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Sample records for two-semester introductory calculus

  1. A Two-Semester Precalculus/Calculus I Sequence: A Case Study

    Science.gov (United States)

    Axtell, Michael

    2006-01-01

    The article begins by highlighting recent trends and concerns in post-secondary Calculus and Precalculus education. The main purpose of the article is to discuss the transition to a new Precalculus/Calculus I two-semester course at Wabash College, a small liberal arts college for men. Three years of data from the earlier, traditional, sequence are…

  2. A Two-Semester Precalculus/Calculus I Sequence: A Case Study

    Science.gov (United States)

    Axtell, Michael

    2006-01-01

    The article begins by highlighting recent trends and concerns in post-secondary Calculus and Precalculus education. The main purpose of the article is to discuss the transition to a new Precalculus/Calculus I two-semester course at Wabash College, a small liberal arts college for men. Three years of data from the earlier, traditional, sequence are…

  3. Development and Implementation of a Two-Semester Introductory Organic-Bioorganic Chemistry Sequence: Conclusions from the First Six Years

    Science.gov (United States)

    Goess, Brian C.

    2014-01-01

    A two-semester second-year introductory organic chemistry sequence featuring one semester of accelerated organic chemistry followed by one semester of bioorganic chemistry is described. Assessment data collected over a six-year period reveal that such a course sequence can facilitate student mastery of fundamental organic chemistry in the first…

  4. Development and Implementation of a Two-Semester Introductory Organic-Bioorganic Chemistry Sequence: Conclusions from the First Six Years

    Science.gov (United States)

    Goess, Brian C.

    2014-01-01

    A two-semester second-year introductory organic chemistry sequence featuring one semester of accelerated organic chemistry followed by one semester of bioorganic chemistry is described. Assessment data collected over a six-year period reveal that such a course sequence can facilitate student mastery of fundamental organic chemistry in the first…

  5. Analyzing Conceptual Gains in Introductory Calculus with Interactively-Engaged Teaching Styles

    Science.gov (United States)

    Thomas, Matthew

    2013-01-01

    This dissertation examines the relationship between an instructional style called Interactive-Engagement (IE) and gains on a measure of conceptual knowledge called the Calculus Concept Inventory (CCI). The data comes from two semesters of introductory calculus courses (Fall 2010 and Spring 2011), consisting of a total of 482 students from the…

  6. Introductory analysis a deeper view of calculus

    CERN Document Server

    Bagby, Richard J

    2000-01-01

    Introductory Analysis addresses the needs of students taking a course in analysis after completing a semester or two of calculus, and offers an alternative to texts that assume that math majors are their only audience. By using a conversational style that does not compromise mathematical precision, the author explains the material in terms that help the reader gain a firmer grasp of calculus concepts.* Written in an engaging, conversational tone and readable style while softening the rigor and theory* Takes a realistic approach to the necessary and accessible level of abstraction for the secondary education students* A thorough concentration of basic topics of calculus* Features a student-friendly introduction to delta-epsilon arguments * Includes a limited use of abstract generalizations for easy use* Covers natural logarithms and exponential functions* Provides the computational techniques often encountered in basic calculus

  7. Transitioning from Introductory Calculus to Formal Limit Conceptions

    Science.gov (United States)

    Nagle, Courtney

    2013-01-01

    The limit concept is a fundamental mathematical notion both for its practical applications and its importance as a prerequisite for later calculus topics. Past research suggests that limit conceptualizations promoted in introductory calculus are far removed from the formal epsilon-delta definition of limit. In this article, I provide an overview…

  8. Promoting Metacognition in Introductory Calculus-based Physics Labs

    Science.gov (United States)

    Grennell, Drew; Boudreaux, Andrew

    2010-10-01

    In the Western Washington University physics department, a project is underway to develop research-based laboratory curriculum for the introductory calculus-based course. Instructional goals not only include supporting students' conceptual understanding and reasoning ability, but also providing students with opportunities to engage in metacognition. For the latter, our approach has been to scaffold reflective thinking with guided questions. Specific instructional strategies include analysis of alternate reasoning presented in fictitious dialogues and comparison of students' initial ideas with their lab group's final, consensus understanding. Assessment of student metacognition includes pre- and post- course data from selected questions on the CLASS survey, analysis of written lab worksheets, and student opinion surveys. CLASS results are similar to a traditional physics course and analysis of lab sheets show that students struggle to engage in a metacognitive process. Future directions include video studies, as well as use of additional written assessments adapted from educational psychology.

  9. Calculus introductory theory and applications in physical and life science

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    Johnson, R M

    1995-01-01

    This lucid and balanced introduction for first year engineers and applied mathematicians conveys the clear understanding of the fundamentals and applications of calculus, as a prelude to studying more advanced functions. Short and fundamental diagnostic exercises at the end of each chapter test comprehension before moving to new material.Provides a clear understanding of the fundamentals and applications of calculus, as a prelude to studying more advanced functionsIncludes short, useful diagnostic exercises at the end of each chapter

  10. Transversality of Electromagnetic Waves in the Calculus-Based Introductory Physics Course

    Science.gov (United States)

    Burko, Lior M.

    2008-01-01

    Introductory calculus-based physics textbooks state that electromagnetic waves are transverse and list many of their properties, but most such textbooks do not bring forth arguments why this is so. Both physical and theoretical arguments are at a level appropriate for students of courses based on such books, and could be readily used by…

  11. History-Infused Lessons in Introductory Calculus at the Secondary Level: Students' Learning and Perceptions

    Science.gov (United States)

    Poh, Wei Beng; Dindyal, Jaguthsing

    2016-01-01

    A history-infused lesson package developed by a team of teachers in a professional learning community was used to teach introductory calculus in a secondary school. First, we report a quasi-experimental design that showed that students in the experimental group performed significantly better than students in the control group. Second, we report on…

  12. Success in Introductory Calculus: The Role of High School and Pre-Calculus Preparation

    Science.gov (United States)

    Ayebo, Abraham; Ukkelberg, Sarah; Assuah, Charles

    2017-01-01

    Calculus at the college level has significant potential to serve as a pump for increasing the number of students majoring in STEM fields. It is a foundation course for all STEM majors and, if mastered well, should provide students with a positive and successful first-year experience and gateway into more advanced courses. Studies have shown that a…

  13. Instructors’ reasons for choosing problem features in a calculus-based introductory physics course

    Directory of Open Access Journals (Sweden)

    Edit Yerushalmi

    2010-08-01

    Full Text Available This study investigates how the beliefs and values of physics faculty influence their choice of physics problems for their students in an introductory physics course. The study identifies the goals these instructors have for their students, the problem features they believe facilitate those goals, and how those features correspond to problems they choose to use in their classes. This analysis comes from an artifact-based interview of 30 physics faculty teaching introductory calculus-based physics at a wide variety of institutions. The study concludes that instructors’ goals and the problem features they believe support those goals align with research-based curricular materials intended to develop competent problem solvers. However, many of these instructors do not use the beneficial problem features because they believe these features conflict with a more powerful set of values concerned with clarity of presentation and minimizing student stress, especially on exams.

  14. Capsule calculus

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    Ritow, Ira

    2003-01-01

    This brief introductory text presents the basic principles of calculus from the engineering viewpoint. Excellent either as a refresher or as an introductory course, it focuses on developing familiarity with the basic principles rather than presenting detailed proofs.Topics include differential calculus, in terms of differentiation and elementary differential equations; integral calculus, in simple and multiple integration forms; time calculus; equations of motion and their solution; complex variables; complex algebra; complex functions; complex and operational calculus; and simple and inverse

  15. Calculus

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    Jones, Patrick

    2014-01-01

    Practice makes perfect-and helps deepen your understanding of calculus 1001 Calculus Practice Problems For Dummies takes you beyond the instruction and guidance offered in Calculus For Dummies, giving you 1001 opportunities to practice solving problems from the major topics in your calculus course. Plus, an online component provides you with a collection of calculus problems presented in multiple-choice format to further help you test your skills as you go. Gives you a chance to practice and reinforce the skills you learn in your calculus courseHelps you refine your understanding of calculusP

  16. Calculus

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    Grossman, Stanley I

    1981-01-01

    Calculus, Second Edition discusses the techniques and theorems of calculus. This edition introduces the sine and cosine functions, distributes ?-? material over several chapters, and includes a detailed account of analytic geometry and vector analysis.This book also discusses the equation of a straight line, trigonometric limit, derivative of a power function, mean value theorem, and fundamental theorems of calculus. The exponential and logarithmic functions, inverse trigonometric functions, linear and quadratic denominators, and centroid of a plane region are likewise elaborated. Other topics

  17. Calculus

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    Larson, Ron

    2014-01-01

    The Larson CALCULUS program has a long history of innovation in the calculus market. It has been widely praised by a generation of students and professors for its solid and effective pedagogy that addresses the needs of a broad range of teaching and learning styles and environments. Each title is just one component in a comprehensive calculus course program that carefully integrates and coordinates print, media, and technology products for successful teaching and learning.

  18. Newton's Bridge Learning Community: Can Student Learning in Introductory Physics and Calculus be a Pathway to Undergraduate Research?

    Science.gov (United States)

    Li, Eugene

    2014-03-01

    A pathway to undergraduate research for freshman level physics through interdisciplinary pairings of physics and calculus courses is examined. Through ``pairing courses,'' active learning approaches, and jointly constructed inquiry-based course activities, students formulate and investigate a ``research problem.'' Some effects of a student-peer-mentor program is also examined. The use of technology incorporated into ``theme-focused'' activities is outlined. Some of the technological components include the iPad, Vernier sensors with related software, and introductory MATLAB. This presentation analyzes some of the outcomes of the learning community pairing of calculus-based Physics I (Mechanics and Heat) and Math (Calculus II), called a ``A Journey Across Newton's Bridge,'' and also the follow-up course pairing calculus-based Physics II (Electricity and Magnetism) and Multi-variable calculus called ``Multi-Dimensional Experiences'' which are being offered at Montgomery College. Acknowledge support of the Department of Physics, Engineering and Geoscience, Montgomery College, Noyce TPOD-STEM, and GT-STEP Grants.

  19. Calculus

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    Spivak, Michael

    2006-01-01

    Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis.

  20. Calculus

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    Grossman, Stanley I

    1984-01-01

    Calculus, Third Edition emphasizes the techniques and theorems of calculus, including many applied examples and exercises in both drill and applied-type problems.This book discusses shifting the graphs of functions, derivative as a rate of change, derivative of a power function, and theory of maxima and minima. The area between two curves, differential equations of exponential growth and decay, inverse hyperbolic functions, and integration of rational functions are also elaborated. This text likewise covers the fluid pressure, ellipse and translation of axes, graphing in polar coordinates, pro

  1. Calculus

    CERN Document Server

    Zandy, Bernard V

    2003-01-01

    We take great notes-and make learning a snap When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core Calculus concepts-from functions, limits, and derivatives to differentials, integration, and definite integrals- and get the best possible grade. At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented writers who know how to cut to the chase- and zero in on the essential information you need to succeed.

  2. The Benefits of Completing Homework for Students with Different Aptitudes in an Introductory Electricity and Magnetism Course

    Science.gov (United States)

    2015-02-17

    college- level physics course principally because both physics professors and students believe to be an effective tool for learning physics. That is...our research. All students at the United States Air Force Academy (USAF A) take two semesters of calculus-based introductory physics, although only...spring semester . In this study, we chose to focus on the E&M course rather than the Mechanics course for several reasons - students have had less high

  3. Calculus refresher

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    Klaf, A A

    1956-01-01

    This book is unique in English as a refresher for engineers, technicians, and students who either wish to brush up their calculus or find parts of calculus unclear. It is not an ordinary textbook. It is, instead, an examination of the most important aspects of integral and differential calculus in terms of the 756 questions most likely to occur to the technical reader. It provides a very easily followed presentation and may also be used as either an introductory or supplementary textbook. The first part of this book covers simple differential calculus, with constants, variables, functions, inc

  4. Computer Graphics as an Instructional Aid in an Introductory Differential Calculus Course

    Directory of Open Access Journals (Sweden)

    Tapan Kumar Tiwari

    2007-02-01

    Full Text Available Mathematicians in general claim that the Computer Algebra Systems (CAS provide an excellent tool for illustrating calculus concepts. They caution, however, against heavy dependency on the CAS for all computational purposes without the mastery of the procedures involved. This study examined the effect of using the graphical and numerical capabilities of Mathematica as a supplemental instructional tool in enhancing the conceptual knowledge and problem solving abilities of students in a differential calculus course. Topics of differential calculus were introduced by the traditional lecture method to both the control and experimental groups comprised of students enrolled in two sections of the Business and Life Sciences I course. Mathematica was used only by the students of the experimental group to reinforce and illustrate the concepts developed by the traditional method. A content analysis was conducted using the qualitative data obtained from students’ explanations of the derivative of a function. The quantitative data, the students’ test scores, were analyzed using ANCOVA. The results showed that students in the experimental group scored higher than students in the control group on both the conceptual and the computational parts of the examination. The qualitative analysis results revealed that, compared to the control group, a higher percentage of students in the experimental group had a better understanding of the derivative.

  5. Putting Differentials Back into Calculus

    Science.gov (United States)

    Dray, Tevian; Manogue, Corrine A.

    2010-01-01

    We argue that the use of differentials in introductory calculus courses is useful and provides a unifying theme, leading to a coherent view of the calculus. Along the way, we meet several interpretations of differentials, some better than others.

  6. Putting Differentials Back into Calculus

    Science.gov (United States)

    Dray, Tevian; Manogue, Corrine A.

    2010-01-01

    We argue that the use of differentials in introductory calculus courses is useful and provides a unifying theme, leading to a coherent view of the calculus. Along the way, we meet several interpretations of differentials, some better than others.

  7. Calculus Demonstrations Using MATLAB

    Science.gov (United States)

    Dunn, Peter K.; Harman, Chris

    2002-01-01

    The note discusses ways in which technology can be used in the calculus learning process. In particular, five MATLAB programs are detailed for use by instructors or students that demonstrate important concepts in introductory calculus: Newton's method, differentiation and integration. Two of the programs are animated. The programs and the…

  8. Data Analysis Activities and Problems for the Computer Science Major in a Post-calculus Introductory Statistics Course

    OpenAIRE

    Sanchez, Juana

    2011-01-01

    The material presented here is a very small subset of problems currently being prepared for a larger instructional improvement pro ject funded by the Office of Instructional Development (OID) at UCLA. The objective of the pro ject is to create a manual with data sets and contextual problems for Computer Science majors that will complement the textbooks used in the calculus-based upper-division Applied Statistics course. More than one third of the students in this course are from Computer Sci...

  9. The absolute differential calculus (calculus of tensors)

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    Levi-Civita, Tullio

    2013-01-01

    Written by a towering figure of twentieth-century mathematics, this classic examines the mathematical background necessary for a grasp of relativity theory. Tullio Levi-Civita provides a thorough treatment of the introductory theories that form the basis for discussions of fundamental quadratic forms and absolute differential calculus, and he further explores physical applications.Part one opens with considerations of functional determinants and matrices, advancing to systems of total differential equations, linear partial differential equations, algebraic foundations, and a geometrical intro

  10. Two-dimensional calculus

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    Osserman, Robert

    2011-01-01

    The basic component of several-variable calculus, two-dimensional calculus is vital to mastery of the broader field. This extensive treatment of the subject offers the advantage of a thorough integration of linear algebra and materials, which aids readers in the development of geometric intuition. An introductory chapter presents background information on vectors in the plane, plane curves, and functions of two variables. Subsequent chapters address differentiation, transformations, and integration. Each chapter concludes with problem sets, and answers to selected exercises appear at the end o

  11. Technical calculus with analytic geometry

    CERN Document Server

    Gersting, Judith L

    2010-01-01

    This well-thought-out text, filled with many special features, is designed for a two-semester course in calculus for technology students with a background in college algebra and trigonometry. The author has taken special care to make the book appealing to students by providing motivating examples, facilitating an intuitive understanding of the underlying concepts involved, and by providing much opportunity to gain proficiency in techniques and skills.Initial chapters cover functions and graphs, straight lines and conic sections, new coordinate systems, the derivative, using the derivative, in

  12. Development of a two semester sequence of design oriented composites courses

    Energy Technology Data Exchange (ETDEWEB)

    O`Toole, B.J. [Univ. of Nevada, Las Vegas, NV (United States). Mechanical Engineering Dept.

    1994-12-31

    A two semester series of courses has been developed which is an introduction to the design, analysis, manufacturing and testing of composite materials. The courses are ideal for junior or senior level undergraduates or beginning graduate students who have no prior knowledge of composites. This sequence is well suited for a department starting to add composites courses and research activity and which only has a small number of interested faculty members. Some of the topics in this two course sequence are taken and condensed from six separate courses taught at the University of Delaware. Details about suitable textbooks, software, handouts, and resource requirements are discussed. The first course, titled ``Introduction to Composite Materials,`` includes an overview of material systems and manufacturing methods, microstructural analysis, lamina and laminate theory, design procedures, failure analysis, and the use of composites analysis software. The class is divided into small groups of students and each group selects a project early in the semester. The objective of each group is to learn enough to make competent decisions about material selection, fabrication techniques (prototype and mass production), laminate design, and failure prediction. The second course is a laboratory course which includes experimental procedures for material characterization, wet lay-up, mold making, autoclave, and pultrusion manufacturing techniques, and the design of experiments for component testing, The design groups expand their previous paper studies by fabricating and testing their designs. Student reaction has been positive and teaching these courses is enjoyable but time consuming.

  13. Calculus Unlimited

    OpenAIRE

    Marsden, Jerrold; Weinstein, Alan J.

    1981-01-01

    Purpose: This book is intended to supplement our text, Calculus (Benjamin/Cummings, 1980), or virtually any other calculus text (see page vii, How To Use This Book With Your Calculus Text). As the title Calculus Unlimited implies, this text presents an alternative treatment of calculus using the method of exhaustion for the derivative and integral in place of limits. With the aid of this method, a definition of the derivative may be introduced in the first lecture of a calculus course for stu...

  14. Infinitesimal calculus

    CERN Document Server

    Henle, James M

    2003-01-01

    Introducing calculus at the basic level, this text covers hyperreal numbers and hyperreal line, continuous functions, integral and differential calculus, fundamental theorem, infinite sequences and series, infinite polynomials, more. 1979 edition.

  15. Introductory real analysis

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    Kolmogorov, A N; Silverman, Richard A

    1975-01-01

    Self-contained and comprehensive, this elementary introduction to real and functional analysis is readily accessible to those with background in advanced calculus. It covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, and much more. 350 problems. 1970 edition.

  16. Calculus light

    CERN Document Server

    Friedman, Menahem

    2011-01-01

    Another Calculus book? As long as students find calculus scary, the failure rate in mathematics is higher than in all other subjects, and as long as most people mistakenly believe that only geniuses can learn and understand mathematics, there will always be room for a new book of Calculus. We call it Calculus Light. This book is designed for a one semester course in ""light"" calculus -- mostly single variable, meant to be used by undergraduate students without a wide mathematical background and who do not major in mathematics but study subjects such as engineering, biology or management infor

  17. Operational calculus

    CERN Document Server

    Boehme, Thomas K

    1987-01-01

    Operational Calculus, Volume II is a methodical presentation of operational calculus. An outline of the general theory of linear differential equations with constant coefficients is presented. Integral operational calculus and advanced topics in operational calculus, including locally integrable functions and convergence in the space of operators, are also discussed. Formulas and tables are included.Comprised of four sections, this volume begins with a discussion on the general theory of linear differential equations with constant coefficients, focusing on such topics as homogeneous and non-ho

  18. Multivariate calculus and geometry

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    Dineen, Seán

    2014-01-01

    Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations. In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions. Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.

  19. Tuplix Calculus

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    Bergstra, J A; van der Zwaag, M B

    2007-01-01

    We introduce a calculus for tuplices, which are expressions that generalize matrices and vectors. Tuplices have an underlying data type for quantities that are taken from a zero-totalized field. We start with the core tuplix calculus CTC for entries and tests, which are combined using conjunctive composition. We define a standard model and prove that CTC is relatively complete with respect to it. The core calculus is extended with operators for choice, information hiding, scalar multiplication, clearing and encapsulation. We provide two examples of applications; one on incremental financial budgeting, and one on modular financial budget design.

  20. Tuplix Calculus

    Directory of Open Access Journals (Sweden)

    J.A. Bergstra

    2008-01-01

    Full Text Available We introduce a calculus for tuplices, which are expressions that generalize matrices and vectors. Tuplices have an underlying data type for quantities that are taken from a zero-totalized field. We start with the core tuplix calculus CTC for entries and tests, which are combined using conjunctive composition. We define a standard model and prove that CTC is relatively complete with respect to it. The core calculus is extended with operators for choice, information hiding, scalar multiplication, clearing and encapsulation. We provide two examples of applications; one on incremental financial budgeting, and one on modular financial budget design.

  1. Continuation calculus

    Directory of Open Access Journals (Sweden)

    Bram Geron

    2013-09-01

    Full Text Available Programs with control are usually modeled using lambda calculus extended with control operators. Instead of modifying lambda calculus, we consider a different model of computation. We introduce continuation calculus, or CC, a deterministic model of computation that is evaluated using only head reduction, and argue that it is suitable for modeling programs with control. It is demonstrated how to define programs, specify them, and prove them correct. This is shown in detail by presenting in CC a list multiplication program that prematurely returns when it encounters a zero. The correctness proof includes termination of the program. In continuation calculus we can model both call-by-name and call-by-value. In addition, call-by-name functions can be applied to call-by-value results, and conversely.

  2. ESeal Calculus: A Secure Mobile Calculus

    Institute of Scientific and Technical Information of China (English)

    Peng Rong; Chen Xin-meng; Liu Ping

    2003-01-01

    The ESeal Calculus is a secure mobile calculus based on Seal Calculus. By using open-channels, ESeal Calculus makes it possible to communicate between any two arbitrary seals with some secure restrictions. It improves the expression ability and efficiency of Seal calculus without losing security.

  3. Metaphor Clusters: Characterizing Instructor Metaphorical Reasoning on Limit Concepts in College Calculus

    Science.gov (United States)

    Patel, Rita Manubhai; McCombs, Paul; Zollman, Alan

    2014-01-01

    Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student…

  4. Advanced calculus

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    Nickerson, HK; Steenrod, NE

    2011-01-01

    ""This book is a radical departure from all previous concepts of advanced calculus,"" declared the Bulletin of the American Mathematics Society, ""and the nature of this departure merits serious study of the book by everyone interested in undergraduate education in mathematics."" Classroom-tested in a Princeton University honors course, it offers students a unified introduction to advanced calculus. Starting with an abstract treatment of vector spaces and linear transforms, the authors introduce a single basic derivative in an invariant form. All other derivatives - gradient, divergent, curl,

  5. Advanced calculus

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    Widder, David V

    2012-01-01

    This classic text by a distinguished mathematician and former Professor of Mathematics at Harvard University, leads students familiar with elementary calculus into confronting and solving more theoretical problems of advanced calculus. In his preface to the first edition, Professor Widder also recommends various ways the book may be used as a text in both applied mathematics and engineering.Believing that clarity of exposition depends largely on precision of statement, the author has taken pains to state exactly what is to be proved in every case. Each section consists of definitions, theorem

  6. CLEP calculus

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    Hill, Gregory

    2013-01-01

    Earn College Credit with REA's Test Prep for CLEP* Calculus Everything you need to pass the exam and get the college credit you deserve.Our test prep for CLEP* Calculus and the free online tools that come with it, will allow you to create a personalized CLEP* study plan that can be customized to fit you: your schedule, your learning style, and your current level of knowledge.Here's how it works:Diagnostic exam at the REA Study Center focuses your studyOur online diagnostic exam pinpoints your strengths and shows you exactly where you need to focus your study. Armed with this information, you

  7. Advanced calculus of a single variable

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    Geveci, Tunc

    2016-01-01

    This advanced undergraduate textbook is based on a one-semester course on single variable calculus that the author has been teaching at San Diego State University for many years. The aim of this classroom-tested book is to deliver a rigorous discussion of the concepts and theorems that are dealt with informally in the first two semesters of a beginning calculus course. As such, students are expected to gain a deeper understanding of the fundamental concepts of calculus, such as limits (with an emphasis on ε-δ definitions), continuity (including an appreciation of the difference between mere pointwise and uniform continuity), the derivative (with rigorous proofs of various versions of L’Hôpital’s rule) and the Riemann integral (discussing improper integrals in-depth, including the comparison and Dirichlet tests). Success in this course is expected to prepare students for more advanced courses in real and complex analysis and this book will help to accomplish this. The first semester of advanced calculus...

  8. Formal calculus and umbral calculus

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    Robinson, Thomas J

    2009-01-01

    In this paper we use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus and we introduce and study certain operators generalizing the classical umbral shifts. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a short, elementary proof which naturally arose as a motivating computation related to a certain crucial "associativity" property of an important class of vertex operator algebras. Very similar (somewhat forgotten) proofs had appeared by the 19-th century, of course without any motivation related to vertex operator algebras. Using this formula, we derive certain results, including especially the calculation of certain adjoint operators, of the classical umbral calculus. This is, roughly speaking, a reversal of the logical development of some standard treatments, which have obtained formulas for the higher derivatives of a composite function, most not...

  9. Modern calculus and analytic geometry

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    Silverman, Richard A

    2012-01-01

    A self-contained text for an introductory course, this volume places strong emphasis on physical applications. Key elements of differential equations and linear algebra are introduced early and are consistently referenced, all theorems are proved using elementary methods, and numerous worked-out examples appear throughout. The highly readable text approaches calculus from the student's viewpoint and points out potential stumbling blocks before they develop. A collection of more than 1,600 problems ranges from exercise material to exploration of new points of theory - many of the answers are fo

  10. Advanced calculus

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    Friedman, Avner

    2007-01-01

    This rigorous two-part treatment advances from functions of one variable to those of several variables. Intended for students who have already completed a one-year course in elementary calculus, it defers the introduction of functions of several variables for as long as possible, and adds clarity and simplicity by avoiding a mixture of heuristic and rigorous arguments.The first part explores functions of one variable, including numbers and sequences, continuous functions, differentiable functions, integration, and sequences and series of functions. The second part examines functions of several

  11. Matrix calculus

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    Bodewig, E

    1959-01-01

    Matrix Calculus, Second Revised and Enlarged Edition focuses on systematic calculation with the building blocks of a matrix and rows and columns, shunning the use of individual elements. The publication first offers information on vectors, matrices, further applications, measures of the magnitude of a matrix, and forms. The text then examines eigenvalues and exact solutions, including the characteristic equation, eigenrows, extremum properties of the eigenvalues, bounds for the eigenvalues, elementary divisors, and bounds for the determinant. The text ponders on approximate solutions, as well

  12. Advanced calculus

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    Fitzpatrick, Patrick M

    2009-01-01

    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 Euclide

  13. Calculus diaries

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    Ouellette,, Jennifer

    2011-01-01

    Jennifer Ouellette never took maths in the sixth form, mostly because she – like most of us – assumed she wouldn't need it much in real life. But then the English graduate, now an award-winning science-writer, had a change of heart and decided to revisit the equations and formulas that had haunted her youth. The Calculus Diaries is the fun and fascinating account of a year spent confronting her numbers-phobia head on. With wit and verve, Ouellette explains how she discovered that maths could apply to everything from petrol mileages to dieting, rollercoaster rides to winning in Las Vegas.

  14. Calculator calculus

    CERN Document Server

    McCarty, George

    1982-01-01

    How THIS BOOK DIFFERS This book is about the calculus. What distinguishes it, however, from other books is that it uses the pocket calculator to illustrate the theory. A computation that requires hours of labor when done by hand with tables is quite inappropriate as an example or exercise in a beginning calculus course. But that same computation can become a delicate illustration of the theory when the student does it in seconds on his calculator. t Furthermore, the student's own personal involvement and easy accomplishment give hi~ reassurance and en­ couragement. The machine is like a microscope, and its magnification is a hundred millionfold. We shall be interested in limits, and no stage of numerical approximation proves anything about the limit. However, the derivative of fex) = 67.SgX, for instance, acquires real meaning when a student first appreciates its values as numbers, as limits of 10 100 1000 t A quick example is 1.1 , 1.01 , 1.001 , •••• Another example is t = 0.1, 0.01, in the functio...

  15. Renal calculus

    CERN Document Server

    Pyrah, Leslie N

    1979-01-01

    Stone in the urinary tract has fascinated the medical profession from the earliest times and has played an important part in the development of surgery. The earliest major planned operations were for the removal of vesical calculus; renal and ureteric calculi provided the first stimulus for the radiological investigation of the viscera, and the biochemical investigation of the causes of calculus formation has been the training ground for surgeons interested in metabolic disorders. It is therefore no surprise that stone has been the subject of a number of monographs by eminent urologists, but the rapid development of knowledge has made it possible for each one of these authors to produce something new. There is still a technical challenge to the surgeon in the removal of renal calculi, and on this topic we are always glad to have the advice of a master craftsman; but inevitably much of the interest centres on the elucidation of the causes of stone formation and its prevention. Professor Pyrah has had a long an...

  16. Student understanding of calculus within physics and mathematics classrooms

    Science.gov (United States)

    Christensen, Warren; Thompson, John

    2010-03-01

    The earliest results in Physics Education Research demonstrated the challenges facing students in understanding the graphical interpretations of slope, derivative, and area under curves in the context of kinematics. As part of ongoing research on mathematical challenges that may underlie documented physics difficulties, we developed and administered a brief survey on single- and multivariable calculus concepts to students within physics and mathematics classrooms at both the introductory and advanced levels. Initial findings among students in multivariable calculus show that as many as one in five students encounter some type of difficulty when asked to rank the slopes at five different points along a single path. We will present further data on the extent to which students in a first semester calculus course and an introductory calculus-based physics course encounter similar challenges.

  17. Propositional Calculus in Coq

    OpenAIRE

    van Doorn, Floris

    2015-01-01

    I formalize important theorems about classical propositional logic in the proof assistant Coq. The main theorems I prove are (1) the soundness and completeness of natural deduction calculus, (2) the equivalence between natural deduction calculus, Hilbert systems and sequent calculus and (3) cut elimination for sequent calculus.

  18. Propositional Calculus in Coq

    OpenAIRE

    van Doorn, Floris

    2015-01-01

    I formalize important theorems about classical propositional logic in the proof assistant Coq. The main theorems I prove are (1) the soundness and completeness of natural deduction calculus, (2) the equivalence between natural deduction calculus, Hilbert systems and sequent calculus and (3) cut elimination for sequent calculus.

  19. Calculus problems

    CERN Document Server

    Baronti, Marco; van der Putten, Robertus; Venturi, Irene

    2016-01-01

    This book, intended as a practical working guide for students in Engineering, Mathematics, Physics, or any other field where rigorous calculus is needed, includes 450 exercises. Each chapter starts with a summary of the main definitions and results, which is followed by a selection of solved exercises accompanied by brief, illustrative comments. A selection of problems with indicated solutions rounds out each chapter. A final chapter explores problems that are not designed with a single issue in mind but instead call for the combination of a variety of techniques, rounding out the book’s coverage. Though the book’s primary focus is on functions of one real variable, basic ordinary differential equations (separation of variables, linear first order and constant coefficients ODEs) are also discussed. The material is taken from actual written tests that have been delivered at the Engineering School of the University of Genoa. Literally thousands of students have worked on these problems, ensuring their real-...

  20. Baseline training in cognitive and psychodynamic psychotherapy during a psychologist training program. Exploring client outcomes in therapies of one or two semesters.

    Science.gov (United States)

    Dennhag, Inga; Armelius, Bengt-Åke

    2012-01-01

    This effectiveness study explored the outcomes of 187 clients seen by 187 students undergoing baseline training in psychotherapy. Clients reduced their symptoms (SCL-90) and increased their positive self-image (SASB introject) during the therapy. Multiple regression analyses showed no differences between the cognitive and the psychodynamic training approaches and no differences between one and two semesters duration of the therapies. However, 2-3% of variance in end states was accounted for by the interaction between the variables, indicating a moderating effect of duration in the two approaches. Outcomes for clients in the cognitive training approach were significantly better with two semesters than with one semester, while there was no such difference in the psychodynamic approach. Consequences for baseline training are discussed.

  1. The Association of Precollege Use of Calculators with Student Performance in College Calculus

    Science.gov (United States)

    Mao, Yi; White, Tyreke; Sadler, Philip M.; Sonnert, Gerhard

    2017-01-01

    This study investigates how the use of calculators during high school mathematics courses is associated with student performance in introductory college calculus courses in the USA. Data were drawn from a nationally representative sample of 7087 students enrolled in college calculus at 134 colleges and universities. They included information about…

  2. Student Achievement in College Calculus, Louisiana State University 1967-1968.

    Science.gov (United States)

    Scannicchio, Thomas Henry

    An investigation of freshmen achievement in an introductory calculus course was performed on the basis of high school mathematics background to find predictors of college calculus grades. Overall high school academic achievement, overall high school mathematics achievement, number of high school mathematics units, pattern of college preparatory…

  3. Tablet PC: A Preliminary Report on a Tool for Teaching Calculus

    Science.gov (United States)

    Gorgievski, Nicholas; Stroud, Robert; Truxaw, Mary; DeFranco, Thomas

    2005-01-01

    This study examined students' perceptions of the Tablet PC as an instructional tool for teaching Calculus. A thirteen item survey was developed by the researchers and administered to 103 students in an introductory Calculus course at a large university in the Northeast of the United States. The purpose of this survey was to collect data regarding…

  4. The Association of Precollege Use of Calculators with Student Performance in College Calculus

    Science.gov (United States)

    Mao, Yi; White, Tyreke; Sadler, Philip M.; Sonnert, Gerhard

    2017-01-01

    This study investigates how the use of calculators during high school mathematics courses is associated with student performance in introductory college calculus courses in the USA. Data were drawn from a nationally representative sample of 7087 students enrolled in college calculus at 134 colleges and universities. They included information about…

  5. Calculus of one variable

    CERN Document Server

    Grossman, Stanley I

    1986-01-01

    Calculus of One Variable, Second Edition presents the essential topics in the study of the techniques and theorems of calculus.The book provides a comprehensive introduction to calculus. It contains examples, exercises, the history and development of calculus, and various applications. Some of the topics discussed in the text include the concept of limits, one-variable theory, the derivatives of all six trigonometric functions, exponential and logarithmic functions, and infinite series.This textbook is intended for use by college students.

  6. Introductory Overviews

    NARCIS (Netherlands)

    Jakeman, A.J.; Hamilton, S.H.; Athanasiadis, I.N.; Pierce, S.A.

    2015-01-01

    Introductory Overview articles are designed to provide introductory level background to key themes and topics that caters to the eclectic readership of EMS. It is envisaged that these articles will help to break down barriers to shared understanding and dialogue within multidisciplinary teams, and m

  7. On the refinement calculus

    CERN Document Server

    Vickers, Trevor

    1992-01-01

    On the Refinement Calculus gives one view of the development of the refinement calculus and its attempt to bring together - among other things - Z specifications and Dijkstra's programming language. It is an excellent source of reference material for all those seeking the background and mathematical underpinnings of the refinement calculus.

  8. From Random Walks to Brownian Motion, from Diffusion to Entropy: Statistical Principles in Introductory Physics

    Science.gov (United States)

    Reeves, Mark

    2014-03-01

    will present a number of these exercises, with particular focus on the hands-on experiments done by the students, and will give examples of the tangible material that our students work with throughout the two-semester sequence of their course on introductory physics with a bio focus. Supported by NSF DUE.

  9. Students' Exploratory Thinking about a Nonroutine Calculus Task

    Science.gov (United States)

    Nabb, Keith

    2013-01-01

    In this article on introductory calculus, intriguing questions are generated that can ignite an appreciation for the subject of mathematics. These questions open doors to advanced mathematical thinking and harness many elements of research-oriented mathematics. Such questions also offer greater incentives for students to think and reflect.…

  10. Students' Exploratory Thinking about a Nonroutine Calculus Task

    Science.gov (United States)

    Nabb, Keith

    2013-01-01

    In this article on introductory calculus, intriguing questions are generated that can ignite an appreciation for the subject of mathematics. These questions open doors to advanced mathematical thinking and harness many elements of research-oriented mathematics. Such questions also offer greater incentives for students to think and reflect.…

  11. Fundamentals of calculus

    CERN Document Server

    Morris, Carla C

    2015-01-01

    Fundamentals of Calculus encourages students to use power, quotient, and product rules for solutions as well as stresses the importance of modeling skills.  In addition to core integral and differential calculus coverage, the book features finite calculus, which lends itself to modeling and spreadsheets.  Specifically, finite calculus is applied to marginal economic analysis, finance, growth, and decay.  Includes: Linear Equations and FunctionsThe DerivativeUsing the Derivative Exponential and Logarithmic Functions Techniques of DifferentiationIntegral CalculusIntegration TechniquesFunctions

  12. Calculus II For Dummies

    CERN Document Server

    Zegarelli, Mark

    2012-01-01

    An easy-to-understand primer on advanced calculus topics Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. Calculus II For Dummies offers expert instruction, advice, and tips to help second semester calculus students get a handle on the subject and ace their exams. It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals. This hands-on guide also covers sequences and series, wit

  13. Introductory Chemistry

    OpenAIRE

    Baron, Mark; Gonzalez-Rodriguez, Jose; Stevens, Gary; Gray, Nathan; Atherton, Thomas; Winn, Joss

    2010-01-01

    Teaching and Learning resources for the 1st Year Introductory Chemistry course (Forensic Science). 30 credits. These are Open Educational Resources (OER), made available for re-use under a Creative Commons license.

  14. Introductory remarks

    NARCIS (Netherlands)

    Kiefer, Friedemann; Schulte-Merker, Stefan

    2014-01-01

    This introductory section briefly highlights the subsequent chapters in the context of recent findings and open questions in lymphatic vessel biology. It aims to provide a quick overview and orientation in the contents of this monograph collection.

  15. Using Isomorphic Problems to Learn Introductory Physics

    CERN Document Server

    Lin, Shih-Yin

    2016-01-01

    In this study, we examine introductory physics students' ability to perform analogical reasoning between two isomorphic problems which employ the same underlying physics principles but have different surface features. Three hundred and sixty two students from a calculus-based and an algebra-based introductory physics course were given a quiz in the recitation in which they had to first learn from a solved problem provided and take advantage of what they learned from it to solve another problem (which we call the quiz problem) which was isomorphic. Previous research suggests that the multiple-concept quiz problem is challenging for introductory students. Students in different recitation classes received different interventions in order to help them discern and exploit the underlying similarities of the isomorphic solved and quiz problems. We also conducted think-aloud interviews with four introductory students in order to understand in-depth the difficulties they had and explore strategies to provide better sc...

  16. Understanding Introductory Students' Application of Integrals in Physics from Multiple Perspectives

    Science.gov (United States)

    Hu, Dehui

    2013-01-01

    Calculus is used across many physics topics from introductory to upper-division level college courses. The concepts of differentiation and integration are important tools for solving real world problems. Using calculus or any mathematical tool in physics is much more complex than the straightforward application of the equations and algorithms that…

  17. Noncausal stochastic calculus

    CERN Document Server

    Ogawa, Shigeyoshi

    2017-01-01

    This book presents an elementary introduction to the theory of noncausal stochastic calculus that arises as a natural alternative to the standard theory of stochastic calculus founded in 1944 by Professor Kiyoshi Itô. As is generally known, Itô Calculus is essentially based on the "hypothesis of causality", asking random functions to be adapted to a natural filtration generated by Brownian motion or more generally by square integrable martingale. The intention in this book is to establish a stochastic calculus that is free from this "hypothesis of causality". To be more precise, a noncausal theory of stochastic calculus is developed in this book, based on the noncausal integral introduced by the author in 1979. After studying basic properties of the noncausal stochastic integral, various concrete problems of noncausal nature are considered, mostly concerning stochastic functional equations such as SDE, SIE, SPDE, and others, to show not only the necessity of such theory of noncausal stochastic calculus but ...

  18. The Safe Lambda Calculus

    CERN Document Server

    Blum, William

    2009-01-01

    Safety is a syntactic condition of higher-order grammars that constrains occurrences of variables in the production rules according to their type-theoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition to the setting of the simply-typed lambda calculus. In contrast to the original definition of safety, our calculus does not constrain types (to be homogeneous). We show that in the safe lambda calculus, there is no need to rename bound variables when performing substitution, as variable capture is guaranteed not to happen. We also propose an adequate notion of beta-reduction that preserves safety. In the same vein as Schwichtenberg's 1976 characterization of the simply-typed lambda calculus, we show that the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a characterization of representable word functions. We then study the ...

  19. Student Research in an Introductory Psychology Course: Outcomes of Two Experiential Learning Projects and Implications for Instruction of Human Subjects Research

    Science.gov (United States)

    Downey, Christina A.

    2013-01-01

    The present study describes student learning and personal outcomes associated with learning research methods in introductory psychology, via one of two semester-long projects: one involving performing naturalistic observation of the behavior of community members, and the other involving performing a 60-minute interview of local veterans regarding…

  20. Infinitary lambda calculus

    NARCIS (Netherlands)

    Kennaway, J.R.; Klop, J.W.; Sleep, M.R.; Vries, F.-J. de

    1995-01-01

    In a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems. In this paper we perform the same task for the lambda calculus. From the viewpoint of infinitary rewriting, the Böhm model of the lambda calculus can be seen as an infinitary term model

  1. The stochastic quality calculus

    DEFF Research Database (Denmark)

    Zeng, Kebin; Nielson, Flemming; Nielson, Hanne Riis

    2014-01-01

    We introduce the Stochastic Quality Calculus in order to model and reason about distributed processes that rely on each other in order to achieve their overall behaviour. The calculus supports broadcast communication in a truly concurrent setting. Generally distributed delays are associated...

  2. Initialized Fractional Calculus

    Science.gov (United States)

    Lorenzo, Carl F.; Hartley, Tom T.

    2000-01-01

    This paper demonstrates the need for a nonconstant initialization for the fractional calculus and establishes a basic definition set for the initialized fractional differintegral. This definition set allows the formalization of an initialized fractional calculus. Two basis calculi are considered; the Riemann-Liouville and the Grunwald fractional calculi. Two forms of initialization, terminal and side are developed.

  3. Essential calculus with applications

    CERN Document Server

    Silverman, Richard A

    1989-01-01

    Rigorous but accessible text introduces undergraduate-level students to necessary background math, then clear coverage of differential calculus, differentiation as a tool, integral calculus, integration as a tool, and functions of several variables. Numerous problems and a supplementary section of ""Hints and Answers."" 1977 edition.

  4. Calculus of variations and nonlinear partial differential equations

    CERN Document Server

    Marcellini, Paolo

    2008-01-01

    This volume provides the texts of lectures given by L. Ambrosio, L. Caffarelli, M. Crandall, L.C. Evans, N. Fusco at the Summer course held in Cetraro (Italy) in 2005. These are introductory reports on current research by world leaders in the fields of calculus of variations and partial differential equations. The topics discussed are transport equations for nonsmooth vector fields, homogenization, viscosity methods for the infinite Laplacian, weak KAM theory and geometrical aspects of symmetrization. A historical overview of all CIME courses on the calculus of variations and partial differential equations is contributed by Elvira Mascolo.

  5. Calculus with applications

    CERN Document Server

    Lax, Peter D

    2014-01-01

    This new edition of Lax, Burstein, and Lax's Calculus with Applications and Computing offers meaningful explanations of the important theorems of single variable calculus. Written with students in mathematics, the physical sciences, and engineering in mind, and revised with their help, it shows that the themes of calculation, approximation, and modeling are central to mathematics and the main ideas of single variable calculus. This edition brings the innovation of the first edition to a new generation of students. New sections in this book use simple, elementary examples to show that when applying calculus concepts to approximations of  functions, uniform convergence is more natural and easier to use than point-wise convergence. As in the original, this edition includes material that is essential for students in science and engineering, including an elementary introduction to complex numbers and complex-valued functions, applications of calculus to modeling vibrations and population dynamics, and an introduc...

  6. Discrete fractional calculus

    CERN Document Server

    Goodrich, Christopher

    2015-01-01

    This text provides the first comprehensive treatment of the discrete fractional calculus. Experienced researchers will find the text useful as a reference for discrete fractional calculus and topics of current interest. Students who are interested in learning about discrete fractional calculus will find this text to provide a useful starting point. Several exercises are offered at the end of each chapter and select answers have been provided at the end of the book. The presentation of the content is designed to give ample flexibility for potential use in a myriad of courses and for independent study. The novel approach taken by the authors includes a simultaneous treatment of the fractional- and integer-order difference calculus (on a variety of time scales, including both the usual forward and backwards difference operators). The reader will acquire a solid foundation in the classical topics of the discrete calculus while being introduced to exciting recent developments, bringing them to the frontiers of the...

  7. Introductory Persian.

    Science.gov (United States)

    Stilo, Donald L.; And Others

    This introductory text for Persian was designed for use in intensive Peace Corps training, with emphasis on the field of teaching English as a foreign language. The text includes an introduction which gives the language instructor a brief outline of how to teach the course. Each lesson consists of a dialogue followed by pattern practices of…

  8. Impact of Calculus Reform in a Liberal Arts Calculus Course.

    Science.gov (United States)

    Brosnan, Patricia A.; Ralley, Thomas G.

    This report describes the changes in a freshman-level calculus course that occurred as a consequence of adopting the Harvard Consortium Calculus text. The perspective is that of the lecturer. The course is intended as an introduction to calculus for liberal arts students, that is, students who will not be expected to use calculus as a mathematical…

  9. Hermeneutic operative calculus

    Science.gov (United States)

    Ramakrishnan, Sivakumar; Isawasan, Pradeep; Mohanan, Vasuky

    2014-07-01

    The predicate calculus used currently by mathematical logic in computer science, philosophy and linguistic was found to be too restrictive and inadequate for describing the grammar of natural and artificial language. Therefore many higher order logics have been developed to overcome the limitation of predicate calculus. In this paper a new representation of logic using mathematical principles has been developed for the natural language called Hermeneutic Operative Calculus. This Hermeneutic Operative Calculus is a new language interpretive calculus developed to account for the syntactic, semantic and pragmatic features of natural language and allows removing the restrictions of any particular natural language in the semantic field its map out. The logic of Hermeneutic Operative Calculus capable of represent the syntactic and semantic of factual information of a natural language precisely in any language. The logic of this Hermeneutic Operative Calculus has two different forms of operations called object and meta-operations. The object operation allow for listing the various objects, picturing the various propositions and so forth. The meta-operation would specify what cannot be specified by the object operation like semantical stances of a proposition. The basic operative processes of linguistics and cognitive logic will be mathematically conceptualized and elaborated in this paper.

  10. Effect of Written Presentation on Performance in Introductory Physics

    Science.gov (United States)

    Stewart, John; Ballard, Shawn

    2010-01-01

    This study examined the written work of students in the introductory calculus-based electricity and magnetism course at the University of Arkansas. The students' solutions to hourly exams were divided into a small set of countable features organized into three major categories, mathematics, language, and graphics. Each category was further divided…

  11. Effect of Written Presentation on Performance in Introductory Physics

    Science.gov (United States)

    Stewart, John; Ballard, Shawn

    2010-01-01

    This study examined the written work of students in the introductory calculus-based electricity and magnetism course at the University of Arkansas. The students' solutions to hourly exams were divided into a small set of countable features organized into three major categories, mathematics, language, and graphics. Each category was further divided…

  12. Introductory Life Science Mathematics and Quantitative Neuroscience Courses

    Science.gov (United States)

    Duffus, Dwight; Olifer, Andrei

    2010-01-01

    We describe two sets of courses designed to enhance the mathematical, statistical, and computational training of life science undergraduates at Emory College. The first course is an introductory sequence in differential and integral calculus, modeling with differential equations, probability, and inferential statistics. The second is an…

  13. Calculus for dummies

    CERN Document Server

    Ryan, Mark

    2014-01-01

    Slay the calculus monster with this user-friendly guide Calculus For Dummies, 2nd Edition makes calculus manageable-even if you're one of the many students who sweat at the thought of it. By breaking down differentiation and integration into digestible concepts, this guide helps you build a stronger foundation with a solid understanding of the big ideas at work. This user-friendly math book leads you step-by-step through each concept, operation, and solution, explaining the ""how"" and ""why"" in plain English instead of math-speak. Through relevant instruction and practical examples, you'll s

  14. Calculus of variations

    CERN Document Server

    Elsgolc, L E; Stark, M

    1961-01-01

    Calculus of Variations aims to provide an understanding of the basic notions and standard methods of the calculus of variations, including the direct methods of solution of the variational problems. The wide variety of applications of variational methods to different fields of mechanics and technology has made it essential for engineers to learn the fundamentals of the calculus of variations. The book begins with a discussion of the method of variation in problems with fixed boundaries. Subsequent chapters cover variational problems with movable boundaries and some other problems; sufficiency

  15. Generalized Gaussian Error Calculus

    CERN Document Server

    Grabe, Michael

    2010-01-01

    For the first time in 200 years Generalized Gaussian Error Calculus addresses a rigorous, complete and self-consistent revision of the Gaussian error calculus. Since experimentalists realized that measurements in general are burdened by unknown systematic errors, the classical, widespread used evaluation procedures scrutinizing the consequences of random errors alone turned out to be obsolete. As a matter of course, the error calculus to-be, treating random and unknown systematic errors side by side, should ensure the consistency and traceability of physical units, physical constants and physical quantities at large. The generalized Gaussian error calculus considers unknown systematic errors to spawn biased estimators. Beyond, random errors are asked to conform to the idea of what the author calls well-defined measuring conditions. The approach features the properties of a building kit: any overall uncertainty turns out to be the sum of a contribution due to random errors, to be taken from a confidence inter...

  16. A Logical Process Calculus

    Science.gov (United States)

    Cleaveland, Rance; Luettgen, Gerald; Bushnell, Dennis M. (Technical Monitor)

    2002-01-01

    This paper presents the Logical Process Calculus (LPC), a formalism that supports heterogeneous system specifications containing both operational and declarative subspecifications. Syntactically, LPC extends Milner's Calculus of Communicating Systems with operators from the alternation-free linear-time mu-calculus (LT(mu)). Semantically, LPC is equipped with a behavioral preorder that generalizes Hennessy's and DeNicola's must-testing preorder as well as LT(mu's) satisfaction relation, while being compositional for all LPC operators. From a technical point of view, the new calculus is distinguished by the inclusion of: (1) both minimal and maximal fixed-point operators and (2) an unimple-mentability predicate on process terms, which tags inconsistent specifications. The utility of LPC is demonstrated by means of an example highlighting the benefits of heterogeneous system specification.

  17. Computing for calculus

    CERN Document Server

    Christensen, Mark J

    1981-01-01

    Computing for Calculus focuses on BASIC as the computer language used for solving calculus problems.This book discusses the input statement for numeric variables, advanced intrinsic functions, numerical estimation of limits, and linear approximations and tangents. The elementary estimation of areas, numerical and string arrays, line drawing algorithms, and bisection and secant method are also elaborated. This text likewise covers the implicit functions and differentiation, upper and lower rectangular estimates, Simpson's rule and parabolic approximation, and interpolating polynomials. Other to

  18. Discounted Duration Calculus

    DEFF Research Database (Denmark)

    Ody, Heinrich; Fränzle, Martin; Hansen, Michael Reichhardt

    2016-01-01

    To formally reason about the temporal quality of systems discounting was introduced to CTL and LTL. However, these logic are discrete and they cannot express duration properties. In this work we introduce discounting for a variant of Duration Calculus. We prove decidability of model checking...... for a useful fragment of discounted Duration Calculus formulas on timed automata under mild assumptions. Further, we provide an extensive example to show the usefulness of the fragment....

  19. The stack calculus

    OpenAIRE

    Alberto Carraro; Thomas Ehrhard; Antonino Salibra

    2013-01-01

    We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry-Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without any restriction. Its type system enforces strong normalization of expressions and it is a sound and complete system for full implicational Classical Logic. We give a very simple denotational semantics which allows easy calculations of the interpretation of ...

  20. The stack calculus

    Directory of Open Access Journals (Sweden)

    Alberto Carraro

    2013-03-01

    Full Text Available We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry-Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without any restriction. Its type system enforces strong normalization of expressions and it is a sound and complete system for full implicational Classical Logic. We give a very simple denotational semantics which allows easy calculations of the interpretation of expressions.

  1. Introductory statistics

    CERN Document Server

    Ross, Sheldon M

    2005-01-01

    In this revised text, master expositor Sheldon Ross has produced a unique work in introductory statistics. The text's main merits are the clarity of presentation, contemporary examples and applications from diverse areas, and an explanation of intuition and ideas behind the statistical methods. To quote from the preface, ""It is only when a student develops a feel or intuition for statistics that she or he is really on the path toward making sense of data."" Ross achieves this goal through a coherent mix of mathematical analysis, intuitive discussions and examples.* Ross's clear writin

  2. Introductory statistics

    CERN Document Server

    Ross, Sheldon M

    2010-01-01

    In this 3rd edition revised text, master expositor Sheldon Ross has produced a unique work in introductory statistics. The text's main merits are the clarity of presentation, contemporary examples and applications from diverse areas, and an explanation of intuition and ideas behind the statistical methods. Concepts are motivated, illustrated and explained in a way that attempts to increase one's intuition. To quote from the preface, ""It is only when a student develops a feel or intuition for statistics that she or he is really on the path toward making sense of data."" Ross achieves this

  3. Symmetric π—Calculus

    Institute of Scientific and Technical Information of China (English)

    傅育熙

    1998-01-01

    An alternative presentation of the π-calculus is given.This version of the π-calculus is symmetric in the sense that communications are symmetric and there is no difference between input and output prefixes.The point of the symmetric π-calculus is that it has no abstract names.The set of closed names is therefore homogeneous.The π-calculus can be fully embedded into the symmetric π-calculus.The symmetry changes the emphasis of the communication mechanism of the π-calculus and opens up possibility for further variations.

  4. Biomedical Applications for Introductory Physics

    Science.gov (United States)

    Tuszynski, J. A.; Dixon, J. M.

    2001-12-01

    Can be utilized in either Algebra or Calculus-based courses and is available either as a standalone text or as a supplement for books like Cutnell PHYSICS, 5e or Halliday, Resnick, & Walker FUNDAMENTALS OF PHYSICS, 6e. Math level is Algebra & Trigonometry; however, a few examples require the use of integration and differentiation. Unlike competing supplements, Tuszinski offers both a wealth of engaging biomedical applications as well as quantitative problem-solving. The quantitative problem-solving is presented in the form of worked examples and homework problems. The quantitative problem-solving is presented in the form of worked examples and homework problems. The standard organization facilitates the integration of the material into most introductory courses.

  5. Reducing the failure rate in introductory physics classes

    Science.gov (United States)

    Saul, Jeff; Coulombe, Patrick; Lindell, Rebecca

    2017-01-01

    Calculus-based introductory physics courses are often among the most difficult at many colleges and universities. With the national movement to increase STEM majors, the introductory calculus-based courses need to be less of a weed-out course and more of a course that propels students forward into successful majors. This talk discusses two approaches to reduce DFW rates and improve student retention: studio courses and parachute courses. Studio courses integrate lecture/laboratory into one course where the primary mode of instruction is small group activities. Typically, any students enrolled in the college or university can enroll in a studio version of the course. Parachute courses on the other hand, focus on the poor performing students. Designed so that students not doing well in an introductory physics course can switch into the parachute class mid-semester without harm to their GPA. In addition, the parachute course focuses on helping students build the knowledge and skills necessary for success when retaking the calculus-based Physics course. The studio course format has been found to reduce DFW rates at several universities by 40-60% compared with separate lecture and laboratory format versions of the same courses, while parachutes courses were less successful. At one university, the parachute course succeeded in helping 80% of students maintain their GPA, but only helped 20% successfully pass the calculus-based physics course.

  6. A development calculus for specifications

    Institute of Scientific and Technical Information of China (English)

    李未

    2003-01-01

    A first order inference system, named R-calculus, is defined to develop the specifications.This system intends to eliminate the laws which are not consistent with users' requirements. TheR-calculus consists of the structural rules, an axiom, a cut rule, and the rules for logical connectives.Some examples are given to demonstrate the usage of the R-calculus. Furthermore, the propertiesregarding reachability and completeness of the R-calculus are formally defined and proved.

  7. Topology, calculus and approximation

    CERN Document Server

    Komornik, Vilmos

    2017-01-01

    Presenting basic results of topology, calculus of several variables, and approximation theory which are rarely treated in a single volume, this textbook includes several beautiful, but almost forgotten, classical theorems of Descartes, Erdős, Fejér, Stieltjes, and Turán. The exposition style of Topology, Calculus and Approximation follows the Hungarian mathematical tradition of Paul Erdős and others. In the first part, the classical results of Alexandroff, Cantor, Hausdorff, Helly, Peano, Radon, Tietze and Urysohn illustrate the theories of metric, topological and normed spaces. Following this, the general framework of normed spaces and Carathéodory's definition of the derivative are shown to simplify the statement and proof of various theorems in calculus and ordinary differential equations. The third and final part is devoted to interpolation, orthogonal polynomials, numerical integration, asymptotic expansions and the numerical solution of algebraic and differential equations. Students of both pure an...

  8. Noncommutative operational calculus

    Directory of Open Access Journals (Sweden)

    Henry E. Heatherly

    1999-12-01

    Full Text Available Oliver Heaviside's operational calculus was placed on a rigorous mathematical basis by Jan Mikusinski, who constructed an algebraic setting for the operational methods. In this paper, we generalize Mikusi'{n}ski's methods to solve linear ordinary differential equations in which the unknown is a matrix- or linear operator-valued function. Because these functions can be zero-divisors and do not necessarily commute, Mikusi'{n}ski's one-dimensional calculus cannot be used. The noncommuative operational calculus developed here,however, is used to solve a wide class of such equations. In addition, we provide new proofs of existence and uniqueness theorems for certain matrix- and operator valued Volterra integral and integro-differential equations. Several examples are given which demonstrate these new methods.

  9. Introductory Comments

    Directory of Open Access Journals (Sweden)

    Jolanta Sujecka

    2016-12-01

    Full Text Available Introductory Comments The fifth yearly volume of the Colloquia Humanistica comprises a thematic section on Nation, Natsiya, Ethnie. The subject it discusses has thus far received little attention as a research problem in the Slavia Orthodoxa, the Slavia Romana, the Balkans but also in Central and Eastern Europe.   Uwagi wstępne Piąty numer rocznika "Colloquia Humanistica" przedstawia dział tematyczny, poświęcony kategoriom narodu, nacji i etni. Temat ten, w takiej perspektywie, którą proponujemy, nie spotkał się dotąd z należytym namysłem badawczym w sferze Slavia Otrhodoxa, Slavia Romana i na Bałkanach, jak też w Europie Środkowo-Wschodniej.

  10. The History of the Calculus

    Science.gov (United States)

    Harding, Simon; Scott, Paul

    2004-01-01

    Calculus is a mathematical concept that is fundamental to how we understand the world around us. Whether it is in the world of technology, finance, astronomy, sociology, medicine, calculus in one form or another can be found. This brief article describes the origins of calculus in Greece, further developments by Newton and Leibniz, and the…

  11. The dagger lambda calculus

    Directory of Open Access Journals (Sweden)

    Philip Atzemoglou

    2014-12-01

    Full Text Available We present a novel lambda calculus that casts the categorical approach to the study of quantum protocols into the rich and well established tradition of type theory. Our construction extends the linear typed lambda calculus with a linear negation of "trivialised" De Morgan duality. Reduction is realised through explicit substitution, based on a symmetric notion of binding of global scope, with rules acting on the entire typing judgement instead of on a specific subterm. Proofs of subject reduction, confluence, strong normalisation and consistency are provided, and the language is shown to be an internal language for dagger compact categories.

  12. From calculus to analysis

    CERN Document Server

    Pedersen, Steen

    2015-01-01

    This textbook features applications including a proof of the Fundamental Theorem of Algebra, space filling curves, and the theory of irrational numbers.  In addition to the standard results of advanced calculus, the book  contains several interesting applications of these results. The text is intended to form a bridge between calculus and analysis. It is based on the authors lecture notes used and revised nearly every year over the last decade. The book contains numerous illustrations and cross references throughout, as well as exercises with solutions at the end of each section

  13. A calculus for quality

    DEFF Research Database (Denmark)

    Nielson, Hanne Riis; Nielson, Flemming; Vigo, Roberto

    2013-01-01

    A main challenge of programming component-based software is to ensure that the components continue to behave in a reasonable manner even when communication becomes unreliable. We propose a process calculus, the Quality Calculus, for programming software components where it becomes natural to plan...... for default behaviour in case the ideal behaviour fails due to unreliable communication and thereby to increase the quality of service offered by the systems. The development is facilitated by a SAT-based robustness analysis to determine whether or not the code is vulnerable to unreliable communication...

  14. The calculus primer

    CERN Document Server

    Schaaf, William L

    2011-01-01

    Comprehensive but concise, this introduction to differential and integral calculus covers all the topics usually included in a first course. The straightforward development places less emphasis on mathematical rigor, and the informal manner of presentation sets students at ease. Many carefully worked-out examples illuminate the text, in addition to numerous diagrams, problems, and answers.Bearing the needs of beginners constantly in mind, the treatment covers all the basic concepts of calculus: functions, derivatives, differentiation of algebraic and transcendental functions, partial different

  15. A Giant Urethral Calculus.

    Science.gov (United States)

    Sigdel, G; Agarwal, A; Keshaw, B W

    2014-01-01

    Urethral calculi are rare forms of urolithiasis. Majority of the calculi are migratory from urinary bladder or upper urinary tract. Primary urethral calculi usually occur in presence of urethral stricture or diverticulum. In this article we report a case of a giant posterior urethral calculus measuring 7x3x2 cm in a 47 years old male. Patient presented with acute retention of urine which was preceded by burning micturition and dribbling of urine for one week. The calculus was pushed in to the bladder through the cystoscope and was removed by suprapubic cystolithotomy.

  16. Calculus of variations

    CERN Document Server

    Gelfand, I M

    2000-01-01

    Based on a series of lectures given by I. M. Gelfand at Moscow State University, this book actually goes considerably beyond the material presented in the lectures. The aim is to give a treatment of the elements of the calculus of variations in a form both easily understandable and sufficiently modern. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws.The reader who merely wishes to become familiar with the most basic concepts and methods of the calculus of variations need on

  17. Schaum's outline of calculus

    CERN Document Server

    Ayres, Frank

    1999-01-01

    Students can gain a thorough understanding of differential and integral calculus with this powerful study tool. They'll also find the related analytic geometry much easier. The clear review of algebra and geometry in this edition will make calculus easier for students who wish to strengthen their knowledge in these areas. Updated to meet the emphasis in current courses, this new edition of a popular guide­­--more than 104,000 copies were bought of the prior edition--­­includes problems and examples using graphing calculators.

  18. Obstacles to Mathematization in Introductory Physics

    CERN Document Server

    Brahmia, S; Kanim, S E

    2016-01-01

    Recent studies have demonstrated that although physics students are generally successful executing mathematical procedures in context, they struggle with the use of mathematical concepts for sense making. University physics instructors often note that their students struggle with basic algebraic reasoning, a foundation on which more advanced mathematical thinking rests. However, little systematic research has been done to measure and categorize difficulties in this population. This paper describes a large-scale study (N > 600) designed to investigate trends in student reasoning with ratio and proportion, quantification, and symbolizing within the calculus-based introductory physics course. Although the assessment items require mathematical reasoning typically taught at the middle school level in mathematics courses, we find success rates of about 50% among calculus-based physics students. For many of these students, numerical complexity and physical context interferes with basic arithmetic reasoning. We argue...

  19. Duration Calculus: Logical Foundations

    DEFF Research Database (Denmark)

    Hansen, Michael Reichhardt; Chaochen, Zhou

    1997-01-01

    The Duration Calculus (abbreviated DC) represents a logical approach to formal design of real-time systems, where real numbers are used to model time and Boolean valued functions over time are used to model states and events of real-time systems. Since it introduction, DC has been applied to many...

  20. Lacroix and the calculus

    CERN Document Server

    Domingues, João Caramalho

    2008-01-01

    Silvestre François Lacroix (Paris, 1765 - ibid., 1843) was a most influential mathematical book author. His most famous work is the three-volume Traité du calcul différentiel et du calcul intégral (1797-1800; 2nd ed. 1810-1819) – an encyclopedic appraisal of 18th-century calculus which remained the standard reference on the subject through much of the 19th century, in spite of Cauchy's reform of the subject in the 1820's. Lacroix and the Calculus is the first major study of Lacroix’s large Traité. It uses the unique and massive bibliography given by Lacroix to explore late 18th-century calculus, and the way it is reflected in Lacroix’s account. Several particular aspects are addressed in detail, including: the foundations of differential calculus, analytic and differential geometry, conceptions of the integral, and types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions). Lacroix’s large Traité... was a...

  1. Provability Calculus of Constructions

    DEFF Research Database (Denmark)

    Nyblad, Kasten

    This thesis presents a type system, Provability Calculus of Constructions (PCoC) that can be used for the formalization of logic. In a theorem prover based on the system, the user can extend the prover with new inference rules in a logically consistent manner. This is done by representing PCo...

  2. A Virtual Class Calculus

    DEFF Research Database (Denmark)

    Ernst, Erik; Ostermann, Klaus; Cook, William Randall

    2006-01-01

    model for virtual classes has been a long-standing open question. This paper presents a virtual class calculus, vc, that captures the essence of virtual classes in these full-fledged programming languages. The key contributions of the paper are a formalization of the dynamic and static semantics of vc...

  3. Calculus Courses' Assessment Data

    Science.gov (United States)

    Pauna, Matti

    2017-01-01

    In this paper we describe computer-aided assessment methods used in online Calculus courses and the data they produce. The online learning environment collects a lot of time-stamped data about every action a student makes. Assessment data can be harnessed into use as a feedback, predictor, and recommendation facility for students and instructors.…

  4. From Calculating to Calculus

    Science.gov (United States)

    Steckroth, Jeffrey J.

    2010-01-01

    For nearly three decades, during which the author taught everything from basic algebra to advanced placement calculus, the author thought of himself as a secondary school mathematics teacher. The notion of teaching elementary school math never appealed to him because of its simplicity. The author stresses that anyone could teach children to count,…

  5. From Calculating to Calculus

    Science.gov (United States)

    Steckroth, Jeffrey J.

    2010-01-01

    For nearly three decades, during which the author taught everything from basic algebra to advanced placement calculus, the author thought of himself as a secondary school mathematics teacher. The notion of teaching elementary school math never appealed to him because of its simplicity. The author stresses that anyone could teach children to count,…

  6. Probabilistic Mu-Calculus

    DEFF Research Database (Denmark)

    Larsen, Kim Guldstrand; Mardare, Radu Iulian; Xue, Bingtian

    2016-01-01

    We introduce a version of the probabilistic µ-calculus (PMC) built on top of a probabilistic modal logic that allows encoding n-ary inequational conditions on transition probabilities. PMC extends previously studied calculi and we prove that, despite its expressiveness, it enjoys a series of good...

  7. Fractional calculus in bioengineering.

    Science.gov (United States)

    Magin, Richard L

    2004-01-01

    Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub

  8. Community structure in introductory physics course networks

    CERN Document Server

    Traxler, Adrienne L

    2015-01-01

    Student-to-student interactions are foundational to many active learning environments, but are most often studied using qualitative methods. Network analysis tools provide a quantitative complement to this picture, allowing researchers to describe the social interactions of whole classrooms as systems. Past results from introductory physics courses have suggested a sharp division in the formation of social structure between large lecture sections and small studio classroom environments. Extending those results, this study focuses on calculus-based introductory physics courses at a large public university with a heavily commuter and nontraditional student population. Community detection network methods are used to characterize pre- and post-course collaborative structure in several sections, and differences are considered between small and large classes. These results are compared with expectations from earlier findings, and comment on implications for instruction and further study.

  9. Using Analogies to Learn Introductory Physics

    CERN Document Server

    Lin, Shih-Yin

    2016-01-01

    Identifying the relevant physics principles is a central component of problem solving. A major goal of most introductory physics courses is to help students discern the deep similarities between problems based upon the physics principles so that they can transfer what they learned by solving one problem to solve another problem which involves the same principle. We conducted an investigation in which 251 calculus- and algebra-based introductory physics students were asked explicitly in the recitation quiz to learn from a solved problem and then solve another problem that has different surface features but the same underlying physics principles. We find that many students were able to discern the deep similarities between the problems. When the solved problem was provided, students were likely to invoke the correct principles; however, more scaffolding is needed to help students apply these principles correctly.

  10. Introduction to stochastic analysis and Malliavin calculus

    CERN Document Server

    Prato, Giuseppe

    2014-01-01

    This volume presents an introductory course on differential stochastic equations and Malliavin calculus. The material of the book has grown out of a series of courses delivered at the Scuola Normale Superiore di Pisa (and also at the Trento and Funchal Universities) and has been refined over several years of teaching experience in the subject. The lectures are addressed to a reader who is familiar with basic notions of measure theory and functional analysis. The first part is devoted to the Gaussian measure in a separable Hilbert space, the Malliavin derivative, the construction of the Brownian motion and Itô's formula. The second part deals with differential stochastic equations and their connection with parabolic problems. The third part provides an introduction to the Malliavin calculus. Several applications are given, notably the Feynman-Kac, Girsanov and Clark-Ocone formulae, the Krylov-Bogoliubov and Von Neumann theorems. In this third edition several small improvements are added and a new section devo...

  11. Assessing expertise in introductory physics using categorization task

    Directory of Open Access Journals (Sweden)

    Andrew Mason

    2011-10-01

    Full Text Available The ability to categorize problems based upon underlying principles, rather than surface features or contexts, is considered one of several proxy predictors of expertise in problem solving. With inspiration from the classic study by Chi, Feltovich, and Glaser, we assess the distribution of expertise among introductory physics students by asking three introductory physics classes, each with more than a hundred students, to categorize mechanics problems based upon similarity of solution. We compare their categorization with those of physics graduate students and faculty members. To evaluate the effect of problem context on students’ ability to categorize, two sets of problems were developed for categorization. Some problems in one set included those available from the prior study by Chi et al. We find a large overlap between calculus-based introductory students and graduate students with regard to their categorizations that were assessed as “good.” Our findings, which contrast with those of Chi et al., suggest that there is a wide distribution of expertise in mechanics among introductory and graduate students. Although the categorization task is conceptual, introductory students in the calculus-based course performed better than those in the algebra-based course. Qualitative trends in categorization of problems are similar between the non-Chi problems and problems available from the Chi study used in our study although the Chi problems used are more difficult on average.

  12. ESeal Calculus: A Secure Mobile Calculus

    Institute of Scientific and Technical Information of China (English)

    PengRong; UuPing

    2003-01-01

    The ESeal Calculus is a secure mobile calculus based on Seal Calculus. By using open-channels,ESeal Calculus makes it possible to communicate between any two arbitrary seals with some secure restrictions. It improves the expression ability and efficiency of Seal calculus without losing security.

  13. Gravitational Wave Detection in the Introductory Lab

    Science.gov (United States)

    Burko, Lior M.

    2017-01-01

    Great physics breakthroughs are rarely included in the introductory physics course. General relativity and binary black hole coalescence are no different, and can be included in the introductory course only in a very limited sense. However, we can design activities that directly involve the detection of GW150914, the designation of the Gravitation Wave signal detected on September 14, 2015, thereby engage the students in this exciting discovery directly. The activities naturally do not include the construction of a detector or the detection of gravitational waves. Instead, we design it to include analysis of the data from GW150914, which includes some interesting analysis activities for students of the introductory course. The same activities can be assigned either as a laboratory exercise or as a computational project for the same population of students. The analysis tools used here are simple and available to the intended student population. It does not include the sophisticated analysis tools, which were used by LIGO to carefully analyze the detected signal. However, these simple tools are sufficient to allow the student to get important results. We have successfully assigned this lab project for students of the introductory course with calculus at Georgia Gwinnett College.

  14. Using isomorphic problems to learn introductory physics

    Directory of Open Access Journals (Sweden)

    Shih-Yin Lin

    2011-08-01

    Full Text Available In this study, we examine introductory physics students’ ability to perform analogical reasoning between two isomorphic problems which employ the same underlying physics principles but have different surface features. Three hundred sixty-two students from a calculus-based and an algebra-based introductory physics course were given a quiz in the recitation in which they had to first learn from a solved problem provided and take advantage of what they learned from it to solve another problem (which we call the quiz problem which was isomorphic. Previous research suggests that the multiple-concept quiz problem is challenging for introductory students. Students in different recitation classes received different interventions in order to help them discern and exploit the underlying similarities of the isomorphic solved and quiz problems. We also conducted think-aloud interviews with four introductory students in order to understand in depth the difficulties they had and explore strategies to provide better scaffolding. We found that most students were able to learn from the solved problem to some extent with the scaffolding provided and invoke the relevant principles in the quiz problem. However, they were not necessarily able to apply the principles correctly. Research suggests that more scaffolding is needed to help students in applying these principles appropriately. We outline a few possible strategies for future investigation.

  15. The Malliavin calculus

    CERN Document Server

    Bell, Denis R

    2006-01-01

    This introduction to Malliavin's stochastic calculus of variations is suitable for graduate students and professional mathematicians. Author Denis R. Bell particularly emphasizes the problem that motivated the subject's development, with detailed accounts of the different forms of the theory developed by Stroock and Bismut, discussions of the relationship between these two approaches, and descriptions of a variety of applications.The first chapter covers enough technical background to make the subsequent material accessible to readers without specialized knowledge of stochastic analysis. Succe

  16. Pre-calculus essentials

    CERN Document Server

    Woodward, Ernest

    2012-01-01

    REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Pre-Calculus reviews sets, numbers, operations and properties, coordinate geometry, fundamental algebraic topics, solving equations and inequalities, functions, trigonometry, exponents

  17. Boolean integral calculus

    Science.gov (United States)

    Tucker, Jerry H.; Tapia, Moiez A.; Bennett, A. Wayne

    1988-01-01

    The concept of Boolean integration is developed, and different Boolean integral operators are introduced. Given the changes in a desired function in terms of the changes in its arguments, the ways of 'integrating' (i.e. realizing) such a function, if it exists, are presented. The necessary and sufficient conditions for integrating, in different senses, the expression specifying the changes are obtained. Boolean calculus has applications in the design of logic circuits and in fault analysis.

  18. Boolean integral calculus

    Science.gov (United States)

    Tucker, Jerry H.; Tapia, Moiez A.; Bennett, A. Wayne

    1988-01-01

    The concept of Boolean integration is developed, and different Boolean integral operators are introduced. Given the changes in a desired function in terms of the changes in its arguments, the ways of 'integrating' (i.e. realizing) such a function, if it exists, are presented. The necessary and sufficient conditions for integrating, in different senses, the expression specifying the changes are obtained. Boolean calculus has applications in the design of logic circuits and in fault analysis.

  19. Inductive situation calculus

    OpenAIRE

    Denecker, Marc; Ternovska, Eugenia

    2004-01-01

    Temporal reasoning has always been a major test case for knowledge representation formalisms. In this paper, we develop an inductive variant of the situation calculus in ED-logic, classical logic extended with inductive definitions. This logic has been proposed recently and is an extension of classical logic. It allows for a uniform representation of various forms of definitions, including monotone inductive definitions and non-monotone forms of inductive definitions such as iterated inductio...

  20. Calculus III essentials

    CERN Document Server

    REA, Editors of

    2012-01-01

    REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Calculus III includes vector analysis, real valued functions, partial differentiation, multiple integrations, vector fields, and infinite series.

  1. Understanding the calculus

    OpenAIRE

    Tall, David

    1985-01-01

    A number of significant changes have have occurred recently that give us a golden opportunity to review the teaching of calculus. The most obvious is the arrival of the microcomputer in the mathematics classroom, allowing graphic demonstrations and individual investigations into the mathematical ideas. But equally potent are new\\ud insights into mathematics and mathematics education that suggest new ways of approaching the subject.\\ud In this article I shall consider some of the difficulties ...

  2. Calculus I essentials

    CERN Document Server

    2012-01-01

    REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Calculus I covers functions, limits, basic derivatives, and integrals.

  3. Calculus with vectors

    CERN Document Server

    Treiman, Jay S

    2014-01-01

    Calculus with Vectors grew out of a strong need for a beginning calculus textbook for undergraduates who intend to pursue careers in STEM. fields. The approach introduces vector-valued functions from the start, emphasizing the connections between one-variable and multi-variable calculus. The text includes early vectors and early transcendentals and includes a rigorous but informal approach to vectors. Examples and focused applications are well presented along with an abundance of motivating exercises. All three-dimensional graphs have rotatable versions included as extra source materials and may be freely downloaded and manipulated with Maple Player; a free Maple Player App is available for the iPad on iTunes. The approaches taken to topics such as the derivation of the derivatives of sine and cosine, the approach to limits, and the use of "tables" of integration have been modified from the standards seen in other textbooks in order to maximize the ease with which students may comprehend the material. Additio...

  4. Effect of written presentation on performance in introductory physics

    OpenAIRE

    Shawn Ballard; John Stewart

    2010-01-01

    This study examined the written work of students in the introductory calculus-based electricity and magnetism course at the University of Arkansas. The students’ solutions to hourly exams were divided into a small set of countable features organized into three major categories, mathematics, language, and graphics. Each category was further divided into subfeatures. The total number of features alone explained more than 30% of the variance in exam scores and from 9% to 15% of the variance in c...

  5. Effect of written presentation on performance in introductory physics

    OpenAIRE

    Shawn Ballard; John Stewart

    2010-01-01

    This study examined the written work of students in the introductory calculus-based electricity and magnetism course at the University of Arkansas. The students’ solutions to hourly exams were divided into a small set of countable features organized into three major categories, mathematics, language, and graphics. Each category was further divided into subfeatures. The total number of features alone explained more than 30% of the variance in exam scores and from 9% to 15% of the variance in c...

  6. Introduction to the operational calculus

    CERN Document Server

    Berg, Lothar

    2013-01-01

    Introduction to the Operational Calculus is a translation of ""Einfuhrung in die Operatorenrechnung, Second Edition."" This book deals with Heaviside's interpretation, on the Laplace integral, and on Jan Mikusinki's fundamental work ""Operational Calculus."" Throughout the book, basic algebraic concepts appear as aids to understanding some relevant points of the subject. An important field for research in analysis is asymptotic properties. This text also discusses examples to show the potentialities in applying operational calculus that run beyond ordinary differential equations with constant

  7. Recent Progress in Regge Calculus

    OpenAIRE

    1997-01-01

    While there has been some advance in the use of Regge calculus as a tool in numerical relativity, the main progress in Regge calculus recently has been in quantum gravity. After a brief discussion of this progress, attention is focussed on two particular, related aspects. Firstly, the possible definitions of diffeomorphisms or gauge transformations in Regge calculus are examined and examples are given. Secondly, an investigation of the signature of the simplicial supermetric is described. Thi...

  8. k-Schur functions and affine Schubert calculus

    CERN Document Server

    Lam, Thomas; Morse, Jennifer; Schilling, Anne; Shimozono, Mark; Zabrocki, Mike

    2014-01-01

    This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry. This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with ...

  9. Polynomial Calculus: Rethinking the Role of Calculus in High Schools

    Science.gov (United States)

    Grant, Melva R.; Crombie, William; Enderson, Mary; Cobb, Nell

    2016-01-01

    Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in…

  10. Early Vector Calculus: A Path through Multivariable Calculus

    Science.gov (United States)

    Robertson, Robert L.

    2013-01-01

    The divergence theorem, Stokes' theorem, and Green's theorem appear near the end of calculus texts. These are important results, but many instructors struggle to reach them. We describe a pathway through a standard calculus text that allows instructors to emphasize these theorems. (Contains 2 figures.)

  11. Polynomial Calculus: Rethinking the Role of Calculus in High Schools

    Science.gov (United States)

    Grant, Melva R.; Crombie, William; Enderson, Mary; Cobb, Nell

    2016-01-01

    Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in…

  12. Early Vector Calculus: A Path through Multivariable Calculus

    Science.gov (United States)

    Robertson, Robert L.

    2013-01-01

    The divergence theorem, Stokes' theorem, and Green's theorem appear near the end of calculus texts. These are important results, but many instructors struggle to reach them. We describe a pathway through a standard calculus text that allows instructors to emphasize these theorems. (Contains 2 figures.)

  13. Polynomial calculus: rethinking the role of calculus in high schools

    Science.gov (United States)

    Grant, Melva R.; Crombie, William; Enderson, Mary; Cobb, Nell

    2016-08-01

    Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.

  14. Calculus in the Middle School?

    Science.gov (United States)

    Barger, Rita H.; McCoy, Ann C.

    2010-01-01

    This article presents an example of how middle school teachers can lay a foundation for calculus. Although many middle school activities connect directly to calculus concepts, the authors have decided to look in depth at only one: the concept of change. They will show how teachers can lead their students to see and appreciate the calculus…

  15. The Calculus of a Vase

    Science.gov (United States)

    Scherger, Nicole

    2012-01-01

    Of the most universal applications in integral calculus are those involved with finding volumes of solids of revolution. These profound problems are typically taught with traditional approaches of the disk and shell methods, after which most calculus curriculums will additionally cover arc length and surfaces of revolution. Even in these visibly…

  16. Student Attitudes and Calculus Reform.

    Science.gov (United States)

    Bookman, Jack; Friedman, Charles P.

    1998-01-01

    Compares the attitudes about mathematics of students from traditionally taught calculus classes and those taught in a "reformed" calculus course. Reports that one to two years after, reform students felt significantly more that they understood how math was used and that they had been required to understand math rather than to memorize formulas.…

  17. Calculus in the Middle School?

    Science.gov (United States)

    Barger, Rita H.; McCoy, Ann C.

    2010-01-01

    This article presents an example of how middle school teachers can lay a foundation for calculus. Although many middle school activities connect directly to calculus concepts, the authors have decided to look in depth at only one: the concept of change. They will show how teachers can lead their students to see and appreciate the calculus…

  18. The Calculus of a Vase

    Science.gov (United States)

    Scherger, Nicole

    2012-01-01

    Of the most universal applications in integral calculus are those involved with finding volumes of solids of revolution. These profound problems are typically taught with traditional approaches of the disk and shell methods, after which most calculus curriculums will additionally cover arc length and surfaces of revolution. Even in these visibly…

  19. Calculus problems and solutions

    CERN Document Server

    Ginzburg, A

    2011-01-01

    Ideal for self-instruction as well as for classroom use, this text helps students improve their understanding and problem-solving skills in analysis, analytic geometry, and higher algebra. More than 1,200 problems appear in the text, with concise explanations of the basic notions and theorems to be used in their solution. Many are followed by complete answers; solutions for the others appear at the end of the book. Topics include sequences, functions of a single variable, limit of a function, differential calculus for functions of a single variable, fundamental theorems and applications of dif

  20. Advanced calculus problem solver

    CERN Document Server

    REA, Editors of

    2012-01-01

    Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies.Here in this highly useful reference is the finest overview of advanced calculus currently av

  1. Calculus super review

    CERN Document Server

    2012-01-01

    Get all you need to know with Super Reviews! Each Super Review is packed with in-depth, student-friendly topic reviews that fully explain everything about the subject. The Calculus I Super Review includes a review of functions, limits, basic derivatives, the definite integral, combinations, and permutations. Take the Super Review quizzes to see how much you've learned - and where you need more study. Makes an excellent study aid and textbook companion. Great for self-study!DETAILS- From cover to cover, each in-depth topic review is easy-to-follow and easy-to-grasp - Perfect when preparing for

  2. Rapid Conversion of Traditional Introductory Physics Sequences to an Activity-Based Format

    Science.gov (United States)

    Yoder, Garett; Cook, Jerry

    2014-01-01

    The Department of Physics at EKU [Eastern Kentucky University] with support from the National Science Foundations Course Curriculum and Laboratory Improvement Program has successfully converted our entire introductory physics sequence, both algebra-based and calculus-based courses, to an activity-based format where laboratory activities,…

  3. Exploring the Relationship between Self-Efficacy and Retention in Introductory Physics

    Science.gov (United States)

    Sawtelle, Vashti; Brewe, Eric; Kramer, Laird H.

    2012-01-01

    The quantitative results of Sources of Self-Efficacy in Science Courses-Physics (SOSESC-P) are presented as a logistic regression predicting the passing of students in introductory Physics with Calculus I, overall as well as disaggregated by gender. Self-efficacy as a theory to explain human behavior change [Bandura [1977] "Psychological…

  4. Rapid Conversion of Traditional Introductory Physics Sequences to an Activity-Based Format

    Science.gov (United States)

    Yoder, Garett; Cook, Jerry

    2014-01-01

    The Department of Physics at EKU [Eastern Kentucky University] with support from the National Science Foundations Course Curriculum and Laboratory Improvement Program has successfully converted our entire introductory physics sequence, both algebra-based and calculus-based courses, to an activity-based format where laboratory activities,…

  5. Tale of Two Curricula: The Performance of 2000 Students in Introductory Electromagnetism

    Science.gov (United States)

    Kohlmyer, Matthew A.; Caballero, Marcos D.; Catrambone, Richard; Chabay, Ruth W.; Ding, Lin; Haugan, Mark P.; Marr, M. Jackson; Sherwood, Bruce A.; Schatz, Michael F.

    2009-01-01

    The performance of over 2000 students in introductory calculus-based electromagnetism (E&M) courses at four large research universities was measured using the Brief Electricity and Magnetism Assessment (BEMA). Two different curricula were used at these universities: a traditional E&M curriculum and the Matter & Interactions (M&I)…

  6. Tale of Two Curricula: The Performance of 2000 Students in Introductory Electromagnetism

    Science.gov (United States)

    Kohlmyer, Matthew A.; Caballero, Marcos D.; Catrambone, Richard; Chabay, Ruth W.; Ding, Lin; Haugan, Mark P.; Marr, M. Jackson; Sherwood, Bruce A.; Schatz, Michael F.

    2009-01-01

    The performance of over 2000 students in introductory calculus-based electromagnetism (E&M) courses at four large research universities was measured using the Brief Electricity and Magnetism Assessment (BEMA). Two different curricula were used at these universities: a traditional E&M curriculum and the Matter & Interactions (M&I) curriculum. At…

  7. Assessing Expertise in Introductory Physics Using Categorization Task

    CERN Document Server

    Mason, Andrew

    2016-01-01

    The ability to categorize problems based upon underlying principles, rather than surface features or contexts, is considered one of several proxy predictors of expertise in problem solving. With inspiration from the classic study by Chi, Feltovich, and Glaser, we assess the distribution of expertise among introductory physics students by asking three introductory physics classes, each with more than a hundred students, to categorize mechanics problems based upon similarity of solution. We compare their categorization with those of physics graduate students and faculty members. To evaluate the effect of problem context on students' ability to categorize, two sets of problems were developed for categorization. Some problems in one set included those available from the prior study by Chi et al. We find a large overlap between calculus-based introductory students and graduate students with regard to their categorizations that were assessed as "good." Our findings, which contrast with those of Chi et al., suggest ...

  8. A Simple Acronym for Doing Calculus: CAL

    Science.gov (United States)

    Hathaway, Richard J.

    2008-01-01

    An acronym is presented that provides students a potentially useful, unifying view of the major topics covered in an elementary calculus sequence. The acronym (CAL) is based on viewing the calculus procedure for solving a calculus problem P* in three steps: (1) recognizing that the problem cannot be solved using simple (non-calculus) techniques;…

  9. Leveraging Prior Calculus Study with Embedded Review

    Science.gov (United States)

    Nikolov, Margaret C.; Withers, Wm. Douglas

    2016-01-01

    We propose a new course structure to address the needs of college students with previous calculus study but no course validations as an alternative to repeating the first year of calculus. Students are introduced directly to topics from Calculus III unpreceded by a formal review of topics from Calculus I or II, but with additional syllabus time…

  10. Open Calculus: A Free Online Learning Environment

    Science.gov (United States)

    Korey, Jane; Rheinlander, Kim; Wallace, Dorothy

    2007-01-01

    Dartmouth College mathematicians have developed a free online calculus course called "Open Calculus." Open Calculus is an exportable distance-learning/self-study environment for learning calculus including written text, nearly 4000 online homework problems and instructional videos. The paper recounts the evaluation of course elements since 2000 in…

  11. Leveraging Prior Calculus Study with Embedded Review

    Science.gov (United States)

    Nikolov, Margaret C.; Withers, Wm. Douglas

    2016-01-01

    We propose a new course structure to address the needs of college students with previous calculus study but no course validations as an alternative to repeating the first year of calculus. Students are introduced directly to topics from Calculus III unpreceded by a formal review of topics from Calculus I or II, but with additional syllabus time…

  12. Pre-Calculus For Dummies

    CERN Document Server

    Kuang, Yang

    2012-01-01

    The fun and easy way to learn pre-calculus Getting ready for calculus but still feel a bit confused? Have no fear. Pre-Calculus For Dummies is an un-intimidating, hands-on guide that walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. With this guide's help you'll quickly and painlessly get a handle on all of the concepts - not just the number crunching - and understand how to perform all pre-calc tasks, from graphing to tackling proofs. You'll also get a new appreciation for

  13. Mathematics for physics with calculus

    CERN Document Server

    Das, Biman

    2005-01-01

    Designed for students who plan to take or who are presently taking calculus-based physics courses. This book will develop necessary mathematical skills and help students gain the competence to use precalculus, calculus, vector algebra, vector calculus, and the statistical analysis of experimental data. Students taking intermediate physics, engineering, and other science courses will also find the book useful-and will be able to use the book as a mathematical resource for these intermediate level courses. The book emphasizes primarily the use of mathematical techniques and mathematical concepts in Physics and does not go into their rigorous developments.

  14. Stochastic Calculus of Wrapped Compartments

    CERN Document Server

    Coppo, Mario; Drocco, Maurizio; Grassi, Elena; Troina, Angelo; 10.4204/EPTCS.28.6

    2010-01-01

    The Calculus of Wrapped Compartments (CWC) is a variant of the Calculus of Looping Sequences (CLS). While keeping the same expressiveness, CWC strongly simplifies the development of automatic tools for the analysis of biological systems. The main simplification consists in the removal of the sequencing operator, thus lightening the formal treatment of the patterns to be matched in a term (whose complexity in CLS is strongly affected by the variables matching in the sequences). We define a stochastic semantics for this new calculus. As an application we model the interaction between macrophages and apoptotic neutrophils and a mechanism of gene regulation in E.Coli.

  15. The Guarded Lambda-Calculus

    DEFF Research Database (Denmark)

    Clouston, Ranald; Bizjak, Aleš; Grathwohl, Hans Bugge;

    2016-01-01

    -former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always......We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type...

  16. The calculus a genetic approach

    CERN Document Server

    Toeplitz, Otto

    2007-01-01

    When first published posthumously in 1963, this book presented a radically different approach to the teaching of calculus.  In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique a

  17. Advanced calculus a transition to analysis

    CERN Document Server

    Dence, Thomas P

    2010-01-01

    Designed for a one-semester advanced calculus course, Advanced Calculus explores the theory of calculus and highlights the connections between calculus and real analysis -- providing a mathematically sophisticated introduction to functional analytical concepts. The text is interesting to read and includes many illustrative worked-out examples and instructive exercises, and precise historical notes to aid in further exploration of calculus. Ancillary list: * Companion website, Ebook- http://www.elsevierdirect.com/product.jsp?isbn=9780123749550 * Student Solutions Manual- To come * Instructor

  18. Assessing college students' retention and transfer from calculus to physics

    Science.gov (United States)

    Cui, Lili

    Many introductory calculus-based physics students have difficulties when solving physics problems involving calculus. This study investigates students' retention and transfer from calculus to physics. While retention is the ability to recall your knowledge at a later point in time, transfer of learning is defined as the ability to apply what one has learned in one situation to a different situation. In this dissertation we propose a theoretical framework to assess students' transfer of learning in the context of problem solving. We define two kinds of transfer---horizontal transfer and vertical transfer. Horizontal transfer involves applying previously learned ideas in a problem. Vertical transfer involves constructing new ideas to solve the problem. Students need to employ both horizontal and vertical transfer when they solve any problem. This framework evolves through this research and provides a lens that enables us to examine horizontal and vertical transfer. Additionally, this proposed framework offers researchers a vocabulary to describe and assess transfer of learning in any problem solving context. We use a combination of qualitative and quantitative methods to examine transfer in the context of problem solving. The participants in this study were students enrolled in a second-semester physics course taken by future engineers and physicists, calculus instructors and physics instructors. A total of 416 students' exam sheets were collected and reviewed. Statistical methods were used to analyze the quantitative data. A total of 28 students and nine instructors were interviewed. The video and audio recordings were transcribed and analyzed in light of the aforementioned theoretical framework. A major finding from this study is that a majority of students possess the requisite calculus skills, yet have several difficulties in applying them in the context of physics. These difficulties included: deciding the appropriate variable and limits of integration; not

  19. Intuitionistic fuzzy calculus

    CERN Document Server

    Lei, Qian

    2017-01-01

    This book offers a comprehensive and systematic review of the latest research findings in the area of intuitionistic fuzzy calculus. After introducing the intuitionistic fuzzy numbers’ operational laws and their geometrical and algebraic properties, the book defines the concept of intuitionistic fuzzy functions and presents the research on the derivative, differential, indefinite integral and definite integral of intuitionistic fuzzy functions. It also discusses some of the methods that have been successfully used to deal with continuous intuitionistic fuzzy information or data, which are different from the previous aggregation operators focusing on discrete information or data. Mainly intended for engineers and researchers in the fields of fuzzy mathematics, operations research, information science and management science, this book is also a valuable textbook for postgraduate and advanced undergraduate students alike.

  20. Feynman integral calculus

    CERN Document Server

    Smirnov, Vladimir A

    2006-01-01

    The problem of evaluating Feynman integrals over loop momenta has existed from the early days of perturbative quantum field theory. The goal of the book is to summarize those methods for evaluating Feynman integrals that have been developed over a span of more than fifty years. `Feynman Integral Calculus' characterizes the most powerful methods in a systematic way. It concentrates on the methods that have been employed recently for most sophisticated calculations and illustrates them with numerous examples, starting from very simple ones and progressing to nontrivial examples. It also shows how to choose adequate methods and combine them in a non-trivial way. This is a textbook version of the previous book (Evaluating Feynman integrals, STMP 211) of the author. Problems and solutions have been included, Appendix G has been added, more details have been presented, recent publications on evaluating Feynman integrals have been taken into account and the bibliography has been updated.

  1. Stochastic calculus with infinitesimals

    CERN Document Server

    Herzberg, Frederik

    2013-01-01

    Stochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the book.

  2. Stochastic calculus and applications

    CERN Document Server

    Cohen, Samuel N

    2015-01-01

    Completely revised and greatly expanded, the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integrals as used by systems theorists, electronic engineers and, more recently, those working in quantitative and mathematical finance. Building upon the original release of this title, this text will be of great interest to research mathematicians and graduate students working in those fields, as well as quants in the finance industry. New features of this edition include: End of chapter exercises; New chapters on basic measure theory and Backward SDEs; Reworked proofs, examples and explanatory material; Increased focus on motivating the mathematics; Extensive topical index. "Such a self-contained and complete exposition of stochastic calculus and applications fills an existing gap in the literature. The book can be recommended for first-year graduate studies. It will be useful for all who intend to wo...

  3. Shape Calculus. A Spatial Mobile Calculus for 3D Shapes

    Directory of Open Access Journals (Sweden)

    E. Bartocci

    2010-01-01

    Full Text Available We present a bio-inspired calculus for describing 3D shapes moving in a space. A shape forms a 3D process when combined with a behaviour. Behaviours are specified with a timed CCS-like process algebra using a notion of channel to naturally model binding sites on the surface of shapes. The calculus embeds collision detection and response, binding of compatible 3D processes and split of composed 3D processes.

  4. Synthesizing controllers from duration calculus

    DEFF Research Database (Denmark)

    Fränzle, Martin

    1996-01-01

    Duration Calculus is a logic for reasoning about requirements for real-time systems at a high level of abstraction from operational detail, which qualifies it as an interesting starting point for embedded controller design. Such a design activity is generally thought to aim at a control device...... the physical behaviours of which satisfy the requirements formula, i.e. the refinement relation between requirements and implementations is taken to be trajectory inclusion. Due to the abstractness of the vocabulary of Duration Calculus, trajectory inclusion between control requirements and controller designs...... for embedded controller design and exploit this fact for developing an automatic procedure for controller synthesis from specifications formalized in Duration Calculus. As far as we know, this is the first positive result concerning feasibility of automatic synthesis from dense-time Duration Calculus....

  5. Cartooning in Algebra and Calculus

    Science.gov (United States)

    Moseley, L. Jeneva

    2014-01-01

    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 understanding.

  6. A Formal Calculus for Categories

    DEFF Research Database (Denmark)

    Cáccamo, Mario José

    This dissertation studies the logic underlying category theory. In particular we present a formal calculus for reasoning about universal properties. The aim is to systematise judgements about functoriality and naturality central to categorical reasoning. The calculus is based on a language which...... extends the typed lambda calculus with new binders to represent universal constructions. The types of the languages are interpreted as locally small categories and the expressions represent functors. The logic supports a syntactic treatment of universality and duality. Contravariance requires a definition...... of universality generous enough to deal with functors of mixed variance. Ends generalise limits to cover these kinds of functors and moreover provide the basis for a very convenient algebraic manipulation of expressions. The equational theory of the lambda calculus is extended with new rules for the definitions...

  7. Testicular calculus: A rare case.

    Science.gov (United States)

    Sen, Volkan; Bozkurt, Ozan; Demır, Omer; Tuna, Burcin; Yorukoglu, Kutsal; Esen, Adil

    2015-01-01

    Testicular calculus is an extremely rare case with unknown etiology and pathogenesis. To our knowledge, here we report the third case of testicular calculus. A 31-year-old man was admitted to our clinic with painful solid mass in left testis. After diagnostic work-up for a possible testicular tumour, he underwent inguinal orchiectomy and histopathologic examination showed a testicular calculus. Case hypothesis: Solid testicular lesions in young adults generally correspond to testicular cancer. Differential diagnosis should be done carefully. Future implications: In young adults with painful and solid testicular mass with hyperechogenic appearance on scrotal ultrasonography, testicular calculus must be kept in mind in differential diagnosis. Further reports on this topic may let us do more clear recommendations about the etiology and treatment of this rare disease.

  8. Differential calculus and its applications

    CERN Document Server

    Field, Michael J

    2013-01-01

    Based on undergraduate courses in advanced calculus, the treatment covers a wide range of topics, from soft functional analysis and finite-dimensional linear algebra to differential equations on submanifolds of Euclidean space. 1976 edition.

  9. Cartooning in Algebra and Calculus

    Science.gov (United States)

    Moseley, L. Jeneva

    2014-01-01

    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 understanding.

  10. Neutrosophic Precalculus and Neutrosophic Calculus

    OpenAIRE

    Florentin Smarandache

    2015-01-01

    Neutrosophic Analysis is a generalization of Set Analysis, which in its turn is a generalization of Interval Analysis. Neutrosophic Precalculus is referred to indeterminate staticity, while Neutrosophic Calculus is the mathematics of indeterminate change. The Neutrosophic Precalculus and Neutrosophic Calculus can be developed in many ways, depending on the types of indeterminacy one has and on the methods used to deal with such indeterminacy. In this book, the author presents a few examples o...

  11. Decidability of Mean Value Calculus

    Institute of Scientific and Technical Information of China (English)

    LI Xiaoshan

    1999-01-01

    Mean Value Calculus (MVC)[1] is a real-time logicwhich can be used to specify and verify real-time systems[2]. As aconservative extension of Duration Calculus (DC)[3], MVC increasesthe expressive power but keeps the properties of DC. In this paper wepresent decidability results of MVC. An interesting result is that propositional MVC with chop star operator is still decidable, which develops the results of[4]and[5].

  12. Neutrosophic Precalculus and Neutrosophic Calculus

    OpenAIRE

    Florentin Smarandache

    2015-01-01

    Neutrosophic Analysis is a generalization of Set Analysis, which in its turn is a generalization of Interval Analysis. Neutrosophic Precalculus is referred to indeterminate staticity, while Neutrosophic Calculus is the mathematics of indeterminate change. The Neutrosophic Precalculus and Neutrosophic Calculus can be developed in many ways, depending on the types of indeterminacy one has and on the methods used to deal with such indeterminacy. In this book, the author presents a few examples o...

  13. Foliated stochastic calculus: Harmonic measures

    CERN Document Server

    Catuogno, Pedro J; Ruffino, Paulo R

    2010-01-01

    In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Other results include a decomposition of the Laplacian in terms of the foliated and basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.

  14. ``Riemann equations'' in bidifferential calculus

    Science.gov (United States)

    Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.

    2015-10-01

    We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.

  15. Examining issues of underrepresented minority students in introductory physics

    Science.gov (United States)

    Watkins, Jessica Ellen

    In this dissertation we examine several issues related to the retention of under-represented minority students in physics and science. In the first section, we show that in calculus-based introductory physics courses, the gender gap on the FCI is diminished through the use of interactive techniques, but in lower-level introductory courses, the gap persists, similar to reports published at other institutions. We find that under-represented racial minorities perform similar to their peers with comparable academic preparation on conceptual surveys, but their average exam grades and course grades are lower. We also examine student persistence in science majors; finding a significant relationship between pedagogy in an introductory physics course and persistence in science. In the second section, we look at student end-of-semester evaluations and find that female students rate interactive teaching methods a full point lower than their male peers. Looking more deeply at student interview data, we find that female students report more social issues related to the discussions in class and both male and female students cite feeling pressure to obtain the correct answer to clicker questions. Finally, we take a look an often-cited claim for gender differences in STEM participation: cognitive differences explain achievement differences in physics. We examine specifically the role of mental rotations in physics achievement and problem-solving, viewing mental rotations as a tool that students can use on physics problems. We first look at student survey results for lower-level introductory students, finding a low, but significant correlation between performance on a mental rotations test and performance in introductory physics courses. In contrast, we did not find a significant relationship for students in the upper-level introductory course. We also examine student problem-solving interviews to investigate the role of mental rotations on introductory problems.

  16. Differential Calculus on N-Graded Manifolds

    OpenAIRE

    Sardanashvily, G.; W. Wachowski

    2017-01-01

    The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over N-graded commutative rings and on N-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and on Z2-graded manifolds. We follow the notion of an N-graded manifold as a local-ringed space whose body is a s...

  17. Essential AOP: The A Calculus

    DEFF Research Database (Denmark)

    De Fraine, Bruno; Ernst, Erik; Südholt, Mario

    2012-01-01

    Aspect-oriented programming (AOP) has produced interesting language designs, but also ad hoc semantics that needs clarification. We contribute to this clarification with a calculus that models essential AOP, both simpler and more general than existing formalizations. In AOP, advice may intercept......-oriented code. Two well-known pointcut categories, call and execution, are commonly considered similar.We formally expose their differences, and resolve the associated soundness problem. Our calculus includes type ranges, an intuitive and concise alternative to explicit type variables that allows advice...... to be polymorphic over intercepted methods. We use calculus parameters to cover type safety for a wide design space of other features. Type soundness is verified in Coq....

  18. Essential AOP: The A Calculus

    DEFF Research Database (Denmark)

    De Fraine, Bruno; Ernst, Erik; Südholt, Mario

    2012-01-01

    Aspect-oriented programming (AOP) has produced interesting language designs, but also ad hoc semantics that needs clarification. We contribute to this clarification with a calculus that models essential AOP, both simpler and more general than existing formalizations. In AOP, advice may intercept......-oriented code. Two well-known pointcut categories, call and execution, are commonly considered similar.We formally expose their differences, and resolve the associated soundness problem. Our calculus includes type ranges, an intuitive and concise alternative to explicit type variables that allows advice...... to be polymorphic over intercepted methods. We use calculus parameters to cover type safety for a wide design space of other features. Type soundness is verified in Coq....

  19. Topic Order in Introductory Physics and its Impact on the STEM Curricular Ladder

    Directory of Open Access Journals (Sweden)

    Teresa L Larkin

    2017-02-01

    Full Text Available Introductory physics courses are an important rung on the curricular ladder in STEM. These courses help to strengthen students critical thinking and problem solving skills while simultaneously introducing them to many topics they will explore in more detail in later courses in physics and engineering. For these reasons, introductory physics is a required element on the curricular ladder. Most often, introductory physics is offered as a two-semester sequence with basic mechanics being taught in the first semester and electricity and magnetism in the second. In fact, this curricular sequence has not been altered in decades. Is there a reason for this? There are many other enduring questions that arise pertaining to these foundation courses in physics. These questions include: Does taking the introductory course sequence “out of order” have an impact on student learning in physics? What topics should be taught? When should these topics be taught? What topics could be left out? The list of questions is essentially endless. This paper will address some of these questions in part, through a brief discussion on student learning in a second-semester algebra-based physics course. Connections will also be made to the broader curricular ladder in STEM. To this end, an illustration that makes connections to an engineering statics course will be presented. This discussion will conclude by presenting some broader implications for the larger STEM communities.

  20. The Power of Investigative Calculus Projects

    Science.gov (United States)

    Perrin, John Robert; Quinn, Robert J.

    2008-01-01

    This article describes investigative calculus projects in which students explore a question or problem of their own construction. Three exemplary pieces of student work are showcased. Investigative calculus projects are an excellent way to foster student understanding and interest in calculus. (Contains 4 figures.)

  1. An AP Calculus Classroom Amusement Park

    Science.gov (United States)

    Ferguson, Sarah

    2016-01-01

    Throughout the school year, AP Calculus teachers strive to teach course content comprehensively and swiftly in an effort to finish all required material before the AP Calculus exam. As early May approaches and the AP Calculus test looms, students and teachers nervously complete lessons, assignments, and assessments to ensure student preparation.…

  2. An AP Calculus Classroom Amusement Park

    Science.gov (United States)

    Ferguson, Sarah

    2016-01-01

    Throughout the school year, AP Calculus teachers strive to teach course content comprehensively and swiftly in an effort to finish all required material before the AP Calculus exam. As early May approaches and the AP Calculus test looms, students and teachers nervously complete lessons, assignments, and assessments to ensure student preparation.…

  3. A Calculus for Trust Management

    DEFF Research Database (Denmark)

    Carbone, Marco; Nielsen, Mogens; Sassone, Vladimiro

    2004-01-01

    We introduce ctm, a process calculus which embodies a notion of trust for global computing systems. In ctm each principal (location) is equipped with a policy, which determines its legal behaviour, and with a protocol, which allows interactions between principals and the flow of information from...... principals to policies. We elect to formalise policies using a Datalog-like logic, and to express protocols in the process algebra style. This yields an expressive calculus very suitable for the global computing scenarios, and provides a formalisation of notions such as trust evolution. For ctm we define...

  4. Elementary calculus an infinitesimal approach

    CERN Document Server

    Keisler, H Jerome

    2012-01-01

    This first-year calculus book is centered around the use of infinitesimals, an approach largely neglected until recently for reasons of mathematical rigor. It exposes students to the intuition that originally led to the calculus, simplifying their grasp of the central concepts of derivatives and integrals. The author also teaches the traditional approach, giving students the benefits of both methods.Chapters 1 through 4 employ infinitesimals to quickly develop the basic concepts of derivatives, continuity, and integrals. Chapter 5 introduces the traditional limit concept, using approximation p

  5. Sequent Calculus and Equational Programming

    Directory of Open Access Journals (Sweden)

    Nicolas Guenot

    2015-07-01

    Full Text Available Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums.

  6. Individualized additional instruction for calculus

    Science.gov (United States)

    Takata, Ken

    2010-10-01

    College students enrolling in the calculus sequence have a wide variance in their preparation and abilities, yet they are usually taught from the same lecture. We describe another pedagogical model of Individualized Additional Instruction (IAI) that assesses each student frequently and prescribes further instruction and homework based on the student's performance. Our study compares two calculus classes, one taught with mandatory remedial IAI and the other without. The class with mandatory remedial IAI did significantly better on comprehensive multiple-choice exams, participated more frequently in classroom discussion and showed greater interest in theorem-proving and other advanced topics.

  7. Giant intravesical calculus during pregnancy.

    Science.gov (United States)

    Escobar-del Barco, Laura; Rodriguez-Colorado, Silvia; Dueñas-Garcia, Omar Felipe; Avilez-Cevasco, Juan Carlos

    2008-10-01

    Urolithiasis is commonly found during pregnancy; but the presence of a giant vesical calculus during pregnancy is a very rare entity, associated with several potential obstetric complications. A 25-year-old primigravida at 25 weeks of gestational age was referred to our tertiary care unit because she presented a giant hyperechoic intravesical mass and inability to pass urine with suprapubic pain since 2 days. An open cystolithotomy revealed a huge intravesical calculus. The patient continued with her pregnancy until full term without adverse perinatal outcomes.

  8. Calculus with a quaternionic variable

    Science.gov (United States)

    Schwartz, Charles

    2009-01-01

    Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn in trying to extend our reach to include quaternions. The noncommutativity of the quaternion algebra poses obstacles for the usual manipulations of calculus, but we show in this paper how many of those obstacles can be overcome. The surprising result is that the first order term in the expansion of F(x +δ) is a compact formula involving both F'(x) and [F(x )-F(x∗)]/(x -x∗). This advance in the differential calculus for quaternionic variables also leads us to some progress in studying integration.

  9. A Case Study: Novel Group Interactions through Introductory Computational Physics

    CERN Document Server

    Obsniuk, Michael J; Caballero, Marcos D

    2015-01-01

    With the advent of high-level programming languages capable of quickly rendering three-dimensional simulations, the inclusion of computers as a learning tool in the classroom has become more prevalent. Although work has begun to study the patterns seen in implementing and assessing computation in introductory physics, more insight is needed to understand the observed effects of blending computation with physics in a group setting. In a newly adopted format of introductory calculus-based mechanics, called Projects and Practices in Physics, groups of students work on short modeling projects -- which make use of a novel inquiry-based approach -- to develop their understanding of both physics content and practice. Preliminary analyses of observational data of groups engaging with computation, coupled with synchronized computer screencast, has revealed a unique group interaction afforded by the practices specific to computational physics -- problem debugging.

  10. Vision and change in introductory physics for the life sciences

    CERN Document Server

    Mochrie, S G J

    2015-01-01

    Since 2010, the Yale physics department has offered a novel calculus-based introductory physics for the life science (IPLS) sequence, that re-imagines the IPLS syllabus to include a selection of biologically and medically relevant topics, that are highly meaningful to its audience of biological science and premedical undergraduates. The first semester, in particular, differs considerably from traditional first-semester introductory physics. Here, we highlight the novel aspects of Yale's first-semester course, and describe student feedback about the course, including a comparison between how students evaluate the course and how they evaluate courses with a traditional syllabus, and how students' perceptions of the relevance of physics to biology and medicine are affected by having taken the course.

  11. A Process Calculus for Molecular Interaction Maps

    Directory of Open Access Journals (Sweden)

    Roberto Barbuti

    2009-11-01

    Full Text Available We present the MIM calculus, a modeling formalism with a strong biological basis, which provides biologically-meaningful operators for representing the interaction capabilities of molecular species. The operators of the calculus are inspired by the reaction symbols used in Molecular Interaction Maps (MIMs, a diagrammatic notation used by biologists. Models of the calculus can be easily derived from MIM diagrams, for which an unambiguous and executable interpretation is thus obtained. We give a formal definition of the syntax and semantics of the MIM calculus, and we study properties of the formalism. A case study is also presented to show the use of the calculus for modeling biomolecular networks.

  12. Linear-algebraic lambda-calculus

    CERN Document Server

    Arrighi, P; Arrighi, Pablo; Dowek, Gilles

    2005-01-01

    With a view towards models of quantum computation and/or the interpretation of linear logic, we define a functional language where all functions are linear operators by construction. A small step operational semantic (and hence an interpreter/simulator) is provided for this language in the form of a term rewrite system. The linear-algebraic lambda-calculus hereby constructed is linear in a different (yet related) sense to that, say, of the linear lambda-calculus. These various notions of linearity are discussed in the context of quantum programming languages. KEYWORDS: quantum lambda-calculus, linear lambda-calculus, $\\lambda$-calculus, quantum logics.

  13. A Process Calculus for Molecular Interaction Maps

    CERN Document Server

    Barbuti, Roberto; Milazzo, Paolo; Pardini, Giovanni; Rama, Aureliano; 10.4204/EPTCS.11.3

    2009-01-01

    We present the MIM calculus, a modeling formalism with a strong biological basis, which provides biologically-meaningful operators for representing the interaction capabilities of molecular species. The operators of the calculus are inspired by the reaction symbols used in Molecular Interaction Maps (MIMs), a diagrammatic notation used by biologists. Models of the calculus can be easily derived from MIM diagrams, for which an unambiguous and executable interpretation is thus obtained. We give a formal definition of the syntax and semantics of the MIM calculus, and we study properties of the formalism. A case study is also presented to show the use of the calculus for modeling biomolecular networks.

  14. Applying π-Calculus to Practice

    DEFF Research Database (Denmark)

    Abendroth, Jorg

    2003-01-01

    The π-Calculus has been developed to reason about behavioural equivalence. Different notations of equivalence are defined in terms of process interactions, as well as the context of processes. There are various extensions of the π-Calculus, such as the SPI calculus, which has primitives...... modles are instantiated correctly. In this paper we will utilize the to π-Calculus reason about access control policies and mechanism. An equivalence of different policy implementations, as well as access control mechanism will be shown. Finally some experiences regarding the use of π-Calculus...

  15. Stochastic Pi-calculus Revisited

    DEFF Research Database (Denmark)

    Cardelli, Luca; Mardare, Radu Iulian

    2013-01-01

    We develop a version of stochastic Pi-calculus with a semantics based on measure theory. We dene the behaviour of a process in a rate environment using measures over the measurable space of processes induced by structural congruence. We extend the stochastic bisimulation to include the concept of...

  16. Children, Additive Change, and Calculus.

    Science.gov (United States)

    Nemirovsky, Ricardo; And Others

    Students can learn to solve problems of qualitative integration and differentiation independently of their study of formal calculus or algebra. This exploratory study investigated the basic intuitions that elementary school children construct in their daily experience with physical and symbolic change. Elementary school children (n=18) were…

  17. Stochastic calculus and anticommuting variables

    CERN Document Server

    Rogers, A

    1994-01-01

    A theory of integration for anticommuting paths is described. This is combined with standard It\\^o calculus to give a geometric theory of Brownian paths on curved supermanifolds. (Invited lecture given at meeting on `Espaces de Lacets', Institut de Recherche Math\\'ematique Advanc\\'ee, Universit\\'e Louis Pasteur, Strasbourg, June 1994.)

  18. Portfolio Analysis for Vector Calculus

    Science.gov (United States)

    Kaplan, Samuel R.

    2015-01-01

    Classic stock portfolio analysis provides an applied context for Lagrange multipliers that undergraduate students appreciate. Although modern methods of portfolio analysis are beyond the scope of vector calculus, classic methods reinforce the utility of this material. This paper discusses how to introduce classic stock portfolio analysis in a…

  19. Constructivized Calculus in College Mathematics

    Science.gov (United States)

    Lawrence, Barbara Ann

    2012-01-01

    The purpose of this study is to present some of the classical concepts, definitions, and theorems of calculus from the constructivists' point of view in the spirit of the philosophies of L.E.J. Brouwer and Errett Bishop. This presentation will compare the classical statements to the constructivized statements. The method focuses on giving…

  20. Calculus Students' Understanding of Volume

    Science.gov (United States)

    Dorko, Allison; Speer, Natasha M.

    2013-01-01

    Researchers have documented difficulties that elementary school students have in understanding volume. Despite its importance in higher mathematics, we know little about college students' understanding of volume. This study investigated calculus students' understanding of volume. Clinical interview transcripts and written responses to volume…

  1. Reading the World with Calculus

    Science.gov (United States)

    Verzosa, Debbie

    2015-01-01

    It is now increasingly recognized that mathematics is not a neutral value-free subject. Rather, mathematics can challenge students' taken-for-granted realities and promote action. This article describes two issues, namely deforestation and income inequality. These were specifically chosen because they can be related to a range of calculus concepts…

  2. POGIL in the Calculus Classroom

    Science.gov (United States)

    Bénéteau, Catherine; Guadarrama, Zdenka; Guerra, Jill E.; Lenz, Laurie; Lewis, Jennifer E.; Straumanis, Andrei

    2017-01-01

    In this paper, we will describe the experience of the authors in using process-oriented guided inquiry learning (POGIL) in calculus at four institutions across the USA. We will briefly examine how POGIL compares to and fits in with other kinds of inquiry-based learning approaches. In particular, we will first discuss the unique structure of a…

  3. Portfolio Analysis for Vector Calculus

    Science.gov (United States)

    Kaplan, Samuel R.

    2015-01-01

    Classic stock portfolio analysis provides an applied context for Lagrange multipliers that undergraduate students appreciate. Although modern methods of portfolio analysis are beyond the scope of vector calculus, classic methods reinforce the utility of this material. This paper discusses how to introduce classic stock portfolio analysis in a…

  4. Reading the World with Calculus

    Science.gov (United States)

    Verzosa, Debbie

    2015-01-01

    It is now increasingly recognized that mathematics is not a neutral value-free subject. Rather, mathematics can challenge students' taken-for-granted realities and promote action. This article describes two issues, namely deforestation and income inequality. These were specifically chosen because they can be related to a range of calculus concepts…

  5. Constructivized Calculus in College Mathematics

    Science.gov (United States)

    Lawrence, Barbara Ann

    2012-01-01

    The purpose of this study is to present some of the classical concepts, definitions, and theorems of calculus from the constructivists' point of view in the spirit of the philosophies of L.E.J. Brouwer and Errett Bishop. This presentation will compare the classical statements to the constructivized statements. The method focuses on giving…

  6. Complexity and the Fractional Calculus

    Science.gov (United States)

    2013-01-01

    developed in a number of signif- icant ways in the recent past. Sokolov et al. [1] maintain that this calculus was restricted to the field of mathematics... Sokolov , J. Klafter, and A. Blumen, “Fractional kinetics,” Physics Today, vol. 55, no. 11, pp. 48–54, 2002. [2] V. Seshadri and B. J. West, “Fractal

  7. Supragingival calculus: formation and control.

    Science.gov (United States)

    Jin, Ye; Yip, Hak-Kong

    2002-01-01

    Dental calculus is composed of inorganic components and organic matrix. Brushite, dicalcium phosphate dihydrate, octacalcium phosphate, hydroxyapatite, and whitlockite form the mineral part of dental calculus. Salivary proteins selectively adsorb on the tooth surface to form an acquired pellicle. It is followed by the adherence of various oral micro-organisms. Fimbriae, flagella, and some other surface proteins are essential for microbial adherence. Microbial co-aggregation and co-adhesion enable some micro-organisms, which are incapable of adhering, to adhere to the pellicle-coated tooth surface. Once organisms attach to the tooth surface, new genes could be expressed so that mature dental plaque can form and biofilm bacteria assume increased resistance to antimicrobial agents. Supersaturation of saliva and plaque fluid with respect to calcium phosphates is the driving force for plaque mineralization. Both salivary flow rate and plaque pH appear to influence the saturation degree of calcium phosphates. Acidic phospholipids and specific proteolipids present in cell membranes play a key role in microbial mineralization. The roles of crystal growth inhibitors, promoters, and organic acids in calculus formation are discussed. Application of biofilm culture systems in plaque mineralization is concisely reviewed. Anti-calculus agents used--centering on triclosan plus polyvinyl methyl ether/maleic acid copolymer, pyrophosphate plus polyvinyl methyl ether/maleic acid copolymer, and zinc ion-in commercial dentifrices are also discussed in this paper.

  8. Lambda calculus with explicit recursion

    NARCIS (Netherlands)

    Z.M. Ariola (Zena); J.W. Klop (Jan Willem)

    1996-01-01

    textabstractThis paper is concerned with the study of $lambda$-calculus with explicit recursion, namely of cyclic $lambda$-graphs. The starting point is to treat a $lambda$-graph as a system of recursion equations involving $lambda$-terms, and to manipulate such systems in an unrestricted manner,

  9. Advanced calculus of several variables

    CERN Document Server

    Edwards, C H

    1995-01-01

    Modern conceptual treatment of multivariable calculus, emphasizing the interplay of geometry and analysis via linear algebra and the approximation of nonlinear mappings by linear ones. At the same time, ample attention is paid to the classical applications and computational methods. Hundreds of examples, problems and figures. 1973 edition.

  10. Stochastic Pi-calculus Revisited

    DEFF Research Database (Denmark)

    Cardelli, Luca; Mardare, Radu Iulian

    2013-01-01

    We develop a version of stochastic Pi-calculus with a semantics based on measure theory. We dene the behaviour of a process in a rate environment using measures over the measurable space of processes induced by structural congruence. We extend the stochastic bisimulation to include the concept of...

  11. ENERGY CALCULUS IN CHINESE LANGUAGESEGMENTATION

    Institute of Scientific and Technical Information of China (English)

    2000-01-01

    Based on cognitive science, the EnergyCalculus in Chinese language segmentation was presented to eliminate segmentation ambiguity. The notion of "EnergyCost" was advanced to denote the extent of the under-standability of a certain segmentation. EnergyCost function was defined with Z-notation. This approcah is effective to all natural language segmentation.

  12. Qutrit Dichromatic Calculus and Its Universality

    Directory of Open Access Journals (Sweden)

    Quanlong Wang

    2014-12-01

    Full Text Available We introduce a dichromatic calculus (RG for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a counterexample to Ranchin's universality proof, and give another proof by Lie theory that the qudit ZX calculus contains all single qudit unitary transformations, which implies that qudit ZX calculus, with qutrit dichromatic calculus as a special case, is universal for quantum mechanics.

  13. Introductory statistical thermodynamics

    CERN Document Server

    Dalarsson, Nils; Golubovic, Leonardo

    2011-01-01

    Introductory Statistical Thermodynamics is a text for an introductory one-semester course in statistical thermodynamics for upper-level undergraduate and graduate students in physics and engineering. The book offers a high level of detail in derivations of all equations and results. This information is necessary for students to grasp difficult concepts in physics that are needed to move on to higher level courses. The text is elementary, self contained, and mathematically well-founded, containing a number of problems with detailed solutions to help students to grasp the more difficult

  14. Factors Associated with Success in College Calculus II

    Science.gov (United States)

    Rosasco, Margaret E.

    2013-01-01

    Students are entering college having earned credit for college Calculus 1 based on their scores on the College Board's Advanced Placement (AP) Calculus AB exam. Despite being granted credit for college Calculus 1, it is unclear whether these students are adequately prepared for college Calculus 2. College calculus classes are often taught from a…

  15. Factors Associated with Success in College Calculus II

    Science.gov (United States)

    Rosasco, Margaret E.

    2013-01-01

    Students are entering college having earned credit for college Calculus 1 based on their scores on the College Board's Advanced Placement (AP) Calculus AB exam. Despite being granted credit for college Calculus 1, it is unclear whether these students are adequately prepared for college Calculus 2. College calculus classes are often taught from a…

  16. Factors Associated with Success in College Calculus II

    Science.gov (United States)

    Rosasco, Margaret E.

    2013-01-01

    Students are entering college having earned credit for college Calculus 1 based on their scores on the College Board's Advanced Placement (AP) Calculus AB exam. Despite being granted credit for college Calculus 1, it is unclear whether these students are adequately prepared for college Calculus 2. College calculus classes are often taught…

  17. Sustainable careers: Introductory chapter

    NARCIS (Netherlands)

    Heijden, B.I.J.M. van der; Vos, A de

    2015-01-01

    In this introductory chapter we will introduce the concept of ‘sustainable careers’ within the broader framework of contemporary careers. Departing from changes in the career context with regard to the dimensions of time, social space, agency and meaning, we advocate a fresh perspective on careers t

  18. Introductory Spanish IV.

    Science.gov (United States)

    Dade County Public Schools, Miami, FL.

    This introductory conversational course in Spanish is intended primarily for use in the classroom. The student is to learn some general differences between Argentina and the United States and how to better describe his friends. The common verbs "tener,""venir,""ir,""dar," and those that change "e" are presented. Additional uses of "ser" and…

  19. An Introductory Lisp Parser.

    Science.gov (United States)

    Loritz, Donald

    1987-01-01

    Gives a short grammar of the Lisp computer language. Presents an introductory English parser (Simparse) as an example of how to write a parser in Lisp. Lists references for further explanation. Intended as preparation for teachers who may use computer-assisted language instruction in the future. (LMO)

  20. Discrete calculus methods for counting

    CERN Document Server

    Mariconda, Carlo

    2016-01-01

    This book provides an introduction to combinatorics, finite calculus, formal series, recurrences, and approximations of sums. Readers will find not only coverage of the basic elements of the subjects but also deep insights into a range of less common topics rarely considered within a single book, such as counting with occupancy constraints, a clear distinction between algebraic and analytical properties of formal power series, an introduction to discrete dynamical systems with a thorough description of Sarkovskii’s theorem, symbolic calculus, and a complete description of the Euler-Maclaurin formulas and their applications. Although several books touch on one or more of these aspects, precious few cover all of them. The authors, both pure mathematicians, have attempted to develop methods that will allow the student to formulate a given problem in a precise mathematical framework. The aim is to equip readers with a sound strategy for classifying and solving problems by pursuing a mathematically rigorous yet ...

  1. Reductionism and the Universal Calculus

    CERN Document Server

    Sarma, Gopal P

    2016-01-01

    In the seminal essay, "On the unreasonable effectiveness of mathematics in the physical sciences," physicist Eugene Wigner poses a fundamental philosophical question concerning the relationship between a physical system and our capacity to model its behavior with the symbolic language of mathematics. In this essay, I examine an ambitious 16th and 17th-century intellectual agenda from the perspective of Wigner's question, namely, what historian Paolo Rossi calls "the quest to create a universal language." While many elite thinkers pursued related ideas, the most inspiring and forceful was Gottfried Leibniz's effort to create a "universal calculus," a pictorial language which would transparently represent the entirety of human knowledge, as well as an associated symbolic calculus with which to model the behavior of physical systems and derive new truths. I suggest that a deeper understanding of why the efforts of Leibniz and others failed could shed light on Wigner's original question. I argue that the notion o...

  2. Cosmological modelling with Regge calculus

    CERN Document Server

    Liu, Rex G

    2015-01-01

    The late universe's matter distribution obeys the Copernican principle at only the coarsest of scales. The relative importance of such inhomogeneity is still not well understood. Because of the Einstein field equations' non-linear nature, some argue a non-perturbative approach is necessary to correctly model inhomogeneities and may even obviate any need for dark energy. We shall discuss an approach based on Regge calculus, a discrete approximation to general relativity: we shall discuss the Collins--Williams formulation of Regge calculus and its application to two toy universes. The first is a universe for which the continuum solution is well-established, the $\\Lambda$-FLRW universe. The second is an inhomogeneous universe, the `lattice universe' wherein matter consists solely of a lattice of point masses with pure vacuum in between, a distribution more similar to that of the actual universe compared to FLRW universes. We shall discuss both regular lattices and one where one mass gets perturbed.

  3. Brownian motion and stochastic calculus

    CERN Document Server

    Karatzas, Ioannis

    1998-01-01

    This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization). This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large num...

  4. Fractional-calculus diffusion equation

    OpenAIRE

    Ajlouni, Abdul-Wali MS; Al-Rabai'ah, Hussam A

    2010-01-01

    Background Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. Results The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive...

  5. The calculus of telescopic urbanism

    OpenAIRE

    Arabindoo, P.

    2013-01-01

    Developing Amin's invocation of a telescopic urbanism as more than a visual metaphor, this paper seeks to rethink its epistemological and methodological focus, resisting at the same time the tendency to oversimplify the relationship between the different optics he outlines. Threatened by a dominant meta-narrative of a numerically driven calculus, this paper identifies an opportunity in Amin's telescopic urbanism to reject the 'big-data' approach to the city. In this context, it challenges the...

  6. Operational calculus and generalized functions

    CERN Document Server

    Erdelyi, Arthur

    2013-01-01

    This brief monograph by a distinguished professor is based on a mathematics course offered at the California Institute of Technology. The majority of students taking this course were advanced undergraduates and graduate students of engineering. A solid background in advanced calculus is a prerequisite.Topics include elementary and convergence theories of convolution quotients, differential equations involving operator functions, and exponential functions of operators. Tools developed in the preceding chapters are then applied to problems in partial differential equations. Solutions to selected

  7. A Calculus of Evolving Objects

    Directory of Open Access Journals (Sweden)

    M. Dezani-Ciancaglini

    2008-01-01

    Full Text Available The demands of developing modern, highly dynamic applications have led to an increasing interest in dynamic programming languages and mechanisms. Not only must applications evolve over time, but the object models themselves may need to be adapted to the requirements of different run-time contexts. Class-based models and prototype-based models, for example, may need to co-exist to meet the demands of dynamically evolving applications. Multi-dimensional dispatch, fine-grained and dynamic software composition, and run-time evolution of behaviour are further examples of diverse mechanisms which may need to co-exist in a dynamically evolving run-time environment. How can we model the semantics of these highly dynamic features, yet still offer some reasonable safety guarantees?To this end we present an original calculus in which objects can adapt their behaviour at run-time. Both objects and environments are represented by first-class mappings between variables and values. Message sends are dynamically resolved to method calls. Variables may be dynamically bound, making it possible to model a variety of dynamic mechanisms within the same calculus. Despite the highly dynamic nature of the calculus, safety properties are assured by a type assignment system.

  8. Time scales: from Nabla calculus to Delta calculus and vice versa via duality

    OpenAIRE

    Caputo, M. Cristina

    2009-01-01

    In this note we show how one can obtain results from the nabla calculus from results on the delta calculus and vice versa via a duality argument. We provide applications of the main results to the calculus of variations on time scales.

  9. Time scales: from Nabla calculus to Delta calculus and vice versa via duality

    OpenAIRE

    Caputo, M. Cristina

    2009-01-01

    In this note we show how one can obtain results from the nabla calculus from results on the delta calculus and vice versa via a duality argument. We provide applications of the main results to the calculus of variations on time scales.

  10. Qutrit Dichromatic Calculus and Its Universality

    OpenAIRE

    2014-01-01

    We introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a ...

  11. Solutions manual to accompany Fundamentals of calculus

    CERN Document Server

    Morris, Carla C

    2015-01-01

    Solutions Manual to Accompany Fundamentals of Calculus the text that encourages students to use power, quotient, and product rules for solutions as well as stresses the importance of modeling skills.  In addition to core integral and differential calculus coverage, the core book features finite calculus, which lends itself to modeling and spreadsheets.  Specifically, finite calculus is applied to marginal economic analysis, finance, growth, and decay.  Includes: Linear Equations and Functions The Derivative Using the Derivative Exponential and Logarithmic

  12. A Higher-Order Calculus for Categories

    DEFF Research Database (Denmark)

    Cáccamo, Mario José; Winskel, Glynn

    2001-01-01

    A calculus for a fragment of category theory is presented. The types in the language denote categories and the expressions functors. The judgements of the calculus systematise categorical arguments such as: an expression is functorial in its free variables; two expressions are naturally isomorphic...... in their free variables. There are special binders for limits and more general ends. The rules for limits and ends support an algebraic manipulation of universal constructions as opposed to a more traditional diagrammatic approach. Duality within the calculus and applications in proving continuity are discussed...... with examples. The calculus gives a basis for mechanising a theory of categories in a generic theorem prover like Isabelle....

  13. Petri nets semantics ofπ-calculus

    Institute of Scientific and Technical Information of China (English)

    Zhenhua YU; Yuanli CAI; Haiping XU

    2008-01-01

    As π-calculus based on the interleaving semantics cannot depict the true concurrency and has few supporting tools,it is translated into Petri nets.π-calculus is divided into basic elements,sequence,concurrency,choice and recursive modules.These modules are translated into Petri nets to construct a complicated system.Petri nets semantics for π-calculus visualize system structure as well as system behaviors.The structural analysis techniques allow direct qualitative analysis of the system properties on the structure of the nets.Finally,Petri nets semantics for π-calculus are illustrated by applying them to mobile telephone systems.

  14. A Calculus for Context-Awareness

    DEFF Research Database (Denmark)

    Zimmer, Pascal

    2005-01-01

    In order to answer the challenge of pervasive computing, we propose a new process calculus, whose aim is to describe dynamic systems composed of agents able to move and react differently depending on their location. This Context-Aware Calculus features a hierarchical structure similar to mobile...... ambients, and a generic multi-agent synchronization mechanism, inspired from the join-calculus. After general ideas and introduction, we review the full calculus' syntax and semantics, as well as some motivating examples, study its expressiveness, and show how the notion of computation itself can be made...

  15. Pre-calculus workbook for dummies

    CERN Document Server

    Kuang, Yang

    2011-01-01

    Get the confidence and math skills you need to get started with calculus Are you preparing for calculus? This hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in the course. You'll get hundreds of valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every problem. You'll also memorize the most frequently used equations, see how to avoid common mistakes, understand tricky trig proofs, and much more. Pre-Calculus Workbook For Dummies is the perfect tool for anyone who wa

  16. Fractional calculus with applications for nuclear reactor dynamics

    CERN Document Server

    Ray, Santanu Saha

    2015-01-01

    Introduces Novel Applications for Solving Neutron Transport EquationsWhile deemed nonessential in the past, fractional calculus is now gaining momentum in the science and engineering community. Various disciplines have discovered that realistic models of physical phenomenon can be achieved with fractional calculus and are using them in numerous ways. Since fractional calculus represents a reactor more closely than classical integer order calculus, Fractional Calculus with Applications for Nuclear Reactor Dynamics focuses on the application of fractional calculus to describe the physical behavi

  17. Dental Calculus Arrest of Dental Caries

    Science.gov (United States)

    Keyes, Paul H.; Rams, Thomas E.

    2016-01-01

    Background An inverse relationship between dental calculus mineralization and dental caries demineralization on teeth has been noted in some studies. Dental calculus may even form superficial layers over existing dental caries and arrest their progression, but this phenomenon has been only rarely documented and infrequently considered in the field of Cariology. To further assess the occurrence of dental calculus arrest of dental caries, this study evaluated a large number of extracted human teeth for the presence and location of dental caries, dental calculus, and dental plaque biofilms. Materials and methods A total of 1,200 teeth were preserved in 10% buffered formal saline, and viewed while moist by a single experienced examiner using a research stereomicroscope at 15-25× magnification. Representative teeth were sectioned and photographed, and their dental plaque biofilms subjected to gram-stain examination with light microscopy at 100× magnification. Results Dental calculus was observed on 1,140 (95%) of the extracted human teeth, and no dental carious lesions were found underlying dental calculus-covered surfaces on 1,139 of these teeth. However, dental calculus arrest of dental caries was found on one (0.54%) of 187 evaluated teeth that presented with unrestored proximal enamel caries. On the distal surface of a maxillary premolar tooth, dental calculus mineralization filled the outer surface cavitation of an incipient dental caries lesion. The dental calculus-covered carious lesion extended only slightly into enamel, and exhibited a brown pigmentation characteristic of inactive or arrested dental caries. In contrast, the tooth's mesial surface, without a superficial layer of dental calculus, had a large carious lesion going through enamel and deep into dentin. Conclusions These observations further document the potential protective effects of dental calculus mineralization against dental caries. PMID:27446993

  18. Sensors an introductory course

    CERN Document Server

    Kalantar-zadeh, Kourosh

    2013-01-01

    Sensors: An Introductory Course provides an essential reference on the fundamentals of sensors. The book is designed to help readers in developing skills and the understanding required in order to implement a wide range of sensors that are commonly used in our daily lives. This book covers the basic concepts in the sensors field, including definitions and terminologies. The physical sensing effects are described, and devices which utilize these effects are presented. The most frequently used organic and inorganic sensors are introduced and the techniques for implementing them are discussed. This book: Provides a comprehensive representation of the most common sensors and can be used as a reference in relevant fields Presents learning materials in a concise and easy to understand manner Includes examples of how sensors are incorporated in real life measurements Contains detailed figures and schematics to assist in understanding the sensor performance Sensors: An Introductory Course is ideal for university stu...

  19. Introductory statistical inference

    CERN Document Server

    Mukhopadhyay, Nitis

    2014-01-01

    This gracefully organized text reveals the rigorous theory of probability and statistical inference in the style of a tutorial, using worked examples, exercises, figures, tables, and computer simulations to develop and illustrate concepts. Drills and boxed summaries emphasize and reinforce important ideas and special techniques.Beginning with a review of the basic concepts and methods in probability theory, moments, and moment generating functions, the author moves to more intricate topics. Introductory Statistical Inference studies multivariate random variables, exponential families of dist

  20. Introductory graph theory

    CERN Document Server

    Chartrand, Gary

    1984-01-01

    Graph theory is used today in the physical sciences, social sciences, computer science, and other areas. Introductory Graph Theory presents a nontechnical introduction to this exciting field in a clear, lively, and informative style. Author Gary Chartrand covers the important elementary topics of graph theory and its applications. In addition, he presents a large variety of proofs designed to strengthen mathematical techniques and offers challenging opportunities to have fun with mathematics. Ten major topics - profusely illustrated - include: Mathematical Models, Elementary Concepts of Grap

  1. Using Categorization of Problems as an Instructional Tool to Help Introductory Students Learn Physics

    CERN Document Server

    Mason, Andrew

    2016-01-01

    The ability to categorize problems based upon underlying principles, rather than contexts, is considered a hallmark of expertise in physics problem solving. With inspiration from a classic study by Chi, Feltovich, and Glaser, we compared the categorization of 25 introductory mechanics problems based upon similarity of solution by students in large calculus-based introductory courses with physics faculty and Ph.D. students. Here, we summarize the study and suggest that a categorization task, especially when conducted with students working with peers in small groups, can be an effective pedagogical tool to help students in introductory physics courses learn to discern the underlying similarity between problems with diverse contexts but the same underlying physics principles.

  2. Effect of Scaffolding on Helping Introductory Physics Students Solve Quantitative Problems Involving Strong Alternative Conceptions

    CERN Document Server

    Lin, Shih-Yin

    2016-01-01

    It is well-known that introductory physics students often have alternative conceptions that are inconsistent with established physical principles and concepts. Invoking alternative conceptions in quantitative problem-solving process can derail the entire process. In order to help students solve quantitative problems involving strong alternative conceptions correctly, appropriate scaffolding support can be helpful. The goal of this study is to examine how different scaffolding supports involving analogical problem solving influence introductory physics students' performance on a target quantitative problem in a situation where many students' solution process is derailed due to alternative conceptions. Three different scaffolding supports were designed and implemented in calculus-based and algebra-based introductory physics courses to evaluate the level of scaffolding needed to help students learn from an analogical problem that is similar in the underlying principles but for which the problem solving process i...

  3. Care and Feeding of a Paperless, Calculus-based Physics Course

    Science.gov (United States)

    Moore, Christopher; Fuller, Robert; Plano-Clark, Vicki L.; Dunbar, Steven R.

    1997-04-01

    Technology is playing an increasing role in our lives at home, at work, and in the classroom. We have begun a calculus-based introductory physics course to integrate mathematics and multimedia with the traditional physics content. This course relies on the use of technology to teach physics. We formulated the following rule for the conduct of the course: ''No paper is transferred between instructional staff and students that contains course information or assignments for grading.'' Implementing and maintaining this physics course within the context of the instructor goals will be discussed. Preliminary results of feedback from the students and an evaluation team will be presented.

  4. Teaching assistants' performance at identifying common introductory student difficulties in mechanics revealed by the Force Concept Inventory

    Science.gov (United States)

    Maries, Alexandru; Singh, Chandralekha

    2016-06-01

    The Force Concept Inventory (FCI) has been widely used to assess student understanding of introductory mechanics concepts by a variety of educators and physics education researchers. One reason for this extensive use is that many of the items on the FCI have strong distractor choices which correspond to students' alternate conceptions in mechanics. Instruction is unlikely to be effective if instructors do not know the common alternate conceptions of introductory physics students and explicitly take into account students' initial knowledge states in their instructional design. Here, we discuss research involving the FCI to evaluate one aspect of the pedagogical content knowledge of teaching assistants (TAs): knowledge of introductory student alternate conceptions in mechanics as revealed by the FCI. For each item on the FCI, the TAs were asked to identify the most common incorrect answer choice of introductory physics students. This exercise was followed by a class discussion with the TAs related to this task, including the importance of knowing student difficulties in teaching and learning. Then, we used FCI pretest and post-test data from a large population (˜900 ) of introductory physics students to assess the extent to which TAs were able to identify alternate conceptions of introductory students related to force and motion. In addition, we carried out think-aloud interviews with graduate students who had more than two semesters of teaching experience in recitations to examine how they reason about the task. We find that while the TAs, on average, performed better than random guessing at identifying introductory students' difficulties with FCI content, they did not identify many common difficulties that introductory physics students have after traditional instruction. We discuss specific alternate conceptions, the extent to which TAs are able to identify them, and results from the think-aloud interviews that provided valuable information about why TAs sometimes

  5. Teaching assistants’ performance at identifying common introductory student difficulties in mechanics revealed by the Force Concept Inventory

    Directory of Open Access Journals (Sweden)

    Alexandru Maries

    2016-05-01

    Full Text Available The Force Concept Inventory (FCI has been widely used to assess student understanding of introductory mechanics concepts by a variety of educators and physics education researchers. One reason for this extensive use is that many of the items on the FCI have strong distractor choices which correspond to students’ alternate conceptions in mechanics. Instruction is unlikely to be effective if instructors do not know the common alternate conceptions of introductory physics students and explicitly take into account students’ initial knowledge states in their instructional design. Here, we discuss research involving the FCI to evaluate one aspect of the pedagogical content knowledge of teaching assistants (TAs: knowledge of introductory student alternate conceptions in mechanics as revealed by the FCI. For each item on the FCI, the TAs were asked to identify the most common incorrect answer choice of introductory physics students. This exercise was followed by a class discussion with the TAs related to this task, including the importance of knowing student difficulties in teaching and learning. Then, we used FCI pretest and post-test data from a large population (∼900 of introductory physics students to assess the extent to which TAs were able to identify alternate conceptions of introductory students related to force and motion. In addition, we carried out think-aloud interviews with graduate students who had more than two semesters of teaching experience in recitations to examine how they reason about the task. We find that while the TAs, on average, performed better than random guessing at identifying introductory students’ difficulties with FCI content, they did not identify many common difficulties that introductory physics students have after traditional instruction. We discuss specific alternate conceptions, the extent to which TAs are able to identify them, and results from the think-aloud interviews that provided valuable information

  6. Imagine Yourself in This Calculus Classroom

    Science.gov (United States)

    Bryan, Luajean

    2007-01-01

    The efforts to attract students to precalculus, trigonometry, and calculus classes became more successful at the author's school when projects-based classes were offered. Data collection from an untethered hot air balloon flight for calculus students was planned to maximize enrollment. The data were analyzed numerically, graphically, and…

  7. Aspects of Calculus for Preservice Teachers

    Science.gov (United States)

    Fothergill, Lee

    2011-01-01

    The purpose of this study was to compare the perspectives of faculty members who had experience teaching undergraduate calculus and preservice teachers who had recently completed student teaching in regards to a first semester undergraduate calculus course. An online survey was created and sent to recent student teachers and college mathematics…

  8. Educating about Sustainability while Enhancing Calculus

    Science.gov (United States)

    Pfaff, Thomas J.

    2011-01-01

    We give an overview of why it is important to include sustainability in mathematics classes and provide specific examples of how to do this for a calculus class. We illustrate that when students use "Excel" to fit curves to real data, fundamentally important questions about sustainability become calculus questions about those curves. (Contains 5…

  9. Sandboxing in a Distributed Pi-Calculus

    DEFF Research Database (Denmark)

    Hüttel, Hans; Kühnrich, Morten

    2006-01-01

    This paper presents an extension of the Dpi-calculus due to Hennessy and Riely with constructs for signing and authenticating code and for sandboxing. A sort system, built on Milner's sort systems for the polyadic pi-calculus, is presented and proven sound with respect to an error predicate which...

  10. A Cross-National Study of Calculus

    Science.gov (United States)

    Chai, Jun; Friedler, Louis M.; Wolff, Edward F.; Li, Jun; Rhea, Karen

    2015-01-01

    The results from a cross-national study comparing calculus performance of students at East China Normal University (ECNU) in Shanghai and students at the University of Michigan before and after their first university calculus course are presented. Overall, ECNU significantly outperformed Michigan on both the pre- and post-tests, but the Michigan…

  11. Calculus and Success in a Business School

    Science.gov (United States)

    Kim, Dong-gook; Garcia, Fernando; Dey, Ishita

    2012-01-01

    Many business schools or colleges require calculus as a prerequisite for certain classes or for continuing to upper division courses. While there are many studies investigating the relationship between performance in calculus and performance in a single course, such as economics, statistics, and finance, there are very few studies investigating…

  12. Educating about Sustainability while Enhancing Calculus

    Science.gov (United States)

    Pfaff, Thomas J.

    2011-01-01

    We give an overview of why it is important to include sustainability in mathematics classes and provide specific examples of how to do this for a calculus class. We illustrate that when students use "Excel" to fit curves to real data, fundamentally important questions about sustainability become calculus questions about those curves. (Contains 5…

  13. Hybrid Logical Analyses of the Ambient Calculus

    DEFF Research Database (Denmark)

    Bolander, Thomas; Hansen, Rene Rydhof

    2010-01-01

    In this paper, hybrid logic is used to formulate three control flow analyses for Mobile Ambients, a process calculus designed for modelling mobility. We show that hybrid logic is very well-suited to express the semantic structure of the ambient calculus and how features of hybrid logic can...

  14. A Cross-National Study of Calculus

    Science.gov (United States)

    Chai, Jun; Friedler, Louis M.; Wolff, Edward F.; Li, Jun; Rhea, Karen

    2015-01-01

    The results from a cross-national study comparing calculus performance of students at East China Normal University (ECNU) in Shanghai and students at the University of Michigan before and after their first university calculus course are presented. Overall, ECNU significantly outperformed Michigan on both the pre- and post-tests, but the Michigan…

  15. RAMAN-SPECTRA OF HUMAN DENTAL CALCULUS

    NARCIS (Netherlands)

    TSUDA, H; ARENDS, J

    1993-01-01

    Raman spectra of human dental calculus have been observed for the first time by use of micro-Raman spectroscopy. The spectral features of calculus were influenced easily by heating caused by laser irradiation. Therefore, the measurements were carried out at relatively low power (5 mW, 1-mu m spot

  16. Attendance and Attainment in a Calculus Course

    Science.gov (United States)

    Meulenbroek, Bernard; van den Bogaard, Maartje

    2013-01-01

    In this paper the relationship between attendance and attainment in a standard calculus course is investigated. Calculus could in principle be studied without attending lectures due to the wealth of material available (in hardcopy and online). However, in this study we will show that the pass rate of students attending classes regularly (>75%…

  17. RAMAN-SPECTRA OF HUMAN DENTAL CALCULUS

    NARCIS (Netherlands)

    TSUDA, H; ARENDS, J

    1993-01-01

    Raman spectra of human dental calculus have been observed for the first time by use of micro-Raman spectroscopy. The spectral features of calculus were influenced easily by heating caused by laser irradiation. Therefore, the measurements were carried out at relatively low power (5 mW, 1-mu m spot si

  18. Łukasiewicz mu-Calculus

    NARCIS (Netherlands)

    M. Mio (M.); A Simpson

    2013-01-01

    htmlabstractThe paper explores properties of Łukasiewicz mu-calculus, a version of the quantitative/probabilistic modal m-calculus containing both weak and strong conjunctions and disjunctions from Łukasiewicz (fuzzy) logic. We show that this logic encodes the well-known probabilistic temporal

  19. Imagine Yourself in This Calculus Classroom

    Science.gov (United States)

    Bryan, Luajean

    2007-01-01

    The efforts to attract students to precalculus, trigonometry, and calculus classes became more successful at the author's school when projects-based classes were offered. Data collection from an untethered hot air balloon flight for calculus students was planned to maximize enrollment. The data were analyzed numerically, graphically, and…

  20. Anti-calculus and whitening toothpastes

    NARCIS (Netherlands)

    van Loveren, C.; Duckworth, R.M.

    2013-01-01

    In terms of novel formulations, there seems to have been a shift in emphasis from anti-caries/anti-gingivitis to anti-calculus/whitening toothpastes in recent years. The anti-calculus and whitening effects of toothpastes are to some extent based on the same active ingredients: compounds of high

  1. Raise Test Scores: Integrate Biology and Calculus.

    Science.gov (United States)

    Lukens, Jeffrey D.; Feinstein, Sheryl

    This paper presents the results of research that compared the academic achievement of high school students enrolled in an integrated Advanced Placement Biology/Advanced Placement Calculus course with students enrolled in traditional Advanced Placement Biology and Advanced Placement Calculus courses. Study subjects included high school students…

  2. Matlab differential and integral calculus

    CERN Document Server

    Lopez, Cesar

    2014-01-01

    MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Differential and Integral Calculus introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to givi

  3. A course in advanced calculus

    CERN Document Server

    Borden, Robert S

    1997-01-01

    This remarkable undergraduate-level text offers a study in calculus that simultaneously unifies the concepts of integration in Euclidean space while at the same time giving students an overview of other areas intimately related to mathematical analysis. The author achieves this ambitious undertaking by shifting easily from one related subject to another. Thus, discussions of topology, linear algebra, and inequalities yield to examinations of innerproduct spaces, Fourier series, and the secret of Pythagoras. Beginning with a look at sets and structures, the text advances to such topics as lim

  4. AP calculus AB/BC

    CERN Document Server

    Schwartz, Stu

    2013-01-01

    All Access for the AP® Calculus AB & BC Exams Book + Web + Mobile Everything you need to prepare for the Advanced Placement® exam, in a study system built around you! There are many different ways to prepare for an Advanced Placement® exam. What's best for you depends on how much time you have to study and how comfortable you are with the subject matter. To score your highest, you need a system that can be customized to fit you: your schedule, your learning style, and your current level of knowledge. This book, and the free online tools that come with it, will help you personalize your AP® Cal

  5. Calculus with a Quaternionic Variable

    OpenAIRE

    2009-01-01

    Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn in trying to extend our reach to include quaternions. The noncommutativity of the quaternion algebra poses obstacles for the usual manipulations of calculus, but we show in this paper how many of those obstacles can be overcome. The surprising result is that the first order term in the expansion of F(x+delta)is a compact formula involving both F'(x) and [F(x) − F(...

  6. Advanced calculus of several variables

    CERN Document Server

    Kumar, Devendra

    2014-01-01

    ADVANCED CALCULUS OF SEVERAL VARIABLES covers important topics of Transformations and topology on Euclidean in n-space Rn Functions of several variables, Differentiation in Rn, Multiple integrals and Integration in Rn. The topics have been presented in a simple clear and coherent style with a number of examples and exercises. Proofs have been made direct and simple. Unsolved problems just after relevant articles in the form of exercises and typical problems followed by suggestions have been given. This book will help the reader work on the problems of Numerical Analysis, Operations Research, Differential Equations and Engineering applications.

  7. Calculus and analysis in Euclidean space

    CERN Document Server

    Shurman, Jerry

    2016-01-01

    The graceful role of analysis in underpinning calculus is often lost to their separation in the curriculum. This book entwines the two subjects, providing a conceptual approach to multivariable calculus closely supported by the structure and reasoning of analysis. The setting is Euclidean space, with the material on differentiation culminating in the inverse and implicit function theorems, and the material on integration culminating in the general fundamental theorem of integral calculus. More in-depth than most calculus books but less technical than a typical analysis introduction, Calculus and Analysis in Euclidean Space offers a rich blend of content to students outside the traditional mathematics major, while also providing transitional preparation for those who will continue on in the subject. The writing in this book aims to convey the intent of ideas early in discussion. The narrative proceeds through figures, formulas, and text, guiding the reader to do mathematics resourcefully by marshaling the skil...

  8. Reasoning about objects using process calculus techniques

    DEFF Research Database (Denmark)

    Kleist, Josva

    This thesis investigates the applicability of techniques known from the world of process calculi to reason about properties of object-oriented programs. The investigation is performed upon a small object-oriented language - The Sigma-calculus of Abadi and Cardelli. The investigation is twofold: We...... investigate translations of Sigma-calculi into process calculi, with the idea that one should be able to show properties of Sigma-calculus program by showing properties about their translation. We present translations of two Sigma-calculi into Pi-calculi. A translation of the untyped functional Sigma-calculus...... turns out to be insufficient. Based on our experiences, we present a translation of a typed imperative Sigma-calculus, which looks promising. We are able to provide simple proofs of the equivalence of different Sigma-calculus objects using this translation. We use a labelled transition system adapted...

  9. Enriching an effect calculus with linear types

    DEFF Research Database (Denmark)

    Egger, Jeff; Møgelberg, Rasmus Ejlers; Simpson, Alex

    2009-01-01

    We define an ``enriched effect calculus'' by conservatively extending  a type theory for computational effects with primitives from linear logic. By doing so, we obtain a generalisation of linear type theory, intended as a formalism for expressing linear aspects of effects. As a worked example, we...... formulate  linearly-used continuations in the enriched effect calculus. These are captured by a fundamental translation of the enriched effect calculus into itself, which extends existing call-by-value and call-by-name linearly-used CPS translations. We show that our translation is involutive. Full...... completeness results for the various linearly-used CPS translations  follow. Our main results, the conservativity of enriching the effect calculus with linear primitives, and the involution property of the fundamental translation, are proved using a category-theoretic semantics for the enriched effect calculus...

  10. A functional presentation of Pi calculus

    Institute of Scientific and Technical Information of China (English)

    2001-01-01

    From the very beginning process algebra introduced the dichotomy between channels and processes. This dichotomy prevails in all present process calculi.The situation is in contrast to that with lambda calculus which has only one class of entities——the lambda terms. We introduce in this paper a process calculus called Lamp in which channels are process names. The language is more uniform than existing process calculi in two aspects: First it has a unified treatment of channels and processes. There is only one class of syntactical entities——processes. Second it has a unified presentation of both first order and higher order process calculi. The language is functional in the sense that lambda calculus is functional.Two bisimulation equivalences, barbed and closed bisimilarities, are proved to coincide.A natural translation from Pi calculus to Lamp is shown to preserve both operational and algebraic semantics. The relationship between lazy lambda calculus and Lamp is discussed.

  11. A Calculus of Located Entities

    Directory of Open Access Journals (Sweden)

    Adriana Compagnoni

    2014-03-01

    Full Text Available We define BioScapeL, a stochastic pi-calculus in 3D-space. A novel aspect of BioScapeL is that entities have programmable locations. The programmer can specify a particular location where to place an entity, or a location relative to the current location of the entity. The motivation for the extension comes from the need to describe the evolution of populations of biochemical species in space, while keeping a sufficiently high level description, so that phenomena like diffusion, collision, and confinement can remain part of the semantics of the calculus. Combined with the random diffusion movement inherited from BioScape, programmable locations allow us to capture the assemblies of configurations of polymers, oligomers, and complexes such as microtubules or actin filaments. Further new aspects of BioScapeL include random translation and scaling. Random translation is instrumental in describing the location of new entities relative to the old ones. For example, when a cell secretes a hydronium ion, the ion should be placed at a given distance from the originating cell, but in a random direction. Additionally, scaling allows us to capture at a high level events such as division and growth; for example, daughter cells after mitosis have half the size of the mother cell.

  12. Using Dynamic Software to Address Common College Calculus Stumbling Blocks

    Science.gov (United States)

    Seneres, Alice W.; Kerrigan, John A.

    2014-01-01

    There are specific topics in college calculus that can be major stumbling blocks for students. Having taught college calculus for four years to over a thousand students, we observed that even the students who have already taken pre-calculus or calculus during their high school careers had common misunderstandings. Students may remember a technique…

  13. Using Dynamic Software to Address Common College Calculus Stumbling Blocks

    Science.gov (United States)

    Seneres, Alice W.; Kerrigan, John A.

    2014-01-01

    There are specific topics in college calculus that can be major stumbling blocks for students. Having taught college calculus for four years to over a thousand students, we observed that even the students who have already taken pre-calculus or calculus during their high school careers had common misunderstandings. Students may remember a technique…

  14. Relativity in Introductory Physics

    CERN Document Server

    Baylis, W E

    2004-01-01

    A century after its formulation by Einstein, it is time to incorporate special relativity early in the physics curriculum. The approach advocated here employs a simple algebraic extension of vector formalism that generates Minkowski spacetime, displays covariant symmetries, and enables calculations of boosts and spatial rotations without matrices or tensors. The approach is part of a comprehensive geometric algebra with applications in many areas of physics, but only an intuitive subset is needed at the introductory level. The approach and some of its extensions are given here and illustrated with insights into the geometry of spacetime.

  15. Mathematization in introductory physics

    Science.gov (United States)

    Brahmia, Suzanne M.

    Mathematization is central to STEM disciplines as a cornerstone of the quantitative reasoning that characterizes these fields. Introductory physics is required for most STEM majors in part so that students develop expert-like mathematization. This dissertation describes coordinated research and curriculum development for strengthening mathematization in introductory physics; it blends scholarship in physics and mathematics education in the form of three papers. The first paper explores mathematization in the context of physics, and makes an original contribution to the measurement of physics students' struggle to mathematize. Instructors naturally assume students have a conceptual mastery of algebra before embarking on a college physics course because these students are enrolled in math courses beyond algebra. This paper provides evidence that refutes the validity of this assumption and categorizes some of the barriers students commonly encounter with quantification and representing ideas symbolically. The second paper develops a model of instruction that can help students progress from their starting points to their instructor's desired endpoints. Instructors recognize that the introductory physics course introduces new ideas at an astonishing rate. More than most physicists realize, however, the way that mathematics is used in the course is foreign to a large portion of class. This paper puts forth an instructional model that can move all students toward better quantitative and physical reasoning, despite the substantial variability of those students' initial states. The third paper describes the design and testing of curricular materials that foster mathematical creativity to prepare students to better understand physics reasoning. Few students enter introductory physics with experience generating equations in response to specific challenges involving unfamiliar quantities and units, yet this generative use of mathematics is typical of the thinking involved in

  16. Motivation and Study Habits of College Calculus Students: Does Studying Calculus in High School Make a Difference?

    Science.gov (United States)

    Gibson, Megan

    2013-01-01

    Due in part to the growing popularity of the Advanced Placement program, an increasingly large percentage of entering college students are enrolling in calculus courses having already taken calculus in high school. Many students do not score high enough on the AP calculus examination to place out of Calculus I, and many do not take the…

  17. The Impact of Taking a College Pre-Calculus Course on Students' College Calculus Performance

    Science.gov (United States)

    Sonnert, Gerhard; Sadler, Philip M.

    2014-01-01

    Poor performance on placement exams keeps many US students who pursue a STEM (science, technology, engineering, mathematics) career from enrolling directly in college calculus. Instead, they must take a pre-calculus course that aims to better prepare them for later calculus coursework. In the USA, enrollment in pre-calculus courses in two- and…

  18. The Impact of Taking a College Pre-Calculus Course on Students' College Calculus Performance

    Science.gov (United States)

    Sonnert, Gerhard; Sadler, Philip M.

    2014-01-01

    Poor performance on placement exams keeps many US students who pursue a STEM (science, technology, engineering, mathematics) career from enrolling directly in college calculus. Instead, they must take a pre-calculus course that aims to better prepare them for later calculus coursework. In the USA, enrollment in pre-calculus courses in two- and…

  19. Motivation and Study Habits of College Calculus Students: Does Studying Calculus in High School Make a Difference?

    Science.gov (United States)

    Gibson, Megan

    2013-01-01

    Due in part to the growing popularity of the Advanced Placement program, an increasingly large percentage of entering college students are enrolling in calculus courses having already taken calculus in high school. Many students do not score high enough on the AP calculus examination to place out of Calculus I, and many do not take the…

  20. Modern introductory physics

    CERN Document Server

    Holbrow, Charles H; Amato, Joseph C; Galvez, Enrique; Parks, M. Elizabeth

    2010-01-01

    Modern Introductory Physics, 2nd Edition, by Charles H. Holbrow, James N. Lloyd, Joseph C. Amato, Enrique Galvez, and Beth Parks, is a successful innovative text for teaching introductory college and university physics. It is thematically organized to emphasize the physics that answers the fundamental question: Why do we believe in atoms and their properties?  The book provides a sound introduction to basic physical concepts with particular attention to the nineteenth- and twentieth-century physics underlying our modern ideas of atoms and their structure.  After a review of basic Newtonian mechanics, the book discusses early physical evidence that matter is made of atoms.  With a simple model of the atom Newtonian mechanics can explain the ideal gas laws, temperature, and viscosity.  Basic concepts of electricity and magnetism are introduced along with a more complicated model of the atom to account for the observed electrical properties of atoms. The physics of waves---particularly light and x-rays---an...

  1. AP calculus AB & BC crash course

    CERN Document Server

    Rosebush, J

    2012-01-01

    AP Calculus AB & BC Crash Course - Gets You a Higher Advanced Placement Score in Less Time Crash Course is perfect for the time-crunched student, the last-minute studier, or anyone who wants a refresher on the subject. AP Calculus AB & BC Crash Course gives you: Targeted, Focused Review - Study Only What You Need to Know Crash Course is based on an in-depth analysis of the AP Calculus AB & BC course description outline and actual AP test questions. It covers only the information tested on the exams, so you can make the most of your valuable study time. Written by experienced math teachers, our

  2. Recursive sequences in first-year calculus

    Science.gov (United States)

    Krainer, Thomas

    2016-02-01

    This article provides ready-to-use supplementary material on recursive sequences for a second-semester calculus class. It equips first-year calculus students with a basic methodical procedure based on which they can conduct a rigorous convergence or divergence analysis of many simple recursive sequences on their own without the need to invoke inductive arguments as is typically required in calculus textbooks. The sequences that are accessible to this kind of analysis are predominantly (eventually) monotonic, but also certain recursive sequences that alternate around their limit point as they converge can be considered.

  3. Pre-calculus workbook for dummies

    CERN Document Server

    Gilman, Michelle Rose; Neal, Karina

    2009-01-01

    Get the confidence and the math skills you need to get started with calculus! Are you preparing for calculus? This easy-to-follow, hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in your cour sework. You get valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every problem. You'll also memorize the most frequently used equations, see how to avoid common mistakes, understand tricky trig proofs, and much more. 100s of Problems! Detailed, fully worked-out solutions to problem

  4. Sequent Calculus in the Topos of Trees

    DEFF Research Database (Denmark)

    Clouston, Ranald; Goré, Rajeev

    2015-01-01

    of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal logic KM and the intermediate logic LC. We therefore call this logic KMlin . We give a sound and cut-free complete sequent calculus for KMlin via a strategy...... that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence...

  5. Area Regge calculus and continuum limit

    CERN Document Server

    Khatsymovsky, V M

    2002-01-01

    Encountered in the literature generalisations of general relativity to independent area variables are considered, the discrete (generalised Regge calculus) and continuum ones. The generalised Regge calculus can be either with purely area variables or, as we suggest, with area tensor-connection variables. Just for the latter, in particular, we prove that in analogy with corresponding statement in ordinary Regge calculus (by Feinberg, Friedberg, Lee and Ren), passing to the (appropriately defined) continuum limit yields the generalised continuum area tensor-connection general relativity.

  6. Calculus a complete introduction : teach yourself

    CERN Document Server

    Neill, Hugh

    2013-01-01

    Calculus: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using calculus. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all areas of calculus, including functions, gradients, rates of change, differentiation, exponential and logarithmic functions and integration. Everything you will need to know is here in one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions.

  7. Applications of fractional calculus in physics

    CERN Document Server

    2000-01-01

    Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to noninteger orders. While these results have been accumulated over centuries in various branches of mathematics, they have until recently found little appreciation or application in physics and other mathematically oriented sciences. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus.This volume provides an introduction to fractional calculus for physicists, and co

  8. On Some Syntactic Properties of the Modalized Heyting Calculus

    OpenAIRE

    Muravitsky, Alexei

    2016-01-01

    We show that the modalized Heyting calculus introduced by Leo Esakia admits a normal axiomatization. Then, we prove that the inference rules $\\square\\alpha/\\alpha$ and $\\square\\alpha\\rightarrow\\alpha/\\alpha$ are admissible in this calculus. Finally, we show that this calculus and intuitionistic propositional calculus are assertorically equipollent, which leads to a variant of limited separation property for the modalized Heyting calculus.

  9. On Some Syntactic Properties of the Modalized Heyting Calculus

    OpenAIRE

    Muravitsky, Alexei

    2016-01-01

    We show that the modalized Heyting calculus introduced by Leo Esakia admits a normal axiomatization. Then, we prove that the inference rules $\\square\\alpha/\\alpha$ and $\\square\\alpha\\rightarrow\\alpha/\\alpha$ are admissible in this calculus. Finally, we show that this calculus and intuitionistic propositional calculus are assertorically equipollent, which leads to a variant of limited separation property for the modalized Heyting calculus.

  10. From X to Pi; Representing the Classical Sequent Calculus in the Pi-calculus

    CERN Document Server

    van Bakel, Steffen; Vigliotti, Maria Grazia

    2011-01-01

    We study the Pi-calculus, enriched with pairing and non-blocking input, and define a notion of type assignment that uses the type constructor "arrow". We encode the circuits of the calculus X into this variant of Pi, and show that all reduction (cut-elimination) and assignable types are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen's calculus LK, this implies that all proofs in LK have a representation in Pi.

  11. Astrophysical Applications of Fractional Calculus

    Science.gov (United States)

    Stanislavsky, Aleksander A.

    The paradigm of fractional calculus occupies an important place for the macroscopic description of subdiffusion. Its advance in theoretical astrophysics is expected to be very attractive too. In this report we discuss a recent development of the idea to some astrophysical problems. One of them is connected with a random migration of bright points associated with magnetic fields at the solar photosphere. The transport of the bright points has subdiffusive features that require the fractional generalization of the Leighton's model. Another problem is related to the angular distribution of radio beams, being propagated through a medium with random inhomogeneities. The peculiarity of this medium is that radio beams are trapped because of random wave localization. This idea can be useful for the diagnostics of interplanetary and interstellar turbulent media.

  12. A CALCULUS FOR SERVICES INNOVATION

    Institute of Scientific and Technical Information of China (English)

    James M.TIEN; Daniel BERG

    2007-01-01

    Innovation in the services area - especially in the electronic services (e-services) domain - can be systematically developed by first considering the strategic drivers and foci, then the tactical principles and enablers, and finally the operational decision attributes, all of which constitute a process or calculus of services innovation. More specifically, there are four customer drivers (i.e., collaboration,customization, integration and adaptation), three business foci (i.e., creation-focused, solution-focused and competition-focused), six business principles (i.e., reconstruct market boundaries, focus on the big picture not numbers, reach beyond existing demand, get strategic sequence right, overcome organizational hurdles and build execution into strategy), eight technical enablers (i.e., software algorithms, automation, telecommunication, collaboration, standardization, customization,organization, and globalization), and six attributes of decision informatics (i.e., decision-driven,information-based, real-time, continuously-adaptive, customer-centric and computationally-intensive).It should be noted that the four customer drivers are all directed at empowering the individual - that is,at recognizing that the individual can, respectively, contribute in a collaborative situation, receive customized or personalized attention, access an integrated system or process, and obtain adaptive real-time or just-in-time input. The developed process or calculus serves to identify the potential white spaces or blue oceans for innovation. In addition to expanding on current innovations in services and related experiences, white spaces are identified for possible future innovations; they include those that can mitigate the unforeseen consequences or abuses of earlier innovations, safeguard our rights to privacy, protect us from the always-on, interconnected world, provide us with an authoritative search engine, and generate a GDP metric that can adequately measure the growing

  13. The Calculus Concept Readiness (CCR) Instrument: Assessing Student Readiness for Calculus

    CERN Document Server

    Carlson, Marilyn; West, Richard

    2010-01-01

    The Calculus Concept Readiness (CCR) instrument is based on the broad body of mathematics education research that has revealed major understandings, representational abilities, and reasoning abilities students need to construct in precalculus level courses to be successful in calculus. The CCR is a 25-item multiple-choice instrument, and the CCR taxonomy articulates what the CCR assesses. The methodology used to develop and validate the CCR is described and illustrated. Results from administering the CCR as a readiness examination in calculus are provided along with data to guide others in using the CCR as a readiness examination for beginning calculus.

  14. The calculus lifesaver all the tools you need to excel at calculus

    CERN Document Server

    Banner, Adrian

    2009-01-01

    For many students, calculus can be the most mystifying and frustrating course they will ever take. The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it. All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner's popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A's but get only average grades on exams. The complete course will be available for free on the Web in a series of

  15. Effect of written presentation on performance in introductory physics

    Directory of Open Access Journals (Sweden)

    Shawn Ballard

    2010-10-01

    Full Text Available This study examined the written work of students in the introductory calculus-based electricity and magnetism course at the University of Arkansas. The students’ solutions to hourly exams were divided into a small set of countable features organized into three major categories, mathematics, language, and graphics. Each category was further divided into subfeatures. The total number of features alone explained more than 30% of the variance in exam scores and from 9% to 15% of the variance in conceptual posttest scores. If all features and subfeatures are used, between 44% and 49% of the variance in exam scores is explained and between 22% and 28% of the variance in conceptual posttest scores. The use of language is consistently positively correlated with both exam performance and conceptual understanding.

  16. Inward bound/outward bound: Modern introductory physics at Colgate

    Science.gov (United States)

    Holbrow, C. H.; Amato, J. C.

    1997-03-01

    For the past ten years we have been modernizing our calculus-level introductory physics course. The first term is now organized around the theme of atoms: Why do we believe in them? How do we learn about them? The course traces the progress of our understanding of inner space, from the origins of the atomic hypothesis to present day quantum physics. The second term illustrates how physics extends our range of understanding to outer space by applying the principles of classical mechanics to large-scale structures such as stars, galaxies, and the entire Universe. In our approach the three conservation laws receive more emphasis than they do in the traditional course. We believe that this allows a simpler exposition of classical mechanics. Our goals are to convey the excitement and challenge of contemporary physics to our first year students, and at the same time, to build their mathematical skills.

  17. Extending Stochastic Network Calculus to Loss Analysis

    Directory of Open Access Journals (Sweden)

    Chao Luo

    2013-01-01

    Full Text Available Loss is an important parameter of Quality of Service (QoS. Though stochastic network calculus is a very useful tool for performance evaluation of computer networks, existing studies on stochastic service guarantees mainly focused on the delay and backlog. Some efforts have been made to analyse loss by deterministic network calculus, but there are few results to extend stochastic network calculus for loss analysis. In this paper, we introduce a new parameter named loss factor into stochastic network calculus and then derive the loss bound through the existing arrival curve and service curve via this parameter. We then prove that our result is suitable for the networks with multiple input flows. Simulations show the impact of buffer size, arrival traffic, and service on the loss factor.

  18. Brownian motion, martingales, and stochastic calculus

    CERN Document Server

    Le Gall, Jean-François

    2016-01-01

    This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested i...

  19. Fractional Calculus and -Valently Starlike Functions

    Directory of Open Access Journals (Sweden)

    Özkan Öznur

    2009-01-01

    Full Text Available Abstract In this investigation, the authors prove coefficient bounds, distortion inequalities for fractional calculus of a family of multivalent functions with negative coefficients, which is defined by means of a certain nonhomogenous Cauchy-Euler differential equation.

  20. Applying Change of Variable to Calculus Problems

    Science.gov (United States)

    Kachapova, Farida; Kachapov, Ilias

    2011-01-01

    This article describes the technique of introducing a new variable in some calculus problems to help students master the skills of integration and evaluation of limits. This technique is algorithmic and easy to apply.

  1. The origins of Cauchy's rigorous calculus

    CERN Document Server

    Grabiner, Judith V

    2005-01-01

    This text examines the reinterpretation of calculus by Augustin-Louis Cauchy and his peers in the 19th century. These intellectuals created a collection of well-defined theorems about limits, continuity, series, derivatives, and integrals. 1981 edition.

  2. Model-Checking Discrete Duration Calculus

    DEFF Research Database (Denmark)

    Hansen, Michael Reichhardt

    1994-01-01

    Duration calculus was introduced by Chaochen Zhou et al. (1991) as a logic to specify and reason about requirements for real-time systems. It is an extension of interval temporal logic where one can reason about integrated constraints over time-dependent and Boolean valued states without explicit...... mention of absolute time. Several major case studies have shown that duration calculus provides a high level of abstraction for both expressing and reasoning about specifications. Using timed automata one can express how real-time systems can be constructed at a level of detail which is close to an actual...... implementation. We consider in the paper the correctness of timed automata with respect to duration calculus formulae. For a subset of duration calculus, we show that one can automatically verify whether a timed automaton ℳ is correct with respect to a formula 𝒟, abbreviated ℳ|=𝒟, i.e. one...

  3. One Answer to "What Is Calculus?"

    Science.gov (United States)

    Shilgalis, Thomas W.

    1979-01-01

    A number of questions are posed that can be answered with the aid of calculus. These include best value problems, best shape problems, problems involving integration, and growth and decay problems. (MP)

  4. Model-Checking Discrete Duration Calculus

    DEFF Research Database (Denmark)

    Hansen, Michael Reichhardt

    1994-01-01

    Duration calculus was introduced by Chaochen Zhou et al. (1991) as a logic to specify and reason about requirements for real-time systems. It is an extension of interval temporal logic where one can reason about integrated constraints over time-dependent and Boolean valued states without explicit...... mention of absolute time. Several major case studies have shown that duration calculus provides a high level of abstraction for both expressing and reasoning about specifications. Using timed automata one can express how real-time systems can be constructed at a level of detail which is close to an actual...... implementation. We consider in the paper the correctness of timed automata with respect to duration calculus formulae. For a subset of duration calculus, we show that one can automatically verify whether a timed automaton ℳ is correct with respect to a formula 𝒟, abbreviated ℳ|=𝒟, i.e. one...

  5. Applications of Monte Carlo Methods in Calculus.

    Science.gov (United States)

    Gordon, Sheldon P.; Gordon, Florence S.

    1990-01-01

    Discusses the application of probabilistic ideas, especially Monte Carlo simulation, to calculus. Describes some applications using the Monte Carlo method: Riemann sums; maximizing and minimizing a function; mean value theorems; and testing conjectures. (YP)

  6. Matrix calculus for axially symmetric polarized beam.

    Science.gov (United States)

    Matsuo, Shigeki

    2011-06-20

    The Jones calculus is a well known method for analyzing the polarization of a fully polarized beam. It deals with a beam having spatially homogeneous polarization. In recent years, axially symmetric polarized beams, where the polarization is not homogeneous in its cross section, have attracted great interest. In the present article, we show the formula for the rotation of beams and optical elements on the angularly variant term-added Jones calculus, which is required for analyzing axially symmetric beams. In addition, we introduce an extension of the Jones calculus: use of the polar coordinate basis. With this calculus, the representation of some angularly variant beams and optical elements are simplified and become intuitive. We show definitions, examples, and conversion formulas between different notations.

  7. A Temporal Approach to Stochastic Network Calculus

    CERN Document Server

    Xie, Jing; Xie, Min

    2011-01-01

    Stochastic network calculus is a newly developed theory for stochastic service guarantee analysis of computer networks. In the current stochastic network calculus literature, its fundamental models are based on the cumulative amount of traffic or cumulative amount of service. However, there are network scenarios where direct application of such models is difficult. This paper presents a temporal approach to stochastic network calculus. The key idea is to develop models and derive results from the time perspective. Particularly, we define traffic models and service models based on the cumulative packet inter-arrival time and the cumulative packet service time, respectively. Relations among these models as well as with the existing models in the literature are established. In addition, we prove the basic properties of the proposed models, such as delay bound and backlog bound, output characterization, concatenation property and superposition property. These results form a temporal stochastic network calculus an...

  8. A primer on exterior differential calculus

    Directory of Open Access Journals (Sweden)

    Burton D.A.

    2003-01-01

    Full Text Available A pedagogical application-oriented introduction to the cal­culus of exterior differential forms on differential manifolds is presented. Stokes' theorem, the Lie derivative, linear con­nections and their curvature, torsion and non-metricity are discussed. Numerous examples using differential calculus are given and some detailed comparisons are made with their tradi­tional vector counterparts. In particular, vector calculus on R3 is cast in terms of exterior calculus and the traditional Stokes' and divergence theorems replaced by the more powerful exterior expression of Stokes' theorem. Examples from classical continuum mechanics and spacetime physics are discussed and worked through using the language of exterior forms. The numerous advantages of this calculus, over more traditional ma­chinery, are stressed throughout the article. .

  9. Extending stochastic network calculus to loss analysis.

    Science.gov (United States)

    Luo, Chao; Yu, Li; Zheng, Jun

    2013-01-01

    Loss is an important parameter of Quality of Service (QoS). Though stochastic network calculus is a very useful tool for performance evaluation of computer networks, existing studies on stochastic service guarantees mainly focused on the delay and backlog. Some efforts have been made to analyse loss by deterministic network calculus, but there are few results to extend stochastic network calculus for loss analysis. In this paper, we introduce a new parameter named loss factor into stochastic network calculus and then derive the loss bound through the existing arrival curve and service curve via this parameter. We then prove that our result is suitable for the networks with multiple input flows. Simulations show the impact of buffer size, arrival traffic, and service on the loss factor.

  10. A Graph Calculus for Predicate Logic

    Directory of Open Access Journals (Sweden)

    Paulo A. S. Veloso

    2013-03-01

    Full Text Available We introduce a refutation graph calculus for classical first-order predicate logic, which is an extension of previous ones for binary relations. One reduces logical consequence to establishing that a constructed graph has empty extension, i. e. it represents bottom. Our calculus establishes that a graph has empty extension by converting it to a normal form, which is expanded to other graphs until we can recognize conflicting situations (equivalent to a formula and its negation.

  11. Łukasiewicz mu-Calculus

    Directory of Open Access Journals (Sweden)

    Matteo Mio

    2013-08-01

    Full Text Available The paper explores properties of Łukasiewicz mu-calculus, a version of the quantitative/probabilistic modal mu-calculus containing both weak and strong conjunctions and disjunctions from Łukasiewicz (fuzzy logic. We show that this logic encodes the well-known probabilistic temporal logic PCTL. And we give a model-checking algorithm for computing the rational denotational value of a formula at any state in a finite rational probabilistic nondeterministic transition system.

  12. Tuplix Calculus Specifications of Financial Transfer Networks

    CERN Document Server

    Bergstra, J A; van der Zwaag, M B

    2008-01-01

    We study the application of Tuplix Calculus in modular financial budget design. We formalize organizational structure using financial transfer networks. We consider the notion of flux of money over a network, and a way to enforce the matching of influx and outflux for parts of a network. We exploit so-called signed attribute notation to make internal streams visible through encapsulations. Finally, we propose a Tuplix Calculus construct for the definition of data functions.

  13. The Britannica Guide to Analysis and Calculus

    CERN Document Server

    2011-01-01

    The dynamism of the natural world means that it is constantly changing, sometimes rapidly, sometimes gradually. By mathematically interpreting the continuous change that characterizes so many natural processes, analysis and calculus have become indispensable to bridging the divide between mathematics and the sciences. This comprehensive volume examines the key concepts of calculus, providing students with a robust understanding of integration and differentiation. Biographies of important figures will leave readers with an increased appreciation for the sometimes competing theories that informe

  14. Borel functional calculus for quaternionic normal operators

    Science.gov (United States)

    G, Ramesh; P, Santhosh Kumar

    2017-05-01

    In this article, we give an approach to Borel functional calculus for quaternionic normal operators, which are not necessarily bounded. First, we establish the definition of functional calculus for a subclass of quaternion valued Borel functions, and then we extend the same to the class of quaternion valued Borel functions as well as L∞-functions. We also prove spectral mapping theorem as a consequence.

  15. Analysis of student engagement in an online annotation system in the context of a flipped introductory physics class

    Science.gov (United States)

    Miller, Kelly; Zyto, Sacha; Karger, David; Yoo, Junehee; Mazur, Eric

    2016-12-01

    We discuss student participation in an online social annotation forum over two semesters of a flipped, introductory physics course at Harvard University. We find that students who engage in high-level discussion online, especially by providing answers to their peers' questions, make more gains in conceptual understanding than students who do not. This is true regardless of students' physics background. We find that we can steer online interaction towards more productive and engaging discussion by seeding the discussion and managing the size of the sections. Seeded sections produce higher quality annotations and a greater proportion of generative threads than unseeded sections. Larger sections produce longer threads; however, beyond a certain section size, the quality of the discussion decreases.

  16. Introductory physics going soft

    Science.gov (United States)

    Langbeheim, Elon; Livne, Shelly; Safran, Samuel A.; Yerushalmi, Edit

    2012-01-01

    We describe an elective course on soft matter at the level of introductory physics. Soft matter physics serves as a context that motivates the presentation of basic ideas in statistical thermodynamics and their applications. It also is an example of a contemporary field that is interdisciplinary and touches on chemistry, biology, and physics. We outline a curriculum that uses the lattice gas model as a quantitative and visual tool, initially to introduce entropy, and later to facilitate the calculation of interactions. We demonstrate how free energy minimization can be used to teach students to understand the properties of soft matter systems such as the phases of fluid mixtures, wetting of interfaces, self-assembly of surfactants, and polymers. We discuss several suggested activities in the form of inquiry projects which allow students to apply the concepts they have learned to experimental systems.

  17. Differential Calculus on N-Graded Manifolds

    Directory of Open Access Journals (Sweden)

    G. Sardanashvily

    2017-01-01

    Full Text Available The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over N-graded commutative rings and on N-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and on Z2-graded manifolds. We follow the notion of an N-graded manifold as a local-ringed space whose body is a smooth manifold Z. A key point is that the graded derivation module of the structure ring of graded functions on an N-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body Z. Accordingly, the Chevalley–Eilenberg differential calculus on an N-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on N-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of N-graded bundles.

  18. Barbed congruence of the asymmetric chi calculus

    Institute of Scientific and Technical Information of China (English)

    DONG Xiao-ju; FU Yu-xi

    2006-01-01

    The chi calculus is a model of mobile processes. It has evolved from the pi-calculus with motivations from simplification and communication-as-cut-elimination. This paper studies the chi calculus in the framework incorporating asymmetric communication. The major feature of the calculus is the identification of two actions:x/x and τ. The investigation on the barbed bisimilarity shows how the property affects the observational theory.Based on the definition of the barbed bisimilarity, the simulation properties of the barbed bisimilarity are studied. It shows that the algebraic properties of the barbed bisimilarity have changed greatly compared with the chi calculus. Although the definition of the barbed bisimilarity is very simple, the property of closeness under contexts makes it difficult to understand the barbed bisimilarity directly. Therefore an open style definition of the barbed bisimilarity is given, which is a context free description of barbed bisimilarity. Its definition is complex,but it is a well-behaved relation for it coincides with the barbed bisimilarity. It also helps to build an axiomatization system for the barbed congruence. Besides the axioms for the strong barbed bisimilarity, the paper proposes a new tau law and four new update laws for the barbed congruence. Both the operational and algebraic properties of the enriched calculus improve the understanding of the bisimulation behaviors of the model.

  19. Endoscopic vs. tactile evaluation of subgingival calculus.

    Science.gov (United States)

    Osborn, Joy B; Lenton, Patricia A; Lunos, Scott A; Blue, Christine M

    2014-08-01

    Endoscopic technology has been developed to facilitate imagery for use during diagnostic and therapeutic phases of periodontal care. The purpose of this study was to compare the level of subgingival calculus detection using a periodontal endoscope with that of conventional tactile explorer in periodontitis subjects. A convenience sample of 26 subjects with moderate periodontitis in at least 2 quadrants was recruited from the University of Minnesota School of Dentistry to undergo quadrant scaling and root planing. One quadrant from each subject was randomized for tactile calculus detection alone and the other quadrant for tactile detection plus the Perioscope ™ (Perioscopy Inc., Oakland, Cali). A calculus index on a 0 to 3 score was performed at baseline and at 2 post-scaling and root planing visits. Sites where calculus was detected at visit 1 were retreated. T-tests were used to determine within-subject differences between Perioscope™ and tactile measures, and changes in measures between visits. Significantly more calculus was detected using the Perioscope™ vs. tactile explorer for all 3 subject visits (pcalculus detection from baseline to visit 1 were statistically significant for both the Perioscope™ and tactile quadrants (pcalculus detection from visit 1 to visit 2 was only significant for the Perioscope™ quadrant (pcalculus at this visit. It was concluded that the addition of a visual component to calculus detection via the Perioscope™ was most helpful in the re-evaluation phase of periodontal therapy. Copyright © 2014 The American Dental Hygienists’ Association.

  20. Modelling and Analysis of Dynamic Reconfiguration in BP-Calculus

    DEFF Research Database (Denmark)

    Abouzaid, Faisal; Mullins, John; Mazzara, Manuel;

    2012-01-01

    The BP-calculus is a formalism based on the π-calculus and encoded in WS-BPEL. The BP-calculus is intended to specificaly model and verify Service Oriented Applications. One important feature of SOA is the ability to compose services that may dynamically evolve along runtime. Dynamic reconfigurat......The BP-calculus is a formalism based on the π-calculus and encoded in WS-BPEL. The BP-calculus is intended to specificaly model and verify Service Oriented Applications. One important feature of SOA is the ability to compose services that may dynamically evolve along runtime. Dynamic...

  1. Introductory statistics for engineering experimentation

    CERN Document Server

    Nelson, Peter R; Coffin, Marie

    2003-01-01

    The Accreditation Board for Engineering and Technology (ABET) introduced a criterion starting with their 1992-1993 site visits that "Students must demonstrate a knowledge of the application of statistics to engineering problems." Since most engineering curricula are filled with requirements in their own discipline, they generally do not have time for a traditional two semesters of probability and statistics. Attempts to condense that material into a single semester often results in so much time being spent on probability that the statistics useful for designing and analyzing engineering/scientific experiments is never covered. In developing a one-semester course whose purpose was to introduce engineering/scientific students to the most useful statistical methods, this book was created to satisfy those needs. - Provides the statistical design and analysis of engineering experiments & problems - Presents a student-friendly approach through providing statistical models for advanced learning techniques - Cove...

  2. Teaching introductory physics a sourcebook

    CERN Document Server

    Swartz, Clifford E

    1996-01-01

    Introductory physics attracts a wide variety of students, with different backgrounds, levels of preparedness, and academic destinations. To many, the course is one of the most daunting in the science curriclum, full of arcane principles that are difficult to grasp. To others, it is one of the most highly anticipated -the first step on the path to the upper reaches of scientific inquiry. In their years as instructors and as editors of The Physics Teacher, Clifford E. Swartz and the late Thomas Miner developed and encountered many innovative and effective ways of introducing students to the fundamental principles of physics. Teaching Introductory Physics brings these strategies, insights and techniques to you in a unique, convenient volume. This is a reference and a tutorial book for teachers of an introductory physics course at any level. It has review articles on most of the topics of introductory physics, providing background information and suggestions about presentation and relative importance. Whether you...

  3. Introductory Statistics in the Garden

    Science.gov (United States)

    Wagaman, John C.

    2017-01-01

    This article describes four semesters of introductory statistics courses that incorporate service learning and gardening into the curriculum with applications of the binomial distribution, least squares regression and hypothesis testing. The activities span multiple semesters and are iterative in nature.

  4. News clippings for introductory astronomy

    Science.gov (United States)

    Bobrowsky, Matthew

    1999-09-01

    Most students entering our introductory astronomy course for nonscience majors arrive not merely lacking scientific facts-they also have misconceptions about the nature of science, and many have a handicapping ``science anxiety'' (in addition to math anxiety). So I have added a ``current science'' requirement to our introductory course. Each student must compile a file of five astronomy news articles taken from readily available sources.

  5. A MOOC for Introductory Physics

    Science.gov (United States)

    Schatz, Michael

    2014-03-01

    We describe an effort to develop and to implement a college-level introductory physics (mechanics) MOOC that offers bona fide laboratory experiences. We also discuss efforts to use MOOC curricular materials to ``flip'' the classroom in a large lecture introductory physics course offered on-campus at Georgia Tech. Preliminary results of assessments and surveys from both MOOC and on-campus students will be presented.

  6. Electronic Algebra and Calculus Tutor

    Directory of Open Access Journals (Sweden)

    Larissa Fradkin

    2012-06-01

    Full Text Available Modern undergraduates join science and engineering courses with poorer mathematical background than most contemporaries of the current faculty had when they were freshers. The problem is very acute in the United Kingdom but more and more countries adopt less resource intensive models of teaching and the problem spreads. University tutors and lecturers spend more and more time covering the basics. However, most of them still rely on traditional methods of delivery which presuppose that learners have a good memory and considerable time to practice, so that they can memorize disjointed facts and discover for themselves various connections between the underlying concepts. These suppositions are particularly unrealistic when dealing with a large number of undergraduates who are ordinary learners with limited mathematics background. The first author has developed a teaching system that allows such adult learners achieve relatively deep learning of mathematics – and remarkably quickly – through a teacher-guided (often called Socratic dialog, which aims at the frequent reinforcement of basic mathematical abstractions through Eulerian sequencing. These ideas have been applied to create a prototype of a Cognitive Mathematics Tutoring System aimed at teaching basic mathematics to University freshers., an electronic Personal Algebra and Calculus Tutor (e- PACT.

  7. Student Use of a Single Lecture Video in a Flipped Introductory Mechanics Course

    CERN Document Server

    Aiken, John M; Douglas, Scott S; Greco, Edwin F; Thoms, Brian D; Caballero, Marcos D; Schatz, Michael F

    2014-01-01

    In the Fall of 2013, Georgia Tech offered a 'flipped' calculus-based introductory mechanics class as an alternative to the traditional large-enrollment lecture class. This class flipped instruction by introducing new material outside of the classroom through pre-recorded, lecture videos. Video lectures constituted students' initial introduction to course material. We analyze how students engaged with online lecture videos via 'clickstream' data, consisting of time-stamped interactions (plays, pauses, seeks, etc.) with the online video player. Analysis of these events has shown that students may be focusing on elements of the video that facilitate a 'correct' solution.

  8. Comparing the efficacy of multimedia modules with traditional textbooks for learning introductory physics content

    CERN Document Server

    Stelzer, Timothy; Mestre, Jose; Brookes, David T

    2008-01-01

    A clinical study was performed comparing the efficacy of multimedia learning modules with traditional textbooks for the first few topics of a calculus based introductory electricity and magnetism course. Students were randomly assigned to three different groups experiencing different presentations of the material; one group received the multimedia learning module presentations and the other two received the presentations via written text. All students were then tested on their learning immediately following the presentations as well as two weeks later. The students receiving the multimedia learning modules performed significantly better than the students experiencing the text-based presentations on both tests.

  9. Ancient DNA analysis of dental calculus.

    Science.gov (United States)

    Weyrich, Laura S; Dobney, Keith; Cooper, Alan

    2015-02-01

    Dental calculus (calcified tartar or plaque) is today widespread on modern human teeth around the world. A combination of soft starchy foods, changing acidity of the oral environment, genetic pre-disposition, and the absence of dental hygiene all lead to the build-up of microorganisms and food debris on the tooth crown, which eventually calcifies through a complex process of mineralisation. Millions of oral microbes are trapped and preserved within this mineralised matrix, including pathogens associated with the oral cavity and airways, masticated food debris, and other types of extraneous particles that enter the mouth. As a result, archaeologists and anthropologists are increasingly using ancient human dental calculus to explore broad aspects of past human diet and health. Most recently, high-throughput DNA sequencing of ancient dental calculus has provided valuable insights into the evolution of the oral microbiome and shed new light on the impacts of some of the major biocultural transitions on human health throughout history and prehistory. Here, we provide a brief historical overview of archaeological dental calculus research, and discuss the current approaches to ancient DNA sampling and sequencing. Novel applications of ancient DNA from dental calculus are discussed, highlighting the considerable scope of this new research field for evolutionary biology and modern medicine.

  10. Anti-calculus and whitening toothpastes.

    Science.gov (United States)

    van Loveren, Cor; Duckworth, Ralph M

    2013-01-01

    In terms of novel formulations, there seems to have been a shift in emphasis from anti-caries/anti-gingivitis to anti-calculus/whitening toothpastes in recent years. The anti-calculus and whitening effects of toothpastes are to some extent based on the same active ingredients: compounds of high affinity for tooth mineral. Due to this affinity, crystal growth may be hindered (anti-calculus) and chromophores be displaced (whitening). Besides these common ingredients, both types of toothpaste may contain agents specifically aimed at each condition. Clinical studies have shown that these active ingredients can be successfully formulated in fluoride toothpastes to give significant reductions in supragingival calculus and stain formation and facilitate their removal. Some of the ingredients are formulated in toothpastes that additionally contain anti-plaque and anti-gingivitis ingredients, making these toothpastes (together with the fluoride) truly multi-functional. The development of these products is not straightforward because of interaction between formulation components and because the active ingredients must maintain their beneficial characteristics during the shelf life of the paste. Neither a therapeutic benefit (in terms of less gingivitis or less caries) nor a societal benefit (in terms of less treatment demand) has been demonstrated as a result of the anti-calculus and whitening effects of toothpastes.

  11. More calculus of a single variable

    CERN Document Server

    Mercer, Peter R

    2014-01-01

    This book goes beyond the basics of a first course in calculus to reveal the power and richness of the subject. Standard topics from calculus — such as the real numbers, differentiation and integration, mean value theorems, the exponential function — are reviewed and elucidated before digging into a deeper exploration of theory and applications, such as the AGM inequality, convexity, the art of integration, and explicit formulas for π. Further topics and examples are introduced through a plethora of exercises that both challenge and delight the reader. While the reader is thereby exposed to the many threads of calculus, the coherence of the subject is preserved throughout by an emphasis on patterns of development, of proof and argumentation, and of generalization. More Calculus of a Single Variable is suitable as a text for a course in advanced calculus, as a supplementary text for courses in analysis, and for self-study by students, instructors, and, indeed, all connoisseurs of ingenious calculations.

  12. The Case for Infusing Quantitative Literacy into Introductory Geoscience Courses

    Directory of Open Access Journals (Sweden)

    Jennifer M. Wenner

    2009-01-01

    Full Text Available We present the case for introductory geoscience courses as model venues for increasing the quantitative literacy (QL of large numbers of the college-educated population. The geosciences provide meaningful context for a number of fundamental mathematical concepts that are revisited several times in a single course. Using some best practices from the mathematics education community surrounding problem solving, calculus reform, pre-college mathematics and five geoscience/math workshops, geoscience and mathematics faculty have identified five pedagogical ideas to increase the QL of the students who populate introductory geoscience courses. These five ideas include techniques such as: place mathematical concepts in context, use multiple representations, use technology appropriately, work in groups, and do multiple-day, in-depth problems that place quantitative skills in multiple contexts. We discuss the pedagogical underpinnings of these five ideas and illustrate some ways that the geosciences represent ideal places to use these techniques. However, the inclusion of QL in introductory courses is often met with resistance at all levels. Faculty who wish to include quantitative content must use creative means to break down barriers of public perception of geoscience as qualitative, administrative worry that enrollments will drop and faculty resistance to change. Novel ways to infuse QL into geoscience classrooms include use of web-based resources, shadow courses, setting clear expectations, and promoting quantitative geoscience to the general public. In order to help faculty increase the QL of geoscience students, a community-built faculty-centered web resource (Teaching Quantitative Skills in the Geosciences houses multiple examples that implement the five best practices of QL throughout the geoscience curriculum. We direct faculty to three portions of the web resource: Teaching Quantitative Literacy, QL activities, and the 2006 workshop website

  13. Bladder calculus resulting from an intravesical translocation of ...

    African Journals Online (AJOL)

    Bladder calculus resulting from an intravesical translocation of intrauterine ... AFRICAN JOURNALS ONLINE (AJOL) · Journals · Advanced Search · USING AJOL ... translocation and secondary calculus formation is a very rare complication.

  14. Calculus of multivariate functions: it's application in business | Awen ...

    African Journals Online (AJOL)

    Calculus of multivariate functions: it's application in business. ... AFRICAN JOURNALS ONLINE (AJOL) · Journals · Advanced Search · USING AJOL · RESOURCES ... Calculus of multivariate functions is a mathematical concept that has to do ...

  15. Student difficulties in translating between mathematical and graphical representations in introductory physics

    CERN Document Server

    Lin, Shih-Yin; Singh, Chandralekha

    2016-01-01

    We investigate introductory physics students' difficulties in translating between mathematical and graphical representations and the effect of scaffolding on students' performance. We gave a typical problem that can be solved using Gauss's law involving a spherically symmetric charge distribution (a conducting sphere concentric with a conducting spherical shell) to 95 calculus-based introductory physics students. We asked students to write a mathematical expression for the electric field in various regions and asked them to graph the electric field. We knew from previous experience that students have great difficulty in graphing the electric field. Therefore, we implemented two scaffolding interventions to help them. Students who received the scaffolding support were either (1) asked to plot the electric field in each region first (before having to plot it as a function of distance from the center of the sphere) or (2) asked to plot the electric field in each region after explicitly evaluating the electric fi...

  16. Peaks and decays of student knowledge in an introductory E&M course

    Directory of Open Access Journals (Sweden)

    Eleanor C. Sayre

    2009-02-01

    Full Text Available A common format for assessment of learning is pretesting and post-testing. In this study, we collect student test data several times per week throughout a course, allowing for the measurement of the changes in student knowledge with a time resolution on the order of a few days. To avoid the possibility of test-retest effects, separate and quasirandom subpopulations of students are tested on a variety of tasks. We report on data taken in a calculus-based introductory E&M class populated primarily by engineering majors. Unsurprisingly for a traditional introductory course, there is little change in many conceptual questions. However, the data suggest that some student performance peaks and decays rapidly during a quarter, a pattern consistent with memory research yet unmeasurable by pretesting and post-testing. In addition, it appears that some course topics can interfere with prior knowledge, decreasing performance on questions related to earlier topics in the course.

  17. On realizations of exterior calculus with dN = 0

    Science.gov (United States)

    Abramov, V.

    1998-11-01

    We study realizations of the q-exterior calculus with exterior differential d satisfying d N = 0, N > 2 on the free associative algebra with one generator and on the generalized Clifford algebras. Analogs of the notions of connection and curvature are discussed in the case of the q-exterior calculus on the generalized Clifford algebra. We show that the q-exterior calculus on the free associative algebra with one generator is related to q-calculus on the braided line.

  18. RARE CASE OF GIANT VESICAL CALCULUS

    Directory of Open Access Journals (Sweden)

    Deepak Ramraj

    2015-02-01

    Full Text Available Giant vesical calculus is a rare entity. Vesical calculi can be primary (stones form de novo in bladder or secondary to the migrated renal calculi, chronic UTI, bladder outlet obstruction, bladder diverticulum or carcinoma, foreign body and neurogenic bladder. We report a case of an 85year old male patient who presented with history of recurrent episodes of burning micturition, pain abdomen, straining at micturition and diminished stream. Ultrasonography and X ray KUB showed a large vesical calculus. Patient underwent a n Open Cystolithomy and a large calculus of size 9x13cm weighing 310gms was removed. Bladder wall hypertrophy was seen with signs of inflammation. Bladder mucosal biopsy was taken which was normal on histopathological examination. Post - operative recovery was uneventful

  19. White noise calculus and Fock space

    CERN Document Server

    Obata, Nobuaki

    1994-01-01

    White Noise Calculus is a distribution theory on Gaussian space, proposed by T. Hida in 1975. This approach enables us to use pointwise defined creation and annihilation operators as well as the well-established theory of nuclear space.This self-contained monograph presents, for the first time, a systematic introduction to operator theory on fock space by means of white noise calculus. The goal is a comprehensive account of general expansion theory of Fock space operators and its applications. In particular,first order differential operators, Laplacians, rotation group, Fourier transform and their interrelations are discussed in detail w.r.t. harmonic analysis on Gaussian space. The mathematical formalism used here is based on distribution theory and functional analysis , prior knowledge of white noise calculus is not required.

  20. A MATLAB companion for multivariable calculus

    CERN Document Server

    Cooper, Jeffery

    2001-01-01

    Offering a concise collection of MatLab programs and exercises to accompany a third semester course in multivariable calculus, A MatLab Companion for Multivariable Calculus introduces simple numerical procedures such as numerical differentiation, numerical integration and Newton''s method in several variables, thereby allowing students to tackle realistic problems. The many examples show students how to use MatLab effectively and easily in many contexts. Numerous exercises in mathematics and applications areas are presented, graded from routine to more demanding projects requiring some programming. Matlab M-files are provided on the Harcourt/Academic Press web site at http://www.harcourt-ap.com/matlab.html.* Computer-oriented material that complements the essential topics in multivariable calculus* Main ideas presented with examples of computations and graphics displays using MATLAB * Numerous examples of short code in the text, which can be modified for use with the exercises* MATLAB files are used to implem...

  1. Fractional Calculus in Wave Propagation Problems

    CERN Document Server

    Mainardi, Francesco

    2012-01-01

    Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.

  2. Standardization in resource lambda-calculus

    Directory of Open Access Journals (Sweden)

    Maurizio Dominici

    2012-11-01

    Full Text Available The resource calculus is an extension of the lambda-calculus allowing to model resource consumption. It is intrinsically non-deterministic and has two general notions of reduction – one parallel, preserving all the possible results as a formal sum, and one non-deterministic, performing an exclusive choice at every step. We prove that the non-deterministic reduction enjoys a notion of standardization, which is the natural extension with respect to the similar one in classical lambda-calculus. The full parallel reduction only enjoys a weaker notion of standardization instead. The result allows an operational characterization of may-solvability, which has been introduced and already characterized (from the syntactical and logical points of view by Pagani and Ronchi Della Rocca.

  3. Fuzzy relational calculus theory, applications and software

    CERN Document Server

    Peeva, Ketty

    2004-01-01

    This book examines fuzzy relational calculus theory with applications in various engineering subjects. The scope of the text covers unified and exact methods with algorithms for direct and inverse problem resolution in fuzzy relational calculus. Extensive engineering applications of fuzzy relation compositions and fuzzy linear systems (linear, relational and intuitionistic) are discussed. Some examples of such applications include solutions of equivalence, reduction and minimization problems in fuzzy machines, pattern recognition in fuzzy languages, optimization and inference engines in textile and chemical engineering, etc. A comprehensive overview of the authors' original work in fuzzy relational calculus is also provided in each chapter. The attached CD-Rom contains a toolbox with many functions for fuzzy calculations, together with an original algorithm for inverse problem resolution in MATLAB. This book is also suitable for use as a textbook in related courses at advanced undergraduate and graduate level...

  4. Formalizing BPEL-TC Through ?-Calculus

    Directory of Open Access Journals (Sweden)

    Preeti Marwaha

    2013-07-01

    Full Text Available WS-BPEL is way to define business processes that interact with external entities through webservice operations using WSDL. We have proposed BPEL-TC, an extension to existing WS-BPEL whichuses temporally customized Web Services (WSDL-TC as a model for process decomposition and assembly.WSDL-TC handles both backward compatible and incompatible changes and also maintains variousversions of the artifacts that results due to changes over time and customizations desired by the users. Inthis paper, we are using pi-calculus to formalize Business Process Execution Language- TemporalCustomization (BPEL-TC process. π -calculus is a model of computation for concurrent systems alongwith changing connectivity of interactive systems. Pi-calculus is an extension of the process algebra CCS,with added mobility to CCS while preserving its algebraic properties.

  5. Enhancing Students’ Understanding in Calculus Trough Writing

    Directory of Open Access Journals (Sweden)

    Noraini Idris

    2009-02-01

    Full Text Available The purpose of this study was to investigate the effects of using writing activities on students’ understanding and achievement in Calculus. The design of this study was quasi-experimental. The subjects of this study consisted of two secondary schools in one of the states in Malaysia. Each school was assigned one intact class of Form Four to be the experimental group and another one intact class as the control. The experimental group learned mathematics by using the writing activities for five weeks, while the control group learned mathematics by using traditional whole-class instruction. A 20-item Calculus Achievement test was designed with reliability .87. The findings showed that the experimental group exhibited significantly greater improvement on calculus achievement. The students showed positive reaction towards the use of writing. Findings of this study provide information to schools to take advantage of writing activities to promote understanding.

  6. Research of Semantic Comparison between χ-calculus and π-calculus%χ-演算与π-演算的语义比较研究

    Institute of Scientific and Technical Information of China (English)

    徐林; 傅育熙

    2000-01-01

    Through the comparison of syntactic structure,operational semantics and algebraic semantics between χ-calculus and π-calculus, this paper concludes that χ-calculus has more succinct syntactic structure,more explicit operational semantics,more intuitionistic algebraic semantics and more favorable algebraic property. And a translation from π-calculus to χ-calculus is presented.

  7. Computer Managed Instruction Homework Modules for Calculus I.

    Science.gov (United States)

    Goodman-Petrushka, Sharon; Roitberg, Yael

    This booklet contains 11 modules (290 multiple-choice items) designed for use in the first course of a three-course calculus sequence using the textbook "Calculus with Analytic Geometry" (Dennis G. Zill). In each module, relevant sections of the textbook are identified for users. It can, however, be used in conjunction with any calculus textbook.…

  8. Improving Calculus II and III through the Redistribution of Topics

    Science.gov (United States)

    George, C. Yousuf; Koetz, Matt; Lewis, Heather A.

    2016-01-01

    Three years ago our mathematics department rearranged the topics in second and third semester calculus, moving multivariable calculus to the second semester and series to the third semester. This paper describes the new arrangement of topics, and how it could be adapted to calculus curricula at different schools. It also explains the benefits we…

  9. Science 101: How Do We Use Calculus in Science?

    Science.gov (United States)

    Robertson, Bill

    2014-01-01

    How is calculus used in science? That might seem like an odd question to answer in a magazine intended primarily for elementary school teachers. After all, how much calculus gets used in elementary science? Here the author guesses that quite a few readers of this column do not know a whole lot about calculus and have not taken a course in…

  10. A Transition Course from Advanced Placement to College Calculus

    Science.gov (United States)

    Lucas, Timothy A.; Spivey, Joseph

    2011-01-01

    In the Spring of 2007, a group of highly motivated mathematics graduate students conducted a review of Duke's Calculus curriculum. They focused on two main problems. The first problem is the result of a very positive trend: a growing number of students are earning AP credit for Calculus I in high school. However, this results in Calculus II…

  11. Science 101: How Do We Use Calculus in Science?

    Science.gov (United States)

    Robertson, Bill

    2014-01-01

    How is calculus used in science? That might seem like an odd question to answer in a magazine intended primarily for elementary school teachers. After all, how much calculus gets used in elementary science? Here the author guesses that quite a few readers of this column do not know a whole lot about calculus and have not taken a course in…

  12. Improving Calculus II and III through the Redistribution of Topics

    Science.gov (United States)

    George, C. Yousuf; Koetz, Matt; Lewis, Heather A.

    2016-01-01

    Three years ago our mathematics department rearranged the topics in second and third semester calculus, moving multivariable calculus to the second semester and series to the third semester. This paper describes the new arrangement of topics, and how it could be adapted to calculus curricula at different schools. It also explains the benefits we…

  13. A Transition Course from Advanced Placement to College Calculus

    Science.gov (United States)

    Lucas, Timothy A.; Spivey, Joseph

    2011-01-01

    In the Spring of 2007, a group of highly motivated mathematics graduate students conducted a review of Duke's Calculus curriculum. They focused on two main problems. The first problem is the result of a very positive trend: a growing number of students are earning AP credit for Calculus I in high school. However, this results in Calculus II…

  14. A Calculus of Communicating Systems with Label Passing

    DEFF Research Database (Denmark)

    Engberg, Uffe Henrik; Nielsen, Mogens

    Milner's Calculus of Communicating Systems (CCS) is extended with a mechanism for label passing - as an attempt to remedy some of the shortcomings of CCS w.r.t. dynamic change of agent interconnections. In the extended calculus, restriction is viewed formally as a binder, and the calculus allows...

  15. Mu-calculus-based deontic logic for regular actions

    NARCIS (Netherlands)

    Broersen, Jan; Wieringa, Roelf J.; Meyer, John-Jules; Demolombe, R.; Hilpinen, R.

    This paper introduces deontic logic of regular actions as a fragment of the modal mu calculus Semantic characterizations of deontic notions for regular actions are given in terms of conditions on mu calculus structures and mu calculus formulas capturing this semantics are constructed

  16. Computer Managed Instruction Homework Modules for Calculus I.

    Science.gov (United States)

    Goodman-Petrushka, Sharon; Roitberg, Yael

    This booklet contains 11 modules (290 multiple-choice items) designed for use in the first course of a three-course calculus sequence using the textbook "Calculus with Analytic Geometry" (Dennis G. Zill). In each module, relevant sections of the textbook are identified for users. It can, however, be used in conjunction with any calculus textbook.…

  17. A Historical Perspective on Teaching and Learning Calculus

    Science.gov (United States)

    Doorman, Michiel; van Maanen, Jan

    2008-01-01

    Calculus is one of those topics in mathematics where the algorithmic manipulation of symbols is easier than understanding the underlying concepts. Around 1680 Leibniz invented a symbol system for calculus that codifies and simplifies the essential elements of reasoning. The calculus of Leibniz brings within the reach of an ordinary student…

  18. A BRUTUS Logic for a Spi-Calculus Dialect

    NARCIS (Netherlands)

    Gnesi, S.; Latella, D.; Lenzini, G.

    2000-01-01

    In the field of process algebras, the spi-calculus, a modified version of the π-calculus with encryption primitives, is indicated as an expressive specification language for cryptographic protocols. In spi-calculus basic security properties, such as secrecy and integrity can be formalized as may-tes

  19. TWO-PHASE EJECTOR of CARBON DIOXIDE HEAT PUMP CALCULUS

    Directory of Open Access Journals (Sweden)

    Sit B.M.

    2010-12-01

    Full Text Available It is presented the calculus of the two-phase ejector for carbon dioxide heat pump. The method of calculus is based on the method elaborated by S.M. Kandil, W.E. Lear, S.A. Sherif, and is modified taking into account entrainment ratio as the input for the calculus.

  20. A BRUTUS Logic for a Spi-Calculus Dialect

    NARCIS (Netherlands)

    Gnesi, S.; Latella, D.; Lenzini, Gabriele

    2000-01-01

    In the field of process algebras, the spi-calculus, a modified version of the π-calculus with encryption primitives, is indicated as an expressive specification language for cryptographic protocols. In spi-calculus basic security properties, such as secrecy and integrity can be formalized as

  1. Investigating the Use of Mastery-Style Online Homework Exercises in Introductory Algebra-based Mechanics in a Controlled Clinical Study

    CERN Document Server

    Evans, William R

    2016-01-01

    Homework in introductory physics represents an important part of a student's learning experience; therefore choosing the manner in which homework is presented merits investigation. We performed three rounds of clinical trials comparing the effects of mastery-style homework vs traditional-style homework with students in both algebra-based and calculus-based introductory mechanics. Results indicate a benefit from mastery-style over traditional-style homework, principally for weaker students who are less familiar with the material being covered and on questions that are nearer transfer to the study materials.

  2. An Executable Calculus for Service Choreography

    Science.gov (United States)

    Besana, Paolo; Barker, Adam

    The Lightweight Coordination Calculus (LCC) is a compact choreography language based on process calculus. LCC is a directly executable specification and can therefore be dynamically distributed to a group of peers for enactment at run-time; this offers flexibility and allows peers to coordinate in open systems without prior knowledge of an interaction. This paper contributes to the body of choreography research by proposing two extensions to LCC covering parallel composition and choreography abstraction. These language extensions are evaluated against a subset of the Service Interaction Patterns, a benchmark in the process modelling community.

  3. The lambda sigma calculus and strong normalization

    DEFF Research Database (Denmark)

    Schack-Nielsen, Anders; Schürmann, Carsten

    Explicit substitution calculi can be classified into several dis- tinct categories depending on whether they are confluent, meta-confluent, strong normalization preserving, strongly normalizing, simulating, fully compositional, and/or local. In this paper we present a variant of the λσ-calculus......, which satisfies all seven conditions. In particular, we show how to circumvent Mellies counter-example to strong normalization by a slight restriction of the congruence rules. The calculus is implemented as the core data structure of the Celf logical framework. All meta-theoretic aspects of this work...

  4. A sequent calculus for signed interval logic

    DEFF Research Database (Denmark)

    Rasmussen, Thomas Marthedal

    2001-01-01

    We propose and discuss a complete sequent calculus formulation for Signed Interval Logic (SIL) with the chief purpose of improving proof support for SIL in practice. The main theoretical result is a simple characterization of the limit between decidability and undecidability of quantifier-free SIL....... We present a mechanization of SIL in the generic proof assistant Isabelle and consider techniques for automated reasoning. Many of the results and ideas of this report are also applicable to traditional (non-signed) interval logic and, hence, to Duration Calculus....

  5. Safety versus Security in the Quality Calculus

    DEFF Research Database (Denmark)

    Nielson, Hanne Riis; Nielson, Flemming

    2013-01-01

    Safety and security are both needed for ensuring that cyber-physical systems live up to expectations, but often an intelligent trade-off is called for, because sometimes it is impossible to obtain optimal safety at the same time as optimal security. In the context of the Quality Calculus we develop...... be performed in highly trusted contexts. This is potentially too demanding and the Quality Calculus is therefore extended with a primitive for endorsing data to a higher trust level (accepting violations of the explicit flow) and for temporarily asserting a higher trust in the context (accepting violations...

  6. Probabilistic Analysis of the Quality Calculus

    DEFF Research Database (Denmark)

    Nielson, Hanne Riis; Nielson, Flemming

    2013-01-01

    We consider a fragment of the Quality Calculus, previously introduced for defensive programming of software components such that it becomes natural to plan for default behaviour in case the ideal behaviour fails due to unreliable communication. This paper develops a probabilistically based trust...... analysis supporting the Quality Calculus. It uses information about the probabilities that expected input will be absent in order to determine the trustworthiness of the data used for controlling the distributed system; the main challenge is to take accord of the stochastic dependency between some...

  7. A Cone Pseudo-differential Calculus

    Institute of Scientific and Technical Information of China (English)

    2000-01-01

    @@  The calculus of pseudo-differential operators on singular spaces and theconcept of ellipti-city in operator algebras on manifolds with singularitieshave become an enormous challenge for analysists. The so-called cone algebras(with discrete and continuous asymptotics) are investigated by manymathematicians, especially by B. W. Schulze, who developed and enrichedcone and edge pseudo-differential calculus, see Schulze[4-7], Rempel and Schulze [2, 3]. In this note,we construct a cone pseudo-differentialcalculus for operators which respect conormal asymptotics of a prescribedasymptotic type.

  8. Hybrid Logical Analyses of the Ambient Calculus

    DEFF Research Database (Denmark)

    Bolander, Thomas; Hansen, René Rydhof

    2007-01-01

    In this paper, hybrid logic is used to formulate a rational reconstruction of a previously published control flow analysis for the mobile ambients calculus and we further show how a more precise flow-sensitive analysis, that takes the ordering of action sequences into account, can be formulated...... in a natural way. We show that hybrid logic is very well suited to express the semantic structure of the ambient calculus and how features of hybrid logic can be exploited to reduce the "administrative overhead" of the analysis specification and thus simplify it. Finally, we use HyLoTab, a fully automated...

  9. Projects for calculus the language of change

    CERN Document Server

    Stroyan, Keith D

    1999-01-01

    Projects for Calculus is designed to add depth and meaning to any calculus course. The fifty-two projects presented in this text offer the opportunity to expand the use and understanding of mathematics. The wide range of topics will appeal to both instructors and students. Shorter, less demanding projects can be managed by the independent learner, while more involved, in-depth projects may be used for group learning. Each task draws on special mathematical topics and applications from subjects including medicine, engineering, economics, ecology, physics, and biology.Subjects including:* Medicine* Engineering* Economics* Ecology* Physics* Biology

  10. Neutrix Calculus and Finite Quantum Field Theory

    CERN Document Server

    Ng, Y J

    2004-01-01

    In general, quantum field theories require regularizations and infinite renormalizations due to ultraviolet divergences in their loop calculations. Furthermore, perturbation series in theories like QED are not convergent series, but are asymptotic series in their interaction couplings. We propose to apply neutrix calculus, developed by van der Corput and Hadamard in connection with asymptotic series, to tackle divergent integrals, yielding finite renormalizations for the parameters in quantum field theories. We observe that quantum gravity theories are rendered more manageable, and that both renormalizable field theories and effective field theories can be accommodated in the framework of neutrix calculus.

  11. Multivariable dynamic calculus on time scales

    CERN Document Server

    Bohner, Martin

    2016-01-01

    This book offers the reader an overview of recent developments of multivariable dynamic calculus on time scales, taking readers beyond the traditional calculus texts. Covering topics from parameter-dependent integrals to partial differentiation on time scales, the book’s nine pedagogically oriented chapters provide a pathway to this active area of research that will appeal to students and researchers in mathematics and the physical sciences. The authors present a clear and well-organized treatment of the concept behind the mathematics and solution techniques, including many practical examples and exercises.

  12. Sequent Calculus in the Topos of Trees

    DEFF Research Database (Denmark)

    Clouston, Ranald; Goré, Rajeev

    2015-01-01

    Nakano’s “later” modality, inspired by Gödel-Löb provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment...... that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence...

  13. Calculus on manifolds a modern approach to classical theorems of advanced calculus

    CERN Document Server

    Spivak, Michael D

    1965-01-01

    This little book is especially concerned with those portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approa

  14. A Paradox in the Metatheory of the Classical Predicate Calculus

    CERN Document Server

    Boyce, Stephen

    2009-01-01

    This paper shows that the metatheory of the classical, first-order predicate calculus is subject to paradox. It is shown that an interpretation M of the language of the calculus is definable within this metatheory such that: a formula of the calculus F(x) is satisfied at a certain denumerable sequence s of elements of the domain of M if and only if F(x) is not satisfied at s. Since the conclusion is absurd, the hypothesis that the metatheory provides a reliable account of the calculus should be rejected. The calculus may be unfit for purpose since the possibility of unsound inferences cannot be excluded.

  15. Detection, removal and prevention of calculus: Literature Review

    Directory of Open Access Journals (Sweden)

    Deepa G. Kamath

    2014-01-01

    Full Text Available Dental plaque is considered to be a major etiological factor in the development of periodontal disease. Accordingly, the elimination of supra- and sub-gingival plaque and calculus is the cornerstone of periodontal therapy. Dental calculus is mineralized plaque; because it is porous, it can absorb various toxic products that can damage the periodontal tissues. Hence, calculus should be accurately detected and thoroughly removed for adequate periodontal therapy. Many techniques have been used to identify and remove calculus deposits present on the root surface. The purpose of this review was to compile the various methods and their advantages for the detection and removal of calculus.

  16. Developing and validating a conceptual survey to assess introductory physics students’ understanding of magnetism

    Science.gov (United States)

    Li, Jing; Singh, Chandralekha

    2017-03-01

    Development of validated physics surveys on various topics is important for investigating the extent to which students master those concepts after traditional instruction and for assessing innovative curricula and pedagogies that can improve student understanding significantly. Here, we discuss the development and validation of a conceptual multiple-choice survey related to magnetism suitable for introductory physics courses. The survey was developed taking into account common students’ difficulties with magnetism concepts covered in introductory physics courses found in our investigation and the incorrect choices to the multiple-choice questions were designed based upon those common student difficulties. After the development and validation of the survey, it was administered to introductory physics students in various classes in paper-pencil format before and after traditional lecture-based instruction in relevant concepts. We compared the performance of students on the survey in the algebra-based and calculus-based introductory physics courses before and after traditional lecture-based instruction in relevant magnetism concepts. We discuss the common difficulties of introductory physics students with magnetism concepts we found via the survey. We also administered the survey to upper-level undergraduates majoring in physics and PhD students to benchmark the survey and compared their performance with those of traditionally taught introductory physics students for whom the survey is intended. A comparison with the base line data on the validated magnetism survey from traditionally taught introductory physics courses and upper-level undergraduate and PhD students discussed in this paper can help instructors assess the effectiveness of curricula and pedagogies which is especially designed to help students integrate conceptual and quantitative understanding and develop a good grasp of the concepts. In particular, if introductory physics students’ average

  17. BFKL Pomeron calculus: Nucleus-nucleus scattering

    Energy Technology Data Exchange (ETDEWEB)

    Contreras, Carlos, E-mail: carlos.contreras@usm.cl [Departamento de Fisica, Universidad Tecnica Federico Santa Maria, Avda. Espana 1680, and Centro Cientifico-Tecnologico de Valparaiso, Casilla 110-V, Valparaiso (Chile); Levin, Eugene, E-mail: leving@post.tau.ac.il [Departamento de Fisica, Universidad Tecnica Federico Santa Maria, Avda. Espana 1680, and Centro Cientifico-Tecnologico de Valparaiso, Casilla 110-V, Valparaiso (Chile); Department of Particle Physics, School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978 (Israel); Miller, Jeremy S., E-mail: jeremy.miller@ist.utl.pt [Department of Particle Physics, School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978 (Israel); CENTRA, Departamento de Fisica, Instituto Superior Tecnico (IST), Av. Rovisco Pais, 1049-001 Lisboa (Portugal)

    2012-04-15

    In this paper the action of the BFKL Pomeron calculus is rewritten in momentum representation, and the equations of motion for nucleus-nucleus collisions are derived, in this representation. We found the semiclassical solutions to these equations, outside of the saturation domain. Inside this domain these equations reduce to the set of delay differential equations, and their asymptotic solutions are derived.

  18. Discrete Calculus as a Bridge between Scales

    Science.gov (United States)

    Degiuli, Eric; McElwaine, Jim

    2012-02-01

    Understanding how continuum descriptions of disordered media emerge from the microscopic scale is a fundamental challenge in condensed matter physics. In many systems, it is necessary to coarse-grain balance equations at the microscopic scale to obtain macroscopic equations. We report development of an exact, discrete calculus, which allows identification of discrete microscopic equations with their continuum equivalent [1]. This allows the application of powerful techniques of calculus, such as the Helmholtz decomposition, the Divergence Theorem, and Stokes' Theorem. We illustrate our results with granular materials. In particular, we show how Newton's laws for a single grain reproduce their continuum equivalent in the calculus. This allows introduction of a discrete Airy stress function, exactly as in the continuum. As an application of the formalism, we show how these results give the natural mean-field variation of discrete quantities, in agreement with numerical simulations. The discrete calculus thus acts as a bridge between discrete microscale quantities and continuous macroscale quantities. [4pt] [1] E. DeGiuli & J. McElwaine, PRE 2011. doi: 10.1103/PhysRevE.84.041310

  19. On Online Assignments in a Calculus Class

    Science.gov (United States)

    Jungic, Veselin; Kent, Deborah; Menz, Petra

    2012-01-01

    In this paper, we describe our experience with the creation and utilization of online assignments for several calculus classes at Simon Fraser University (SFU). We present our findings regarding available software by considering the needs and perspectives of the instructors, students, and administrators. We provide a list of questions that guide…

  20. Nonlinear Young integrals via fractional calculus

    OpenAIRE

    Hu, Yaozhong (1961-); Le, Khoa

    2015-01-01

    For H\\"older continuous functions $W(t,x)$ and $\\varphi_t$, we define nonlinear integral $\\int_a^b W(dt, \\varphi_t)$ via fractional calculus. This nonlinear integral arises naturally in the Feynman-Kac formula for stochastic heat equations with random coefficients. We also define iterated nonlinear integrals.

  1. Boolean integral calculus for digital systems

    Science.gov (United States)

    Tucker, J. H.; Tapia, M. A.; Bennett, A. W.

    1985-01-01

    The concept of Boolean integration is introduced and developed. When the changes in a desired function are specified in terms of changes in its arguments, then ways of 'integrating' (i.e., realizing) the function, if it exists, are presented. Boolean integral calculus has applications in design of logic circuits.

  2. A Higher-Order Calculus for Categories

    DEFF Research Database (Denmark)

    Cáccamo, Mario José; Winskel, Glynn

    2001-01-01

    in their free variables. There are special binders for limits and more general ends. The rules for limits and ends support an algebraic manipulation of universal constructions as opposed to a more traditional diagrammatic approach. Duality within the calculus and applications in proving continuity are discussed...

  3. Exposing Calculus Students to Advanced Mathematics

    Science.gov (United States)

    Griffiths, Barry J.; Haciomeroglu, Erhan Selcuk

    2014-01-01

    To ensure the competitiveness of the USA in the global economy, and its role as a leader in science and engineering, it is important to cultivate the next generation of home grown mathematicians. However, while universities across the USA offer calculus classes to thousands of undergraduate students each year, very few of them go on to major in…

  4. A Functional Calculus for Quotient Bounded Operators

    Directory of Open Access Journals (Sweden)

    Sorin Mirel Stoian

    2006-12-01

    Full Text Available If (X, P is a sequentially locally convex space, then a quotient bounded operator T beloging to QP is regular (in the sense of Waelbroeck if and only if it is a bounded element (in the sense of Allan of algebra QP. The classic functional calculus for bounded operators on Banach space is generalized for bounded elements of algebra QP.

  5. Sharks, Minnows, and Wheelbarrows: Calculus Modeling Projects

    Science.gov (United States)

    Smith, Michael D.

    2011-01-01

    The purpose of this article is to present two very active applied modeling projects that were successfully implemented in a first semester calculus course at Hollins University. The first project uses a logistic equation to model the spread of a new disease such as swine flu. The second project is a human take on the popular article "Do Dogs Know…

  6. Exploring Flipped Classroom Instruction in Calculus III

    Science.gov (United States)

    Wasserman, Nicholas H.; Quint, Christa; Norris, Scott A.; Carr, Thomas

    2017-01-01

    In an undergraduate Calculus III class, we explore the effect of "flipping" the instructional delivery of content on both student performance and student perceptions. Two instructors collaborated to determine daily lecture notes, assigned the same homework problems, and gave identical exams; however, compared to a more traditional…

  7. Teaching Calculus with Wolfram|Alpha

    Science.gov (United States)

    Dimiceli, Vincent E.; Lang, Andrew S. I. D.; Locke, LeighAnne

    2010-01-01

    This article describes the benefits and drawbacks of using Wolfram|Alpha as the platform for teaching calculus concepts in the lab setting. It is a result of our experiences designing and creating an entirely new set of labs using Wolfram|Alpha. We present the reasoning behind our transition from using a standard computer algebra system (CAS) to…

  8. A TENTATIVE GUIDE, DIFFERENTIAL AND INTEGRAL CALCULUS.

    Science.gov (United States)

    BRANT, VINCENT; GERARDI, WILLIAM

    THE COURSE IS INTENDED TO GO BEYOND THE REQUIREMENTS OF THE ADVANCED PLACEMENT PROGRAM IN MATHEMATICS AS DESIGNED BY THE COLLEGE ENTRANCE EXAMINATION BOARD. THE ADVANCED PLACEMENT PROGRAM CONSISTS OF A 1-YEAR COURSE COMBINING ANALYTIC GEOMETRY AND CALCULUS. PRESUPPOSED HERE ARE--A SEMESTER COURSE IN ANALYTIC GEOMETRY AND A THOROUGH KNOWLEDGE OF…

  9. Are Homeschoolers Prepared for College Calculus?

    Science.gov (United States)

    Wilkens, Christian P.; Wade, Carol H.; Sonnert, Gerhard; Sadler, Philip M.

    2015-01-01

    Homeschooling in the United States has grown considerably over the past several decades. This article presents findings from the Factors Influencing College Success in Mathematics (FICSMath) survey, a national study of 10,492 students enrolled in tertiary calculus, including 190 students who reported homeschooling for a majority of their high…

  10. A robust interpretation of duration calculus

    DEFF Research Database (Denmark)

    Franzle, M.; Hansen, Michael Reichhardt

    2005-01-01

    Calculus (DC), our findings are that the robust interpretation of DC is equivalent to a multi-valued interpretation that uses the real numbers as semantic domain and assigns Lipschitz-continuous interpretations to all operators of DC. Furthermore, this continuity permits approximation between discrete...

  11. A Temporal Concurrent Constraint Programming Calculus

    DEFF Research Database (Denmark)

    Palamidessi, Catuscia; Valencia Posso, Frank Darwin

    2001-01-01

    The tcc model is a formalism for reactive concurrent constraint programming. In this paper we propose a model of temporal concurrent constraint programming which adds to tcc the capability of modeling asynchronous and non-deterministic timed behavior. We call this tcc extension the ntcc calculus...

  12. Some Factors Effected Student's Calculus Learning Outcome

    Science.gov (United States)

    Rajagukguk, Wamington

    2016-01-01

    The purpose of this study is to determine the factors effected calculus learning outcome of the student. This study was conducted with 176 respondents, which were selected randomly. The data were obtained by questionnaire, and then analyzed by using multiple regressions, and correlation, at level of a = 0.05. The findings showed there is the…

  13. Are Homeschoolers Prepared for College Calculus?

    Science.gov (United States)

    Wilkens, Christian P.; Wade, Carol H.; Sonnert, Gerhard; Sadler, Philip M.

    2015-01-01

    Homeschooling in the United States has grown considerably over the past several decades. This article presents findings from the Factors Influencing College Success in Mathematics (FICSMath) survey, a national study of 10,492 students enrolled in tertiary calculus, including 190 students who reported homeschooling for a majority of their high…

  14. Flipping a Calculus Class: One Instructor's Experience

    Science.gov (United States)

    Palmer, Katrina

    2015-01-01

    This paper describes one instructor's experiences during a year of flipping four calculus classes. The first exploration attempts to understand student expectations of a math class and their preference towards a flipped classroom. The second examines success of students from a flipped classroom, and the last investigates relationships with student…

  15. Enabling quaternion derivatives: the generalized HR calculus.

    Science.gov (United States)

    Xu, Dongpo; Jahanchahi, Cyrus; Took, Clive C; Mandic, Danilo P

    2015-08-01

    Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

  16. Using Discovery in the Calculus Class

    Science.gov (United States)

    Shilgalis, Thomas W.

    1975-01-01

    This article shows how two discoverable theorems from elementary calculus can be presented to students in a manner that assists them in making the generalizations themselves. The theorems are the mean value theorems for derivatives and for integrals. A conjecture is suggested by pictures and then refined. (Author/KM)

  17. Exploring Flipped Classroom Instruction in Calculus III

    Science.gov (United States)

    Wasserman, Nicholas H.; Quint, Christa; Norris, Scott A.; Carr, Thomas

    2017-01-01

    In an undergraduate Calculus III class, we explore the effect of "flipping" the instructional delivery of content on both student performance and student perceptions. Two instructors collaborated to determine daily lecture notes, assigned the same homework problems, and gave identical exams; however, compared to a more traditional…

  18. Teaching Calculus Students How to Study.

    Science.gov (United States)

    Boelkins, Matthew R.; Pfaff, Thomas J.

    1998-01-01

    Addresses the problem of poor study habits in calculus students and presents techniques to teach students how to study consistently and effectively. Concludes that many students greatly appreciate the added structure, work harder than in previous courses, and witness newfound success as a consequence. (Author/ASK)

  19. Supercalculators and University Entrance Calculus Examinations.

    Science.gov (United States)

    Hong, Ye Yoon; Thomas, Mike; Kiernan, Christine

    2000-01-01

    Investigates whether the use of computer algebra systems could provide a significant advantage to students taking standard university entrance calculus examinations. Indicates that supercalculators would probably provide a significant advantage, particularly for lower-achieving students. Demonstrates that it is possible to write questions in which…

  20. A Note on Discrete Mathematics and Calculus.

    Science.gov (United States)

    O'Reilly, Thomas J.

    1987-01-01

    Much of the current literature on the topic of discrete mathematics and calculus during the first two years of an undergraduate mathematics curriculum is cited. A relationship between the recursive integration formulas and recursively defined polynomials is described. A Pascal program is included. (Author/RH)

  1. The Inductive Applications of Probability Calculus

    Directory of Open Access Journals (Sweden)

    Corrado Gini

    2015-06-01

    Full Text Available The Author goes back to Founders of Probability calculus to investigate their original interpretation of the probability measure in the applications of the probability theory to real problems. The Author puts in evidence some misunderstandings related to the inversion of deductions derived by the use of probability distributions for investigating the causes of events.

  2. Enabling quaternion derivatives: the generalized HR calculus

    Science.gov (United States)

    Xu, Dongpo; Jahanchahi, Cyrus; Took, Clive C.; Mandic, Danilo P.

    2015-01-01

    Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis. PMID:26361555

  3. Students' Difficulties with Vector Calculus in Electrodynamics

    Science.gov (United States)

    Bollen, Laurens; van Kampen, Paul; De Cock, Mieke

    2015-01-01

    Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven…

  4. Optimal control and the calculus of variations

    CERN Document Server

    Pinch, Enid R

    1993-01-01

    This introduction to optimal control theory is intended for undergraduate mathematicians and for engineers and scientists with some knowledge of classical analysis. It includes sections on classical optimization and the calculus of variations. All the important theorems are carefully proved. There are many worked examples and exercises for the reader to attempt.

  5. A Planar Calculus for Infinite Index Subfactors

    Science.gov (United States)

    Penneys, David

    2013-05-01

    We develop an analog of Jones' planar calculus for II 1-factor bimodules with arbitrary left and right von Neumann dimension. We generalize to bimodules Burns' results on rotations and extremality for infinite index subfactors. These results are obtained without Jones' basic construction and the resulting Jones projections.

  6. Is Calculus an Appropriate High School Course?

    Science.gov (United States)

    Rash, Agnes M.

    1977-01-01

    Discusses some alternatives to calculus as an advanced high school course which will prepare students for college level work, improve their background in algebra, geometry and trigonometry, and introduce new and interesting material of a more advanced nature. (Author/RK)

  7. Teaching Calculus with Wolfram|Alpha

    Science.gov (United States)

    Dimiceli, Vincent E.; Lang, Andrew S. I. D.; Locke, LeighAnne

    2010-01-01

    This article describes the benefits and drawbacks of using Wolfram|Alpha as the platform for teaching calculus concepts in the lab setting. It is a result of our experiences designing and creating an entirely new set of labs using Wolfram|Alpha. We present the reasoning behind our transition from using a standard computer algebra system (CAS) to…

  8. A Typed Functional Calculus With State

    NARCIS (Netherlands)

    Rensink, Arend; Müllhäuser, M.

    1997-01-01

    We extend the simple typed \\lambda-calculus with statements. A statement (which can also be thought of as a method or transition) is an abstraction similar to function abstraction: it can be instantiated by providing it with a source state, whereafter it yields a pair of values consisting of an

  9. Bladder calculus presenting as excessive masturbation.

    Science.gov (United States)

    De Alwis, A C D; Senaratne, A M R D; De Silva, S M P D; Rodrigo, V S D

    2006-09-01

    Masturbation in childhood is a normal behaviour which most commonly begins at 2 months of age, and peaks at 4 years and in adolescence. However excessive masturbation causes anxiety in parents. We describe a boy with a bladder calculus presenting as excessive masturbation.

  10. Global calculus in local BRST cohomology

    CERN Document Server

    Giachetta, G; Sardanashvily, G

    2000-01-01

    The construction of local BRST cohomology is extended to an arbitrary affine bundle. Cohomology of the variational complex on the infinite order jet space of a smooth fibre bundle is computed. This provides a solution of the global inverse problem of the calculus of variations in Lagrangian field theory.

  11. Dogs Don't Need Calculus

    Science.gov (United States)

    Bolt, Mike

    2010-01-01

    Many optimization problems can be solved without resorting to calculus. This article develops a new variational method for optimization that relies on inequalities. The method is illustrated by four examples, the last of which provides a completely algebraic solution to the problem of minimizing the time it takes a dog to retrieve a thrown ball,…

  12. Using Matlab in a Multivariable Calculus Course.

    Science.gov (United States)

    Schlatter, Mark D.

    The benefits of high-level mathematics packages such as Matlab include both a computer algebra system and the ability to provide students with concrete visual examples. This paper discusses how both capabilities of Matlab were used in a multivariate calculus class. Graphical user interfaces which display three-dimensional surfaces, contour plots,…

  13. Partial differential equations and calculus of variations

    CERN Document Server

    Leis, Rolf

    1988-01-01

    This volume contains 18 invited papers by members and guests of the former Sonderforschungsbereich in Bonn (SFB 72) who, over the years, collaborated on the research group "Solution of PDE's and Calculus of Variations". The emphasis is on existence and regularity results, on special equations of mathematical physics and on scattering theory.

  14. Exposing Calculus Students to Advanced Mathematics

    Science.gov (United States)

    Griffiths, Barry J.; Haciomeroglu, Erhan Selcuk

    2014-01-01

    To ensure the competitiveness of the USA in the global economy, and its role as a leader in science and engineering, it is important to cultivate the next generation of home grown mathematicians. However, while universities across the USA offer calculus classes to thousands of undergraduate students each year, very few of them go on to major in…

  15. Students' Difficulties with Vector Calculus in Electrodynamics

    Science.gov (United States)

    Bollen, Laurens; van Kampen, Paul; De Cock, Mieke

    2015-01-01

    Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven…

  16. A Stochastic Broadcast Pi-Calculus

    DEFF Research Database (Denmark)

    Song, Lei; Nielson, Flemming; Nielsen, Bo Friis

    2011-01-01

    In this paper we propose a stochastic broadcast PI-calculus which can be used to model server-client based systems where synchronization is always governed by only one participant. Therefore, there is no need to determine the joint synchronization rates. We also take immediate transitions...

  17. A Temporal Concurrent Constraint Programming Calculus

    DEFF Research Database (Denmark)

    Palamidessi, Catuscia; Valencia Posso, Frank Darwin

    2001-01-01

    The tcc model is a formalism for reactive concurrent constraint programming. In this paper we propose a model of temporal concurrent constraint programming which adds to tcc the capability of modeling asynchronous and non-deterministic timed behavior. We call this tcc extension the ntcc calculus...

  18. Expressing First-Order π-Calculus in Higher-Order Calculus of Communicating Systems

    Institute of Scientific and Technical Information of China (English)

    Xian Xu

    2009-01-01

    In the study of process calculi, encoding between different calculi is an effective way to compare the expressive power of calculi and can shed light on the essence of where the difference lies. Thomsen and Sangiorgi have worked on the higher-order calculi (higher-order Calculus of Communicating Systems (CCS) and higher-order It-calculus, respectively) and the encoding from and to first-order π-calculus. However a fully abstract encoding of first-order π-calculus with higher-order CCS is not available up-today. This is what we intend to settle in this paper. We follow the encoding strategy, first proposed by Thomsen, of translating first-order π-calculus into Plain CHOCS. We show that the encoding strategy is fully abstract with respect to early bisimilarity (first-order π-calculus) and wired bisimilarity (Plain CHOCS) (which is a bisimulation defined on wired processes only sending and receiving wires), that is the core of the encoding strategy. Moreover from the fact that the wired bisimilarity is contained by the well-established context bisimilarity, we secure the soundness of the encoding, with respect to early bisimilarity and context bisimilarity. We use index technique to get around all the technical details to reach these main results of this paper. Finally, we make some discussion on our work and suggest some future work.

  19. Characteristics of subgingival calculus detection by multiphoton fluorescence microscopy

    Science.gov (United States)

    Tung, Oi-Hong; Lee, Shyh-Yuan; Lai, Yu-Lin; Chen, How-Foo

    2011-06-01

    Subgingival calculus has been recognized as a major cause of periodontitis, which is one of the main chronic infectious diseases of oral cavities and a principal cause of tooth loss in humans. Bacteria deposited in subgingival calculus or plaque cause gingival inflammation, function deterioration, and then periodontitis. However, subgingival calculus within the periodontal pocket is a complicated and potentially delicate structure to be detected with current dental armamentaria, namely dental x-rays and dental probes. Consequently, complete removal of subgingival calculus remains a challenge to periodontal therapies. In this study, the detection of subgingival calculus employing a multiphoton autofluorescence imaging method was characterized in comparison with a one-photon confocal fluorescence imaging technique. Feasibility of such a system was studied based on fluorescence response of gingiva, healthy teeth, and calculus with and without gingiva covered. The multiphoton fluorescence technology perceived the tissue-covered subgingival calculus that cannot be observed by the one-photon confocal fluorescence method.

  20. Fractional calculus in bioengineering, part 3.

    Science.gov (United States)

    Magin, Richard L

    2004-01-01

    Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub

  1. Success in introductory college physics: The role of high school preparation

    Science.gov (United States)

    Sadler, Philip M.; Tai, Robert H.

    2001-03-01

    High school teachers and college physics professors differ in their beliefs concerning the extent to which a high school physics course prepares students for college physics success. In this study of 1,933 introductory college physics students, demographic and schooling factors account for a large fraction of the variation in college physics grades at 18 colleges and universities from around the nation. Controlling for student backgrounds, taking a high school physics course has a modestly positive relationship with the grade earned in introductory college physics. More rigorous preparation, including calculus and 2 years of high school physics, predicts higher grades. Students who had high school courses that spent more time on fewer topics, concepts, problems, and labs performed much better in college than those who raced through more content in a textbook-centered course. College professors should recognize that a substantial fraction of the variation observed in the performance of students they teach can be explained by the range in effectiveness of their pre-college preparation, not simply innate ability. Although students without a high school physics course often do well in college physics, they are more likely to be academically stronger, with more educated parents, having previously taken calculus, and taking physics in their sophomore or junior year in college.

  2. Characterizing, modeling, and addressing gender disparities in introductory college physics

    Science.gov (United States)

    Kost-Smith, Lauren Elizabeth

    2011-12-01

    The underrepresentation and underperformance of females in physics has been well documented and has long concerned policy-makers, educators, and the physics community. In this thesis, we focus on gender disparities in the first- and second-semester introductory, calculus-based physics courses at the University of Colorado. Success in these courses is critical for future study and careers in physics (and other sciences). Using data gathered from roughly 10,000 undergraduate students, we identify and model gender differences in the introductory physics courses in three areas: student performance, retention, and psychological factors. We observe gender differences on several measures in the introductory physics courses: females are less likely to take a high school physics course than males and have lower standardized mathematics test scores; males outscore females on both pre- and post-course conceptual physics surveys and in-class exams; and males have more expert-like attitudes and beliefs about physics than females. These background differences of males and females account for 60% to 70% of the gender gap that we observe on a post-course survey of conceptual physics understanding. In analyzing underlying psychological factors of learning, we find that female students report lower self-confidence related to succeeding in the introductory courses (self-efficacy) and are less likely to report seeing themselves as a "physics person". Students' self-efficacy beliefs are significant predictors of their performance, even when measures of physics and mathematics background are controlled, and account for an additional 10% of the gender gap. Informed by results from these studies, we implemented and tested a psychological, self-affirmation intervention aimed at enhancing female students' performance in Physics 1. Self-affirmation reduced the gender gap in performance on both in-class exams and the post-course conceptual physics survey. Further, the benefit of the self

  3. Concept Maps in Introductory Statistics

    Science.gov (United States)

    Witmer, Jeffrey A.

    2016-01-01

    Concept maps are tools for organizing thoughts on the main ideas in a course. I present an example of a concept map that was created through the work of students in an introductory class and discuss major topics in statistics and relationships among them.

  4. Failure Rates in Introductory Programming

    DEFF Research Database (Denmark)

    Bennedsen, Jens; Caspersen, Michael Edelgaard

    2007-01-01

    It is a common conception that CS1 is a very difficult course and that failure rates are high. However, until now there has only been anecdotal evidence for this claim. This article reports on a survey among institutions around the world regarding failure rates in introductory programming courses...

  5. Student Expectations in Introductory Physics.

    Science.gov (United States)

    Redish, Edward F.; Saul, Jeffery M.; Steinberg, Richard N.

    Students' understanding of what science is about and how it is done and their expectations as to what goes on in a science course play a powerful role in what they can get out of introductory college physics. This is particularly true when there is a large gap between what the students expect to do and what the instructor expects them to do. This…

  6. Semantics in Teaching Introductory Physics.

    Science.gov (United States)

    Williams, H. Thomas

    1999-01-01

    Contends that the large vocabulary used for precise purposes in physics contains many words that have related but potentially confusing meanings in everyday usage. Analyzes the treatment of Newton's Laws of Motion in several well-known introductory textbooks for evidence of inconsistent language use. Makes teaching suggestions. (Contains 11…

  7. Fourier Analysis in Introductory Physics

    Science.gov (United States)

    Huggins, Elisha

    2007-01-01

    In an after-dinner talk at the fall 2005 meeting of the New England chapter of the AAPT, Professor Robert Arns drew an analogy between classical physics and Classic Coke. To generations of physics teachers and textbook writers, classical physics was the real thing. Modern physics, which in introductory textbooks "appears in one or more extra…

  8. Information Literacy in Introductory Biology

    Science.gov (United States)

    Firooznia, Fardad; Andreadis, Debra K.

    2006-01-01

    Incorporating information literacy exercises into the science curriculum will help students to navigate through the myriad of information available in different formats, and to become better scientific thinkers and writers. Here we describe how we incorporated such exercises in introductory biology and evaluate their effectiveness in achieving our…

  9. Online and Face-to-Face Language Learning: A Comparative Analysis of Oral Proficiency in Introductory Spanish

    Directory of Open Access Journals (Sweden)

    Dianne Burke Moneypenny

    2016-07-01

    Full Text Available The primary resistance to online foreign language teaching often involves questions of spoken mastery of second language. In order to address this concern, this research comparatively assesses undergraduate students’ oral proficiency in online and face-to-face Spanish classes, while taking into account students’ previous second language experience. The sample consisted of 90 undergraduate students, both online and face-to-face, who completed the Versant test at the end of the two semester sequence of Introductory Spanish. The Versant test assessed pronunciation, vocabulary, sentence formation, and fluency as factors of oral proficiency and calculated an overall oral proficiency score. T-test, one-way analysis of variance (ANOVA, and Kruskal-Wallis H test were used to determine whether there were any significant differences between online and face-to-face students’ proficiency scores. The results indicate that there were no statistically significant differences in oral proficiency at the introductory level between students who completed Spanish online and those that participated in face-to-face courses.

  10. Fractional calculus in bioengineering, part 2.

    Science.gov (United States)

    Magin, Richard L

    2004-01-01

    Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub

  11. Ups and downs of using ``kitchen sink'' experiments in an introductory fluid mechanics class

    Science.gov (United States)

    Kaye, Nigel

    2015-11-01

    Both positive and negative experiences from two semesters of using take home ``kitchen sink'' experiments in an introductory civil engineering fluid mechanics class are reported. Four different experimental assignments were given each semester to groups of four students. The students were tasked with using common household equipment to measure various properties of fluids or fluid flows. These included the density of cooking oil, the exit velocity from a garden hose, and the mass flux of air from a compressed air can. Students were given minimal guidance on how to do the measurements and each measurement had to be done in at least two different ways. The labs were used to relate their course work to everyday situations and was also used as a platform for discussing experimental uncertainty and error propagation in calculations. In general the students successfully completed each task using at least one method. Finding a second method sometimes proved problematic. The presentation will discuss the logistics of running the program and the positive and negative aspects from the instructor viewpoint. A summary of student feedback on the labs will also be presented. Links to resources for those interested in implementing such a program will be provided.

  12. Differential calculus in normed linear spaces

    CERN Document Server

    Mukherjea, Kalyan

    2007-01-01

    This book presents Advanced Calculus from a geometric point of view: instead of dealing with partial derivatives of functions of several variables, the derivative of the function is treated as a linear transformation between normed linear spaces. Not only does this lead to a simplified and transparent exposition of "difficult" results like the Inverse and Implicit Function Theorems but also permits, without any extra effort, a discussion of the Differential Calculus of functions defined on infinite dimensional Hilbert or Banach spaces.The prerequisites demanded of the reader are modest: a sound understanding of convergence of sequences and series of real numbers, the continuity and differentiability properties of functions of a real variable and a little Linear Algebra should provide adequate background for understanding the book. The first two chapters cover much of the more advanced background material on Linear Algebra (like dual spaces, multilinear functions and tensor products.) Chapter 3 gives an ab ini...

  13. A Simplified Stabilizer ZX-calculus

    Directory of Open Access Journals (Sweden)

    Miriam Backens

    2017-01-01

    Full Text Available The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent the same quantum evolution or quantum state. We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the essential axioms which potentially capture fundamental structures of quantum mechanics. We thus give a significantly smaller set of axioms and prove that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the language, can in fact be derived. In particular, we show that the additional symbol and one of the rules which had been recently introduced to keep track of scalars (diagrams with no inputs or outputs are not necessary.

  14. Intersection Logic in sequent calculus style

    CERN Document Server

    Della Rocca, Simona Ronchi; Stavrinos, Yiorgos; Veneti, Anastasia; 10.4204/EPTCS.45.2

    2011-01-01

    The intersection type assignment system has been designed directly as deductive system for assigning formulae of the implicative and conjunctive fragment of the intuitionistic logic to terms of lambda-calculus. But its relation with the logic is not standard. Between all the logics that have been proposed as its foundation, we consider ISL, which gives a logical interpretation of the intersection by splitting the intuitionistic conjunction into two connectives, with a local and global behaviour respectively, being the intersection the local one. We think ISL is a logic interesting by itself, and in order to support this claim we give a sequent calculus formulation of it, and we prove that it enjoys the cut elimination property.

  15. Affine connection form of Regge calculus

    CERN Document Server

    Khatsymovsky, V M

    2015-01-01

    Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein-Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the 3-simplices which play a role of a general discrete connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4,R) of the connection matrices. As a result, we have some action invariant w. r. t. arbitrary change of coordinates of the vertices (and related GL(4,R) transformations in...

  16. Some Applications of Fractional Calculus in Engineering

    Directory of Open Access Journals (Sweden)

    J. A. Tenreiro Machado

    2010-01-01

    Full Text Available Fractional Calculus (FC goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades, due to the progress in the area of chaos that revealed subtle relationships with the FC concepts. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses FC in the study of system dynamics and control. In this perspective, this paper investigates the use of FC in the fields of controller tuning, legged robots, redundant robots, heat diffusion, and digital circuit synthesis.

  17. GIANT VESICAL CALCULUS – A CASE REPORT

    Directory of Open Access Journals (Sweden)

    Hanumanthaiah

    2014-06-01

    Full Text Available Until 20th century, bladder stones were one of the most prevalent disorders among the poor class and the incidence was especially high in childhood and adolescent. 1 The decrease in incidence of bladder calculi is attributed mainly to dietary and nutritional progress especially in children. 2 A solitary bladder calculus is usual, although multiple stones are found in 25% of cases. 3 Bladder stones are rare, and they constitute about 5% of all urinary stones, 4, 5 it is classified as migrated from upper urinary tract, primary idiopathic, or secondary calculi. 6 Bladder stones are managed by Extracorporeal Shockwave Lithotripsy (ESWL, endourology procedures, or open surgery. We report an unusual case of giant vesical calculus weighing 600grams in a 55 year old female with no evidence of hematuria, urinary retention, and dysuria.

  18. Attendance and attainment in a Calculus course

    Science.gov (United States)

    Meulenbroek, Bernard; van den Bogaard, Maartje

    2013-10-01

    In this paper the relationship between attendance and attainment in a standard calculus course is investigated. Calculus could in principle be studied without attending lectures due to the wealth of material available (in hardcopy and online). However, in this study we will show that the pass rate of students attending classes regularly (>75% of the classes) is much higher than the pass rate of students attending fewer classes. We use a logistic model to investigate whether this correlation is significant. We will argue why we believe that this correlation between attendance and attainment is causal, i.e. why it is necessary for most students to attend classes in order to (improve their chances to) pass the exam.

  19. Operator calculus - the exterior differential complex

    CERN Document Server

    Harrison, Jenny

    2011-01-01

    This paper and its sequels lay the groundwork for an operator calculus based on a spectral pair ('B,O) where 'B is a complete locally convex topological vector space of "differential chains" and O is an algebra of continuous operators acting on 'B. The topological dual of 'B is isomorphic to the classical Fr\\'echet space B of differential forms with uniform bounds on each of its directional derivatives. In a sequel H. Pugh and the author show that 'B is not generally reflexive. Since basic operators sufficient for a full calculus described in this paper, and important products are closed in 'B, there is little need for the larger double dual space B'. The covariant, constructive viewpoint of chains takes precedence over the contravariant, abstract viewpoint of cochains. In other words, chains come first. Applications include the first proof of a solution to Plateau's problem for soap films, solving a two hundred year old problem.

  20. Proof-relevant pi-calculus

    Directory of Open Access Journals (Sweden)

    Roly Perera

    2015-07-01

    Full Text Available Formalising the pi-calculus is an illuminating test of the expressiveness of logical frameworks and mechanised metatheory systems, because of the presence of name binding, labelled transitions with name extrusion, bisimulation, and structural congruence. Formalisations have been undertaken in a variety of systems, primarily focusing on well-studied (and challenging properties such as the theory of process bisimulation. We present a formalisation in Agda that instead explores the theory of concurrent transitions, residuation, and causal equivalence of traces, which has not previously been formalised for the pi-calculus. Our formalisation employs de Bruijn indices and dependently-typed syntax, and aligns the "proved transitions" proposed by Boudol and Castellani in the context of CCS with the proof terms naturally present in Agda's representation of the labelled transition relation. Our main contributions are proofs of the "diamond lemma" for residuation of concurrent transitions and a formal definition of equivalence of traces up to permutation of transitions.

  1. Exposing calculus students to advanced mathematics

    Science.gov (United States)

    Griffiths, Barry J.; Selcuk Haciomeroglu, Erhan

    2014-07-01

    To ensure the competitiveness of the USA in the global economy, and its role as a leader in science and engineering, it is important to cultivate the next generation of home grown mathematicians. However, while universities across the USA offer calculus classes to thousands of undergraduate students each year, very few of them go on to major in mathematics. This paper posits that one of the main reasons is that the mathematical community does not expose calculus students to the beauty and complexity of upper-level mathematics, and that by doing so before they fully commit to their programme of study, the number of students with a qualification in mathematics can be increased. The results show a significant increase in the number of students planning to add a minor in mathematics, and an increased likelihood among freshmen and sophomores to change their major.

  2. Students’ difficulties with vector calculus in electrodynamics

    Directory of Open Access Journals (Sweden)

    Laurens Bollen

    2015-11-01

    Full Text Available Understanding Maxwell’s equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell’s equations in electrodynamics.

  3. Semiclassical dynamics and magnetic Weyl calculus

    Energy Technology Data Exchange (ETDEWEB)

    Lein, Maximilian Stefan

    2011-01-19

    Weyl quantization and related semiclassical techniques can be used to study conduction properties of crystalline solids subjected to slowly-varying, external electromagnetic fields. The case where the external magnetic field is constant, is not covered by existing theory as proofs involving usual Weyl calculus break down. This is the regime of the so-called quantum Hall effect where quantization of transverse conductance is observed. To rigorously derive semiclassical equations of motion, one needs to systematically develop a magnetic Weyl calculus which contains a semiclassical parameter. Mathematically, the operators involved in the analysis are magnetic pseudodifferential operators, a topic which by itself is of interest for the mathematics and mathematical physics community alike. Hence, we will devote two additional chapters to further understanding of properties of those operators. (orig.)

  4. Thunks and the λ-calculus

    DEFF Research Database (Denmark)

    Hatcliff, John; Danvy, Olivier

    1997-01-01

    Thirty-five years ago, thunks were used to simulate call-by-name under call-by-value in Algol 60. Twenty years ago, Plotkin presented continuation-based simulations of call-by-name under call-by-value and vice versa in the λ-calculus. We connect all three of these classical simulations by factori......Thirty-five years ago, thunks were used to simulate call-by-name under call-by-value in Algol 60. Twenty years ago, Plotkin presented continuation-based simulations of call-by-name under call-by-value and vice versa in the λ-calculus. We connect all three of these classical simulations...

  5. Thunks and the λ-calculus

    DEFF Research Database (Denmark)

    Hatcliff, John; Danvy, Olivier

    1996-01-01

    Thirty-five years ago, thunks were used to simulate call-by-name under call-by-value in Algol 60. Twenty years ago, Plotkin presented continuation-based simulations of call-by-name under call-by-value and vice versa in the λ-calculus. We connect all three of these classical simulations by factori......Thirty-five years ago, thunks were used to simulate call-by-name under call-by-value in Algol 60. Twenty years ago, Plotkin presented continuation-based simulations of call-by-name under call-by-value and vice versa in the λ-calculus. We connect all three of these classical simulations...

  6. Effect of scaffolding on helping introductory physics students solve quantitative problems involving strong alternative conceptions

    Science.gov (United States)

    Lin, Shih-Yin; Singh, Chandralekha

    2015-12-01

    It is well known that introductory physics students often have alternative conceptions that are inconsistent with established physical principles and concepts. Invoking alternative conceptions in the quantitative problem-solving process can derail the entire process. In order to help students solve quantitative problems involving strong alternative conceptions correctly, appropriate scaffolding support can be helpful. The goal of this study is to examine how different scaffolding supports involving analogical problem-solving influence introductory physics students' performance on a target quantitative problem in a situation where many students' solution process is derailed due to alternative conceptions. Three different scaffolding supports were designed and implemented in calculus-based and algebra-based introductory physics courses involving 410 students to evaluate the level of scaffolding needed to help students learn from an analogical problem that is similar in the underlying principles involved but for which the problem-solving process is not derailed by alternative conceptions. We found that for the quantitative problem involving strong alternative conceptions, simply guiding students to work through the solution of the analogical problem first was not enough to help most students discern the similarity between the two problems. However, if additional scaffolding supports that directly helped students examine and repair their knowledge elements involving alternative conceptions were provided, e.g., by guiding students to contemplate related issues and asking them to solve the targeted problem on their own first before learning from the analogical problem provided, students were more likely to discern the underlying similarities between the problems and avoid getting derailed by alternative conceptions when solving the targeted problem. We also found that some scaffolding supports were more effective in the calculus-based course than in the algebra

  7. A stochastic maximum principle via Malliavin calculus

    OpenAIRE

    Øksendal, Bernt; Zhou, Xun Yu; Meyer-Brandis, Thilo

    2008-01-01

    This paper considers a controlled It\\^o-L\\'evy process where the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.

  8. Data analysis recipes: Probability calculus for inference

    OpenAIRE

    Hogg, David W.

    2012-01-01

    In this pedagogical text aimed at those wanting to start thinking about or brush up on probabilistic inference, I review the rules by which probability distribution functions can (and cannot) be combined. I connect these rules to the operations performed in probabilistic data analysis. Dimensional analysis is emphasized as a valuable tool for helping to construct non-wrong probabilistic statements. The applications of probability calculus in constructing likelihoods, marginalized likelihoods,...

  9. Double dumb-bell calculus in childhood.

    Science.gov (United States)

    Joshi, Prashant; Sarda, Dinesh; Ahmad, Ashraf; Kothari, Paras

    2009-01-01

    An eight-year old male was admitted with complaints of right scrotal swelling, dysuria and intermittent retention of urine for 10 days. On per-rectal examination, a hard mass was palpable in the posterior urethra. An X-ray (KUB) of the abdomen revealed a double dumb-bell calculus at the base of bladder, extending into the posterior urethra. A cystolithotomy via the suprapubic approach was successfully curative.

  10. Double dumb-bell calculus in childhood

    Directory of Open Access Journals (Sweden)

    Joshi Prashant

    2009-01-01

    Full Text Available An eight-year old male was admitted with complaints of right scrotal swelling, dysuria and intermittent retention of urine for 10 days. On per-rectal examination, a hard mass was palpable in the posterior urethra. An X-ray (KUB of the abdomen revealed a double dumb-bell calculus at the base of bladder, extending into the posterior urethra. A cystolithotomy via the suprapubic approach was successfully curative.

  11. Data analysis recipes: Probability calculus for inference

    CERN Document Server

    Hogg, David W

    2012-01-01

    In this pedagogical text aimed at those wanting to start thinking about or brush up on probabilistic inference, I review the rules by which probability distribution functions can (and cannot) be combined. I connect these rules to the operations performed in probabilistic data analysis. Dimensional analysis is emphasized as a valuable tool for helping to construct non-wrong probabilistic statements. The applications of probability calculus in constructing likelihoods, marginalized likelihoods, posterior probabilities, and posterior predictions are all discussed.

  12. Predicateμ-Calculus for Mobile Ambients

    Institute of Scientific and Technical Information of China (English)

    Hui-Min Lin

    2005-01-01

    Ambient logics have been proposed to describe properties for mobile agents which may evolve over time as well as space. This paper takes a predicate-based approach to extending an ambient logic with recursion, yielding a predicate t-calculus in which fixpoint formulas are formed using predicate variables. An algorithm is developed for model checking finite-control mobile ambients against formulas of the logic, providing the first decidability result for model checking a spatial logic with recursion.

  13. GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION

    Institute of Scientific and Technical Information of China (English)

    陈泽乾

    2002-01-01

    A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators (e)t and its dual, creation operators (e)*t.

  14. Introduction to the calculus of variations

    CERN Document Server

    Sagan, Hans

    1992-01-01

    Excellent text provides basis for thorough understanding of the problems, methods and techniques of the calculus of variations and prepares readers for the study of modern optimal control theory. Treatment limited to extensive coverage of single integral problems in one and more unknown functions. Carefully chosen variational problems and over 400 exercises. ""Should find wide acceptance as a text and reference.""-American Mathematical Monthly. 1969 edition. Bibliography.

  15. Index Calculus in the Trace Zero Variety

    OpenAIRE

    Gorla, Elisa; Massierer, Maike

    2014-01-01

    International audience; We discuss how to apply Gaudry's index calculus algorithm for abelian varieties to solve the discrete logarithm problem in the trace zero variety of an elliptic curve. We treat in particular the practically relevant cases of field extensions of degree 3 or 5. Our theoretical analysis is compared to other algorithms present in the literature, and is complemented by results from a prototype implementation.

  16. A Calculus for Higher Spin Interactions

    CERN Document Server

    Joung, Euihun; Waldron, Andrew

    2013-01-01

    Higher spin theories can be efficiently described in terms of auxiliary St\\"uckelberg or projective space field multiplets. By considering how higher spin models couple to scale, these approaches can be unified in a conformal geometry/tractor calculus framework. We review these methods and apply them to higher spin vertices to obtain a generating function for massless, massive and partially massless three-point interactions.

  17. Index matrices towards an augmented matrix calculus

    CERN Document Server

    Atanassov, Krassimir T

    2014-01-01

    This book presents the very concept of an index matrix and its related augmented matrix calculus in a comprehensive form. It mostly illustrates the exposition with examples related to the generalized nets and intuitionistic fuzzy sets which are examples of an extremely wide array of possible application areas. The present book contains the basic results of the author over index matrices and some of its open problems with the aim to stimulating more researchers to start working in this area.

  18. Students' difficulties with vector calculus in electrodynamics

    OpenAIRE

    2015-01-01

    Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing ca...

  19. Covariant Calculus for Effective String Theories

    OpenAIRE

    Dass, N. D. Hari; Matlock, Peter

    2007-01-01

    A covariant calculus for the construction of effective string theories is developed. Effective string theory, describing quantum string-like excitations in arbitrary dimension, has in the past been constructed using the principles of conformal field theory, but not in a systematic way. Using the freedom of choice of field definition, a particular field definition is made in a systematic way to allow an explicit construction of effective string theories with manifest exact conformal symmetry. ...

  20. Feynman's operational calculus and beyond noncommutativity and time-ordering

    CERN Document Server

    Johnson, George W; Nielsen, Lance

    2015-01-01

    This book is aimed at providing a coherent, essentially self-contained, rigorous and comprehensive abstract theory of Feynman's operational calculus for noncommuting operators. Although it is inspired by Feynman's original heuristic suggestions and time-ordering rules in his seminal 1951 paper An operator calculus having applications in quantum electrodynamics, as will be made abundantly clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman's work. Hence, the second part of the main title of this book. The basic properties of the operational calculus are developed and certain algebraic and analytic properties of the operational calculus are explored. Also, the operational calculus will be seen to possess some pleasant stability properties. Furthermore, an evolution equation and a generalized integral equation obeyed by the operational calculus are discussed and connections wi...

  1. The Sustainability of Dental Calculus for Archaeological Research

    DEFF Research Database (Denmark)

    Mackie, Meaghan Emma; Radini, Anita; Speller, Camilla

    Dental calculus is a mineralized plaque biofilm formed by microbiota of the oral microbiome. Until recently, the information potential of dental calculus for archaeological study was not fully realised and it was often discarded. However, it is now recognized that dental calculus entombs...... and preserves valuable microfossils and biomolecules within its matrix. While the analysis of calculus is usually destructive, judicious sampling of relatively small quantities of material can provide unique information on ancient health and diet. Additionally, dental calculus is not classified as human tissue......, but as an ectopic growth, and in some cases may provide an alternative to the destructive analysis of human skeletal remains. We present a case study applying microscopy and shotgun proteomic methods to Roman Age individuals to demonstrate the potential of even minute quantities of dental calculus to entrap...

  2. CLINICO-BACTERIOLOGICAL STUDY OF VESICAL CALCULUS

    Directory of Open Access Journals (Sweden)

    Pushpendra

    2016-05-01

    Full Text Available BACKGROUND Vesical calculi are the most common manifestation of lower urinary tract lithiasis. Urinary infections play an important role in aetiopathogenesis of vesical calculi. OBJECTIVE Aim of this study was proposed to establish the bacteriology of stone and urine in an attempt to evaluate the role of infection in the formation of stone. Associated factors like age, sex, site of infection, obstruction, diet were also evaluated. DESIGN Prospective cohort study. METHODS The patients were admitted in surgical ward as provisional diagnosed cases of vesical calculus, were subjected to investigations including CBC, RBS, urine analysis, renal function test, x-ray KUB region and ultrasonography. Patients who were fit for surgery, various surgical procedures were done. Gross examination and core culture of stone was done to establish their aetiology. RESULTS Ninety-four patients with vesical calculus were evaluated. Incidence of vesical calculus was 1.13%. Majority of cases were from rural areas (92.55%. Urinary tract infection was present in 37.2% of cases, majority of cases urine culture was positive (30.95%. Core culture of stone was positive in 18 cases (25.17%. E. coli was the predominant organism both in urine culture (19.04% and core culture of stone (25.71%. CONCLUSIONS There is significant association regarding the presence of vesical calculi and the development of urinary infections. E. coli was the predominant organism found both in urine and core culture of stone.

  3. CAS Introductory Course in Italy

    CERN Multimedia

    2008-01-01

    The CERN Accelerator School’s introductory course is a great success. This year the CERN Accelerator School held its "Introduction to Accelerator Physics" course in Frascati, Italy, from 2-14 November in collaboration with the University of Rome "La Sapienza" and the INFN Frascati National Laboratory. The Introductory level course is particularly important since, for the majority of participants, it is the first opportunity to discover the various aspects of accelerator physics. For this school the programme had been significantly revised in order to take into account the new trends currently being developed in the field, thus putting more emphasis on linacs, synchrotron light sources and free-electron lasers. The school was a resounding success with 115 participants of more than 23 nationalities. Feedback from the students praised the expertise of the lecturers, the high standard of the lectures as well as the excellent organizati...

  4. Introductory physics: The new scholasticism

    CERN Document Server

    Hogg, D W; Hogg, David W.; Mahajan, Sanjoy

    2004-01-01

    Most introductory physics textbooks neglect air resistance in situations where an astute student can observe that it dominates the dynamics. We give examples from many books. Using dimensional analysis we discuss the significance of air resistance. Our discussion can be used to mitigate the baleful influence of these textbooks. Incorrectly neglecting air resistance is just one of their many unphysical teachings. Shouldn't we require at a minimum that our textbooks be physically correct?

  5. Introductory remarks on complex multiplication

    Directory of Open Access Journals (Sweden)

    Harvey Cohn

    1982-01-01

    Full Text Available Complex multiplication in its simplest form is a geometric tiling property. In its advanced form it is a unifying motivation of classical mathematics from elliptic integrals to number theory; and it is still of active interest. This interrelation is explored in an introductory expository fashion with emphasis on a central historical problem, the modular equation between j(z and j(2z.

  6. Using an Isomorphic Problem Pair to Learn Introductory Physics: Transferring from a Two-step Problem to a Three-step Problem

    CERN Document Server

    Lin, Shih-Yin

    2016-01-01

    In this study, we examine introductory physics students' ability to perform analogical reasoning between two isomorphic problems which employ the same underlying physics principles but have different surface features. Three hundred and eighty two students from a calculus-based and an algebra-based introductory physics course were asked to learn from a solved problem provided and take advantage of what they learned from it to solve another isomorphic problem (which we call the quiz problem). The solved problem provided has two sub-problems while the quiz problem has three sub-problems, which is known to be challenging for introductory students from previous research. In addition to the solved problem, students also received extra scaffolding supports that were intended to help them discern and exploit the underlying similarities of the isomorphic solved and quiz problems. The results suggest that students had great difficulty in transferring what they learned from a 2-step problem to a 3-step problem. Although...

  7. Symbolic calculus for Toeplitz operators with half-forms

    OpenAIRE

    Charles, L.

    2006-01-01

    This paper is devoted to the use of half-form bundles in the symbolic calculus of Berezin-Toeplitz operators on Kahler manifolds. We state the Bohr-Sommerfeld conditions and relate them to the functional calculus of Toeplitz operators, a trace formula and the characteristic classes in deformation quantization. We also develop the symbolic calculus of Lagrangian sections, with the crucial estimate of the subprincipal terms.

  8. Application of Quantum Process Calculus to Higher Dimensional Quantum Protocols

    Directory of Open Access Journals (Sweden)

    Simon J. Gay

    2014-07-01

    Full Text Available We describe the use of quantum process calculus to describe and analyze quantum communication protocols, following the successful field of formal methods from classical computer science. We have extended the quantum process calculus to describe d-dimensional quantum systems, which has not been done before. We summarise the necessary theory in the generalisation of quantum gates and Bell states and use the theory to apply the quantum process calculus CQP to quantum protocols, namely qudit teleportation and superdense coding.

  9. Pre-calculus 1,001 practice problems for dummies

    CERN Document Server

    Sterling, Mary Jane; Sterling

    2014-01-01

    Prepare for calculus the smart way, with customizable pre-calculus practice 1,001 Pre-Calculus Practice Problems For Dummies offers 1,001 opportunities to gain confidence in your math skills. Much more than a workbook, this study aid provides pre-calculus problems ranked from easy to advanced, with detailed explanations and step-by-step solutions for each one. The companion website gives you free online access to all 1,001 practice problems and solutions, and you can track your progress and ID where you should focus your study time. Accessible on the go by smart phone, tablet, o

  10. On the Expressive Power of Polyadic Synchronisation in π- calculus

    DEFF Research Database (Denmark)

    Carbone, Marco; Maffeis, Sergio

    2002-01-01

    We extend the pi-calculus with polyadic synchronisation, a generalisation of the communication mechanism which allows channel names to be composite. We show that this operator embeds nicely in the theory of pi-calculus, we suggest that it permits divergence-free encodings of distributed calculi......, and we show that a limited form of polyadic synchronisation can be encoded weakly in pi-calculus. After showing that matching cannot be derived in pi-calculus, we compare the expressivity of polyadic synchronisation, mixed choice and matching. In particular we show that the degree of synchronisation...

  11. Topology,randomness and noise in process calculus

    Institute of Scientific and Technical Information of China (English)

    YING Mingsheng

    2007-01-01

    Formal models of communicating and concurrent systems are one of the most important topics in formal methods,and process calculus is one of the most successful formal models of communicating and concurrent systems.In the previous works,the author systematically studied topology in process calculus,probabilistic process calculus and pi-calculus with noisy channels in order to describe approximate behaviors of communicating and concurrent systems as well as randonmess and noise in them.This article is a brief survey of these works.

  12. The Problem of Differential Calculus on Quantum Groups

    CERN Document Server

    Delius, G W

    1996-01-01

    The bicovariant differential calculi on quantum groups of Woronowicz have the drawback that their dimensions do not agree with that of the corresponding classical calculus. In this paper we discuss the first-order differential calculus which arises from a simple quantum Lie algebra. This calculus has the correct dimension and is shown to be bicovariant and complete. But it does not satisfy the Leibniz rule. For sl_n this approach leads to a differential calculus which satisfies a simple generalization of the Leibniz rule.

  13. Renal vein thrombosis mimicking urinary calculus: a dilemma of diagnosis.

    Science.gov (United States)

    Wang, Yimin; Chen, Shanwen; Wang, Wei; Liu, Jianyong; Jin, Baiye

    2015-07-02

    Renal vein thrombosis (RVT) with flank pain, and hematuria, is often mistaken with renal colic originating from ureteric or renal calculus. Especially in young and otherwise healthy patients, clinicians are easily misled by clinical presentation and calcified RVT. A 38-year-old woman presented with flank pain and hematuria suggestive of renal calculus on ultrasound. She underwent extracorporeal shock wave lithotripsy that failed, leading to the recommendation that percutaneous lithotomy was necessary to remove the renal calculus. In preoperative view of the unusual shape of the calculus without hydronephrosis, noncontrast computed tomography was taken and demonstrated left ureteric calculus. However computed tomography angiography revealed, to our surprise, a calcified RVT that was initially thought to be a urinary calculus. This case shows that a calcified RVT might mimic a urinary calculus on conventional ultrasonography and ureteric calculus on noncontrast computed tomography. Subsequent computed tomography angiography disclosed that a calcified RVT caused the imaging findings, thus creating a potentially dangerous clinical pitfall. Hence, it is suggested that the possibility of a RVT needs to be considered in the differential diagnosis whenever one detects an uncommon shape for a urinary calculus.

  14. Timed Operational Semantics and Well-Formedness of Shape Calculus

    Directory of Open Access Journals (Sweden)

    E. Bartocci

    2010-01-01

    Full Text Available The Shape Calculus is a bio-inspired calculus for describing 3D shapes moving in a space. A shape forms a 3D process when combined with a behaviour. Behaviours are specified with a timed CCS-like process algebra using a notion of channel that models naturally binding sites on the surface of shapes. In this paper, the full formal timed operational semantics of the calculus is provided, together with examples that illustrate the use of the calculus in a well-known biological scenario. Moreover, a result of well-formedness about the evolution of a given network of well-formed 3D processes is proved.

  15. Everyday calculus discovering the hidden math all around us

    CERN Document Server

    Fernandez, Oscar E

    2014-01-01

    Calculus. For some of us, the word conjures up memories of ten-pound textbooks and visions of tedious abstract equations. And yet, in reality, calculus is fun, accessible, and surrounds us everywhere we go. In Everyday Calculus, Oscar Fernandez shows us how to see the math in our coffee, on the highway, and even in the night sky. Fernandez uses our everyday experiences to skillfully reveal the hidden calculus behind a typical day's events. He guides us through how math naturally emerges from simple observations-how hot coffee cools down, for example-and in discussions of over fifty familia

  16. Motivation for Achievement and Attitudes toward Mathematics Instruction in a Required Calculus Course at the Norwegian University of Science and Technology

    Directory of Open Access Journals (Sweden)

    Donna Sundre

    2012-01-01

    Full Text Available This study from the Norwegian University of Science and Technology (NTNU examines students’ learning goals and attitudes toward mathematics in a first-year calculus course in undergraduate engineering education. Achievement motivation research using the Achievement Goal Questionnaire (AGQ is advanced from current literature with two additions: (1 a course specific context using introductory college calculus students, and (2 participation of Norwegian students.Pre- and posttest measures of attitudes indicate that students do change learning goals over time, unfortunately opposite to the instructors’ aspirations. A significant increase in “Mastery Avoidance” and “Work Avoidance” was accompanied with a drop in “Mastery Approach” and “Performance Approach”. Variables such as value, motivation and enjoyment decreased along with a significant drop in self-confidence.

  17. Feynman's Operational Calculus and the Stochastic Functional Calculus in Hilbert Space

    OpenAIRE

    Jefferies, Brian

    2010-01-01

    Let $A_1, A_2$ be bounded linear operators acting on a Banach space $E$. A pair $(\\mu_1, \\mu_2)$ of continuous probability measures on $[0,1]$ determines a functional calculus $f \\rightarrowtail f_{\\mu1,|mu2}(A_1, A_2)$ for analytic functions $f$ by weighting all possible orderings of operator products of $A_1$ and $A_2$ via the probability measures $\\mu_1$ and $\\mu_2$. For example, $f \\rightarrowtail f_{\\mu,\\mu}(A_1, A_2)$ is the Weyl functional calculus with equally weighted operator produc...

  18. Thermal physics in the introductory physics course: Why and how to teach it from a unified atomic perspective

    Science.gov (United States)

    Reif, Frederick

    1999-12-01

    Heat and thermodynamics are traditionally taught in the introductory physics course from a predominantly macroscopic point of view. However, it is advantageous to adopt a more modern approach that systematically builds on students' knowledge of the atomic structure of matter and of elementary mechanics. By focusing on the essential physics without requiring more than elementary classical mechanics, this approach can be made sufficiently simple to be readily teachable during five or six weeks of an ordinary calculus-based introductory physics course. This approach can be highly unified, using atomic considerations to infer the properties of macroscopic systems while also enabling thermodynamic analyses independent of specific atomic models. Furthermore, this integrated point of view provides a deeper physical understanding of basic concepts (such as internal energy, heat, entropy, and absolute temperature) and of important phenomena (such as equilibrium, fluctuations, and irreversibility).

  19. The Development of an Inquiry-based Curriculum Specifically for the Introductory Algebra-based Physics Course

    CERN Document Server

    Thacker, B; Eligon, A M; Diaz, Abel; Eligon, Ann Marie; Thacker, Beth

    2007-01-01

    We discuss an inquiry-based curriculum that has been developed specifically for the introductory algebra-based physics course, taking into account the needs, backgrounds, learning styles and career goals of the students in that class. The course is designed to be taught in a laboratory-based environment, however parts of the materials can be used in other settings. As instructors we found ourselves drawing on materials developed for the calculus-based course and for other populations (materials developed for pre-service teachers, for example), parts of which were appropriate, but not a complete curriculum as we would like to teach it, developed specifically for students in the introductory algebra-based physics course. So we have modified and adapted parts of existing materials and integrated them with our own new units and our own format, creating a course aimed specifically at these students.

  20. Reflection on problem solving in introductory and advanced physics

    Science.gov (United States)

    Mason, Andrew J.

    Reflection is essential in order to learn from problem solving. This thesis explores issues related to how reflective students are and how we can improve their capacity for reflection on problem solving. We investigate how students naturally reflect in their physics courses about problem solving and evaluate strategies that may teach them reflection as an integral component of problem-solving. Problem categorization based upon similarity of solution is a strategy to help them reflect about the deep features of the problems related to the physics principles involved. We find that there is a large overlap between the introductory and graduate students in their ability to categorize. Moreover, introductory students in the calculus-based courses performed better categorization than those in the algebra-based courses even though the categorization task is conceptual. Other investigations involved exploring if reflection could be taught as a skill on individual and group levels. Explicit self-diagnosis in recitation investigated how effectively students could diagnose their own errors on difficult problems, how much scaffolding was necessary for this purpose, and how effective transfer was to other problems employing similar principles. Difficulty in applying physical principles and difference between the self-diagnosed and transfer problems affected performance. We concluded that a sustained intervention is required to learn effective problem-solving strategies. Another study involving reflection on problem solving with peers suggests that those who reflected with peers drew more diagrams and had a larger gain from the midterm to final exam. Another study in quantum mechanics involved giving common problems in midterm and final exams and suggested that advanced students do not automatically reflect on their mistakes. Interviews revealed that even advanced students often focus mostly on exams rather than learning and building a robust knowledge structure. A survey was