours. Over the previous weekend students spend approximately 2 hours reading the book, taking short multiple choice conceptual quizzes, and viewing videos covering the material. In class, students use Learning Catalytics to work through homework problems in groups, guided by the instructor and one learning assistant. Using these interventions, we have reduced the student DWF rate (the fraction of students receiving a D or lower in the class) from an historical average of 35 to 40% to less than 20%.
For several years measures of student performance have been investigated in the first semester of the two-semester algebra-based physics course. Preand post-tests using the Force Concept Inventory (FCI) and course grades were used as measures of student performance and preparation. In addition, information on high school preparation in physics and mathematics, student demographics (college major, classification, SAT math scores, etc.), and grades earned were collected. Results will be presented that indicate high school physics preparation, college major, and SAT math scores strongly correlate with grade earned, FCI pre-instruction scores, and FCI normalized gains. Using the current investigation and institutional data from the work of Coletta and Phillips1, it will be shown that a strong correlation exists between FCI pre-instruction scores and normalized gain. 1. V.P. Coletta and J.A. Phillips, Am. J. Phys 73, 1172 (2005). *Work sponsored by PhysTEC and supported by grants from the National Science Foundation and FIPSE.
Enhancing the well known and widely used map algebra proposed by Dr. Charles Dana Tomlin [1] with the time dimension is an ongoing research topic. The efficient processing of large time series of raster, 3D raster and vector datasets, e. g. raster datasets for temperature or precipitations on continental scale, requires a sophisticated spatio-temporal algebra that is capable of handling datasets with different temporal granularities and spatio-temporal extents. With the temporal enabled GRASS GIS [2] and the GRASS GIS Temporal Framework new spatio-temporal data types are available in GRASS GIS 7, called space time datasets. These space time datasets represent time series of raster, 3D raster and vector map layers. Furthermore the temporal framework provides a wide range of functionalities to support the implementation of a temporal algebra. While spatial capabilities of GRASS GIS are used to perform the spatial processing of the time stamped map layers that are registered in a space time dataset, the temporal processing is provided by the GRASS GIS temporal framework that supports time intervals and time instances. Mixing time instance and time intervals as well as gaps, overlapping or inclusion of intervals and instances is possible. Hence this framework allows an arbitrary layout of the time dimension. We implemented two ways to process space time datasets with arbitrary temporal layout, the temporal topology and the granularity based spatio-temporal algebra. The algebra provides the functionality to define complex spatio-temporal topological operators that process time and space in a single expression. The algebra includes methods to select map layers from space time datasets based on their temporal relations, to temporally shift time stamped map layers, to create temporal buffer and to snap time instances of time stamped map layers to create a valid temporal topology. In addition spatio-temporal operations can be evaluated within conditional statements. These
The initial development of a testlet-basedalgebra test was previously reported (Wainer and Lewis, 1990). This account provides the details of this excursion into the use of hierarchical testlets and validity-based scoring. A pretest of two 15-item hierarchical testlets was carried out in which examinees' performance on a 4-item subset of each…
The history of Current Algebra is reviewed up to the appearance of the Adler-Weisberger sum rule. Particular emphasis is given to the role of current algebra in the historical struggle in strong interaction physics of elementary particles between field theory and the S-matrix approachbased on dispersion relations. The question as to whether some particles are truly fundamental or all hadrons are bound or resonant states of one another played an important role in this struggle and is thus also regarded.
In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraicapproach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
In this work, we consider a D-dimensional ( β, β^' -two-parameters deformed Heisenberg algebra, which was introduced by Kempf et al. The angular-momentum operator in the presence of a minimal length scale based on the Kempf-Mann-Mangano algebra is obtained in the special case of β^' = 2β up to the first order over the deformation parameter β . It is shown that each of the components of the modified angular-momentum operator, commutes with the modified operator {L}2 . We find the magnetostatic field in the presence of a minimal length. The Zeeman effect in the deformed space is studied and also Lande's formula for the energy shift in the presence of a minimal length is obtained. We estimate an upper bound on the isotropic minimal length.
Tablets such as the iPad represent the next iteration of technologies that hold promise to facilitate learning, particularly in mathematics. In the case of algebra, tablets have the potential to bring the curriculum to life by easily linking to supporting materials and they allow an interactive experience whereby manipulation of one type of…
The electronic medium suitable for mathematics learning and teaching is often designed with a notebook interface provided in a computer algebra system. Such a notebook interface facilitates a workspace for mathematical activities along with an online help system. In this paper, the proposed feature is implemented in the Mathematica's notebook…
We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraicapproach oscillWe describe a novel interface-sharpening approach for efficient numerical resolution of a compressible homogeneous two-phase flow governed by a quasi-conservative five-equation model of Allaire et al. (2001) [1]. The algorithm uses a semi-discrete wave propagation method to find approximate solution of this model numerically. In the algorithm, in regions near the interfaces where two different fluid components are present within a cell, the THINC (Tangent of Hyperbola for INterface Capturing) scheme is used as a basis for the reconstruction of a sub-grid discontinuity of volume fractions at each cell edge, and it is complemented by a homogeneous-equilibrium-consistent technique that is derived to ensure a consistent modeling of the other interpolated physical variables in the model. In regions away from the interfaces where the flow is single phase, standard reconstruction scheme such as MUSCL or WENO can be used for obtaining high-order interpolated states. These reconstructions are then used as the initial data for Riemann problems, and the resulting fluctuations form the basis for the spatial discretization. Time integration of the algorithm is done by employing a strong stability-preserving Runge-Kutta method. Numerical results are shown for sample problems with the Mie-Grüneisen equation of state for characterizing the materials of interests in both one and two space dimensions that demonstrate the feasibility of the proposed method for interface-sharpening of compressible two-phase flow. To demonstrate the competitiveness of our approach, we have also included results obtained using the anti-diffusion interface sharpening method.
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically basedalgebraThe skull base surgery is one of the most demanding surgeries. There are different structures that can be injured easily, by operating in the skull base. It is very important for the neurosurgeon to choose the right approach in order to reach the lesion without harming the other intact structures. Due to the pioneering work of Cushing, Hirsch, Yasargil, Krause, Dandy and other dedicated neurosurgeons, it is possible to address the tumor and other lesions in the anterior, the mid-line and the posterior cranial base. With the transsphenoidal, the frontolateral, the pterional and the lateral suboccipital approach nearly every region of the skull base is exposable. In the current state many different skull baseapproaches are described for various neurosurgical diseases during the last 20 years. The selection of an approach may differ from country to country, e.g., in the United States orbitozygomaticotomy for special lesions of the anterior skull base or petrosectomy for clivus meningiomas, are found more frequently than in Europe. The reason for writing the review was the question: Are there keyhole approaches with which someone can deal with a vast variety of lesions in the neurosurgical field? In my opinion the different surgical approaches mentioned above cover almost 95% of all skull base tumors and lesions. In the following text these approaches will be described. These approaches are: 1) pterional approach 2) frontolateral approach 3) transsphenoidal approach 4) suboccipital lateral approach These approaches can be extended and combined with each other. In the following we want to enhance this philosophy. PMID:20602753
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
One of the hallmarks of linear coupling is the resonant exchange of oscillation amplitude between the horizontal and vertical planes when the difference between the unperturbed tunes is close to an integer. The standard derivation of this phenomenon (known as the difference resonance) can be found, for example, in the classic papers of Guignard [1, 2]. One starts with an uncoupled lattice and adds a linear perturbation that couples the two planes. The equations of motion are expressed in hamiltonian form. As the difference between the unperturbed tunes approaches an integer, one finds that the perturbing terms in the hamiltonian can be divided into terms that oscillate slowly and ones that oscillate rapidly. The rapidly oscillating terms are discarded or transformed to higher order with an appropriate canonical transformation. The resulting approximate hamiltonian gives equations of motion that clearly exhibit the exchange of oscillation amplitude between the two planes. If, instead of the hamiltonian, one is given the four-by-four matrix for one turn around a synchrotron, then one has the complete solution for the turn-by-turn (TBT) motion. However, the conditions for the phenomenon of amplitude exchange are not obvious from a casual inspection of the matrix. These conditions and those that give rise to the related sum resonance are identified in this report. The identification is made using the well known formalism of Edwards and Teng [3, 4, 5] and, in particular, the normalized coupling matrix of Sagan and Rubin [6]. The formulae obtained are general in that no particular hamiltonian or coupling elements are assumed. The only assumptions are that the one-turn matrix is symplectic and that it has distinct eigenvalues on the unit circle in the complex plane. Having identified the conditions of the one-turn matrix that give rise to the resonances, we focus on the difference resonance and apply the formulae to the evolution of the horizontal and vertical emittancesDescribes how to use homemade tiles, sketches, and the box method to reach a broader group of students for successful algebra learning. Provides a list of concepts appropriate for such an approach. (KHR)We show how to obtain the necessary structures for statistical analysis of the folllowing orthogonal models Y˜(1 μ +∑i Xiβi ,∑j σj2Mj+σ2I ) . These structures rely on the existence of Jordan algebras, in the sequence of [24], [8], [12], [9], [5] and [10].
A significant hurdle confronts the software reuser attempting to select candidate components from a software repository - discriminating between those components without resorting to inspection of the implementation(s). We outline a mixed classification/axiomatic approach to this problem based upon our lattice-based faceted classification technique and Guttag and Horning's algebraic specification techniques. This approach selects candidates by natural language-derived classification, by their interfaces, using signatures, and by their behavior, using axioms. We briefly outline our problem domain and related work. Lattice-based faceted classifications are described; the reader is referred to surveys of the extensive literature for algebraic specification techniques. Behavioral support for reuse queries is presented, followed by the conclusions.
Presents methods for using literature to develop algebraic thinking in an environment that connects algebra to various situations. Activities are based on the book "Anno's Magic Seeds" with additional resources listed. Students express a constant function, exponential function, and a recursive function in their own words as well as writing about…
The purpose of this study was to understand how students' attitudes were connected to their mathematics learning. This investigation was specific to web-based homework in developmental courses in the community college environment. The mixed-methods approach was used to analyze the relationship between students' attitudes and mathematical…
This note describes some experiments on simple, dense linear algebra algorithms. These experiments show that the CRAY X-MP is capable of small-grain multitasking arising from standard implementations of LU and Cholesky decomposition. The implementation described here provides the ''fastest'' execution rate for LU decomposition, 718 MFLOPS for a matrix of order 1000.We introduce an alternative wayFor most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…In this work, we discuss algebraic and analytic approaches for dynamic tomography. We present a framework of dynamic tomography for both algebraic and analytic approaches. We finally present numerical experiments. PMID:18002059This project supported the creation of a computer-based interactive text for linear algebra using guided discovery in a laboratory-based course which emphasized active learning, collaborative learning, and the use of writing. This pedagogical approach had as its goal improved student understanding and retention of the concepts and methods ofPresented is an alternative approach to the traditional teaching of Boolean algebra for secondary school mathematics. The main aim of the approach is to use Boolean algebra to teach pupils such mathematical processes as modeling and axiomatization. A course using the approach is described. (RH)
A procedure is developed for obtaining generalized flip-flop input equations, and a concise method is presented for representing these equations. The procedure is based on solving a four-valued characteristic equation of the flip-flop, and can encompass flip-flops that are too complex to approach intuitively. The technique is presented using Karnaugh maps, but could easily be implemented in software.
This paper describes the scalability of linear algebra kernels based on remote memory access approach. The current approach differs from the other linear algebra algorithms by the explicit use of shared memory and remote memory accessThe proton-rich nucleus 23Al has a ground state just 123 keV below the one-proton emission threshold, and as a result comparatively little is known experimentally about its properties, as with many such nuclei. Theoretical investigations have tended to model exclusively the ground and first one to three excited states known. In this paper, we theoretically model most of the known spectrum, and predict what states may as yet be unobserved. We use the multichannel algebraic scattering method to describe states as resonances of a valence proton coupled to a 22Mg rotor core. Six states with low-excitation energies and defined {J}π are matched, and we make the first prediction of the properties of four others and propound the possible existence of several more.
The student success rate in the algebra-based Introductory General Physics I course at the University of Houston (UH) and across the United States is low in comparison to success rates in other service courses. In order to improve student success rates, we have implemented, in addition to interactive teaching techniques, pre-testing as an early intervention process to identify and remediate at-risk students. The pre-testing includes a math and problem-solving skills diagnostic exam and pre-tests administered prior to all regular exams. Students identified as at risk based on their scores on these pre-tests are given incentives to utilize a tutoring intervention consisting of on-line math tutoring to address math deficiencies and tutoring by graduate Physics Teaching Assistants to address student understanding of the physics concepts. Results from 503 students enrolled in three sections of the course showed that 78% of the students identified as at-risk students by the diagnostic exam who completed the math tutorial successfully completed the course, as compared to 45% of at-risk students who did not complete the math tutorial. Results of the pre-testing before each regular exam showed that all students who were identified as at risk based on pre-test scores had positive gains ranging from 9 -- 32% for the three regular exams. However, the large standard deviations of these gains indicate that they are not statistically significant; therefore, pretesting before exams will not be offer in the course. However, utilization of the math tutorials as remediation will continue to be offered to all sections of the algebra-based course at UH with the goal of significantly improving the overall success rates for the introductory physics coursesapproaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to theWe construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
The present report introduces the QuBiLS-MIDAS software belonging to the ToMoCoMD-CARDD suite for the calculation of three-dimensional molecular descriptors (MDs) based on the two-linear (bilinear), three-linear, and four-linear (multilinear or N-linear) algebraic forms. Thus, it is unique software that computes these tensor-based indices. These descriptors, establish relations for two, three, and four atoms by using several (dis-)similarity metrics or multimetrics, matrix transformations, cutoffs, local calculations and aggregation operators. The theoretical background of these N-linear indices is also presented. The QuBiLS-MIDAS software was developed in the Java programming language and employs the Chemical Development Kit library for the manipulation of the chemical structures and the calculation of the atomic properties. This software is composed by a desktop user-friendly interface and an Abstract Programming Interface library. The former was created to simplify the configuration of the different options of the MDs, whereas the library was designed to allow its easy integration to other software for chemoinformatics applications. This program provides functionalities for data cleaning tasks and for batch processing of the molecular indices. In addition, it offers parallel calculation of the MDs through the use of all available processors in current computers. The studies of complexity of the main algorithms demonstrate that these were efficiently implemented with respect to their trivial implementation. Lastly, the performance tests reveal that this software has a suitable behavior when the amount of processors is increased. Therefore, the QuBiLS-MIDAS software constitutes a useful application for the computation of the molecular indices based on N-linear algebraic maps and it can be used freely to perform chemoinformatics studies. PMID:24889018The second lowest and second highest bases of the discrete positive and negative irreducible representations of su(1, 1) Lie algebra via spherical harmonics are used to construct generalized coherent states. Depending on whether the representation label is an even or odd integer, each of the new coherent states is separated into two different classes. They are constituted by appropriate superpositions of the increasing and decreasing infinite sequences with respect to the m index of the spherical harmonics {Ym2j ± m(θ, phi)}m = mnplusj ± 1±∞ and {Ym2k ± m - 1(θ, phi)}m = mnplusk ± 2±∞, and converge to the known functions. Also the non-oscillating measures to realize the resolution of the identity condition on the unit disk are calculated interactionnot the case for the satellite-clock biases. The third objective of this paper is to make the link between the CD approach and the GNSS methods based on the notion of double difference. In particular, it is shown that the information provided by a maximum set of independent DDs may not reach that of a complete set of CDs. The corresponding defect is analyzed. One of the main results of the corresponding analysis concerns the DD-CD relationship. In particular, it is shown that the DD ambiguities, once they have been fixed and validated, can be used as input data in the `undifferenced CD equations.' The corresponding algebraic operations are described. The satellite pseudo-clock biases can therefore be also obtained via particular methods in which the notion of double differencing is involved.
A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability (see Tempesta, 2010 [40]). Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussedCommutative BCI-algebras can be considered as semilattices whose sections are equipped with certain involutions. A similar view can be applied to commutative BCK-algebras. However, for general BCK-algebras a certain construction was settled by the author and J. Kühr (Miskolc Math. Notes 8:11-21, 2007) showing that they can be considered as structures essentially weaker than semilattices but still with certain involutions in sections. The aim of this paper is to involve a similar approach for BCI-algebras.
The algebraic classification of the Weyl tensor in the arbitrary dimension n is recovered by means of the principal directions of its 'superenergy' tensor. This point of view can be helpful in order to compute the Weyl aligned null directions explicitly, and permits one to obtain the algebraic type of the Weyl tensor by computing the principal eigenvalue of rank-2 symmetric future tensors. The algebraic types compatible with states of intrinsic gravitational radiation can then be explored. The underlying ideas are general, so that a classification of arbitrary tensors in the general dimension can be achieved. student lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approachalgebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…
Korean secondary school students preparing for college learn about a simple algebraic formula for area bounded by a parabola and line. The approach does not seem well-known among American students. It is noted that, while the formula derivations rely on integration, algebra students could use the formulas without proofs. (MP) BerezinThe structure of the low-lying states in the odd- and even-mass A ≈100 Ru isotopes is studied in the framework of two algebraic models. The even-mass Ru nuclei are first described within the interacting boson model 1 (IBM-1). The output of these calculations was then used to calculate the odd-A isotopes within the interacting boson-fermion model 1 (IBFM-1), where a coupling of the odd neutron to the even-even core is considered. The level energies and transition probabilities calculated in the present work are tested against the experimental data. One-nucleon transfer spectroscopic factors as well as electromagnetic moments were also calculated for the odd-A Ru and compared to the experimental values. The transitional character of the isotopes is studied. Most of the low-lying positive-parity states in the odd-A Ru nuclei below 2 MeV are interpreted on the basis of ν d5 /2 and ν g7 /2 configurations. The role of the ν s1 /2 orbital in the nuclear structure of the odd-mass Ru nuclei at low energies is also studied. The negative-parity states are interpreted as ν h11 /2 excitations coupled to the core. The evolution of the IBM-1 and IBFM-1 parameters is discussed.
Commercial reagent kits for the evaluation of leukocyte subsets involve the staining of a panel of up to six tubes using combinations of pre-mixed fluorescein isothiocyanate (FITC) and R-phycoerythrin (PE) conjugated monoclonal antibodies. We describe a rapid method whereby total CD3+ T-cells, CD4+ T-cells (CD3+ CD4+), CD8+ T-cells (CD3+ CD8+), putative gamma delta-receptor-T-cells (CD3+ CD4- CD8-), and T-cells that are CD3+ CD4+ CD8+ as well as B-lymphocytes and NK-cells can be enumerated after staining in a single tube. Whole blood specimens are labelled with a mixture of antibodies: FITC-conjugated antibodies to CD4 and CD19, PE-conjugated antibodies to CD8 and CD16, and either peridinin chlorophyll protein (PerCP) or allophycocyanin (APC) labelling for antibodies to CD3. After recording 20,000 events the data were analysed on the Consort 32 computer system and LYSYS-II (Becton Dickinson, San Jose, CA) and all of the lymphocyte subset values were determined by Boolean algebra using a technique we refer to as Boolean gate analysis (BGA). Our study has shown that BGA is statistically equivalent to SimulSET lymphocyte subset analysis. Furthermore, the procedure reduces the number of tubes required to two with consequential saving in reagents, consumables, and time. PMID:7514523 forHomework is transforming at a rapid rate with continuous advances in educational technology. Computer-based homework, in particular, is gaining popularity across a range of schools, with little empirical evidence on how to optimize student learning. The current aim was to test the effects of different types of feedback on computer-based homework.…
A introductory insight will be shared regarding a 'separation of variables' approach to understanding the relationship between QCD and the origins of cosmological and particle mass. The discussion will then build upon work presented at DFP 2000, focussing on the formal basis for using 3x3x3 matrix algebra as it underlies and extends Dirac notation. A set of restrictions are established which break the multiple symmetries of the 3x3x3 matrix algebra, yielding Standard Model QCD objects and interactions. It will be shown that the 3x3x3 matrix representation unifies the algebra of strong and weak (and by extension, electromagnetic) interactions. A direct correspondence to string theoretic objects is established by considering the string to be partitioned in thirds. Rubik's cube is used as a graphical means of handling algebraic manipulation of 3x3x3 algebra. Further, its potential utility for advancing pedagogical methods through active engagement is discussed. A simulated classroom exercize will be conductedPurpose: The paper aims to provide an overview of the vocabulary for materiality which is used by practice-basedapproaches to organizational knowing. Design/methodology/approach: The overview is theoretically generated and is based on the anthology Knowing in Organizations: A Practice-basedApproach edited by Nicolini, Gherardi and Yanow. The…This paper is concerned with the problem of analysis and optimisation of the inerter-based isolators based on a "uni-axial" single-degree-of-freedom isolation system. In the first part, in order to gain an in-depth understanding of inerter from the prospective of vibration, the frequency responses of both parallel-connected and series-connected inerters are analysed. In the second part, three other inerter-based isolators are introduced and the tuning procedures in both the H∞ optimisation and the H2 optimisation are proposed in an analytical manner. The achieved H2 and H∞ performance of the inerter-based isolators is superior to that achieved by the traditional dynamic vibration absorber (DVA) when the same inertance-to-mass (or mass) ratio is considered. Moreover, the inerter-based isolators have two unique properties, which are more attractive than the traditional DVA: first, the inertance-to-mass ratio of the inerter-based isolators can easily be larger than the mass ratio of the traditional DVA without increasing the physical mass of the whole system; second, there is no need to mount an additional mass on the object to be isolated to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentrationThe central purpose of this research was to assess student learning outcomes in the Algebra I credit recovery course offered at traditional schools in Louisiana and the Algebra I credit recovery course offered online by the Louisiana Virtual School (LVS). It was also conducted to assess the associations of gender, grade level, and socio-economic…Shortly after starting work for the University of Chichester in the School of Teacher Education, the author was planning a session relating to algebra and found herself inspired by an article in MT182: "Algebraic Infants" by Andrews and Sayers (2003). Based on the making of families of "Multilink" animals, Andrews and Sayers (2003) claim that… simulation code for space charge dominated photoemission processes.« less
Urban African American studentsIn this study, I investigated how proficiently seventh-grade students enrolled in two Southwestern schools solve algebra word problems. I analyzed various factors that could affect this proficiency and explored the differences between English Learners (ELs) and native English Primary students (EPs). I collected the data as part of the Animal Watch…This article examines the effectiveness of a technology-basedalgebra curriculum in a wide variety of middle schools and high schools in seven states. Participating schools were matched into similar pairs and randomly assigned to either continue with the current algebra curriculum for 2 years or to adopt Cognitive Tutor Algebra I (CTAI), which…The I CAN Learn[R] Education System is an interactive, self-paced, mastery-based software system that includes the I CAN Learn[R] Fundamentals of Math (5th-6th grade math) curriculum, the I CAN Learn[R] Pre-Algebra curriculum, and the I CAN Learn[R] Algebra curriculum. College algebra credit is also available to students in participating schools…This research project investigates students' development of problem solving schemata while using strategies that facilitate the process of using solved examples to assist with a new problem (case reuse). Focus group learning interviews were used to explore students' perceptions and understanding of several problem solving strategies. Individual clinical interviews were conducted and quantitative examination data were collected to assess students' conceptual understanding, knowledge organization, and problem solving performance on a variety of problem tasks. The study began with a short one-time treatment of two independent, research-based strategies chosen to facilitate case reuse. Exploration of students' perceptions and use of the strategies lead investigators to select one of the two strategies to be implemented over a full semester of focus group interviews. The strategy chosen was structure mapping. Structure maps are defined as visual representations of quantities and their associations. They were created by experts to model the appropriate mental organization of knowledge elements for a given physical concept. Students were asked to use these maps as they were comfortable while problem solving. Data obtained from this phase of our study (Phase I) offered no evidence of improved problem solving schema. The 11 contact hour study was barely sufficient time for students to become comfortable using the maps. A set of simpler strategies were selected for their more explicit facilitation of analogical reasoning, and were used together during two more semester long focus group treatments (Phase II and Phase III of this study). These strategies included the use of a step-by-step process aimed at reducing cognitive load associated with mathematical procedure, direct reflection of principles involved in a given set of problems, and the direct comparison of problem pairs designed to be void of surface similarities (similar objects or object orientations) and sharing
To improve student success in mathematics, the use of research-based interventions is necessary to help secondary students with learning disabilities (LD) access the algebra curriculum. The authors provide an overview of the following research-basedapproaches: explicit instruction, graduated instructional sequence, technology, and graphic…
The q-deformation of the infinite-dimensional n-algebras is investigated. Based on the structure of the q-deformed Virasoro-Witt algebra, we derive a nontrivial q-deformed Virasoro-Witt n-algebra which is nothing but a sh-n-Lie algebra. Furthermore in terms of the pseud-differential operators, we construct the (co)sine n-algebra and the q-deformed S Diff (T2)n-algebra. We find that they are the sh-n-Lie algebras for the n even case. In terms of the magnetic translation operators, an explicit physical realization of the (co)sine n-algebra is givenIn this paper, a 3 × 3-matrix representation of Birman-Wenzl-Murakami (BWM) algebra has been presented. Based on which, unitary matrices A( θ, φ 1, φ 2) and B( θ, φ 1, φ 2) are generated via Yang-Baxterization approach. A Hamiltonian is constructed from the unitary B( θ, φ) matrix. Then we study Berry phase of the Yang-Baxter system, and obtain the relationship between topological parameter and Berry phase.Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number, pre-algebra, and algebra…
The paper is written in the lecture format and dedicated to one of the main basal approaches, the orbitozygomatic approach, that has been widely used by neurosurgeons for several decades. The authors describe the historical background of the approach development and the surgical technique features and also analyze the published data about application of the orbitozygomatic approach in surgery for skull base tumors and cerebral aneurysms. PMID:26529627Purpose: In this paper, we implemented the four-dimensional (4D) digital tomosynthesis (DTS) imaging based on algebraic image reconstruction technique and total-variation minimization method in order to compensate the undersampled projection data and improve the image quality. Methods: The projection data were acquired as supposed the cone-beam computed tomography system in linear accelerator by the Monte Carlo simulation and the in-house 4D digital phantom generation program. We performed 4D DTS based upon simultaneous algebraic reconstruction technique (SART) among the iterative image reconstruction technique and total-variation minimization method (TVMM). To verify the effectiveness of this reconstruction algorithm, we performed systematic simulation studies to investigate the imaging performance. Results: The 4D DTS algorithm based upon the SART and TVMM seems to give better results than that based upon the existing method, or filtered-backprojection. Conclusion: The advanced image reconstruction algorithm for the 4D DTS would be useful to validate each intra-fraction motion during radiation therapy. In addition, it will be possible to give advantage to real-time imaging for the adaptive radiation therapy. This research was supported by Leading Foreign Research Institute Recruitment Program (Grant No.2009-00420) and Basic Atomic Energy Research Institute (BAERI); (Grant No. 2009-0078390) through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning (MSIP) asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly 1 when an outer unity is adjoined to the original algebra. It is shown that the PI-exponents of unital algebras can take any value greater than 2, and the exponents of finite-dimensional unital algebras form a dense subset in the domain \\lbrack 2,∞). Bibliography: 34 titlesThis paper details an inquiry-basedapproach for teaching the basic notions of rings and fields to liberal arts mathematics students. The task sequence seeks to encourage students to identify and comprehend core concepts of introductory abstract algebra by thinking like mathematicians; that is, by investigating an open-ended mathematical context,… multigrid ideas to a number of different situations. Examples are given corresponding to linear elasticity and specifically in the solution of linear systems associated with extended finite elements for fracture problemsSome concepts in semigroup theory are interpreted in discrete structures such as finite lattices, binary relations, and finite semilattices. An algebraicapproach investigatedIn this paper a Clifford algebra-based method is applied to calculate polymer chain conformations. The approach enables the calculation of the position of an atom in space with the knowledge of the bond length (l), valence angle (theta), and rotation angle (phi) of each of the preceding bonds in the chain. Hence, the set of geometrical parameters {l(i),theta(i),phi(i)} yields all the position coordinates p(i) of the main chain atoms. Moreover, the method allows the calculation of side chain conformations and the computation of rotations of chain segments. With these features it is, in principle, possible to generate conformations of any type of chemical structure. This method is proposed as an alternative for the classical approach by matrix algebra. It is more straightforward and its final symbolic representation considerably simpler than that of matrix algebra. Approaches for realistic modeling by means of incorporation of energetic considerations can be combined with it. This article, however, is entirely focused at showing the suitable mathematical framework on which further developments and applications can be built. PMID:18345877
In the 1990s, Kempf and his collaborators Mangano and Mann introduced a D-dimensional (β, β‧)-two-parameter deformed Heisenberg algebra which leads to an isotropic minimal length (\\triangle Xi)\\min = \\hbar √ {Dβ +β '}, \\forall i\\in \\{1, 2, ..., D\\}. In this work, the Lagrangian formulation of a magnetostatic field in three spatial dimensions (D = 3) described by Kempf algebra is presented in the special case of β‧ = 2β up to the first-order over β. We show that at the classical level there is a similarity between magnetostatics in the presence of a minimal length scale (modified magnetostatics) and the magnetostatic sector of the Abelian Lee-Wick model in three spatial dimensions. The integral form of Ampere's law and the energy density of a magnetostatic field in the modified magnetostatics are obtained. Also, the Biot-Savart law in the modified magnetostatics is found. By studying the effect of minimal length corrections to the gyromagnetic moment of the muon, we conclude that the upper bound on the isotropic minimal length scale in three spatial dimensions is 4.42×10-19 m. The relationship between magnetostatics with a minimal length and the Gaete-Spallucci nonlocal magnetostatics [J. Phys. A: Math. Theor. 45, 065401 (2012)] is investigated. shouldOne-equation model of turbulence based on standard equations of k-epsilon model of turbulence, where k is turbulent energy and e is rate of dissipation of k. Derivation of one-equation model motivated partly by inaccuracies of flows computed by some Navier-Stokes-equations-solving algorithms incorporating algebraic models of turbulence. Satisfies need to avoid having to determine algebraic length scalesWith a view toward further nuclear structure applications of approachesbased on quantum-deformed (or q-deformed) algebras, introduced to the authors by Yu.F. Smirnov, we construct a q analog of a boson realization of the symplectic noncompact sp(4, R) algebra together with a q analog of a fermion realization of the symplectic compact sp(4) algebra. The first study, on the q-deformed Sp(4,R) symmetry, is applied to the development of a q analog of the two-dimensional Interacting Boson Model with q-deformed SU(3) the underpinning dynamical symmetry group. An explicit realization in terms of q-tensor operators with respect to the standard su{sub q}(2) algebra is given. The group-subgroup structure of this framework yields the physical interpretation of the generators of the groups under consideration. The second symplectic algebra, the q-deformed sp(4), is applied to studying isovector pairing correlations in atomic nuclei. A specific q deformation of the sp(4) algebra is realized in terms of q deformed fermion creation and annihilation operators of the shell model. The generators of the algebra close on four distinct realizations of the u{sub q}(2) subalgebra. These reductions, which correspond to different types of pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q deformation is based on a comparison of the results for energies of the lowest isovector-paired 0{sup +} states in the deformed and nondeformed cases.
Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
To improve the accuracy and the efficiency of morphodynamic simulations, we present a subgrid basedapproach roughapproach offers a significant improvement at a minimal computational cost.
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
This article describes some problems of the maximum/minimum type, which are generally solved using calculus at secondary school, but which here are solved algebraically. We prove six algebraic properties and then apply them to this kind of problem. This didactic approach allows pupils to solve these problems even at the beginning of secondary…
The use of multiple forms of collaborative instruction to teach integrated algebra makes it difficult for teachers to determine which collaborative form is best suited for the curriculum. An inconsistent approach to integrated algebra instruction at the study school needed to be addressed for the benefit of teacher effectiveness and student…In this paper, we have investigated the positive-parity states in the odd-mass transitional 123‑135Xe isotopes within the framework of the interacting boson-fermion model. Two solvable extended transitional Hamiltonians which are based on SU(1,1) algebra are employed to provide an investigation of quantum phase transition (QPT) between the spherical and deformed gamma — unstable shapes along the chain of Xe isotopes. The low-states energy spectra and B(E2) values for these nuclei have been calculated and compared with the experimental data. The predicted excitation energies and B(E2) transition rates of the odd isotopes are found to agree well with the experimental data. We have also analyzed the critical behavior of even-odd Xe isotopes via Catastrophe Theory in combination with a coherent state formalism to generate energy surfaces and special isotopes which are the best candidates for the critical point are identifiedSupported eText for students with visual impairments in mathematics has a promising, emerging literature base, although little of the existing research focuses on implementation within a classroom setting. This qualitative study sought to understand the use of supported eText to deliver algebra to students with visual impairments enrolled in algebra mathematics courses. The study also sought to explore supported eText in contrast to students' traditional means of accessing an algebra text. The main results suggest supported eText holds potential in terms of delivering mathematics content; however, more research and more reflection on the field is needed regarding this approach as a sole means of presenting text. Implications for teacher professional development and implementation practices are discussed. PMID:26131793
The process algebraapproachThis paper addresses the implementation of complex multiple degree of freedom virtual environments for haptic display. We suggest that a physics basedapproach to rigid body simulation is appropriate for hand tool simulation, but that currently available simulation techniques are not sufficient to guarantee successful implementation. We discuss the desirable features of a virtual environment simulation, specifically highlighting the importance of stability guarantees.
Algebraic structures are a necessary aspect of algebraic thinking for K-12 students and teachers. An approach for introducing the algebraic structure of groups and fields through the arithmetic properties required for solving simple equations is summarized; the collective (not individual) importance of these axioms as a foundation for algebraic…
Algebra progress monitoring is a research-based practice that extends a long history of research in curriculum-based measurement (CBM). This article describes the theoretical foundations and research evidence for algebra progress monitoring, along with critical features of the practice. A detailed description of one practitioner's implementation…John Saxon's incremental development model has been proclaimed as a superior teaching strategy for mathematics. This study evaluated the Saxon approach and textbook using 276 Algebra I students in experimental and control groups. The groups were compared in cognitive and affective areas. Results are presented. (Author/MT)
Report of an experiment which investigated the use of a programed elementary algebra text as a teaching method. The method was evaluated on the basis of student evaluation of the course and the percentage of students achieving a grade of C or better. Results indicated that the use of programed texts was superior to the traditional approach usingHestenes' geometric algebra and Dirac spinors are reviewed and united into a common mathematical formalism, a unification that establishes the Dirac equation as being manifestly covariant under the Lorentz group, and one that needs no matrix representation of the Dirac algebra. New and simple methods of amplitude or ``trace'' calculations are then described. A number of problems are then considered within the context of the new approach, such as relativistic spin projections, new and covariant C and T-transformations and spinors for massless and Majorana fields.
The modeling of power systems has been primarily driven by the commercial power utility industry. These models usually involve the assumption that system bus voltage and frequency are constant. However, in applications such as shipboard power systems this infinite bus assumption is not valid. This thesis investigates the modeling of a synchronous generator and various loads in a modular fashion on a finite bus. The simulation presented allows the interconnection of multiple state-space models via a bus voltage model. The major difficulty encountered in building a model which computes bus voltage at each time step is that bus voltage is a function of current and current derivative terms. Bus voltage is also an input to the state equations which produce the current and current derivatives. This creates an algebraic loop which is a form of implicit differential equation. A routine has been developed by Linda Petzold of Lawrence Livermore Laboratory for solving these types of equations. The routine, called Differential Algebraic System Solver (DASSL), has been implemented in a pre-release version of the software Advanced Continuous Simulation Language (ACSL) and has been made available to the Naval Postgraduate School on a trial basis. An isolated power system is modeled using this software and the DASSL routine. The system response to several dynamic situations is studied and the results are presented.
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods. PMID:21769158Currently: Ops Rev developed and maintains a framework that includes interface-specific language, patterns, and Viewpoints. Ops Rev implements the framework to design MOS 2.0 and its 5 Mission Services. Implementation de-couples interfaces and instances of interaction Future: A Mission MOSE implements the approach and uses the model based artifacts for reviews. The framework extends further into the ground data layers and provides a unified methodologyFormal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. We propose in particular that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuospatial structure of source representations. For example, in the statement "There are 5 elephants for every 3 rhinoceroses," the spatial proximity of 5 and elephants and 3 and rhinoceroses will bias reasoners to write the incorrect expression 5E = 3R, because that expression maintains the spatial relationships encoded in the source representation. In 3 experiments, participants constructed equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation construction and successfully predicts error patterns that are not accounted for by prior explanations, such as the left to right transcription heuristic. PMID:24820674
A teaching intervention study was conducted with sixth grade students to explore the interconnections between students' growing understanding of arithmetic expressions and beginning algebra. Three groups of students were chosen, with two groups receiving instruction in arithmetic and algebra, and one group in algebra without arithmetic. Students…
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, which uses the CAS…
Logical methods are most valuable in the field of Medicine. They are usually based on Boolean algebra and can thus only deal with binary data) (Present)/(Absent)). Use of ternary algebra opens the way to treatment of the triple-state variables ((Present)/(absent)/(Don't know)) frequently encountered in medical context. PMID:816530It is shown that the space of spherical harmonics Ylm(θ ,φ ) whose 2l - m = p - 1 is given, represent irreducibly a cubic deformation of su(2) algebra, the so-called suΦp(2), with deformation function as Φ p(x) = (27)/(2)x2 + 3(7-3p2)x. The irreducible representation spaces are classified in three different bunches, depending on one of values 3k - 2, 3k - 1 and 3k, with k as a positive integer, to be chosen for p. So, three different methods for generating the spectrum of spherical harmonics are presented by using the cubic deformation of su(2). Moreover, it is shown that p plays the role of deformation parameter. teachingPresents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
This text provides information and exercises on arithmetic topics which should be mastered before a student enrolls in an Elementary Algebra course. Section I describes the fundamental properties and relationships of whole numbers, focusing on basic operations, divisibility tests, exponents, order of operations, prime numbers, greatest common < p < 2} and {x in L_p({M},τ)} one can find {a, b in L_p({M},τ)} and contractions {u_n, v_n in {M}} such that {E}_n(x) = a u_n + v_n b quad and quad max big{ |a|_p,|b|_p big} ≤ c_p |x|_p. Moreover, it turns out that {a u_n} and {v_n b} converge in the row/column Hardy spaces {{H}_p^r({M})} and {{H}_p^c({M})} respectively. In particular, this solves a problem posed by the Defant and Junge in 2004. In the case p = 1, our results establish a noncommutative form of the Davis celebrated theorem on the relation betwe en martingale maximal and square functions in L 1, whose noncommutative form has remained open for quite some time. Given {1 ≤ p ≤ 2} , we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of {{E}_n(x)} in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebrasA new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a quantized Dirac-type operator, the two capturing the kinematics of quantum gravity gravity emerges from a discretization of our construction. Certain convergence issues are left unresolved. This paper is the first of two where the second paper [1] is concerned with mathematical details and proofs concerning the spectrum of the holonomy-diffeomorphism algebra.
The purpose of the study reported herein was to identify the common mistakes made by junior secondary students in Hong Kong when learning algebra and to compare teachers' perceptions of students' ability with the results of an algebra test. An algebra test was developed and administered to a sample of students (aged between 13 and 14 years). From the responses of the participating students (N = 815), it was found that students in schools with a higher level of academic achievement had better algebra test results than did those in schools with a lower level of such achievement. Moreover, it was found that a teacher's perception of a student's ability has a correlation with that student's level of achievement. Based on this finding, an instrument that measures teaching effectiveness is discussed. Last but not least, typical errors in algebra are identified, and some ideas for an instructional design based on these findings are discussed.
The purpose of this quasi-experimental quantitative study was to determine if a new Algebra I curriculum resulted in improved student performance on the state Algebra I exam. The treatment group consisted of 383 9th grade Algebra I students who received the college-ready standards-based (CRSB) curricula. The control group consisted of 338 9th…Research to identify validated instructional approaches to teach math to students with LD and those at-risk for failure in both core and supplemental instructional settings is necessary to assist teachers in closing the achievement gaps that exist across the country. The concrete-to-representational-to-abstract instructional sequence (CRA) has…
During the twenty-first century, human habitation of a self-sustaining lunar base could become a reality. To achieve this goal, the occupants will have to have food, water, and an adequate atmosphere within a carefully designed environment. Advanced technology will be employed to support terrestrial life-sustaining processes on the Moon. One approach to a life support system based on food production, waste management and utilization, and product synthesis is outlined. Inputs include an atmosphere, water, plants, biodegradable substrates, and manufacutured materials such as fiberglass containment vessels from lunar resources. Outputs include purification of air and water, food, and hydrogen (H2) generated from methane (CH4). Important criteria are as follows: (1) minimize resupply from Earth; and (2) recycle as efficiently as possible.
The semantically structure of knowledge, based on ontology approaches have been increasingly adopted by several expertise from diverse domains. Recently ontologies have been moved from the philosophical and metaphysics disciplines to be used in the construction of models to describe a specific theory of a domain. The development and the use of ontologies promote the creation of a unique standard to represent concepts within a specific knowledge domain. In the scope of information security systems the use of an ontology to formalize and represent the concepts of security information challenge the mechanisms and techniques currently used. This paper intends to present a conceptual implementation model of an ontology defined in the security domain. The model presented contains the semantic concepts based on the information security standard ISO/IEC_JTC1, and their relationships to other concepts, defined in a subset of the information security domainSynthetic aperture (SA) ultrasound imaging system produces highly accurate axial and lateral displacement estimates; however, low frame rates and large data volumes can hamper its clinical use. This paper describes a real-time SA imaging based ultrasound elastography system that we have recently developed to overcome this limitation. In this system, we implemented both beamforming and 2D cross-correlation echo tracking on Nvidia GTX 480 graphics processing unit (GPU). We used one thread per pixel for beamforming; whereas, one block per pixel was used for echo tracking. We compared the quality of elastograms computed with our real-time system relative to those computed using our standard single threaded elastographic imaging methodology. In all studies, we used conventional measures of image quality such as elastographic signal to noise ratio (SNRe). Specifically, SNRe of axial and lateral strain elastograms computed with real-time system were 36 dB and 23 dB, respectively, which was numerically equal to those computed with our standard approach. We achieved a frame rate of 6 frames per second using our GPU basedapproach for 16 transmits and kernel size of 60 × 60 pixels, which is 400 times faster than that achieved using our standard protocol.
Geometric algebra is based on two simple ideas. First, the area of a rectangle is equal to the product of the lengths of its sides. Second, if a figure is broken apart into several pieces, the sum of the areas of the pieces equals the area of the original figure. Remarkably, these two ideas provide an elegant way to introduce, connect, and…
Explores how students interpret algebraic notation and what teachers can do to support appropriate interpretations. Presents two research-based strategies and concludes that in the face of reform and technological advances, finding a definition for symbol sense takes on added significance. Contains 22 references. (ASK) circuits have been investigated.
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recentlyMost people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,… positive aboveBecause it is often argued that surrogacy should not be treated as contractual, the question arises in which terms this practice might then be couched. In this article, I argue that a phenomenology of surrogacy centering on the notion of trust provides a description that is illuminating from the moral point of view. My thesis is that surrogacy establishes a complex and extended reproductive unit--the "surrogacy triad" consisting of the surrogate mother, the child, and the intending parents--whose constituents are bound together by mutual trustful commitments. Even though a trust-basedapproach does not provide an ultimate answer to whether surrogacy should be sanctioned or prohibited, it allows for at least some practical suggestions. In particular, I will argue that, under certain conditions, surrogacy is tenable within familial or other significant relationships, and I will stress the necessity of acknowledging the new relationships and moral commitments that result from this practice. PMID:26449234
Cancer is a highly complex disease to understand, because it entails multiple cellular physiological systems. The most common cancer treatments are restricted to chemotherapy, radiation and surgery. Moreover, the early recognition and treatment of cancer remains a technological bottleneck. There is an urgent need to develop new and innovative technologies that could help to delineate tumor margins, identify residual tumor cells and micrometastases, and determine whether a tumor has been completely removed or not. Nanotechnology has witnessed significant progress in the past few decades, and its effect is widespread nowadays in every field. Nanoparticles can be modified in numerous ways to prolong circulation, enhance drug localization, increase drug efficacy, and potentially decrease chances of multidrug resistance by the use of nanotechnology. Recently, research in the field of cancer nanotechnology has made remarkable advances. The present review summarizes the application of various nanotechnology-basedapproaches towards the diagnostics and therapeutics of cancer. PMID:22927757
Lobectomy with systematic lymph node sampling or dissection remains the mainstay of treatment of early stage non-small cell lung cancer. The use of video-assisted thoracic surgery (VATS) to perform lobectomy was first reported in 1992. Advantages of VATS include less trauma and pain, shorter chest drainage duration, decreased hospital stay, and preservation of short-term pulmonary function. However, VATS is characterized by loss of binocular vision and a limited maneuverability of thoracoscopic instruments, an unstable camera platform, and poor ergonomics for the surgeon. To overcome these limitations, robotic systems were developed during the last decades. This article reviews the technical aspects of robotic lobectomy using a VATS-basedapproach. PMID:24780417 restrictions and ensure delivery of standard care to septic patients. PMID:26968289
The evolutionary development of a planetary expansionary outpost is considered in the light of both technical and economic issues. The outline of a partnering taxonomy is set forth which encompasses both institutional and temporal issues related to establishing shared interests and investments. The purely technical issues are discussed in terms of the program components which include nonaerospace technologies such as construction engineering. Five models are proposed in which partnership and autonomy for participants are approached in different ways including: (1) the standard customer/provider relationship; (2) a service-provider scenario; (3) the joint venture; (4) a technology joint-development model; and (5) a redundancy model for reduced costs. Based on the assumed characteristics of planetary surface systems the cooperative private/public models are championed with coordinated design by NASA to facilitate outside cooperation.
We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.
Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. PMID:27171890
This study examines the process of change among grade 4 teachers (students aged 9-10 years) who participated in a yearlong Teacher Quality Grant innovation program. The concerns-based adoption model (CBAM), which informed the design and implementation of the program, was used to examine the process of change. Two questions guided theThis paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075In this paper we focus on the main aspects of computer-aided calculations with tensors and present a new computer algebra system Redberry which was specifically designed for algebraic tensor manipulation. We touch upon distinctive features of tensor software in comparison with pure scalar systems, discuss the main approaches used to handle tensorial expressions and present the comparison of Redberry performance with other relevant tools.
Two novel polynomial su(1, 1) algebras for the physical systems with the first and second Pöschl-Teller (PT) potentials are constructed, and their specific representations are presented. Meanwhile, these polynomial su(1, 1) algebras are used as an algebraic technique to solve eigenvalues and eigenfunctions of the Hamiltonians associated with the first and second PT potentials. The algebraicapproach explores an appropriate new pair of raising and lowing operators K-circumflex{sub ±} of polynomial su(1, 1) algebra as a pair of shift operators of our Hamiltonians. In addition, two usual su(1, 1) algebras associated with the first and second PT potentials are derived naturally from the polynomial su(1, 1) algebras built by us.
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L{sup 2} functions defined on (−1,1)×Z and on the sphere S{sup 2}, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in this way the "algebraic special functions" that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L{sup 2} functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to get a Lie group representation defining "algebraic special functions". •The "algebraic special functions" connect Lie algebras and L{sup 2} functions.
In this paper, we present a linear algebra-based derivation of the analytic formula for the sum of the first nth terms of the arithmetico-geometric sequence. Furthermore, the advantage of the derivation is briefly discussed.
This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…approachaticThis work proposed a generalization of the method proposed by the authors: 'A complex network-basedapproachMathematical modeling in Epidemiology is an important tool to understand the ways under which the diseases are transmitted and controlled. The mathematical modeling can be implemented via deterministic or stochastic models. Deterministic models are based on short systems of non-linear ordinary differential equations and the stochastic models are based on very large systems of linear differential equations. Deterministic models admit complete, rigorous and automatic analysis of stability both local and global from which is possible to derive the algebraic expressions for the basic reproductive number and the corresponding epidemic thresholds using computer algebra software. Stochastic models are more difficult to treat and the analysis of their properties requires complicated considerations in statistical mathematics. In this work we propose to use computer algebra software with the aim to solve epidemic stochastic models such as the SIR model and the carrier-borne model. Specifically we use Maple to solve these stochastic models in the case of small groups and we obtain results that do not appear in standard textbooks or in the books updated on stochastic models in epidemiology. From our results we derive expressions which coincide with those obtained in the classical texts using advanced procedures in mathematical statistics. Our algorithms can be extended for other stochastic models in epidemiology and this shows the power of computer algebra software not only for analysis of deterministic models but also for the analysis of stochastic models. We also perform numerical simulations with our algebraic results and we made estimations for the basic parameters as the basic reproductive rate and the stochastic threshold theorem. We claim that our algorithms and results are important tools to control the diseases in a globalized worldWe construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algebras and the category of quantum measure algebras. The sheaf-theoretic semantic transition of quantum structures shifts their physical significance from the orthoposet axiomatization at the level of events, to the sheaf-theoretic gluing conditions at the level of Boolean localization systemsThis programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex numberMany free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-weekFor students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is…
The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not presentPrognostics and Health Management has been becoming an effective technology to increasing efficiency and reducing cost for complex system. As for the two major categories methods, both model-basedapproaches and datadriven approaches have merits and drawbacks. A kind of fusion approaches that integrate model-basedapproaches and data-driven approaches is presented in this paper and fusion structure is detailed to make full use of their advantages and overcome their limitations.
This paper describes our research in automating the reuse process through the use of application domain models. Application domain models are explicit formal representations of the application knowledge necessary to understand, specify, and generate application programs. Furthermore, they provide a unified repository for the operational structure, rules, policies, and constraints of a specific application area. In our approach, domain models are expressed in terms of a transaction-based meta-modeling language. This paper has described in detail the creation and maintenance of hierarchical structures. These structures are created through a process that includes reverse engineering of data models with supplementary enhancement from application experts. Source code is also reverse engineered but is not a major source of domain model instantiation at this time. In the second phase of the software synthesis process, program specifications are interactively synthesized from an instantiated domain model. These specifications are currently integrated into a manual programming process but will eventually be used to derive executable code with mechanically assisted transformations. This research is performed within the context of programming-in-the-large types of systems. Although our goals are ambitious, we are implementing the synthesis system in an incremental manner through which we can realize tangible results. The client/server architecture is capable of supporting 16 simultaneous X/Motif users and tens of thousands of attributes and classes. Domain models have been partially synthesized from five different application areas. As additional domain models are synthesized and additional knowledge is gathered, we will inevitably add to and modify our representation. However, our current experience indicates that it will scale and expand to meet our modeling needs.Geometric algebra and Clifford algebra are important tools to describe and analyze the physics of the world we live in. Although there is enormous empirical evidence that we are living in four dimensional spacetime, mathematical worlds of higher dimensions can be used to present the physical laws of our world in an aesthetical and didactical more appealing way. In physics and mathematics education we are therefore confronted with the question how these high dimensional spaces should be taught. But as an immediate confrontation of students with high dimensional compactified spacetimes would expect too much from them at the beginning of their university studies, it seems reasonable to approach the mathematics and physics of higher dimensions step by step. The first step naturally is the step from four dimensional spacetime of special relativity to a five dimensional spacetime world. As a toy model for this artificial world cosmological special relativity, invented by Moshe Carmeli, can be used. This five dimensional non-compactified approach describes a spacetime which consists not only of one time dimension and three space dimensions. In addition velocity is regarded as a fifth dimension. This model very probably will not represent physics correctly. But it can be used to discuss and analyze the consequences of an additional dimension in a clear and simple way. Unfortunately Carmeli has formulated cosmological special relativity in standard vector notation. Therefore a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of cosmological special relativity is discussed.
Using the Jordan algebras methods, specially the properties of Peirce decomposition and the Frobenius transformation, we compute the coefficients of the zeta functional equation, in the case of Jordan algebras of type II. As particular cases of our result, we can cite the case of studied by Gelbart [Mem. Amer. Math. Soc. 108 (1971)] and Godement and Jacquet [Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972], and the case of studied by Muro [Adv. Stud. Pure Math. 15 (1989) 429]. Let us also mention, that recently, Bopp and Rubenthaler have obtained a more general result on the zeta functional equation by using methods based on the algebraic properties of regular graded algebras which are in one-to-one correspondence with simple Jordan algebras [Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces, IRMA, Strasbourg, 2003]. The method used in this paper is a direct application of specific properties of Jordan algebras of type II.
The current decades are marked not by the development of new molecules for the cure of various diseases but rather the development of new delivery methods for optimum treatment outcome. Nanomedicine is perhaps playing the biggest role in this concern. Nanomedicine offers numerous advantages over conventional drug delivery approaches and is particularly the hot topic in anticancer research. Nanoparticles (NPs) have many unique criteria that enable them to be incorporated in anticancer therapy. This topical review aims to look at the properties and various forms of NPs and their use in anticancer treatment, recent development of the process of identifying new delivery approaches as well as progress in clinical trials with these newer approaches. Although the outcome of cancer therapy can be increased using nanomedicine there are still many disadvantages of using this approach. We aim to discuss all these issues in this review.
Minimally invasive approaches are becoming increasingly popular to access the anterior skull base. With interdisciplinary cooperation, in particular endonasal endoscopic approaches have seen an impressive expansion of indications over the past decades. The more recently described transorbital approaches represent minimally invasive alternatives with a differing spectrum of access corridors. The purpose of the present paper is to discuss transorbital approaches to the anterior skull base in the light of the current literature. The transorbital approaches allow excellent exposure of areas that are difficult to reach like the anterior and posterior wall of the frontal sinus; working angles may be more favorable and the paranasal sinus system can be preserved while exposing the skull base. Because of their minimal morbidity and the cosmetically excellent results, the transorbital approaches represent an important addition to established endonasal endoscopic and open approaches to the anterior skull base. Their execution requires an interdisciplinary team approach. PMID:27453759
This study examined the effectiveness of using laptops to teach and deliver instruction to students. The meta-analytic approach was employed to compare the means of End-of Course Test scores from North Carolina one-to-one high schools during the traditional teaching period and the laptop teaching period in order to determine if there are……
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra. approach in both lectures and tutorials, and how he employed visualization and an emphasis on language to encourage conceptual thinking. We use Tall's framework of three worlds of mathematical thinking to reflect on the effect of these activities in students' learning. An analysis of students' attitudes to the course and their test and examination results help to answer questions about the value of such an approach, suggesting ways forward in teaching linear algebra.Fault detection and identification (FDI) is important in almost all real systems. Fault detection is the supervision of technical processes aimed at detecting undesired or unpermitted states (faults) and taking appropriate actions to avoid dangerous situations, or to ensure efficiency in a system. This dissertation develops and extends fault detection techniques for systems modeled by differential algebraic equations (DAEs). First, a passive, observer-basedapproach is developed and linear filters are constructed to identify faults by filtering residual information. The method presented here uses the least squares completion to compute an ordinary differential equation (ODE) that contains the solution of the DAE and applies the observer directly to this ODE. While observers have been applied to ODE models for the purpose of fault detection in the past, the use of observers on completions of DAEs is a new idea. Moreover, the resulting residuals are modified requiring additional analysis. Robustness with respect to disturbances is also addressed by a novel frequency filtering technique. Active detection, as opposed to passive detection where outputs are passively monitored, allows the injection of an auxiliary control signal to test the system. These algorithms compute an auxiliary input signal guaranteeing fault detection, assuming bounded noise. In the second part of this dissertation, a novel active detection approach for DAE models is developed by taking linear transformations of the DAEs and solving a bi-layer optimization problem. An efficient real-time detection algorithm is also provided, as is the extension to model uncertainty. The existence of a class of problems where the algorithm breaks down is revealed and an alternative algorithm that finds a nearly minimal auxiliary signal is presented. Finally, asynchronous signal design, that is, applying the test signal on a different interval than the observation window, is explored and discussed.
As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18)In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
The invariants of solvable triangular Lie algebras with one nil-independent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartan's method of moving frames and the special technique developed for triangular and closed algebras in Boyko et al (J. Phys. A: Math. Theor. 2007 40 7557). The conjecture of Tremblay and Winternitz (J. Phys. A: Math. Gen. 2001 34 9085) on the number and form of elements in the bases is completed and provedDe la Torre and Deng suggested a resampling-basedapproach for person-fit assessment (PFA). The approach involves the use of the [math equation unavailable] statistic, a corrected expected a posteriori estimate of the examinee ability, and the Monte Carlo (MC) resampling method. The Type I error rate of the approach was closer to the nominal level… approach approachA sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The properties of sequential products on Hilbert space effect algebras are discussed. For a general SEA, relationships between sequential independence, coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center is isomorphic to a fuzzy set system is presented. It is shown that the existence, of a sequential product is a strong restriction that eliminates many effect algebras from being SEA's. For example, there are no finite nonboolean SEA's, A measure of sharpness called the sharpness index is studied. The existence of horizontal sums of SEA's is characterized and examples of horizontal sums and tensor products are presented.
A new method for calculating the curvature tensor is developed and applied to the Scharzschild case. The method employs Clifford algebra and has definite advantages over conventional methods using differential forms or tensor analysis.
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understandingCourses in first year mechanics generally start with an introduction to vector methods which include scalar and vector multiplication1. While the demonstration of the validity of the distributive law for scalar multiplication is straightforward, this is not so for vector multiplication. The latter requires complicated geometrical visualization, so its proof is often skipped1. Neither the commutative nor associative law holds for vector multiplication, so there is no a priori reason that the distributive law should hold. In this paper we present an algebraicapproach to the proof that requires no geometric visualization. It is based on two relations: (1) the distributive law for scalar multiplication and (2) a*(bxc) =c*(axb) =b*(cxa). 1. e.g. C. Kittlel, W.D. Knight, M.A. Ruderman, Mechanics, Berkeley Physics Course Vol. 1, 2nd ed. McGraw Hill, pp34-39.
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.
The various difficulties and cognitive obstacles that students face when they are introduced to algebra are well documented and described in the relevant bibliography. If it is possible, in order to avoid these difficulties, we have adopted the functional approach widening the meaning of algebraic thinking. In this paper, which is part of wider…The "Cognitive Tutor[R] Algebra I" curriculum, published by Carnegie Learning, is an approach that combines algebra textbooks with interactive software. The software is developed around an artificial intelligence model that identifies strengths and weaknesses in each individual student's mastery of mathematical concepts. It then customizes prompts…
Based on positive psychology, self-determination theory and a perspective on teacher quality, this study proposes and examines a strength-basedapproach to teacher professional development. A mixed method pre-test/post-test design was adopted to study perceived outcomes of the approach for 93 teachers of six primary schools in the Netherlands and…
The purpose of this study was to examine the overarching framework of EFL (English as a Foreign Language) reading instructional approach reflected in an EFL secondary school curriculum in Malaysia. Based on such analysis, a comparison was made if Communicative Task-Based Language is the overarching instructional approach for the Malaysian EFL…
Advances in computer science, in particular object oriented programming, and software engineering have had little practical impact on computer algebra systems in the last 30 years. The software design of existing systems is still dominated by ad-hoc memory management, weakly typed algorithm libraries and proprietary domain specific interactive expression interpreters. We discuss a modular approach to computer algebra software: usage of state-of-the-art memory management and run-time systems (e.g. JVM) usage of strongly typed, generic, object oriented programming languages (e.g. Java) and usage of general purpose, dynamic interactive expression interpreters (e.g. Python) To illustrate the workability of this approach, we have implemented and studied computer algebra systems in Java and Scala. In this paper we report on the current state of this work by presenting new examples.).
This paper describes briefly the development and key features of one of the major context-based courses for upper high school students, Salters Advanced Chemistry. It goes on to consider the research evidence on the impact of the course, focusing on teachers' views, and, in particular, on students' affective and cognitive responses. The research…
Web-Based InstructionA variety of approaches exist for combining data from multiple sensors. The model-basedapproachIn this paper, we study three-dimensional (3D) reduced Birman-Murakami-Wenzl (BMW) algebrabased on topological basis theory. Several examples of BMW algebra representations are reviewed. We also discuss a special solution of BMW algebra, which can be used to construct Heisenberg XXZ model. The theory of topological basis provides a useful method to solve quantum spin chain models. It is also shown that the ground state of XXZ spin chain is superposition state of topological basis.
Described are computer program packages requiring little or no knowledge of computer programing for college algebra, calculus, and abstract algebra. Widely available computer algebra systems are listed. (MNS)The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of scalars, namely, within one or the other of the infinitely many reduced power algebras which can replace the usual real numbers R, or complex numbers C. Three possible major advantages in Physics of such a reformulation are: 1) the disappearance of the well known and hard to deal with problem of the so called "infinities in Physics", 2) the possibilitiy to have infinitely many "levels of precision" instead of the only one existing at present, 3) the possibility to model "hierarchies of Planck constants", [2]. Last and not least, the scalars given by reduced power algebras contain as a particular case those obtained by Nonstandard Analysis, yet they are far more simple and easy to deal with, being in fact on the level of a first course in Algebra. A detailed version of this paper can be found in arxiv:0901.4825 PMID:26723166
The modern graphics processing unit (GPU) found in many standard personal computers is a highly parallel math processor capable of nearly 1 TFLOPS peak throughput at a cost similar to a high-end CPU and an excellent FLOPS/watt ratio. High-level linear algebra operations are computationally intense, often requiring O(N3) operations and would seem a natural fit for the processing power of the GPU. Our work is on CULA, a GPU accelerated implementation of linear algebra routines. We present results from factorizations such as LU decomposition, singular value decomposition and QR decomposition along with applications like system solution and least squares
Evidence-based medicine gathers physician's experience and the best scientific evidence to make medical decisions. This proposal has been widely promulgated by medical opinion leaders. Despite a large literature supporting this practice, a formal discussion has not been established regarding its epistemological consequences in daily medical work. The main proposal of evidence-based medicine consists of choosing the best medical decision according to the best available results from scientific studies. Herein, the goal was to highlight inappropriate application of the scientific method used by physics to clinical science. The inaccuracy resides in describing health and disease in strictly numeric equivalents that can be homogenized on a continuous scale. Finally, the authors consider each diseased human being as a complex system, unique and particular, and that this being is defined by an historical background as well as current actual context. Therefore, evidence-based medicine possesses certain limitations that must be recognized in order to to provide better health care to patients. PMID:19753837
In this paper the bosonic string model with fermionic ghosts is considered in the framework of the KN algebra. The authors' attentions are paid to representations of KN algebra and a Clifford algebra of the ghosts. The authors show that a Virasoro-like algebra is obtained from KN algebra when KN algebra has certain antilinear anti-involution, and that it is isomorphic to the usual Virasoro algebra. The authors show that there is an expected relation between a central charge of this Virasoro-like algebra and an anomaly of the combined system.
Character skeleton plays a significant role in character recognition. The strokes of a character may consist of two regions, i.e., singular and regular regions. The intersections and junctions of the strokes belong to singular region, while the straight and smooth parts of the strokes are categorized to regular region. Therefore, a skeletonization method requires two different processes to treat the skeletons in theses two different regions. All traditional skeletonization algorithms are based on the symmetry analysis technique. The major problems of these methods are as follows. 1) The computation of the primary skeleton in the regular region is indirect, so that its implementation is sophisticated and costly. 2) The extracted skeleton cannot be exactly located on the central line of the stroke. 3) The captured skeleton in the singular region may be distorted by artifacts and branches. To overcome these problems, a novel scheme of extracting the skeleton of character based on wavelet transform is presented in this paper. This scheme consists of two main steps, namely: a) extraction of primary skeleton in the regular region and b) amendment processing of the primary skeletons and connection of them in the singular region. A direct technique is used in the first step, where a new wavelet-based symmetry analysis is developed for finding the central line of the stroke directly. A novel method called smooth interpolation is designed in the second step, where a smooth operation is applied to the primary skeleton, and, thereafter, the interpolation compensation technique is proposed to link the primary skeleton, so that the skeleton in the singular region can be produced. Experiments are conducted and positive results are achieved, which show that the proposed skeletonization scheme is applicable to not only binary image but also gray-level image, and the skeleton is robust against noise and affine transform. PMID:17491454
This article presents a review of arts-based and creative approaches to dementia care as an alternative to antipsychotic medications. While use of antipsychotics may be appropriate for some people, the literature highlights the success of creative approaches and the benefits of their lack of negative side effects associated with antipsychotics. The focus is the use of biographical approaches, music, dance and movement to improve wellbeing, enhance social networks, support inclusive practice and enable participation. Staff must be trained to use these approaches. A case study is presented to demonstrate how creative approaches can be implemented in practice and the outcomes that can be expected when used appropriately. PMID:26938607
When forced treatment in mental health care is under consideration, two approaches guide clinicians in their actions: the dominant rights-basedapproach and the relational ethical approach. We hypothesized that nurses with bachelor's degrees differentiate better between the two approaches than nurses without a degree. To test this hypothesis a survey was performed in major Slovenian health institutions. We found that nurses emphasize the importance of ethics and personal values, but 55.4% of all the nurse participants confused the two approaches. The results confirmed our hypothesis and indicate the importance of nurses' formal education, especially when caring for patients with mental illness. PMID:19372123
In this paper, we aim at locating base station (BS) rationally to satisfy the most customs by using the least BSs. An ameliorated GA is proposed to search for the optimum solution. In the algorithm, we mesh the area to be planned according to least overlap length derived from coverage radius, bring into isometric grid encoding method to represent BS distribution as well as its number and develop select, crossover and mutation operators to serve our unique necessity. We also construct our comprehensive object function after synthesizing coverage ratio, overlap ratio, population and geographical conditions. Finally, after importing an electronic map of the area to be planned, a recommended strategy draft would be exported correspondingly. We eventually import HongKong, China to simulate and yield a satisfactoryActive triangulation measurement systems with a rigid geometric configuration are inappropriate for scanning large objects with low measuring tolerances. The reason is that the ratio between the depth recovery error and the lateral extension is a constant that depends on the geometric setup. As a consequence, measuring large areas with low depth recovery error requires the use of multiresolution techniques. We propose a multiresolution technique based on a camera-projector system previously calibrated. The method consists of changing the camera or projector's parameters in order to increase the system depth sensitivity. A subpixel retroprojection error in the self-calibration process and a decrease of approximately one order of magnitude in the depth recovery error can be achieved using the proposed method.
Wound healing in the pediatric patient is of utmost clinical and social importance, since hypertrophic scarring can have aesthetic and psychological sequelae, from early childhood to late adolescence. Wound healing is a well-orchestrated reparative response affecting the damaged tissue at the cellular, tissue, organ, and system scales. While tremendous progress has been made towards understanding wound healing at the individual temporal and spatial scales, its effects across the scales remain severely understudied and poorly understood. Here we discuss the critical need for systems-based computational modeling of wound healing healing in the pediatric patient could open new avenues in treating children with skin disorders such as birth defects, skin cancer, wounds, and burn injuries. PMID:23314298
Enuresis is defined as intermittent urinary incontinence during sleep in a child at least five years of age. Approximately 5% to 10% of all seven-year-olds have enuresis, and an estimated 5 to 7 million children in the United States have enuresis. The pathophysiology of primary nocturnal enuresis involves the inability to awaken from sleep in response to a full bladder, coupled with excessive nighttime urine production or a decreased functional capacity of the bladder. Initial evaluation should include a history, physical examination, and urinalysis. Several conditions, such as constipation, obstructive sleep apnea, diabetes mellitus, diabetes insipidus, chronic kidney disease, and psychiatric disorders, are associated with enuresis. If identified, these conditions should be evaluated and treated. Treatment of primary monosymptomatic enuresis (i.e., the only symptom is nocturnal bed-wetting in a child who has never been dry) begins with counseling the child and parents on effective behavioral modifications. First-line treatments for enuresis include bed alarm therapy and desmopressin. The choice of therapy is based on the child's age and nighttime voiding patterns, and the desires of the child and family. Referral to a pediatric urologist is indicated for children with primary enuresis refractory to standard and combination therapies, and for children with some secondary causes of enuresis, including urinary tract malformations, recurrent urinary tract infections, or neurologic disorders. PMID:25369644
A concise study of ternary and cubic algebras with Z3 grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, S3, and its abelian subgroup Z3 may play an important role in quantum physics. We show then how most of important algebras with Z2 grading can be generalized with ternary composition laws combined with a Z3 grading. We investigate in particular a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, , one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra. An analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator, and some properties of its eigenfunctions are briefly discussed.…
My purpose in this paper is to illustrate the way in which an enactivist methodological approach guided me as I conducted a two-case longitudinal study where the learning of algebra was explored in different contexts throughout time. Three groups of students in two different schools in the city of Puebla, Mexico, were followed from the last year…
The periodicity of Mercury and Venus' transits is regulated by the resonances of the orbital periods of these planets with the Earth's one. A simple algebraicapproach is presented for explaining the resonance of 243 years of Venus and the one of 46 years for Mercury and the transits in May and November for Mercury and June and December for Venus.
In the present study, I examine a modelling strategy as employed by a teacher in the context of an algebra lesson. The actions of this teacher suggest that a modelling approach will have a greater impact on enriching student learning if we do not lose sight of the need to manage associated cognitive loads that could either aid or hinder the… ruleA new mathematical representation is proposed for the configuration space structure induced by recombination, which we call "P-structure." It consists of a mapping of pairs ofapproach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator. PMID:10021760 | 677.169 | 1 |
This Electronic Calculator Teaches Math with Little or no Supervision!
If you are interested in learning and studying math in an engaging and interactive context, the Mathlab Scientific Graphing Calculator is the perfect electronic device. An intelligent scientific calculator that does far more th | 677.169 | 1 |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
The 5 R's Of Math
Basic mathematics formula written on a small elementary blackboard with various paints, crayons and pencils on a school desk.
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Students faced with the daunting task of studying for algebra or calculus or any level of math exam should learn the five Rs of math.
They are: 1.) Recopy the work. Copy all theorems, principles and definitions without condensing and copy the instructors' explanation, too. 2.) Rework model problems over and over until they can be completed without stopping. 3.) Recite – out loud – to explain each step of the problem-solving process. Suggestion: pets make good listeners. 4). Recheck work, and 5.) test for Reasonableness.
This means ensure that the answer makes sense. Students enrolled in college math courses need to block out at least 2 hours of math homework for every hour of class time.
Applying these study tips should help them tackle the tough math problems for a rewarding passing grade pay | 677.169 | 1 |
Overview
Whether you're new to fractions, decimals, and percentages or just brushing up on those topics, CliffsQuickReview Basic Math and Pre-Algebra can help. This guide introduces each topic, defines key terms, and walks you through each sample problem step-by-step. In no time, you'll be ready to tackle other concepts in this book such as
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CliffsQuickReview Basic Math and Pre-Algebra acts as a supplement to your textbook and to classroom lectures. Use this reference in any way that fits your personal style for study and review — you decide what works best with your needs. Here are just a few ways you can search for topics:
Use the free Pocket Guide full of essential information
Get a glimpse of what you'll gain from a chapter by reading through the Chapter Check-In at the beginning of each chapter
Use the Chapter Checkout at the end of each chapter to gauge your grasp of the important information you need to know
Test your knowledge more completely in the CQR Review and look for additional sources of information in the CQR Resource Center
Use the glossary to find key terms fast.
With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades.
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Jerry Bobrow, PhD, is an award-winning teacher and educator. He is a national authority in the field of test preparation. As executive directory of Bobrow Test Preparation Services, Dr. Bobrow has been administering the test preparation programs for most California State Universities for the past 27 years. Dr. Bobrow has authored more than 30 national best-selling test preparation books including Cliffs Preparation Guides for the GRE, GMAT, MSAT, SAT I, CBEST, NTE, ACT, and PPST. Each year he personally lectures to thousands of students on preparing for these important exams.
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a good review book that goes from basic math through pre-algebra without skipping parts or dragging. The explanations are clear, and detailed and there are enough practice questions to make sure you get the point. I would highly suggest this! | 677.169 | 1 |
Variables Project/Basic Algebraic Equations PBL
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Students use inquiry to drive their interest in completing this project. It contains a real-life scenario, algebraic equations, expressions, and variables. Note: teacher should change first word of code to say correct school name (currently, it says creator's school | 677.169 | 1 |
Newton's Method
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This is topic 29 in my AP Calculus curriculum. This PowerPoint guides students through how to apply Newton's Method to approximate roots of functions. This topic is not on the AB or the BC Calculus exams, but may be used after the exam. You may either have students go through the PowerPoint on their own or you may do it as a whole class. It is best used with the corresponding notes that I have created for | 677.169 | 1 |
Written by Manhattan Prepā??s high-caliber GRE instructors, the Number Properties GRE Strategy Guide provides a comprehensive analysis of the properties and rules of integers tested on the GRE. Rebuild the skills you need with thorough coverage of divisibility and primes, odds and evens, positives and negatives, exponents, roots, and more. Each chapter... more...
Written by Manhattan Prepā??s high-caliber GRE instructors, the Fractions, Decimals, & Percents GRE Strategy Guide provides an in-depth look at the variety of GRE questions that test this surprisingly tricky area of math. Learn to see the connections among these part-whole relationships as you hone the skills you need to manipulate digits, decimals,... more...
Written by Manhattan Prepā??s high-caliber GRE instructors, this Guide is really two books in one, thoroughly covering two of the quantitative question types on the GRE. The Guide to Quantitative Comparisons provides you with a framework for understanding what Quantitative Comparisons questions test. Most importantly, it discusses time-saving strategies... more...
Written by Manhattan Prepā??s high-caliber GRE instructors, the Algebra GRE Strategy Guide provides thorough coverage of this fundamental branch of math. Focused on the ways in which Algebra is tested on the GRE, this book will help you grasp core concepts and nuanced approaches for solving every type of algebraic problem. Each chapter provides comprehensive... more...
Written by Manhattan Prepā??s high-caliber GRE instructors, the Geometry GRE Strategy Guide equips you with powerful tools to comprehend and solve every geometry problem on the GRE. Refresh your knowledge of shapes, planes, lines, angles, objects, and more. Learn to understand the concepts and grasp their applications, mastering not only fundamental... more...
Manhattan Prepā??s 500 GRE Math Flash Cards provides practical exposure to the basic math concepts tested on the GRE. Designed to be user-friendly for all students, these cards include easy-to-follow explanations of math concepts that promise to enhance comprehension and build fundamental skills. Try using these cards with Manhattan Prepā??s 2-volume... more...
The Manhattan Prep Logic Games LSAT Strategy Guide is truly cutting-edge. Containing the best of Manhattan Prep?s proven strategies, this book will teach you how to tackle the LSAT games efficiently and flexibly. Beginning with how to recognize each and every game type, the Logic Games LSAT Strategy Guide takes you through the entire solving process.... more... | 677.169 | 1 |
This new edition further prepares future teachers to impact students' mathematical understanding.
Reconceptualizing Mathematics, founded on research and studies of learning and mathematics teaching for many years, is designed for use in classrooms in which students take an active part in learning and experience doing math. The esteemed author team has written the only textbook of its kind to both incorporate aspects of student-centered learning into lessons and model the teaching that will be expected of their students. To this end, the authors provide worthwhile tasks, activities, and support for facilitating discussions.
Quantitative reasoning and problem solving are recurring themes in Reconceptualizing Mathematics. The authors approach problem solving that teaches students to understand the quantities embedded in the situation and how they relate to each other.
Maximize Teaching and Learning with WebAssign Premium Macmillan Learning and WebAssign have partnered to deliver WebAssign Premium – a comprehensive and flexible suite of resources for your math for teachers course. Combining the most widely used online homework platform with authoritative textbook content and Macmillan's esteemed Mathtools, WebAssign Premium extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at | 677.169 | 1 |
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Show More four thousand years. This book will focus on just a small part of the story, in a sense the most recent chapter of it: the mathematics of western Europe from the sixteenth to the nineteenth centuries. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations. Almost every source is given in its original form, not just in the language in which it was first written, but as far as practicable in the layout and typeface in which it was read by contemporaries. This book is designed to provide mathematics undergraduates with some historical background to the material that is now taught universally to students in their final years at school and the first years at college or university: the core subjects of calculus, analysis, and abstract algebra, along with others such as mechanics, probability, and number theory. All of these evolved into their present form in a relatively limited area of western Europe from the mid sixteenth century onwards, and it is there that we find the major writings that relate in a recognizable way to contemporary | 677.169 | 1 |
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Annotation: Clearly defines all the mathematical terms used in primary school and is packed full of colourful photographs, illustrations and real-world examples to bring maths alive. This book is a welcome reference for both students and parents | 677.169 | 1 |
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The aim of this self-study Functional Skills Maths Level 2 course is to ensure that you are equipped with the important mathematical skills that will be useful to you in everyday | 677.169 | 1 |
Linear Functions Unit Project
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For this project students are instructed to create their own business as they learn, or after they have learned, about linear functions. Their business must sell a product or service for a constant rate (providing a constant slope), so that each student is able to write a linear function to model the profit of their business. The project involves students creating tables, graphs, and written descriptions of their functions. It is also a great project to differentiate! Students can make more or less complex functions based upon their readiness level.
This document contains a full description of the project and its requirements, the academic vocabulary the students will learn and use, the questions students must answer in their project presentation, and the 8th grade Common Core Mathematics Standards that the project addresses | 677.169 | 1 |
Easy Learning GCSE Maths Exam Practice Workbook for AQA B includes exam-style practice questions with highlighted grade levels so that students know exactly which grade they are working at, making revising for GCSE Maths easy.
Book Description Collins 017318
Book Description Collins 01247318
Book Description Collins 017318 | 677.169 | 1 |
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