Putting Differentials Back into Calculus
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.
Differential calculus and its applications
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.
Multivariable calculus and differential geometry
Walschap, Gerard
2015-01-01
This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.
Calculus & ordinary differential equations
Pearson, David
1995-01-01
Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.
A primer on exterior differential calculus
Burton D.A.
2003-01-01
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'...
Matlab differential and integral calculus
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
Iterated Differential Forms III: Integral Calculus
Vinogradov, A.M.; Vitagliano, L.
2006-01-01
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra $\\Lambda_{\\infty}$ will be discussed in subsequent notes.
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. .
Calculus of tensors and differential forms
Sinha, Rajnikant
2014-01-01
Calculus of tensors and differential forms is an introductory-level textbook. Through this book, students will familiarize themselves with tools they need in order to use for further study on general relativity and research, such as affine tensors, tensor calculus on manifolds, relative tensors, Lie derivatives, wedge products, differential forms, and Stokes' theorem. The treatment is concrete and in detail, so that abstract concepts do not deter even physics and engineering students. This self contained book requires undergraduate-level calculus of several variables and linear algebra as prerequisite. Fubini's theorem in real analysis, to be used in Stokes' theorem, has been proved earlier than Stokes' theorem so that students don't have to search elsewhere.
Classroom Voting Patterns in Differential Calculus
Cline, Kelly; Zullo, Holly; VonEpps, Lahna
2012-01-01
We study how different sections voted on the same set of classroom voting questions in differential calculus, finding that voting patterns can be used to identify some of the questions that have the most pedagogic value. We use statistics to identify three types of especially useful questions: 1. To identify good discussion questions, we look for…
Differential Calculus in Normed Linear Spaces
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 9; Issue 2. Differential Calculus in Normed Linear Spaces. Harish Seshadri. Book Review Volume 9 Issue 2 February 2004 pp 90-91. Fulltext. Click here to view fulltext PDF. Permanent link: http://www.ias.ac.in/article/fulltext/reso/009/02/0090-0091 ...
Differential calculus in normed linear spaces
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...
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
Malliavin Calculus With Applications to Stochastic Partial Differential Equations
Sanz-Solé, Marta
2005-01-01
Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics.This book presents applications of Malliavin calculus to the analysis of probability laws of solutions to stochastic partial differential equations driven by Gaussian noises that are white in time and coloured in space. The first five chapters introduce the calculus itself
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
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.
Partial differential equations and calculus of variations
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.
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.
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
Generalized Differential Calculus and Applications to Optimization
Rector, Robert Blake Hayden
This thesis contains contributions in three areas: the theory of generalized calculus, numerical algorithms for operations research, and applications of optimization to problems in modern electric power systems. A geometric approach is used to advance the theory and tools used for studying generalized notions of derivatives for nonsmooth functions. These advances specifically pertain to methods for calculating subdifferentials and to expanding our understanding of a certain notion of derivative of set-valued maps, called the coderivative, in infinite dimensions. A strong understanding of the subdifferential is essential for numerical optimization algorithms, which are developed and applied to nonsmooth problems in operations research, including non-convex problems. Finally, an optimization framework is applied to solve a problem in electric power systems involving a smart solar inverter and battery storage system providing energy and ancillary services to the grid.
Differential Calculus on h-Deformed Spaces
Herlemont, Basile; Ogievetsky, Oleg
2017-10-01
We construct the rings of generalized differential operators on the h-deformed vector space of gl-type. In contrast to the q-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of h-deformed differential operators {Diff}_{h},σ(n) is labeled by a rational function σ in n variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings {Diff}_{h},σ(n).
Covariant differential calculus on the quantum exterior vector space
Parashar, Preeti; Soni, S. K.
1992-12-01
We formulate a differential calculus on the quantum exterior vector space spanned by the generators of a non-anticommutative algebra satisfying 10052_2005_Article_BF01559737_TeX2GIFE1.gif begin{gathered} r^{ij} equiv θ ^i θ ^j + B_{kl}^{ij} θ ^k θ ^l = 0i,j = 1,2,...,n. \\ and \\ (θ ^i )^2 = (θ ^j )^2 = ... = (θ ^n )^2 = 0, \\ where B {kl/ij} is the most general matrix defined in terms of complex deformation parameters. Following considerations analogous to those of Wess and Zumino, we are able to exhibit covariance of our calculus under ( n/2)+1 parameter deformation of GL(n) and explicitly check that the non-anticommutative differential calculus satisfies the general constraints given by them, such as the “linear” condition dr ij ≃0 and the “quadratic” condition r ij x n ≃0 where x n = dλ n are the differentials of the variables.
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.
Differentiated Instruction in a Calculus Curriculum for College Students in Taiwan
Chen, Jing-Hua; Chen, Yi-Chou
2018-01-01
Objectives: To explore differentiated instruction within a calculus curriculum. For college students to learn concentration, motivation and the impact of academic achievement; explore the attitudes and ideas of students on differentiated instruction within a calculus curriculum; build up the diversity of mathematics education within varied…
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.
Amdahl, Kenn; Loats, Jim
This book, written for students of calculus, is designed to augment the explanations of concepts covered in a calculus class. It consists of an overview of calculus divided into basic ideas and vocabulary, the process of differential calculus, and integral calculus. The book is intended as a resource to explain the concepts of calculus in everyday…
Robust overlay metrology with differential Mueller matrix calculus.
Chen, Xiuguo; Gu, Honggang; Jiang, Hao; Zhang, Chuanwei; Liu, Shiyuan
2017-04-17
Overlay control is of vital importance to good device performances in semiconductor manufacturing. In this work, the differential Mueller matrix calculus is introduced to investigate the Mueller matrices of double-patterned gratings with overlay displacements, which helps to reveal six elementary optical properties hidden in the Mueller matrices. We find and demonstrate that, among these six elementary optical properties, the linear birefringence and dichroism, LB' and LD', along the ± 45° axes show a linear response to the overlay displacement and are zero when the overlay displacement is absent at any conical mounting. Although the elements from the two 2 × 2 off-diagonal blocks of the Mueller matrix have a similar property to LB' and LD', as reported in the literature, we demonstrate that it is only valid at a special conical mounting with the plane of incidence parallel to grating lines. The better property of LB' and LD' than the Mueller matrix elements of the off-diagonal blocks in the presence of overlay displacement verifies them to be a more robust indicator for the diffraction-based overlay metrology.
Stochastic Calculus and Differential Equations for Physics and Finance
McCauley, Joseph L.
2013-02-01
1. Random variables and probability distributions; 2. Martingales, Markov, and nonstationarity; 3. Stochastic calculus; 4. Ito processes and Fokker-Planck equations; 5. Selfsimilar Ito processes; 6. Fractional Brownian motion; 7. Kolmogorov's PDEs and Chapman-Kolmogorov; 8. Non Markov Ito processes; 9. Black-Scholes, martingales, and Feynman-Katz; 10. Stochastic calculus with martingales; 11. Statistical physics and finance, a brief history of both; 12. Introduction to new financial economics; 13. Statistical ensembles and time series analysis; 14. Econometrics; 15. Semimartingales; References; Index.
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
Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces
Meljanac, Stjepan; Krešić–Jurić, Saša; Martinić, Tea
2017-07-01
This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra g0, we construct a Lie superalgebra g =g0⊕g1 containing noncommutative coordinates and one-forms. We show that g can be extended by a set of generators TAB whose action on the enveloping algebra U (g ) gives the commutation relations between monomials in U (g0 ) and one-forms. Realizations of noncommutative coordinates, one-forms, and the generators TAB as formal power series in a semicompleted Weyl superalgebra are found. In the special case dim(g0 ) =dim(g1 ) , we also find a realization of the exterior derivative on U (g0 ) . The realizations of these geometric objects yield a bicovariant differential calculus on U (g0 ) as a deformation of the standard calculus on the Euclidean space.
Jukic Matic, Ljerka; Dahl, Bettina
2014-01-01
This paper reports a study on retention of differential and integral calculus concepts of a second-year student of physical chemistry at a Danish university. The focus was on what knowledge the student retained 14 months after the course and on what effect beliefs about mathematics had on the retention. We argue that if a student can quickly…
Balankin, Alexander S.; Bory-Reyes, Juan; Shapiro, Michael
2016-02-01
One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil-Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals.
Differential reflectometry versus tactile sense detection of subgingival calculus in dentistry
Shakibaie, Fardad; Walsh, Laurence J.
2012-10-01
Detecting dental calculus is clinically challenging in dentistry. This study used typodonts with extracted premolar and molar teeth and simulated gingival tissue to compare the performance of differential reflectometry and periodontal probing. A total of 30 extracted teeth were set in an anatomical configuration in stone to create three typodonts. Clear polyvinyl siloxane impression material was placed to replicate the periodontal soft tissues. Pocket depths ranged from 10 to 15 mm. The three models were placed in a phantom head, and an experienced dentist assessed the presence of subgingival calculus first using the DetecTar (differential reflectometry) and then a periodontal probe. Scores from these two different methods were compared to the gold standard (direct examination of the root surface using 20× magnification) to determine the accuracy and reproducibility. Differential reflectometry was more accurate than tactile assessment (79% versus 60%), and its reproducibility was also higher (Cohen kappa 0.54 versus 0.39). Both methods performed better on single rooted premolar teeth than on multirooted teeth. These laboratory results indicate that differential reflectometry allows more accurate and reproducible detection of subgingival calculus than conventional probing, and supports its use for supplementing traditional periodontal examination methods in dental practice.
Operational calculus and differential equations with infinitely smooth coefficients
Directory of Open Access Journals (Sweden)
Dennis Nemzer
1990-01-01
Full Text Available A subring MF of the field of Mikusiński operators is constructed as a countable union space. Some topological properties of MF are investigated. Then, the product of an infinitely differentiable function and an element of MF is given and is used to investigate operational equations with infinitely smooth coefficients.
Climate Modeling in the Calculus and Differential Equations Classroom
Kose, Emek; Kunze, Jennifer
2013-01-01
Students in college-level mathematics classes can build the differential equations of an energy balance model of the Earth's climate themselves, from a basic understanding of the background science. Here we use variable albedo and qualitative analysis to find stable and unstable equilibria of such a model, providing a problem or perhaps a…
The Frolicher-Nijenhuis Calculus in Synthetic Differential Geometry
Nishimura, Hirokazu
2008-01-01
Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.
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
Differential and Integral Calculus in careers of technical sciences. Specificities of their teaching
Directory of Open Access Journals (Sweden)
Iván Javier Villamar-Alvarado
2017-12-01
Full Text Available The Differential and Integral Calculus has great relevance for the professionals of the technical sciences since it provides them with a solid theoretical-conceptual body to process information, to use models that simulate real processes, to solve technical problems, to work in multidisciplinary projects and to communicate with precision. In spite of this relevance, there are numerous international dissatisfactions related to their learning by training engineers. The objective was to unveil the specificities of the teaching-learning process of this discipline that favor a successful appropriation of its content by the future engineers. The result was to unveil the didactic need to resolve the dialectical contradiction that manifests itself between the systematization of the engineering functionality of the aforementioned content and its contextualized interdisciplinary generalization. As a consequence, the need arises to create new didactic proposals that overcome this contradiction, as a way to perfect the teaching-learning process of this discipline in the engineering careers.
The conceptual basis of mathematics in cardiology: (II). Calculus and differential equations.
Bates, Jason H T; Sobel, Burton E
2003-04-01
This is the second in a series of four articles developed for the readers of Coronary Artery Disease. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to
Essential calculus with applications
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.
Zhukovsky, K
2014-01-01
We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.
Directory of Open Access Journals (Sweden)
K. Zhukovsky
2014-01-01
Full Text Available We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.
A Study in the Use of Maniplatives to Teach Topics in Differential and Integral Calculus
Rivero, Maria de Lourdes
2016-01-01
The use of mathematics manipulatives for the elementary grades is well-studied, but little research exists on their value for teaching calculus students. This project studied the role of physical manipulatives on student learning in two high school calculus classes. It explored the effect of two lessons taught with manipulatives, and compared two lessons on the same topic, one taught in the traditional way and the other incorporating the use of manipulatives. In evaluating the teaching method...
Hansbo, Peter; Larson, Mats G.
2017-10-01
We employ surface differential calculus to derive models for Kirchhoff plates including in-plane membrane deformations. We also extend our formulation to structures of plates. For solving the resulting set of partial differential equations, we employ a finite element method based on elements that are continuous for the displacements and discontinuous for the rotations, using C^0-elements for the discretisation of the plate as well as for the membrane deformations. Key to the formulation of the method is a convenient definition of jumps and averages of forms that are d-linear in terms of the element edge normals.
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
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
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
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
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
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-...
Introduction to the operational calculus
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
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
Investigating the Fundamental Theorem of Calculus
Johnson, Heather L.
2010-01-01
The fundamental theorem of calculus, in its simplified complexity, connects differential and integral calculus. The power of the theorem comes not merely from recognizing it as a mathematical fact but from using it as a systematic tool. As a high school calculus teacher, the author developed and taught lessons on this fundamental theorem that were…
Bergstra, J.A.; Ponse, A.; van der Zwaag, M.B.
2008-01-01
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
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.
Topology, calculus and approximation
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...
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.
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.
Differential calculus for p-norms of complex-valued vector functions with applications
Kohaupt, L.
2002-08-01
For complex-valued n-dimensional vector functions t|->s(t), supposed to be sufficiently smooth, the differentiability properties of the mapping t|->[short parallel]s(t)[short parallel]p at every point are investigated, where [short parallel]·[short parallel]p is the usual vector norm in resp. , for p[set membership, variant][1,[infinity
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...
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
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,
Testicular calculus: A rare case.
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.
Fractional calculus in bioengineering.
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
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.
Arán Filippetti, Vanessa; Richaud, María Cristina
2017-10-01
Though the relationship between executive functions (EFs) and mathematical skills has been well documented, little is known about how both EFs and IQ differentially support diverse math domains in primary students. Inconsistency of results may be due to the statistical techniques employed, specifically, if the analysis is conducted with observed variables, i.e., regression analysis, or at the latent level, i.e., structural equation modeling (SEM). The current study explores the contribution of both EFs and IQ in mathematics through an SEM approach. A total of 118 8- to 12-year-olds were administered measures of EFs, crystallized (Gc) and fluid (Gf) intelligence, and math abilities (i.e., number production, mental calculus and arithmetical problem-solving). Confirmatory factor analysis (CFA) offered support for the three-factor solution of EFs: (1) working memory (WM), (2) shifting, and (3) inhibition. Regarding the relationship among EFs, IQ and math abilities, the results of the SEM analysis showed that (i) WM and age predict number production and mental calculus, and (ii) shifting and sex predict arithmetical problem-solving. In all of the SEM models, EFs partially or totally mediated the relationship between IQ, age and math achievement. These results suggest that EFs differentially supports math abilities in primary-school children and is a more significant predictor of math achievement than IQ level.
Generalized vector calculus on convex domain
Agrawal, Om P.; Xu, Yufeng
2015-06-01
In this paper, we apply recently proposed generalized integral and differential operators to develop generalized vector calculus and generalized variational calculus for problems defined over a convex domain. In particular, we present some generalization of Green's and Gauss divergence theorems involving some new operators, and apply these theorems to generalized variational calculus. For fractional power kernels, the formulation leads to fractional vector calculus and fractional variational calculus for problems defined over a convex domain. In special cases, when certain parameters take integer values, we obtain formulations for integer order problems. Two examples are presented to demonstrate applications of the generalized variational calculus which utilize the generalized vector calculus developed in the paper. The first example leads to a generalized partial differential equation and the second example leads to a generalized eigenvalue problem, both in two dimensional convex domains. We solve the generalized partial differential equation by using polynomial approximation. A special case of the second example is a generalized isoperimetric problem. We find an approximate solution to this problem. Many physical problems containing integer order integrals and derivatives are defined over arbitrary domains. We speculate that future problems containing fractional and generalized integrals and derivatives in fractional mechanics will be defined over arbitrary domains, and therefore, a general variational calculus incorporating a general vector calculus will be needed for these problems. This research is our first attempt in that direction.
Solutions manual to accompany Fundamentals of calculus
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
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
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.
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...
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...
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.
Calculus problems and solutions
Ginzburg, Abraham
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
Calculus and analysis in Euclidean space
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...
Stochastic calculus with infinitesimals
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.
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.
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.
Calculus a complete introduction : teach yourself
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.
Applications of fractional calculus in physics
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
Brownian motion, martingales, and stochastic calculus
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...
The Britannica Guide to Analysis and Calculus
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
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
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.
Bickelhaupt, Sebastian; Steudle, Franziska; Paech, Daniel; Mlynarska, Anna; Kuder, Tristan Anselm; Lederer, Wolfgang; Daniel, Heidi; Freitag, Martin; Delorme, Stefan; Schlemmer, Heinz-Peter; Laun, Frederik Bernd
2017-01-01
To evaluate a fractional order calculus (FROC) model in diffusion weighted imaging to differentiate between malignant and benign breast lesions in breast cancer screening work-up using recently introduced parameters (βFROC, DFROC and μFROC). This retrospective analysis within a prospective IRB-approved study included 51 participants (mean 58.4 years) after written informed consent. All patients had suspicious screening mammograms and indication for biopsy. Prior to biopsy, full diagnostic contrast-enhanced MRI examination was acquired including diffusion-weighted-imaging (DWI, b = 0,100,750,1500 s/mm2). Conventional apparent diffusion coefficient Dapp and FROC parameters (βFROC, DFROC and μFROC) as suggested further indicators of diffusivity components were measured in benign and malignant lesions. Receiver operating characteristics (ROC) were calculated to evaluate the diagnostic performance of the parameters. 29/51 patients histopathologically revealed malignant lesions. The analysis revealed an AUC for Dapp of 0.89 (95% CI 0.80-0.98). For FROC derived parameters, AUC was 0.75 (0.60-0.89) for DFROC, 0.59 (0.43-0.75) for βFROC and 0.59 (0.42-0.77) for μFROC. Comparison of the AUC curves revealed a significantly higher AUC of Dapp compared to the FROC parameters DFROC (p = 0.009), βFROC (p = 0.003) and μFROC (p = 0.001). In contrast to recent description in brain tumors, the apparent diffusion coefficient Dapp showed a significantly higher AUC than the recently proposed FROC parameters βFROC, DFROC and μFROC for differentiating between malignant and benign breast lesions. This might be related to the intrinsic high heterogeneity within breast tissue or to the lower maximal b-value used in our study.
Directory of Open Access Journals (Sweden)
Sebastian Bickelhaupt
Full Text Available To evaluate a fractional order calculus (FROC model in diffusion weighted imaging to differentiate between malignant and benign breast lesions in breast cancer screening work-up using recently introduced parameters (βFROC, DFROC and μFROC.This retrospective analysis within a prospective IRB-approved study included 51 participants (mean 58.4 years after written informed consent. All patients had suspicious screening mammograms and indication for biopsy. Prior to biopsy, full diagnostic contrast-enhanced MRI examination was acquired including diffusion-weighted-imaging (DWI, b = 0,100,750,1500 s/mm2. Conventional apparent diffusion coefficient Dapp and FROC parameters (βFROC, DFROC and μFROC as suggested further indicators of diffusivity components were measured in benign and malignant lesions. Receiver operating characteristics (ROC were calculated to evaluate the diagnostic performance of the parameters.29/51 patients histopathologically revealed malignant lesions. The analysis revealed an AUC for Dapp of 0.89 (95% CI 0.80-0.98. For FROC derived parameters, AUC was 0.75 (0.60-0.89 for DFROC, 0.59 (0.43-0.75 for βFROC and 0.59 (0.42-0.77 for μFROC. Comparison of the AUC curves revealed a significantly higher AUC of Dapp compared to the FROC parameters DFROC (p = 0.009, βFROC (p = 0.003 and μFROC (p = 0.001.In contrast to recent description in brain tumors, the apparent diffusion coefficient Dapp showed a significantly higher AUC than the recently proposed FROC parameters βFROC, DFROC and μFROC for differentiating between malignant and benign breast lesions. This might be related to the intrinsic high heterogeneity within breast tissue or to the lower maximal b-value used in our study.
Fundamental theorem of Wiener calculus
Directory of Open Access Journals (Sweden)
Chull Park
1990-01-01
Full Text Available In this paper we define and develop a theory of differentiation in Wiener space C[0,T]. We then proceed to establish a fundamental theorem of the integral calculus for C[0,T]. First of all, we show that the derivative of the indefinite Wiener integral exists and equals the integrand functional. Secondly, we show that certain functionals defined on C[0,T] are equal to the indefinite integral of their Wiener derivative.
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 ...
Lenczewski, Romuald
2001-01-01
By introducing a color filtration to the multiplicity space, we extend the quantum Ito calculus on multiple symmetric Fock space to the framework of filtered adapted biprocesses. In this new notion of adaptedness,``classical'' time filtration makes the integrands similar to adapted processes, whereas ``quantum'' color filtration produces their deviations from adaptedness. An important feature of this calculus, which we call filtered stochastic calculus, is that it provides an explicit interpo...
Brownian motion and stochastic calculus
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...
Toward lattice fractional vector calculus
Tarasov, Vasily E.
2014-09-01
An analog of fractional vector calculus for physical lattice models is suggested. We use an approach based on the models of three-dimensional lattices with long-range inter-particle interactions. The lattice analogs of fractional partial derivatives are represented by kernels of lattice long-range interactions, where the Fourier series transformations of these kernels have a power-law form with respect to wave vector components. In the continuum limit, these lattice partial derivatives give derivatives of non-integer order with respect to coordinates. In the three-dimensional description of the non-local continuum, the fractional differential operators have the form of fractional partial derivatives of the Riesz type. As examples of the applications of the suggested lattice fractional vector calculus, we give lattice models with long-range interactions for the fractional Maxwell equations of non-local continuous media and for the fractional generalization of the Mindlin and Aifantis continuum models of gradient elasticity.
On flipping the classroom in large first year calculus courses
Jungić, Veselin; Kaur, Harpreet; Mulholland, Jamie; Xin, Cindy
2015-05-01
Over the course of two years, 2012--2014, we have implemented a 'flipping' the classroom approach in three of our large enrolment first year calculus courses: differential and integral calculus for scientists and engineers. In this article we describe the details of our particular approach and share with the reader some experiences of both instructors and students.
On Flipping the Classroom in Large First Year Calculus Courses
Jungic, Veselin; Kaur, Harpreet; Mulholland, Jamie; Xin, Cindy
2015-01-01
Over the course of two years, 2012-2014, we have implemented a "flipping" the classroom approach in three of our large enrolment first year calculus courses: differential and integral calculus for scientists and engineers. In this article we describe the details of our particular approach and share with the reader some experiences of…
Modelling the Landing of a Plane in a Calculus Lab
Morante, Antonio; Vallejo, Jose A.
2012-01-01
We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics. (Contains 5 figures and 1 note.)
Initialized Fractional Calculus
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.
Sauerheber, Richard D.
2012-01-01
Methods of teaching the Calculus are presented in honour of Sir Isaac Newton, by discussing an extension of his original proofs and discoveries. The methods, requested by Newton to be used that reflect the historical sequence of the discovered Fundamental Theorems, allow first-time students to grasp quickly the basics of the Calculus from its…
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...
Problems in Hydrodynamics and Partial Differential Equations.
CALCULUS OF VARIATIONS, RESEARCH MANAGEMENT ), NONLINEAR DIFFERENTIAL EQUATIONS, PARTIAL DIFFERENTIAL EQUATIONS, BOUNDARY VALUE PROBLEMS, EQUATIONS, INEQUALITIES, MEASURE THEORY , INTEGRALS, ABSTRACTS
Introduction to Integral Calculus Systematic Studies with Engineering Applications for Beginners
Rohde, Ulrich L; Poddar, Ajay K; Ghosh, A K
2011-01-01
An accessible introduction to the fundamentals of calculus needed to solve current problems in engineering and the physical sciences I ntegration is an important function of calculus, and Introduction to Integral Calculus combines fundamental concepts with scientific problems to develop intuition and skills for solving mathematical problems related to engineering and the physical sciences. The authors provide a solid introduction to integral calculus and feature applications of integration, solutions of differential equations, and evaluation methods. With logical organization coupled with cle
Renal vein thrombosis mimicking urinary calculus: a dilemma of diagnosis.
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.
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...... with the outputs and at the same time the inputs impose constraints on the waiting times. Consequently, the expected inputs may not be available when needed and therefore the calculus allows to express the absence of data.The communication delays are expressed by general distributions and the resulting semantics...
Modern calculus and analytic geometry
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
Spinor calculus on 5-dimensional spacetimes
Energy Technology Data Exchange (ETDEWEB)
Gomez-Lobo, Alfonso Garcia-Parrado [Centro de Matematica, Universidade do Minho. 4710-057 Braga (Portugal); Martin-Garcia, Jose M, E-mail: alfonso@math.uminho.p, E-mail: jose.martin-garcia@obspm.f [Laboratoire Univers et Theories, Observatoire de Paris, CNRS, University Paris Diderot, 5 place Jules Janssen, 92190 Meudon (France)
2010-05-01
We explain how Penrose's spinor calculus of 4-dimensional Lorentzian geometry is implemented in a 5-dimensional Lorentzian manifold. A number of issues, such as the essential spin algebra, the spin covariant derivative and the algebro-differential properties of the curvature spinors are discussed.
Symbolic boundary calculus and feedback operators
DEFF Research Database (Denmark)
Pedersen, Michael
1991-01-01
The recent developments in microlocal analysis and pseudodifferential boundary calculus are well suited tools in the investigation of a large number of problems occurring in control theory for partial differential equations. We explain some of the basic ideas of a pseudodifferential model...
Mathematics for engineers III vector calculus
Baumann, Gerd
2011-01-01
This book is part of a four-volume textbook on Engineering Mathematics for undergraduates. Volume III treats vector calculus and differential equations of higher order. The text uses Mathematica as a tool to discuss and to solve examples from mathematics. The basic use of this language is demonstrated by examples.
Students' Difficulties with Vector Calculus in Electrodynamics
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…
An introductory calculus-based mechanics investigation
Allen, Bradley
2017-11-01
One challenge for the introductory physics teacher is incorporating calculus techniques into the laboratory setting. It can be difficult to strike a balance between presenting an experimental task for which calculus is essential and making the mathematics accessible to learners who may be apprehensive about applying it. One-dimensional kinematics is a common way to do so and has been discussed in this publication. This article outlines a two-dimensional kinematics experiment that is straightforward for students to perform, but for which the ideas of differentiation and maximizing are necessary to justify the results theoretically. It also leads to a satisfying result that ties into a derivation which students may recognize.
Multivariable dynamic calculus on time scales
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.
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.
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....
Encoding the Factorisation Calculus
Directory of Open Access Journals (Sweden)
Reuben N. S. Rowe
2015-08-01
Full Text Available Jay and Given-Wilson have recently introduced the Factorisation (or SF- calculus as a minimal fundamental model of intensional computation. It is a combinatory calculus containing a special combinator, F, which is able to examine the internal structure of its first argument. The calculus is significant in that as well as being combinatorially complete it also exhibits the property of structural completeness, i.e. it is able to represent any function on terms definable using pattern matching on arbitrary normal forms. In particular, it admits a term that can decide the structural equality of any two arbitrary normal forms. Since SF-calculus is combinatorially complete, it is clearly at least as powerful as the more familiar and paradigmatic Turing-powerful computational models of Lambda Calculus and Combinatory Logic. Its relationship to these models in the converse direction is less obvious, however. Jay and Given-Wilson have suggested that SF-calculus is strictly more powerful than the aforementioned models, but a detailed study of the connections between these models is yet to be undertaken. This paper begins to bridge that gap by presenting a faithful encoding of the Factorisation Calculus into the Lambda Calculus preserving both reduction and strong normalisation. The existence of such an encoding is a new result. It also suggests that there is, in some sense, an equivalence between the former model and the latter. We discuss to what extent our result constitutes an equivalence by considering it in the context of some previously defined frameworks for comparing computational power and expressiveness.
White noise calculus and Fock space
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.
A MATLAB companion for multivariable calculus
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...
Convex functions and some inequalities in terms of the Non-Newtonian Calculus
Unluyol, Erdal; Salas, Seren; Iscan, Imdat
2017-04-01
Differentiation and integration are basic operations of calculus and analysis. Indeed, they are many versions of the subtraction and addition operations on numbers, respectively. From 1967 till 1970 Michael Grossman and Robert Katz [1] gave definitions of a new kind of derivative and integral, converting the roles of subtraction and addition into division and multiplication, and thus establish a new calculus, called Non-Newtonian Calculus. So, in this paper, it is investigated to the convex functions and some inequalities in terms of Non-Newtonian Calculus. Then we compare with the Newtonian and Non-Newtonian Calculus.
Calculus Technique of Integration by Parts, Correlated with a Geometric Picture
Fromhold, Albert T., Jr.
2005-01-01
The method of integration by parts is one of the most useful in integral calculus. Among the most important applications is the integration of differentials involving products, differentials in involving logarithms, and differentials involving inverse circular functions.
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.
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.
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...
Fractional calculus in bioengineering, part 3.
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
Fractional and multivariable calculus model building and optimization problems
Mathai, A M
2017-01-01
This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models. Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations. The first chapter begins with an introduction to fractional calculus moving on to discuss fractional integrals, fractional derivatives, fractional differential equations and their solutions. Multivariable calculus is covered in the second chapter and introduces the fundamentals of multivariable calculus (multivariable functions, limits and continuity, differentiability, directional derivatives and expansions of multivariable ...
Stochastic integration by parts and functional Itô calculus
Vives, Josep
2016-01-01
This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012). The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes. Rama Cont's notes provide an introduction to the Functional Itô Calculus, a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations. This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to pract...
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...... is innovative in many aspects combining various techniques from topology and model theory....
Calculus Courses' Assessment Data
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.…
Kohatsu, Arturo; Miquel, Montero
2003-01-01
This article is an introduction to Malliavin Calculus for practitioners. We treat one specific application to the calculation of greeks in Finance. We consider also the kernel density method to compute greeks and an extension of the Vega index called the local vega index.
An introduction to differential geometry
Willmore, T J
2012-01-01
This text employs vector methods to explore the classical theory of curves and surfaces. Topics include basic theory of tensor algebra, tensor calculus, calculus of differential forms, and elements of Riemannian geometry. 1959 edition.
Fractional calculus in bioengineering, part 2.
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
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...... case studies and it has been extended in several directions. The aim of this paper is to provide a thorough presentation of the logic.......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...
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.
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
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
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.
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...
Fractional vector calculus and fractional Maxwell’s equations
Tarasov, Vasily E.
2008-11-01
The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered.
Polynomial Calculus: Rethinking the Role of Calculus in High Schools
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…
Early Vector Calculus: A Path through Multivariable Calculus
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.)
Miniature endoscopic optical coherence tomography for calculus detection.
Kao, Meng-Chun; Lin, Chun-Li; Kung, Che-Yen; Huang, Yi-Fung; Kuo, Wen-Chuan
2015-08-20
The effective treatment of periodontitis involves the detection and removal of subgingival dental calculus. However, subgingival calculus is more difficult to detect than supragingival calculus because it is firmly attached to root surfaces within periodontal pockets. To achieve a smooth root surface, clinicians often remove excessive amounts of root structure because of decreased visibility. In addition, enamel pearl, a rare type of ectopic enamel formation on the root surface, can easily be confused with dental calculus in the subgingival environment. In this study, we developed a fiber-probe swept-source optical coherence tomography (SSOCT) technique and combined it with the quantitative measurement of an optical parameter [standard deviation (SD) of the optical coherence tomography (OCT) intensity] to differentiate subgingival calculus from sound enamel, including enamel pearl. Two-dimensional circumferential images were constructed by rotating the miniprobe (0.9 mm diameter) while acquiring image lines, and the adjacent lines in each rotation were stacked to generate a three-dimensional volume. In OCT images, compared to sound enamel and enamel pearls, dental calculus showed significant differences (Pcalculus.
Applications of infinitary lambda calculus
Barendregt, H.; Klop, J.W.
2009-01-01
We present an introduction to infinitary lambda calculus, highlighting its main properties. Subsequently we give three applications of infinitary lambda calculus. The first addresses the non-definability of Surjective Pairing, which was shown by the first author not to be definable in lambda
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…
Calculus in the Middle School?
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…
Stoker, J J
2011-01-01
This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis.
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......C as values and data types within PCoC. The new feature of PCoC is that results of the representation of PCoC can be lifted to PCoC itself. The lifting is fully formalized in PCoC, and the logic therefore supports reflection....
Advanced calculus problem solver
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
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
Energy Technology Data Exchange (ETDEWEB)
He, Ji-Huan, E-mail: hejihuan@suda.edu.cn [National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, 199 Ren-ai Road, Suzhou 215123 (China); Elagan, S.K., E-mail: sayed_khalil2000@yahoo.com [Mathematics and Statistics Department, Faculty of Science, Taif University, P.O. 888 (Saudi Arabia); Department of Mathematics, Faculty of Science, Menofiya University, Shebin Elkom (Egypt); Li, Z.B., E-mail: zhengbiaoli@l26.com [College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011 (China)
2012-01-09
The fractional complex transform is suggested to convert a fractional differential equation with Jumarie's modification of Riemann–Liouville derivative into its classical differential partner. Understanding the fractional complex transform and the chain rule for fractional calculus are elucidated geometrically. -- Highlights: ► The chain rule for fractional calculus is invalid, a counter example is given. ► The fractional complex transform is explained geometrically. ► Fractional equations can be converted into differential equations.
Stochastic calculus an introduction through theory and exercises
Baldi, Paolo
2017-01-01
This book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications. It is the only textbook on the subject to include more than two hundred exercises with complete solutions. After explaining the basic elements of probability, the author introduces more advanced topics such as Brownian motion, martingales and Markov processes. The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance. The final chapter provides detailed solutions to all exercises, in some cases presenting various solution techniques together with a discussion of advantages and drawbacks of the methods used. Stochastic Calculus will be particularly useful to advanced undergraduate and graduate students wishing to acquire a solid understanding of the subject through the theory and exercises. Including full mathematical ...
Equations involving Malliavin calculus operators applications and numerical approximation
Levajković, Tijana
2017-01-01
This book provides a comprehensive and unified introduction to stochastic differential equations and related optimal control problems. The material is new and the presentation is reader-friendly. A major contribution of the book is the development of generalized Malliavin calculus in the framework of white noise analysis, based on chaos expansion representation of stochastic processes and its application for solving several classes of stochastic differential equations with singular data involving the main operators of Malliavin calculus. In addition, applications in optimal control and numerical approximations are discussed. The book is divided into four chapters. The first, entitled White Noise Analysis and Chaos Expansions, includes notation and provides the reader with the theoretical background needed to understand the subsequent chapters. In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator, the Skorokhod integral and the Ornstein-Uhlenbeck operator are introdu...
Leveraging Prior Calculus Study with Embedded Review
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…
Colloquium: Fractional calculus view of complexity: A tutorial
West, Bruce J.
2014-10-01
The fractional calculus has been part of the mathematics and science literature for 310 years. However, it is only in the past decade or so that it has drawn the attention of mainstream science as a way to describe the dynamics of complex phenomena with long-term memory, spatial heterogeneity, along with nonstationary and nonergodic statistics. The most recent application encompasses complex networks, which require new ways of thinking about the world. Part of the new cognition is provided by the fractional calculus description of temporal and topological complexity. Consequently, this Colloquium is not so much a tutorial on the mathematics of the fractional calculus as it is an exploration of how complex phenomena in the physical, social, and life sciences that have eluded traditional mathematical modeling become less mysterious when certain historical assumptions such as differentiability are discarded and the ordinary calculus is replaced with the fractional calculus. Exemplars considered include the fractional differential equations describing the dynamics of viscoelastic materials, turbulence, foraging, and phase transitions in complex social networks.
Fractional complex transform for fractional differential equations
National Research Council Canada - National Science Library
Lİ, Zheng Biao; HE, Ji Huan
2010-01-01
Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily...
A generalized nonlocal vector calculus
Alali, Bacim; Liu, Kuo; Gunzburger, Max
2015-10-01
A nonlocal vector calculus was introduced in Du et al. (Math Model Meth Appl Sci 23:493-540, 2013) that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A formulation is developed that provides a more general setting for the nonlocal vector calculus that is independent of particular nonlocal models. It is shown that general nonlocal calculus operators are integral operators with specific integral kernels. General nonlocal calculus properties are developed, including nonlocal integration by parts formula and Green's identities. The nonlocal vector calculus introduced in Du et al. (Math Model Meth Appl Sci 23:493-540, 2013) is shown to be recoverable from the general formulation as a special example. This special nonlocal vector calculus is used to reformulate the peridynamics equation of motion in terms of the nonlocal gradient operator and its adjoint. A new example of nonlocal vector calculus operators is introduced, which shows the potential use of the general formulation for general nonlocal models.
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
The calculus a genetic approach
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
Mathematics for physics with calculus
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.
Advanced calculus a transition to analysis
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
Classroom Integral Calculus: Some Useful Digressions When Teaching Integral Calculus.
Acosta, Daniel J.; Wills, Randall
2002-01-01
Outlines ideas and exercises for two topics that accompany the standard treatment of integral calculus. Emphasizes intuition to help facilitate student comprehension of the definite integral as a limit of Riemann Sums. (Author/MM)
Stochastic calculus and applications
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...
Geometric Error Analysis in Applied Calculus Problem Solving
Usman, Ahmed Ibrahim
2017-01-01
The paper investigates geometric errors students made as they tried to use their basic geometric knowledge in the solution of the Applied Calculus Optimization Problem (ACOP). Inaccuracies related to the drawing of geometric diagrams (visualization skills) and those associated with the application of basic differentiation concepts into ACOP…
vesical gossypiboma mimicking calculus: a report of two
African Journals Online (AJOL)
should be considered in the differential diagnosis of patients with persistent bladder symptoms and intravesical calcification following open prostatectomy. Key Words: gossypiboma, urinary bladder, calculus. INTRODUCTION. Gossypiboma is an iatrogenic mass lesion resulting from retention of a surgical sponge. The word ...
The history of the calculus and its conceptual development
Boyer, Carl B
1959-01-01
Fluent description of the development of both the integral and differential calculus. Early beginnings in antiquity, medieval contributions, and a century of anticipation lead up to a consideration of Newton and Leibniz, the period of indecison that followed them, and the final rigorous formulation that we know today.
Making Implicit Multivariable Calculus Representations Explicit: A Clinical Study
McGee, Daniel; Moore-Russo, Deborah; Martinez-Planell, Rafael
2015-01-01
Reviewing numerous textbooks, we found that in both differential and integral calculus textbooks the authors commonly assume that: (i) students can generalize associations between representations in two dimensions to associations between representations of the same mathematical concept in three dimensions on their own; and (ii) explicit…
New trends in nanotechnology and fractional calculus applications
Baleanu, Dumitru; Machado, JA Tenreiro
2010-01-01
In recent years, fractional calculus has played a major role in various fields such as mechanics, electricity, biology and economics. This book presents the state-of-the-art in the study of fractional systems and the application of fractional differentiation.
Smooth vectors and Weyl-Pedersen calculus for representations of nilpotent Lie groups
Beltita, Ingrid; Beltita, Daniel
2009-01-01
We present some recent results on smooth vectors for unitary irreducible representations of nilpotent Lie groups. Applications to the Weyl-Pedersen calculus of pseudo-differential operators with symbols on the coadjoint orbits are also discussed.
Cartooning in Algebra and Calculus
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.
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...
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....
Catwalk: First-Semester Calculus.
Speiser, Bob; Walter, Chuck
1994-01-01
Describes the use of time-lapse photographs of a running cat as a model to investigate the concepts of function and derivative in a college calculus course. Discusses student difficulties and implications for teachers. (MKR)
Neutrosophic Precalculus and Neutrosophic Calculus
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...
Generalized calculus with applications to matter and forces
Campos, L M B C
2014-01-01
Combining mathematical theory, physical principles, and engineering problems, Generalized Calculus with Applications to Matter and Forces examines generalized functions, including the Heaviside unit jump and the Dirac unit impulse and its derivatives of all orders, in one and several dimensions. The text introduces the two main approaches to generalized functions: (1) as a nonuniform limit of a family of ordinary functions, and (2) as a functional over a set of test functions from which properties are inherited. The second approach is developed more extensively to encompass multidimensional generalized functions whose arguments are ordinary functions of several variables. As part of a series of books for engineers and scientists exploring advanced mathematics, Generalized Calculus with Applications to Matter and Forces presents generalized functions from an applied point of view, tackling problem classes such as: •Gauss and Stokes’ theorems in the differential geometry, tensor calculus, and theory of ...
Introduction to stochastic analysis and Malliavin calculus
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...
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....
An AP Calculus Classroom Amusement Park
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.…
Ozturk, Okkes; Yilmazer, Resat
2017-07-01
One of the most popular research interests of science and engineering is the fractional calculus theory in recent times. Discrete fractional calculus (DFC) has also an important position in the fractional calculus. The nabla operator in DFC is practical for the singular differential equations. The purpose of this study is to obtain particular solutions of the radial Schrödinger equation (that is, the most important equation of quantum physics) via nabla DFC operator. These solutions were obtained in the forms of discrete fractional.
Elementary calculus an infinitesimal approach
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
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...
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.
Unusual Giant Prostatic Urethral Calculus
African Journals Online (AJOL)
2010-06-29
Jun 29, 2010 ... They are typically asymptomatic and may be associated with benign prostatic hyperplasia, and prostatic cancer.[1] Primary prostatic urethral calculi are usually associated with urethral strictures, posterior urethral valve or diverticula. Acute urinary retention might result secondary to a large urethral calculus.
Calculus Student Understanding of Continuity
Wangle, Jayleen Lillian
2013-01-01
Continuity is a central concept in calculus. Yet very few students seem to understand the nature of continuity. The research described was conducted in two stages. Students were asked questions in multiple choice and true/false format regarding function, limit and continuity. These results were used to identify participants as strong, weak or…
Unusual Giant Prostatic Urethral Calculus
African Journals Online (AJOL)
2010-06-29
Jun 29, 2010 ... associated with benign prostatic hyperplasia, and prostatic cancer.[1] Primary prostatic urethral calculi are ... Giant vesico-prostatic urethral calculus is uncommon. Urethral stones rarely form primarily in the urethra, ... Prostatic calculi associated with hypertrophy of the gland. Group III. Prostatic calculi that ...
Advanced calculus of several variables
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.
Mathematical Features of the Calculus
Sauerheber, Richard D.
2010-01-01
The fundamental theorems of the calculus describe the relationships between derivatives and integrals of functions. The value of any function at a particular location is the definite derivative of its integral and the definite integral of its derivative. Thus, any value is the magnitude of the slope of the tangent of its integral at that position,…
Reading the World with Calculus
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…
Portfolio Analysis for Vector Calculus
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…
Constructivized Calculus in College Mathematics
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…
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...
POGIL in the Calculus Classroom
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…
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.
Calculus in physics classes at UFRGS: an exploratory study
Directory of Open Access Journals (Sweden)
Maria Cecilia Pereira Santarosa
2011-11-01
Full Text Available This study is part f a larger one whose general objective is to investigate and to develop a new strategy for teaching Differential and Integral Calculus I, specifically for physics majors, through a possible integration with the teaching of General and Experimental Physics I. With the specific objective of identifying physics problem-situations that may help in making sense of the mathematical concepts used in Calculus I, and languages and notations that might be used in the teaching of Calculus to favor physics learning, it was investigates, through an ethnographic study, the may mathematics is transposed to classes of General and Experimental Physics I, in classes of physics courses at the Federal University of Rio Grande do Sul (UFRGS. Some findings of this study confirmed those reported in the literature regarding the teaching and learning process in introductory college physics courses. These findings will subsidize the preparation of potentially meaningful instructional materials that will be used in a second stage of the research designed to investigate the learning of declarative and procedural knowledge in basic college physics under an approach that integrates problem-situation in physics and calculus mathematical concepts.
R-Function Relationships for Application in the Fractional Calculus
Lorenzo, Carl F.; Hartley, Tom T.
2000-01-01
The F-function, and its generalization the R-function, are of fundamental importance in the fractional calculus. It has been shown that the solution of the fundamental linear fractional differential equation may be expressed in terms of these functions. These functions serve as generalizations of the exponential function in the solution of fractional differential equations. Because of this central role in the fractional calculus, this paper explores various intrarelationships of the R-function, which will be useful in further analysis. Relationships of the R-function to the common exponential function, e(t), and its fractional derivatives are shown. From the relationships developed, some important approximations are observed. Further, the inverse relationships of the exponential function, el, in terms of the R-function are developed. Also, some approximations for the R-function are developed.
Guggenheimer, Heinrich W
1977-01-01
This is a text of local differential geometry considered as an application of advanced calculus and linear algebra. The discussion is designed for advanced undergraduate or beginning graduate study, and presumes of readers only a fair knowledge of matrix algebra and of advanced calculus of functions of several real variables. The author, who is a Professor of Mathematics at the Polytechnic Institute of New York, begins with a discussion of plane geometry and then treats the local theory of Lie groups and transformation groups, solid differential geometry, and Riemannian geometry, leading to a
Real quaternionic calculus handbook
Morais, João Pedro; Sprößig, Wolfgang
2014-01-01
Real quaternion analysis is a multi-faceted subject. Created to describe phenomena in special relativity, electrodynamics, spin etc., it has developed into a body of material that interacts with many branches of mathematics, such as complex analysis, harmonic analysis, differential geometry, and differential equations. It is also a ubiquitous factor in the description and elucidation of problems in mathematical physics. In the meantime real quaternion analysis has become a well established branch in mathematics and has been greatly successful in many different directions. This book is based on concrete examples and exercises rather than general theorems, thus making it suitable for an introductory one- or two-semester undergraduate course on some of the major aspects of real quaternion analysis in exercises. Alternatively, it may be used for beginning graduate level courses and as a reference work. With exercises at the end of each chapter and its straightforward writing style the book addresses readers who ...
Factors Associated with Success in College Calculus II
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…
A Polyadic pi-Calculus Approach for the Formal Specification of UML-RT
Directory of Open Access Journals (Sweden)
J. M. Bezerra
2009-01-01
Full Text Available UML-RT is a UML real-time profile that allows modeling event-driven and distributed systems; however it is not a formal specification language. This paper proposes a formal approach for UML-RT through a mapping of the UML-RT communicating elements into the -calculus (or pi-calculus process algebra. The formal approach both captures the intended behavior of the system being modeled and provides a rigorous and nonambiguous system description. Our proposal differentiates from other research work because we map UML-RT to -calculus, and we allow the mapping of dynamic reconfiguration of UML-RT unwired ports. We illustrate the usage and applicability of the mapping through three examples. The first example focuses on explaining the mapping; the second one aims to demonstrate the use of the -calculus definitions to verify system requirements; the third case is an example of mobile processes called Handover protocol.
Measurable Stochastics for Brane Calculus
Directory of Open Access Journals (Sweden)
Giorgio Bacci
2010-10-01
Full Text Available We give a stochastic extension of the Brane Calculus, along the lines of recent work by Cardelli and Mardare. In this presentation, the semantics of a Brane process is a measure of the stochastic distribution of possible derivations. To this end, we first introduce a labelled transition system for Brane Calculus, proving its adequacy w.r.t. the usual reduction semantics. Then, brane systems are presented as Markov processes over the measurable space generated by terms up-to syntactic congruence, and where the measures are indexed by the actions of this new LTS. Finally, we provide a SOS presentation of this stochastic semantics, which is compositional and syntax-driven.
Discrete calculus methods for counting
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 ...
Reductionism and the Universal Calculus
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...
Schwalm, W. A.; Schwalm, M. K.; Giona, M.
1998-03-01
Space is filled with triangulating graph \\calG to serve as a quadrature grid. A discrete analog of the theory of differential forms is constructed using the associated simplical complex. The role of a basis for Λ^p at a point is played by the set of (p+1) -simplices containing a given vertex. Vector difference operations analogous to div, grad and curl, together with corresponding vector identities and exact difference analogs of the Stokes-type theorems, are obtained in terms of the boundary partial and coboundary d. Difference versions of the full vector Maxwell electromagnetic equations are analyzed on a random structure.
Space complexity in polynomial calculus
Czech Academy of Sciences Publication Activity Database
Filmus, Y.; Lauria, M.; Nordström, J.; Ron-Zewi, N.; Thapen, Neil
2015-01-01
Roč. 44, č. 4 (2015), s. 1119-1153 ISSN 0097-5397 R&D Projects: GA AV ČR IAA100190902; GA ČR GBP202/12/G061 Institutional support: RVO:67985840 Keywords : proof complexity * polynomial calculus * lower bounds Subject RIV: BA - General Mathematics Impact factor: 0.841, year: 2015 http://epubs.siam.org/doi/10.1137/120895950
Malliavin Calculus applied to finance
Montero Torralbo, Miquel; Kohatsu-Higa, Arturo
2003-01-01
In this article, we give a brief informal introduction to Malliavin Calculus for newcomers. We apply these ideas to the simulation of Greeks in Finance. First to European-type options where formulas can be computed explicitly and therefore can serve as testing ground. Later, we study the case of Asian options where close formulas are not available, and we also open the view for including more exotic derivatives. The Greeks are computed through Monte Carlo simulation.
Calculus anuria and its remedy.
Amanullah; Khan, Ghayyur; Lal, Shanti; Soomro, Muhammad Iqbal; Jalbani, Malik Hussain
2010-01-01
Urinary lithiases have been a major urological problem. The objective was to determine the out come of treatment for patients with calculus anuria. A descriptive study was conducted at Department of Urology Chandka Medical College Hospital, Larkana from March 2007 to April 2009. All patients with a diagnosis of calculus anuria of all ages and either sex were included in the study. Detailed history, physical examination and examination of genitourinary tract was performed. Investigations included complete blood examination, blood urea, serum creatinine, ultrasonography of KUB area, and X-Ray KUB. Among the 66 patients the cause of anuria was bilateral obstruction by the calculi in 46 cases, unilateral obstruction with small/absent/nephrectomised contralateral kidney in 20 cases. In most of the cases, ureteric catheterisation was done to relieve the obstruction. Five deaths were observed, despite emergency urinary diversion and appropriate treatment. Calculus anuria is a urological emergency. Prompt and early intervention can save the life of patient and prevent to develop chronic renal failure.
Malliavin Calculus in Lévy spaces and Applications to Finance.
Petrou, Evangelia
2008-01-01
The main goal of this paper is to generalize the results of Fournie et al. [7] for markets generated by Lévy processes. For this reason we extend the theory of Malliavin calculus to provide the tools that are necessary for the calculation of the sensitivities, such as differentiability results for the solution of a stochastic differential equation.
Time scales: from Nabla calculus to Delta calculus and vice versa via duality
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.
Pre-calculus workbook for dummies
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
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...
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....
Fractional Calculus in Hydrologic Modeling: A Numerical Perspective
Energy Technology Data Exchange (ETDEWEB)
David A. Benson; Mark M. Meerschaert; Jordan Revielle
2012-01-01
Fractional derivatives can be viewed either as a handy extension of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Levy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Levy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.
Hochstadt, Harry
2012-01-01
Modern approach to differential equations presents subject in terms of ideas and concepts rather than special cases and tricks which traditional courses emphasized. No prerequisites needed other than a good calculus course. Certain concepts from linear algebra used throughout. Problem section at end of each chapter.
Dental Calculus Arrest of Dental Caries.
Keyes, Paul H; Rams, Thomas E
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. 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. 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. These observations further document the potential protective effects of dental calculus mineralization against dental caries.
Generalized Functions for the Fractional Calculus
Lorenzo, Carl F.; Hartley, Tom T.
1999-01-01
Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.
Programming Language Concepts - The Lambda Calculus Approach
Fokkinga, M.M.; Asveld, P.R.J.; Nijholt, Antinus
1987-01-01
The Lambda Calculus is a formal system, originally intended as a tool in the foundation of mathematics, but mainly used to study the concepts of algorithm and effective computability. Recently, the Lambda Calculus and related systems acquire attention from Computer Science for another reason too:
Algorithms, The λ Calculus and Programming
Indian Academy of Sciences (India)
IAS Admin
I develop the intuition behind the ¸ calculus and connect it to computer programming through some basic examples. This fills an often felt gap that teachers and students find between the for- mal structure of the ¸ calculus, and the principles and practice of functional programming. Fortran, Pascal, C and C++ are some of the ...
Imagine Yourself in This Calculus Classroom
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…
Improving student learning in calculus through applications
Young, C. Y.; Georgiopoulos, M.; Hagen, S. C.; Geiger, C. L.; Dagley-Falls, M. A.; Islas, A. L.; Ramsey, P. J.; Lancey, P. M.; Straney, R. A.; Forde, D. S.; Bradbury, E. E.
2011-07-01
Nationally only 40% of the incoming freshmen Science, Technology, Engineering and Mathematics (STEM) majors are successful in earning a STEM degree. The University of Central Florida (UCF) EXCEL programme is a National Science Foundation funded STEM Talent Expansion Programme whose goal is to increase the number of UCF STEM graduates. One of the key requirements for STEM majors is a strong foundation in Calculus. To improve student learning in calculus, the EXCEL programme developed two special courses at the freshman level called Applications of Calculus I (Apps I) and Applications of Calculus II (Apps II). Apps I and II are one-credit classes that are co-requisites for Calculus I and II. These classes are teams taught by science and engineering professors whose goal is to demonstrate to students where the calculus topics they are learning appear in upper level science and engineering classes as well as how faculty use calculus in their STEM research programmes. This article outlines the process used in producing the educational materials for the Apps I and II courses, and it also discusses the assessment results pertaining to this specific EXCEL activity. Pre- and post-tests conducted with experimental and control groups indicate significant improvement in student learning in Calculus II as a direct result of the application courses.
RAMAN-SPECTRA OF HUMAN DENTAL CALCULUS
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
Educating about Sustainability while Enhancing Calculus
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…
An Introductory Calculus-Based Mechanics Investigation
Allen, Bradley
2017-01-01
One challenge for the introductory physics teacher is incorporating calculus techniques into the laboratory setting. It can be difficult to strike a balance between presenting an experimental task for which calculus is essential and making the mathematics accessible to learners who may be apprehensive about applying it. One-dimensional kinematics…
Calculus and Success in a Business School
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…
A Cross-National Study of Calculus
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…
Laparoscopic Cholecystectomy in Chronic Calculus Cholecystitis
Directory of Open Access Journals (Sweden)
Prakash Sapkota
2013-12-01
Full Text Available Introduction: Laparoscopic cholecystectomy has clearly become the choice over open cholecystectomy in the treatment of hepatobiliary disease since its introduction by Mouret in 1987. This study evaluates a series of patients with chronic calculus cholecystitis who were treated with laparoscopic and open cholecystectomy and assesses the outcomes of both techniques. Objective: To evaluate the efficacy of laparoscopic vs open cholecystectomy in chronic calculus cholecystitis and establish the out-comes of this treatment modality at Lumbini Medical College and Teaching Hospital. Methods: This was a retrospective analysis over a one-year period (January 1, 2012 to December 31, 2012, per-formed by single surgeon at Lumbini Medical College and Teaching Hospital located midwest of Nepal. 166 patients underwent surgical treatment for chronic calculus cholecystitis. Patients included were only chronic calculus cholecystitis proven histopathologocally and the rest were excluded. Data was collected which included patients demographics, medical history, presentation, complications, conversion rates from laparoscopic. cholecystectomy to open cholecystectomy, operative and postoperative time. Results: Patients treated with laparoscopic cholecystectomy for chronic calculus cholecystitis had shorter operating times and length of stay compared to patients treated with open cholecystectomy for chronic calculus cholecystitis. Conversion rates were 3.54% in chronic calculus cholecystitis during the study period. Complications were also lower in patients who underwent laparoscopic cholecystectomy versus open cholecystectomy for cholelithiasis. Conclusions: Laparoscopic cholecystectomy appears to be a reliable, safe, and cost-effective treatment modality for chronic calculus cholecystitis.
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...
Thematization of the Calculus Graphing Schema
Cooley, Laurel; Baker, Bernadette; Trigueros, Maria
2003-01-01
This article is the result of an investigation of students' conceptualizations of calculus graphing techniques after they had completed at least two semesters of calculus. The work and responses of 27 students to a series of questions that solicit information about the graphical implications of the first derivative, second derivative, continuity,…
Anti-calculus and whitening toothpastes
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
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...
Technical calculus with analytic geometry
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
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
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
Bridging the Vector Calculus Gap
Dray, Tevian; Manogue, Corinne
2003-05-01
As with Britain and America, mathematicians and physicists are separated from each other by a common language. In a nutshell, mathematics is about functions, but physics is about things. For the last several years, we have led an NSF-supported effort to "bridge the vector calculus gap" between mathematics and physics. The unifying theme we have discovered is to emphasize geometric reasoning, not (just) algebraic computation. In this talk, we will illustrate the language differences between mathematicians and physicists, and how we are trying reconcile them in the classroom. For further information about the project go to: http://www.physics.orst.edu/bridge
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...
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...
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.
Using Dynamic Software to Address Common College Calculus Stumbling Blocks
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…
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…
The Impact of Taking a College Pre-Calculus Course on Students' College Calculus Performance
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…
Pre-calculus workbook for dummies
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
AP calculus AB & BC crash course
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
On Some Syntactic Properties of the Modalized Heyting Calculus
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.
Vector difference calculus for physical lattice models
Schwalm, W.; Moritz, B.; Giona, M.; Schwalm, M.
1999-01-01
A vector difference calculus is developed for physical models defined on a general triangulating graph G, which may be a regular or an extremely irregular lattice, using discrete field quantities roughly analogous to differential forms. The role of the space Λp of p-forms at a point is taken on by the linear space generated at a graph vertex by the geometrical p-simplices which contain it. The vector operations divergence, gradient, and curl are developed using the boundary ∂ and coboundary d. Dot, cross, and scalar products are defined in such a way that discrete analogs of the vector integral theorems, including theorems of Gauss-Ostrogradski, Stokes, and Green, as well as most standard vector identities hold exactly, not as approximations to a continuum limit. Physical conservation laws for the models become theorems satisfied by the discrete fields themselves. Three discrete lattice models are constructed as examples, namely a discrete version of the Maxwell equations, the Navier-Stokes equation for incompressible flow, and the Navier linearized model for a homogeneous, isotropic elastic medium. Weight factors needed for obtaining quantitative agreement with continuum calculations are derived for the special case of a regular triangular lattice. Green functions are developed using a generalized Helmholtz decomposition of the fields.
Numerical Convergence of Discrete Exterior Calculus on Arbitrary Surface Meshes
Mohamed, Mamdouh S.
2018-02-13
Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially on curved surfaces. This paper presents numerical evidences demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.
Fractional-calculus diffusion equation.
Ajlouni, Abdul-Wali Ms; Al-Rabai'ah, Hussam A
2010-05-21
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. 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 system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis.
Fractional-calculus diffusion equation
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 system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. Conclusions The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. PMID:20492677
Tensor calculus with open-source software: the SageManifolds project
Gourgoulhon, Eric; Mancini, Marco
2014-01-01
The SageManifolds project aims at extending the mathematics software system Sage towards differential geometry and tensor calculus. As Sage itself, it is free, open-source and is based on the Python programming language. We discuss here some details of the implementation, which relies on Sage's category pattern, and present a concrete example of use.
Fractional calculus approach to study temperature distribution within a spinning satellite
Directory of Open Access Journals (Sweden)
Jyotindra C. Prajapati
2016-09-01
Full Text Available This paper deals with the temperature distribution within spinning satellites and problem is formulated in terms of fractional differential equation. Applying fractional calculus approach, solution of this equation is obtained in terms of Wright generalized hypergeometric function, a generalization of exponential function.
A Case Study of Student and Instructor Reactions to a Calculus E-Book
Bode, Martina; Khorami, Mehdi; Visscher, Daniel
2014-01-01
This article details the results of testing an e-book in two differential calculus classes. Although we, as math instructors, were drawn to the components of the e-book that promote conceptual understanding--such as the interactive figures--the students reported liking the assessment support most. We found that students were initially excited…
What Do Croatian Pre-Service Teachers Remember from Their Calculus Course?
Jukic, Ljerka; Brückler, Franka Miriam
2014-01-01
This paper reports a study on retention of core concepts in differential and integral calculus by examining the knowledge of two pre-service mathematics students. The study is conducted using a mixed method approach and the obtained data were analyzed using theory of three worlds of mathematics. The results showed that having good understanding of…
The calculus lifesaver all the tools you need to excel at calculus
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
The origins of Cauchy's rigorous calculus
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.
Generalized Laplace Transforms and Extended Heaviside Calculus
Deakin, Michael A. B.
2008-01-01
An extended Heaviside calculus proposed by Peraire in a recent paper is similar to a generalization of the Laplace transform proposed by the present author. This similarity will be illustrated by analysis of an example supplied by Peraire.
Introductory analysis a deeper view of calculus
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
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.
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...
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.
Borel functional calculus for quaternionic normal operators
Ramesh, G.; Santhosh Kumar, P.
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.
Application of exterior calculus to waveguides
Ferraro, Rafael
2009-01-01
Exterior calculus is a powerful tool to search for solutions to the electromagnetic field equations, whose strength can be better appreciated when applied to work out non-trivial configurations. Here we show how to exploit this machinery to obtain the electromagnetic TM and TE modes in hollow cylindrical waveguides. The proper use of exterior calculus and Lorentz boosts will straightforwardly lead to such solutions and the respective power transmitted along the waveguide.
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.
Endoscopic vs. tactile evaluation of subgingival calculus.
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.
On certain fractional calculus operators involving generalized Mittag-Leffler function
Directory of Open Access Journals (Sweden)
Dinesh Kumar
2016-06-01
Full Text Available The object of this paper is to establish certain generalized fractional integration and differentiation involving generalized Mittag-Leffler function defined by Salim and Faraj [25]. The considered generalized fractional calculus operators contain the Appell's function $F_3$ [2, p.224] as kernel and are introduced by Saigo and Maeda [23]. The Marichev-Saigo-Maeda fractional calculus operators are the generalization of the Saigo fractional calculus operators. The established results provide extensions of the results given by Gupta and Parihar [3], Saxena and Saigo [30], Samko et al. [26]. On account of the general nature of the generalized Mittag-Leffler function and generalized Wright function, a number of known results can be easily found as special cases of our main results.
Mohamed, Mamdouh S; Samtaney, Ravi
2015-01-01
A conservative discretization of incompressible Navier-Stokes equations on simplicial meshes is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the contraction operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second ord...
Mohamed, Mamdouh S.
2016-02-11
A conservative discretization of incompressible Navier–Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.
Schoenly, Joshua E.; Seka, Wolf; Romanos, Georgios; Rechmann, Peter
A desired outcome of scaling and root planing is the complete removal of calculus and infected root tissue and preservation of healthy cementum for rapid healing of periodontal tissues. Conventional periodontal treatments for calculus removal, such as hand instrument scaling and ultrasonic scaling, often deeply scrape the surface of the underlying hard tissue and may leave behind a smear layer. Pulsed lasers emitting at violet wavelengths (specifically, 380 to 400 nm) are a potential alternative treatment since they can selectively ablate dental calculus without ablating pristine hard tissue (i.e., enamel, cementum, and dentin). In this study, light and scanning electron microscopy are used to compare and contrast the efficacy of in vitro calculus removal for several conventional periodontal treatments (hand instruments, ultrasonic scaler, and Er:YAG laser) to calculus removal with a frequency-doubled Ti:sapphire (λ = 400 nm). After calculus removal, enamel and cementum surfaces are investigated for calculus debris and damage to the underlying hard tissue surface. Compared to the smear layer, grooves, and unintentional hard tissue removal typically found using these conventional treatments, calculus removal using the 400-nm laser is complete and selective without any removal of pristine dental hard tissue. Based on these results, selective ablation from the 400-nm laser appears to produce a root surface that would be more suitable for successful healing of periodontal tissues.
QUANTILE CALCULUS AND CENSORED REGRESSION.
Huang, Yijian
2010-06-01
Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile calculus on cumulative probability scale in this article, upon recognizing that probability and time scales do not always have a one-to-one mapping given a survival distribution. These results give rise to a novel estimation procedure for censored quantile regression, based on estimating integral equations. A numerically reliable and efficient Progressive Localized Minimization (PLMIN) algorithm is proposed for the computation. This procedure reduces exactly to the Kaplan-Meier method in the k-sample problem, and to standard uncensored quantile regression in the absence of censoring. Under regularity conditions, the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a Gaussian process. Simulations show good statistical and algorithmic performance. The proposal is illustrated in the application to a clinical study.
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.
Anti-calculus and whitening toothpastes.
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. Copyright © 2013 S. Karger AG, Basel.
A large primary vaginal calculus in a woman with paraplegia.
Avsar, Ayse Filiz; Keskin, Huseyin Levent; Catma, Tuba; Kaya, Basak; Sivaslioglu, Ahmet Akın
2013-01-01
The study aimed to report a primary vaginal stone, an extremely rare entity, without vesicovaginal fistula in a woman with disability. We describe the case of a large primary vaginal calculus in a 22-year-old woman with paraplegia, which, surprisingly, was not diagnosed until she was examined under general anesthesia during a preparation for laparoscopy for an adnexal mass. The stone had not been identified by physical examination with the patient in a recumbent position or by transabdominal ultrasonography and pelvic tomography during the preoperative preparation. Vaginoscopy was not performed because the vagina was completely filled with the mass. As a result of its size and hard consistency, a right-sided episiotomy was performed and a 136-g stone was removed using ring forceps. A vesicovaginal fistula was excluded. There was no evidence of a foreign body or other nidus on the cut section of the stone, and it was determined to be composed of 100% struvite (ammonium magnesium phosphate). Culture of urine obtained via catheter showed Escherichia coli. After the surgical removal of the calculus without complications, a program of intermittent catheterization was started. The follow-up period was uneventful, and the patient was symptom free at 6 months after the operation. We postulate that the calculus formed as a consequence of urinary contamination of the vagina in association with incontinence and prolonged maintenance in a recumbent posture. This report is important because it highlights that, although vaginal stones are very rare, their possibility should be considered in the differential diagnosis of individuals with long-term paraplegia.
Fuzzy relational calculus theory, applications and software
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...
A BRUTUS Logic for a Spi-Calculus Dialect
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
GIANT CALCULUS IN THE MOUTH OF A PARTIALLY EDENTULOUS
African Journals Online (AJOL)
prevention. Report:- This describes the oral condition of a partially edentulous woman with a giant calculus in the mouth. It highlights the effect of such an enormous calculus in the oral cavity. Conclusi0n:- Such gross calculus formation could have resulted from oral hygiene neglect, facilitated by the anatomical position of.
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.
Improving Calculus II and III through the Redistribution of Topics
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…
Giant Calculus In The Mouth Of Partially Edentulous Woman, (Case ...
African Journals Online (AJOL)
Objective: This case report is to create awareness of the presence of giant calculus in the mouth, the possible causes and its prevention. Report: This describes the oral condition of a partially edentulous woman with a giant calculus in the mouth. It highlights the effect of such an enormous calculus in the oral cavity.
A Transition Course from Advanced Placement to College Calculus
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…
Science 101: How Do We Use Calculus in Science?
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…
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...
Projects for calculus the language of change
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
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...
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....
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...
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...
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...
Calculus of Variations, Classical and Modern
Conti, Roberto
2011-01-01
A. Blaquiere: Quelques aspects geometriques des processus optimaux.- C. Castaing: Quelques problemes de mesurabilite lies a la theorie des commandes.- L. Cesari: Existence theorems for Lagrange and Pontryagin problems of the calculus of variations and optimal control of more-dimensional extensions in Sobolev space.- H. Halkin: Optimal control as programming in infinite dimensional spaces.- C. Olech: The range of integrals of a certain class vector-valued functions.- E. Rothe: Weak topology and calculus of variations.- E.O. Roxin: Problems about the set of attainability.
Relationship between fractional calculus and fractional Fourier transform
Zhang, Yanshan; Zhang, Feng; Lu, Mingfeng
2015-09-01
The fractional calculus (FC) deals with integrals and derivatives of arbitrary (i.e., non-integer) order, and shares its origins with classical integral and differential calculus. The fractional Fourier transform (FRFT), which has been found having many applications in optics and other areas, is a generalization of the usual Fourier transform. The FC and the FRFT are two of the most interesting and useful fractional areas. In recent years, it appears many papers on the FC and FRFT, however, few of them discuss the connection of the two fractional areas. We study their relationship. The relational expression between them is deduced. The expectation of interdisciplinary cross fertilization is our motivation. For example, we can use the properties of the FC (non-locality, etc.) to solve the problem which is difficult to be solved by the FRFT in optical engineering; we can also through the physical meaning of the FRFT optical implementation to explain the physical meaning of the FC. The FC and FRFT approaches can be transposed each other in the two fractional areas. It makes that the success of the fractional methodology is unquestionable with a lot of applications, namely in nonlinear and complex system dynamics and image processing.
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,…
Using MathCAD in General Calculus.
Wieder, Sol
1991-01-01
This article utilizes several examples to describe the use of MathCAD, a computer scratchpad for calculations, within a lecture-demonstration method of teaching general calculus. Both hardware requirements and software features are discussed specifically related to the proper design of student worksheets with emphasis on parameter variation. (JJK)
Enabling quaternion derivatives: the generalized HR calculus
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
Inverted Pedagogy in Second Semester Calculus
Kennedy, Ellie; Beaudrie, Brian; Ernst, Dana C.; St. Laurent, Roy
2015-01-01
This study investigates the effects of applying an inverted classroom model in a second-semester calculus course at a large regional university in the southwest during the Spring of 2013. The sample consisted of four class sections with the same instructor, with a total of 173 students; two class sections were in the experimental group, whereas…
Bladder calculus presenting as excessive masturbation.
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.
Optimal control and the calculus of variations
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.
Enabling quaternion derivatives: the generalized HR calculus.
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.
Boolean integral calculus for digital systems
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.
Are Homeschoolers Prepared for College Calculus?
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…
Is Calculus a Failure in Cryptography?
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 21; Issue 3. Is Calculus a Failure in Cryptography? P Vanchinathan. General Article Volume 21 Issue 3 March 2016 pp 239-245. Fulltext. Click here to view fulltext PDF. Permanent link: http://www.ias.ac.in/article/fulltext/reso/021/03/0239-0245. Keywords.
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...
Some Factors Effected Student's Calculus Learning Outcome
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…
an aid to mastering fundamental calculus concepts
African Journals Online (AJOL)
Erna Kinsey
the classroom, they do not indicate why it happened. The research reported here differs from those quoted above in that it was not a quantitative comparative study conducted with two groups of subjects, but an action research study over an extended period of time with students enrolled for a first course in calculus.
Exploring Flipped Classroom Instruction in Calculus III
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…
Sharks, Minnows, and Wheelbarrows: Calculus Modeling Projects
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…
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...
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...
Fractional vector calculus for fractional advection dispersion
Meerschaert, Mark M.; Mortensen, Jeff; Wheatcraft, Stephen W.
2006-07-01
We develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient, divergence, and curl, and a fractional divergence theorem and Stokes theorem. These basic tools are then applied to provide a physical explanation for the fractional advection-dispersion equation for flow in heterogeneous porous media.
On Online Assignments in a Calculus Class
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…
Characteristics of subgingival calculus detection by multiphoton fluorescence microscopy
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.
Graustein, William C
2006-01-01
This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. It also provides an introduction to the study of Riemannian geometry.Suitable for advanced undergraduates and graduate students, the text presupposes a knowledge of calculus. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of
Towards Analysis-Driven Scientific Software Architecture: The Case for Abstract Data Type Calculus
Directory of Open Access Journals (Sweden)
Damian W.I. Rouson
2008-01-01
Full Text Available This article approaches scientific software architecture from three analytical paths. Each path examines discrete time advancement of multiphysics phenomena governed by coupled differential equations. The new object-oriented Fortran 2003 constructs provide a formal syntax for an abstract data type (ADT calculus. The first analysis uses traditional object-oriented software design metrics to demonstrate the high cohesion and low coupling associated with the calculus. A second analysis from the viewpoint of computational complexity theory demonstrates that a more representative bug search strategy than that considered by Rouson et al. (ACM Trans. Math. Soft. 34(1 (2008 reduces the number of lines searched in a code with λ total lines from O(λ2 to O(λ log2 λ , which in turn becomes nearly independent of the overall code size in the context of ADT calculus. The third analysis derives from information theory an argument that ADT calculus simplifies developer communications in part by minimizing the growth in interface information content as developers add new physics to a multiphysics package.
Fundamentals of tensor calculus for engineers with a primer on smooth manifolds
Mühlich, Uwe
2017-01-01
This book presents the fundamentals of modern tensor calculus for students in engineering and applied physics, emphasizing those aspects that are crucial for applying tensor calculus safely in Euclidian space and for grasping the very essence of the smooth manifold concept. After introducing the subject, it provides a brief exposition on point set topology to familiarize readers with the subject, especially with those topics required in later chapters. It then describes the finite dimensional real vector space and its dual, focusing on the usefulness of the latter for encoding duality concepts in physics. Moreover, it introduces tensors as objects that encode linear mappings and discusses affine and Euclidean spaces. Tensor analysis is explored first in Euclidean space, starting from a generalization of the concept of differentiability and proceeding towards concepts such as directional derivative, covariant derivative and integration based on differential forms. The final chapter addresses the role of smooth...
Buyukkilic, F.; Ok Bayrakdar, Z.; Demirhan, D.
2016-02-01
In this study, we investigate the cumulative diminution phenomenon for a physical quantity and a diminution process with a constant acquisition quantity in each step in a viscous medium. We analyze the existence of a dynamical mechanism that underlies the success of fractional calculus compared with standard mathematics for describing stochastic processes by proposing a Fibonacci approach, where we assume that the complex processes evolves cumulatively in fractal space and discrete time. Thus, when the differential-integral order α is attained, this indicates the involvement of the viscosity of the medium in the evolving process. The future value of the diminishing physical quantity is obtained in terms of the Mittag-Leffler function (MLF) and two rheological laws are inferred from the asymptotic limits. Thus, we conclude that the differential-integral calculus of fractional mathematics implicitly embodies the cumulative diminution mechanism that occurs in a viscous medium.
Theory and applications of fractional differential equations
Kilbas, Anatoly A; Trujillo, Juan J; Van Mill, Jan
2006-01-01
This monograph provides the most recent and up-to-date developments on fractional differential and fractional integro-differential equations involving many different potentially useful operators of fractional calculus. The subject of fractional calculus and its applications (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. Some of the areas of prese
Barbu, Viorel
2016-01-01
This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.
Rauhala, U. A.
2013-12-01
Array algebra of photogrammetry and geodesy unified multi-linear matrix and tensor operators in an expansion of Gaussian adjustment calculus to general matrix inverses and solutions of inverse problems to find all, or some optimal, parametric solutions that satisfy the available observables. By-products in expanding array and tensor calculus to handle redundant observables resulted in general theories of estimation in mathematical statistics and fast transform technology of signal processing. Their applications in gravity modeling and system automation of multi-ray digital image and terrain matching evolved into fast multi-nonlinear differential and integral array calculus. Work since 1980's also uncovered closed-form inverse Taylor and least squares Newton-Raphson-Gauss perturbation solutions of nonlinear systems of equations. Fast nonlinear integral matching of array wavelets enabled an expansion of the bundle adjustment to 4-D stereo imaging and range sensing where real-time stereo sequence and waveform phase matching enabled data-to-info conversion and compression on-board advanced sensors. The resulting unified array calculus of spacetime sensing is applicable in virtually any math and engineering science, including recent work in spacetime physics. The paper focuses on geometric spacetime reconstruction from its image projections inspired by unified relativity and string theories. The collinear imaging equations of active object space shutter of special relativity are expanded to 4-D Lorentz transform. However, regular passive imaging and shutter inside the sensor expands the law of special relativity by a quantum geometric explanation of 4-D photogrammetry. The collinear imaging equations provide common sense explanations to the 10 (and 26) dimensional hyperspace concepts of a purely geometric string theory. The 11-D geometric M-theory is interpreted as a bundle adjustment of spacetime images using 2-D or 5-D membrane observables of image, string and
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.
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.)
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.
Linear algebra a first course with applications to differential equations
Apostol, Tom M
2014-01-01
Developed from the author's successful two-volume Calculus text this book presents Linear Algebra without emphasis on abstraction or formalization. To accommodate a variety of backgrounds, the text begins with a review of prerequisites divided into precalculus and calculus prerequisites. It continues to cover vector algebra, analytic geometry, linear spaces, determinants, linear differential equations and more.
Quantum stochastic calculus with maximal operator domains
Lindsay, J. Martin; Attal, Stéphane
2004-01-01
Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for op...
Introduction to the calculus of variations
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.
Index matrices towards an augmented matrix calculus
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.
An application of Malliavin Calculus to Finance
Kohatsu-Higa, Arturo; Montero, Miquel
2001-01-01
In this article, we give a brief informal introduction to Malliavin Calculus for newcomers. We apply these ideas to the simulation of Greeks in Finance. First to European-type options where formulas can be computed explicitly and therefore can serve as testing ground. Later we study the case of Asian options where close formulas are not available. The Greeks are computed through Monte Carlo simulation.
Algebraic difficulties as an obstacle for high school calculus
Kraemer, Karl
2011-01-01
The mistakes in algebraic manipulations often hinder students’ performance on calculus tasks. This observation is supported by the literature review, revealing that students, who are all supposedly strong in mathematics, are experiencing significant difficulties in their university level calculus course. Several researchers suggest that the biggest hurdle in first year calculus is the significant lack of proficiency with high school algebra. However, most studies do not itemize what such prof...
Mukherjee, Amiya
2015-01-01
This book presents a systematic and comprehensive account of the theory of differentiable manifolds and provides the necessary background for the use of fundamental differential topology tools. The text includes, in particular, the earlier works of Stephen Smale, for which he was awarded the Fields Medal. Explicitly, the topics covered are Thom transversality, Morse theory, theory of handle presentation, h-cobordism theorem, and the generalised Poincaré conjecture. The material is the outcome of lectures and seminars on various aspects of differentiable manifolds and differential topology given over the years at the Indian Statistical Institute in Calcutta, and at other universities throughout India. The book will appeal to graduate students and researchers interested in these topics. An elementary knowledge of linear algebra, general topology, multivariate calculus, analysis, and algebraic topology is recommended.
Complexity and the Fractional Calculus
Directory of Open Access Journals (Sweden)
Pensri Pramukkul
2013-01-01
Full Text Available We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality.
Everyday calculus discovering the hidden math all around us
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
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...... reconfiguration of services increases their availability, but puts accordingly, heavy demands for validation, verification, and evaluation. In this paper we formally model and analyze dynamic reconfigurations and their requirements in BP-calculus and show how reconfigurable components can be modeled using...
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...... dynamic change of scope (of label) in connection with communication. It is proved that algebraic properties of strong (and observational) equivalence for CCS are preserved by the extension. Examples illustrating the expressive power of the calculus and its methods for reasoning are given....
Pre-calculus 1,001 practice problems for dummies
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
An Algorithmic Information Calculus for Causal Discovery and Reprogramming Systems
Zenil, Hector
2017-09-08
We introduce a conceptual framework and an interventional calculus to steer and manipulate systems based on their intrinsic algorithmic probability using the universal principles of the theory of computability and algorithmic information. By applying sequences of controlled interventions to systems and networks, we estimate how changes in their algorithmic information content are reflected in positive/negative shifts towards and away from randomness. The strong connection between approximations to algorithmic complexity (the size of the shortest generating mechanism) and causality induces a sequence of perturbations ranking the network elements by the steering capabilities that each of them is capable of. This new dimension unmasks a separation between causal and non-causal components providing a suite of powerful parameter-free algorithms of wide applicability ranging from optimal dimension reduction, maximal randomness analysis and system control. We introduce methods for reprogramming systems that do not require the full knowledge or access to the system\\'s actual kinetic equations or any probability distributions. A causal interventional analysis of synthetic and regulatory biological networks reveals how the algorithmic reprogramming qualitatively reshapes the system\\'s dynamic landscape. For example, during cellular differentiation we find a decrease in the number of elements corresponding to a transition away from randomness and a combination of the system\\'s intrinsic properties and its intrinsic capabilities to be algorithmically reprogrammed can reconstruct an epigenetic landscape. The interventional calculus is broadly applicable to predictive causal inference of systems such as networks and of relevance to a variety of machine and causal learning techniques driving model-based approaches to better understanding and manipulate complex systems.
Mathematical misconception in calculus 1: Identification and gender difference
Nassir, Asyura Abd; Abdullah, Nur Hidayah Masni; Ahmad, Salimah; Tarmuji, Nor Habibah; Idris, Aminatul Solehah
2017-08-01
A few years of experience of teaching mathematics make us notice that the same types of mistakes are done repeatedly by students. This paper presents an insight into categories of mistakes, how male and female students differ in terms of mistakes that are commonly done and the ability of the students to identify the mistakes. Sample of mistakes were taken from Calculus 1 final exam answer scripts, then it was listed and analyzed. Data analysis revealed that students' misconceptions fall into four categories. The first category is misunderstanding the meaning of brackets, followed by misconception of basic mathematics rules, misconception in notation and misconception in properties of trigonometry. A mistake identification test which consists of ten false mathematical statements was designed based on the mistake done by the previous batch of students that covered topics algebra, trigonometry, index, limit, differentiation and integration. Then, the test was given to students who enrolled in Calculus I course. Respondents of this study were randomly selected among two hundreds engineering students. Data obtained were analyzed using basic descriptive analysis and Chi Square test to capture gender differences in the mistake done for each category. Findings indicate that thirty five percent of the students have the ability to identify the mistakes and make a proper correction for at most two statements. Thirty one percent of the students are able to identify the mistakes but unable to make proper correction. Twenty five percent of the students failed to identify the mistakes in six out of ten false statements. Female students' misconception is more likely in basic mathematics rules compared to male. The findings of this study could serve as baseline information to be stressed in improving teaching and learning mathematics.
Fractional Dynamics Applications of Fractional Calculus to Dynamics of Particles, Fields and Media
Tarasov, Vasily E
2010-01-01
"Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media" presents applications of fractional calculus, integral and differential equations of non-integer orders in describing systems with long-time memory, non-local spatial and fractal properties. Mathematical models of fractal media and distributions, generalized dynamical systems and discrete maps, non-local statistical mechanics and kinetics, dynamics of open quantum systems, the hydrodynamics and electrodynamics of complex media with non-local properties and memory are considered. This book is intended to meet the needs of scientists and graduate students in physics, mechanics and applied mathematics who are interested in electrodynamics, statistical and condensed matter physics, quantum dynamics, complex media theories and kinetics, discrete maps and lattice models, and nonlinear dynamics and chaos. Dr. Vasily E. Tarasov is a Senior Research Associate at Nuclear Physics Institute of Moscow State University and...
Parametric Analysis of a Heavy Metal Sorption Isotherm Based on Fractional Calculus
Directory of Open Access Journals (Sweden)
Enrico M. Gomes
2013-01-01
Full Text Available Heavy metals are widely recognized as being hazardous to human health and environmentally aggressive. The literature reports different approaches for lead removal, for example, water hyacinths. Heavy metal sorption isotherm modeling represents an important tool towards the study of equilibrium conditions. Fractional calculus represents a novel approach and a growing research field for process modeling, based on derivatives of arbitrary order. Recently, a novel isotherm based on fractional calculus was proposed for lead sorption using water hyacinth (Eichhornia crassipes. This paper reports a general procedure on error analysis and its influence on parameter estimation. It was applied to mathematical models based on fractional differential equations, focusing on a heavy metal novel isotherm sorption model. Parameter variance was calculated by using two different approaches (with the complete Hessian matrix and with a simplified Hessian matrix, and joint parameter confidence regions were generated, being successfully able to show that the fractional nature of the model is statistically valid.
Vector 33: A reduce program for vector algebra and calculus in orthogonal curvilinear coordinates
Harper, David
1989-06-01
This paper describes a package with enables REDUCE 3.3 to perform algebra and calculus operations upon vectors. Basic algebraic operations between vectors and between scalars and vectors are provided, including scalar (dot) product and vector (cross) product. The vector differential operators curl, divergence, gradient and Laplacian are also defined, and are valid in any orthogonal curvilinear coordinate system. The package is written in RLISP to allow algebra and calculus to be performed using notation identical to that for operations. Scalars and vectors can be mixed quite freely in the same expression. The package will be of interest to mathematicians, engineers and scientists who need to perform vector calculations in orthogonal curvilinear coordinates.
On a fractal LC-electric circuit modeled by local fractional calculus
Yang, Xiao-Jun; Machado, J. A. Tenreiro; Cattani, Carlo; Gao, Feng
2017-06-01
A non-differentiable model of the LC-electric circuit described by a local fractional differential equation of fractal dimensional order is addressed in this article. From the fractal electrodynamics point of view, the relaxation oscillator, defined on Cantor sets in LC-electric circuit, and its exact solution using the local fractional Laplace transform are obtained. Comparative results among local fractional derivative, Riemann-Liouville fractional derivative and conventional derivative are discussed. Local fractional calculus is proposed as a new tool suitable for the study of a large class of electric circuits.
Elementary differential geometry
Pressley, Andrew
2001-01-01
Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there The book will provide an invaluable resource to all those taking a first course in differential geometry, for their lecture...
Success in Introductory Calculus: The Role of High School and Pre-Calculus Preparation
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…
Improving Student Success in Calculus I Using a Co-Requisite Calculus I Lab
Vestal, Sharon Schaffer; Brandenburger, Thomas; Furth, Alfred
2015-01-01
This paper describes how one university mathematics department was able to improve student success in Calculus I by requiring a co-requisite lab for certain groups of students. The groups of students required to take the co-requisite lab were identified by analyzing student data, including Math ACT scores, ACT Compass Trigonometry scores, and…
Automatic Binding Time Analysis for a Typed Lambda-Calculus
DEFF Research Database (Denmark)
Nielson, Hanne Riis; Nielson, Flemming
1988-01-01
A binding time analysis imposes a distinction between the computations to be performed early (e.g. at compile-time) and those to be performed late (e.g. at run-time). For the lambda-calculus this distinction is formalized by a two-level lambda-calculus. The authors present an algorithm for static...
A Calculus of Circular Proofs and its Categorical Semantics
DEFF Research Database (Denmark)
Santocanale, Luigi
2002-01-01
We present a calculus of “circular proofs”: the graph underlying a proof is not a finite tree but instead it is allowed to contain a certain amount of cycles.The main challenge in developing a theory for the calculus is to define the semantics of proofs, since the usual method by induction...
Transitioning from Introductory Calculus to Formal Limit Conceptions
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…
The Enriched Eﬀect Calculus: Syntax and Semantics
DEFF Research Database (Denmark)
Møgelberg, Rasmus Ejlers; Simpson, Alex; Egger, Jeff
2014-01-01
This article introduces the enriched effect calculus, which extends established type theories for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; e.g. the linear usage of imperative features suc...
A Calculus of Circular Proofs and its Categorical Semantics
DEFF Research Database (Denmark)
Santocanale, Luigi
2002-01-01
We present a calculus of "circular proofs": the graph underlying a proof is not a finite tree but instead it is allowed to contain a certain amount of cycles.The main challenge in developing a theory for the calculus is to define the semantics of proofs, since the usual method by induction...
Calculus: A Computer Oriented Presentation, Part 1 [and] Part 2.
Stenberg, Warren; Walker, Robert J.
Parts one and two of a one-year computer-oriented calculus course (without analytic geometry) are presented. The ideas of calculus are introduced and motivated through computer (i.e., algorithmic) concepts. An introduction to computing via algorithms and a simple flow chart language allows the book to be self-contained, except that material on…
Calculus Students' Early Concept Images of Tangent Lines
Vincent, Brittany; LaRue, Renee; Sealey, Vicki; Engelke, Nicole
2015-01-01
This study explored first-semester calculus students' understanding of tangent lines as well as how students used tangent lines within the context of Newton's method. Task-based interviews were conducted with twelve first-semester calculus students who were asked to verbally describe a tangent line, sketch tangent lines for multiple curves, and…
Partial Fractions in Calculus, Number Theory, and Algebra
Yackel, C. A.; Denny, J. K.
2007-01-01
This paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.
Giant urinary bladder calculus: Case report | Otieno | East African ...
African Journals Online (AJOL)
A vertical calculus weighing more than 100 g is categorised as a giant urinary bladder stone. Giant urinary bladder stones are very rare and very few cases have been reported in English literature and only one case from Africa. This is a case report of a patient with a giant urinary bladder calculus presenting as a rectal ...
Type Inference for Session Types in the Pi-Calculus
DEFF Research Database (Denmark)
Graversen, Eva Fajstrup; Harbo, Jacob Buchreitz; Huttel, Hans
2014-01-01
In this paper we present a direct algorithm for session type inference for the π-calculus. Type inference for session types has previously been achieved by either imposing limitations and restriction on the π-calculus, or by reducing the type inference problem to that for linear types. Our approach...
An Exploration of Definition and Procedural Fluency in Integral Calculus
Grundmeier, Todd A.; Hansen, Jennifer; Sousa, Emily
2006-01-01
A survey was administered to calculus students who had previously been exposed to a course on integral calculus. The purpose of the survey was to explore students' understanding of the definition of a definite integral, their abilities to evaluate definite integrals, and their graphical interpretations of definite integrals. The analysis of…
Prostatic fossa calculus | El Abiad | Pan African Medical Journal
African Journals Online (AJOL)
As part of the evaluation, a plain radiograph was performed and incidentally showed a radiopaque prostatic calculus (Red arrows). A retrograde urethrocystography, performed after a 10-days course of antibiotics, confirmed the presence of an approximately 35 mm non-obstructive calculus occupying almost the whole ...
Visual Thinking and Gender Differences in High School Calculus
Haciomeroglu, Erhan Selcuk; Chicken, Eric
2012-01-01
This study sought to examine calculus students' mathematical performances and preferences for visual or analytic thinking regarding derivative and antiderivative tasks presented graphically. It extends previous studies by investigating factors mediating calculus students' mathematical performances and their preferred modes of thinking. Data were…
Stochastic Model Checking of the Stochastic Quality Calculus
DEFF Research Database (Denmark)
Nielson, Flemming; Nielson, Hanne Riis; Zeng, Kebin
2015-01-01
The Quality Calculus uses quality binders for input to express strategies for continuing the computation even when the desired input has not been received. The Stochastic Quality Calculus adds generally distributed delays for output actions and real-time constraints on the quality binders for input...
Utilizing Microsoft Mathematics in Teaching and Learning Calculus
Oktaviyanthi, Rina; Supriani, Yani
2015-01-01
The experimental design was conducted to investigate the use of Microsoft Mathematics, free software made by Microsoft Corporation, in teaching and learning Calculus. This paper reports results from experimental study details on implementation of Microsoft Mathematics in Calculus, students' achievement and the effects of the use of Microsoft…
Formalization of the Resolution Calculus for First-Order Logic
DEFF Research Database (Denmark)
Schlichtkrull, Anders
2018-01-01
I present a formalization in Isabelle/HOL of the resolution calculus for first-order logic with formal soundness and completeness proofs. To prove the calculus sound, I use the substitution lemma, and to prove it complete, I use Herbrand interpretations and semantic trees. The correspondence...
Secure Data Flow in a Calculus for Context Awareness
DEFF Research Database (Denmark)
Bucur, Doina; Nielsen, Mogens
2008-01-01
We present a Mobile-Ambients-based process calculus to describe context-aware computing in an infrastructure-based Ubiquitous Computing setting. In our calculus, computing agents can provide and discover contextual information and are owners of security policies. Simple access control to contextual...
Effects of Clicker Use on Calculus Students' Mathematics Anxiety
Batchelor, John
2015-01-01
This paper reports the results of a survey study of clicker use and mathematics anxiety among students enrolled in an undergraduate calculus course during the Fall 2013 semester. Students in two large lecture sections of calculus completed surveys at the beginning and end of the course. One class used clickers, whereas the other class was taught…
Coordinating Multiple Representations in a Reform Calculus Textbook
Chang, Briana L.; Cromley, Jennifer G.; Tran, Nhi
2016-01-01
Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions…
Equality and fixpoints in the calculus of structures
DEFF Research Database (Denmark)
Chaudhuri, Kaustuv; Guenot, Nicolas
2014-01-01
The standard proof theory for logics with equality and fixpoints suffers from limitations of the sequent calculus, where reasoning is separated from computational tasks such as unification or rewriting. We propose in this paper an extension of the calculus of structures, a deep inference formalism...
Reflections on Our First Calculus Undergraduate Teaching Assistant
Deshler, Jessica M.
2016-01-01
This article describes some reflections from the first Calculus I undergraduate teaching assistant in our department as she explored the various ways in which she was able to support both novice and experienced Calculus teachers and the effect of her experience on her academic and career plans.
Decidable Fragments of a Higher Order Calculus with Locations
DEFF Research Database (Denmark)
Hüttel, Hans; Godskesen, Jens Christian; Haagensen, Bjørn
2009-01-01
Homer is a higher order process calculus with locations. In this paper we study Homer in the setting of the semantic finite control property, which is a finite reachability criterion that implies decidability of barbed bisimilarity. We show that strong and weak barbed bisimilarity are undecidable...... control π-calculus in Homer....
A Technical Approach in Graphical Study of Calculus Using Maple ...
African Journals Online (AJOL)
We were recently presented with the challenge of rethinking how to teach Calculus to First year IT students as part of an integrated curriculum. We immediately saw this as an opportunity to motivate calculus concepts by using graphical representations and animations. Animation provides motivation, reinforcement, and ...
Calculus Instructors' and Students' Discourses on the Derivative
Park, Jungeun
2011-01-01
Recently, there has been an increasing interest in collegiate mathematics education, especially teaching and learning calculus (e.g., Oehrtman, Carlson, & Thompson, 2008; Speer, Smith, & Horvath, 2010). Of many calculus concepts, the derivative is known as a difficult concept for students to understand because it involves various concepts…
Descartes' Calculus of Subnormals: What Might Have Been
Boudreaux, Gregory Mark; Walls, Jess E.
2013-01-01
Rene Descartes' method for finding tangents (equivalently, subnormals) depends on geometric and algebraic properties of a family of circles intersecting a given curve. It can be generalized to establish a calculus of subnormals, an alternative to the calculus of Newton and Leibniz. Here we prove subnormal counterparts of the well-known…
Use of Technology to Develop Student Intuition in Multivariable Calculus
Kaur, Manmohan
2006-01-01
In order to get undergraduates interested in mathematics, it is essential to involve them in its "discovery". In this paper, we will explain how technology and the knowledge of lower dimensional calculus can be used to help them develop intuition leading to their discovering the first derivative rule in multivariable calculus. (Contains 7 figures.)
An introduction to quantum stochastic calculus
Parthasarathy, KR
2012-01-01
An Introduction to Quantum Stochastic Calculus aims to deepen our understanding of the dynamics of systems subject to the laws of chance both from the classical and the quantum points of view and stimulate further research in their unification. This is probably the first systematic attempt to weave classical probability theory into the quantum framework and provides a wealth of interesting features: The origin of Ito's correction formulae for Brownian motion and the Poisson process can be traced to commutation relations or, equivalently, the uncertainty principle.Quantum stochastic integration
Some Integral Inequalities Using Quantum Calculus Approach
Directory of Open Access Journals (Sweden)
Muhammad Uzair Awan
2017-11-01
Full Text Available The aim of this paper is to introduce a new class of preinvex functions which is called as generalized beta preinvex functions. We show that this class includes some other new classes of preinvex functions. We derive some new integral inequalities using the approach of quantum calculus. These integral inequalities involve generalized preinvex functions and q-Euler-Beta functions. Our results can be viewed as new quantum estimates for trapezoidal like inequalities. Some new special cases are also discussed which can be deduced from the main results of the paper.
A calculus for attribute-based communication
DEFF Research Database (Denmark)
Alrahman, Yehia Abd; De Nicola, Rocco; Loreti, Michele
2015-01-01
The notion of attribute-based communication seems promising to model and analyse systems with huge numbers of interacting components that dynamically adjust and combine their behaviour to achieve specific goals. A basic process calculus, named AbC, is introduced that has as primitive construct...... exactly attribute-based communication and its impact on the above mentioned kind of systems is considered. An AbC system consists of a set of parallel components each of which is equipped with a set of attributes. Communication takes place in a broadcast fashion and communication links among components...
Giant Submandibular Calculus Eroding Oral Cavity Mucosa.
Lim, Eng Haw; Nadarajah, Sanjeevan; Mohamad, Irfan
2017-09-01
Sialolithiasis is the formation of calculi or sialoliths in the salivary gland. It is the most common benign condition of the salivary gland. Sialolithiasis can occur in all salivary glands. The submandibular gland is most commonly affected followed by the parotid gland. Calculi commonly measure less than 10 mm. Calculi of more than 15 mm are termed giant salivary gland calculi and are infrequently reported in the literature. Here, we report a case of unusually large submandibular gland calculus of 5 cm in greatest dimension which caused erosion of the oral cavity.
Calculus for cognitive scientists partial differential equation models
Peterson, James K
2016-01-01
This book shows cognitive scientists in training how mathematics, computer science and science can be usefully and seamlessly intertwined. It is a follow-up to the first two volumes on mathematics for cognitive scientists, and includes the mathematics and computational tools needed to understand how to compute the terms in the Fourier series expansions that solve the cable equation. The latter is derived from first principles by going back to cellular biology and the relevant biophysics. A detailed discussion of ion movement through cellular membranes, and an explanation of how the equations that govern such ion movement leading to the standard transient cable equation are included. There are also solutions for the cable model using separation of variables, as well an explanation of why Fourier series converge and a description of the implementation of MatLab tools to compute the solutions. Finally, the standard Hodgkin - Huxley model is developed for an excitable neuron and is solved using MatLab.
Intitialization, Conceptualization, and Application in the Generalized Fractional Calculus
Lorenzo, Carl F.; Hartley, Tom T.
1998-01-01
This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.
Initialization, conceptualization, and application in the generalized (fractional) calculus.
Lorenzo, Carl F; Hartley, Tom T
2007-01-01
This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.
Calculus detection technologies: where do we stand now?
Archana, V
2014-01-01
Epidemiological studies have implicated dental calculus as an ideal substrate for subgingival microbial colonization. Therefore, the main objective of periodontal therapy is to eliminate the microbial biofilm along with the calculus deposits from the root surface by root surface debridement. Over the past years, a large number of clinical and laboratory studies have been conducted to evaluate the efficacy of calculus removal by various methods. None of these conventional methods or devices was effective in completely eliminating all the calculus from the diseased root surfaces. In this context, a number of newer technologies have been developed to identify and selectively remove the dental calculus. Regarding this fact, the present article highlights a critical review of these devices based on published clinical and experimental data.
Ferguson, Leann J.
2012-01-01
Calculus is an important tool for building mathematical models of the world around us and is thus used in a variety of disciplines, such as physics and engineering. These disciplines rely on calculus courses to provide the mathematical foundation needed for success in their courses. Unfortunately, due to the basal conceptions of what it means to…
Samtaney, Ravi; Mohamed, Mamdouh; Hirani, Anil
2015-11-01
We present examples of numerical solutions of incompressible flow on 2D curved domains. The Navier-Stokes equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. A conservative discretization of Navier-Stokes equations on simplicial meshes is developed based on discrete exterior calculus (DEC). The discretization is then carried out by substituting the corresponding discrete operators based on the DEC framework. By construction, the method is conservative in that both the discrete divergence and circulation are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step. Numerical examples include Taylor vortices on a sphere, Stuart vortices on a sphere, and flow past a cylinder on domains with varying curvature. Supported by the KAUST Office of Competitive Research Funds under Award No. URF/1/1401-01.
Sequent Calculus Representations for Quantum Circuits
Directory of Open Access Journals (Sweden)
Cameron Beebe
2016-06-01
Full Text Available When considering a sequent-style proof system for quantum programs, there are certain elements of quantum mechanics that we may wish to capture, such as phase, dynamics of unitary transformations, and measurement probabilities. Traditional quantum logics which focus primarily on the abstract orthomodular lattice theory and structures of Hilbert spaces have not satisfactorily captured some of these elements. We can start from 'scratch' in an attempt to conceptually characterize the types of proof rules which should be in a system that represents elements necessary for quantum algorithms. This present work attempts to do this from the perspective of the quantum circuit model of quantum computation. A sequent calculus based on single quantum circuits is suggested, and its ability to incorporate important conceptual and dynamic aspects of quantum computing is discussed. In particular, preserving the representation of phase helps illustrate the role of interference as a resource in quantum computation. Interference also provides an intuitive basis for a non-monotonic calculus.
A type system for Continuation Calculus
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Herman Geuvers
2014-09-01
Full Text Available Continuation Calculus (CC, introduced by Geron and Geuvers, is a simple foundational model for functional computation. It is closely related to lambda calculus and term rewriting, but it has no variable binding and no pattern matching. It is Turing complete and evaluation is deterministic. Notions like "call-by-value" and "call-by-name" computation are available by choosing appropriate function definitions: e.g. there is a call-by-value and a call-by-name addition function. In the present paper we extend CC with types, to be able to define data types in a canonical way, and functions over these data types, defined by iteration. Data type definitions follow the so-called "Scott encoding" of data, as opposed to the more familiar "Church encoding". The iteration scheme comes in two flavors: a call-by-value and a call-by-name iteration scheme. The call-by-value variant is a double negation variant of call-by-name iteration. The double negation translation allows to move between call-by-name and call-by-value.
How Students Use Their Knowledge of Calculus in an Engineering Mechanics Course.
Roddick, Cheryl Stitt
This study investigated students' conceptual and procedural understanding of calculus within the context of an engineering mechanics course. Four traditional calculus students were compared with three students from one of the calculus reform projects, Calculus & Mathematica. Task-based interviews were conducted with each participant throughout…
Valuation Semantics for Intuitionic Propositional Calculus and some of its Subcalculi
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Andréa Loparić
2010-04-01
Full Text Available In this paper, we present valuation semantics for the Propositional Intuitionistic Calculus (also called Heyting Calculus and three important subcalculi: the Implicative, the Positive and the Minimal Calculus (also known as Kolmogoroff or Johansson Calculus. Algorithms based in our definitions yields decision methods for these calculi.
A many-sorted calculus based on resolution and paramodulation
Walther, Christoph
1987-01-01
A Many-Sorted Calculus Based on Resolution and Paramodulation emphasizes the utilization of advantages and concepts of many-sorted logic for resolution and paramodulation based automated theorem proving.This book considers some first-order calculus that defines how theorems from given hypotheses by pure syntactic reasoning are obtained, shifting all the semantic and implicit argumentation to the syntactic and explicit level of formal first-order reasoning. This text discusses the efficiency of many-sorted reasoning, formal preliminaries for the RP- and ?RP-calculus, and many-sorted term rewrit
Quantum stochastic calculus and representations of Lie superalgebras
Eyre, Timothy M W
1998-01-01
This book describes the representations of Lie superalgebras that are yielded by a graded version of Hudson-Parthasarathy quantum stochastic calculus. Quantum stochastic calculus and grading theory are given concise introductions, extending readership to mathematicians and physicists with a basic knowledge of algebra and infinite-dimensional Hilbert spaces. The develpment of an explicit formula for the chaotic expansion of a polynomial of quantum stochastic integrals is particularly interesting. The book aims to provide a self-contained exposition of what is known about Z_2-graded quantum stochastic calculus and to provide a framework for future research into this new and fertile area.
A calculus of quality for robustness against unreliable communication
DEFF Research Database (Denmark)
Nielson, Hanne Riis; Nielson, Flemming; Vigo, Roberto
2015-01-01
A main challenge in the development of distributed systems 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....... The framework is illustrated on the design of a fragment of a wireless sensor network, and is substantiated by formal proofs of correctness of the analysis, which relate the original reduction semantics of the calculus to a new semantics with explicit substitutions....
The dental calculus metabolome in modern and historic samples
DEFF Research Database (Denmark)
Velsko, Irina M.; Overmyer, Katherine A.; Speller, Camilla
2017-01-01
present the first comprehensive characterization of the human dental calculus metabolome using a multi-platform approach. Methods: Ultra performance liquid chromatography-tandem mass spectrometry (UPLC–MS/MS) quantified 285 metabolites in modern and historic (200 years old) dental calculus, including...... in the archaeological record. Recent studies have confirmed the survival of authentic ancient DNA and proteins within historic and prehistoric dental calculus, making it a promising substrate for investigating oral microbiome evolution via direct measurement and comparison of modern and ancient specimens. Objective: We...
Calculus, Biology and Medicine: A Case Study in Quantitative Literacy for Science Students
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Kim Rheinlander
2011-01-01
Full Text Available This paper describes a course designed to enhance the numeracy of biology and pre-medical students. The course introduces students with the background of one semester of calculus to systems of nonlinear ordinary differential equations as they appear in the mathematical biology literature. Evaluation of the course showed increased enjoyment and confidence in doing mathematics, and an increased appreciation of the utility of mathematics to science. Students who complete this course are better able to read the research literature in mathematical biology and carry out research problems of their own.
Directory of Open Access Journals (Sweden)
Hoti Lal
2015-11-01
Full Text Available Primary Squamous Cell Carcinoma in the kidney is a rare malignant neoplasm associated with nephrolithiasis, typically monobacterial pyonephrosis and rarely Xanthogranulomatous pyelonephritis. It is an aggressive disease with a poor prognosis mostly due to lack of presenting clinical features like a palpable mass, gross haematuria and pain. We report a case presenting with renal calculus and pyonephrosis managed initially with percutaneous nephrostomy followed by nephrectomy due to complete loss of renal function. Histopathological evaluation revealed poorly differentiated squamous cell carcinoma which is managed by chemotherapy, although initially beneficial, patients later develop disseminated metastatic disease which holds a poor prognosis.
Five-Dimensional Tangent Vectors in Space-Time: IV. Generalization of Exterior Calculus
Krasulin, Alexander
1998-01-01
This part of the series is devoted to the generalization of exterior differential calculus. I give definition to the integral of a five-vector form over a limited space-time volume of appropriate dimension; extend the notion of the exterior derivative to the case of five-vector forms; and formulate the corresponding analogs of the generalized Stokes theorem and of the Poincare theorem about closed forms. I then consider the five-vector generalization of the exterior derivative itself; prove a...
Regge calculus models of closed lattice universes
Liu, Rex G
2016-01-01
This paper examines the behaviour of closed `lattice universes' wherein masses are distributed in a regular lattice on the Cauchy surfaces of closed vacuum universes. Such universes are approximated using a form of Regge calculus originally developed by Collins and Williams to model closed FLRW universes. We consider two types of lattice universes, one where all masses are identical to each other and another where one mass gets perturbed in magnitude. In the unperturbed universe, we consider the possible arrangements of the masses in the Regge Cauchy surfaces and demonstrate that the model will only be stable if each mass lies within some spherical region of convergence. We also briefly discuss the existence of Regge models that are dual to the ones we have considered. We then model a perturbed lattice universe and demonstrate that the model's evolution is well-behaved, with the expansion increasing in magnitude as the perturbation is increased.
Tensor calculus for engineers and physicists
de Souza Sánchez Filho, Emil
2016-01-01
This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of N dimensional spaces. The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step. Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without need...
Introduction to calculus and classical analysis
Hijab, Omar
2016-01-01
This completely self-contained text is intended either for a course in honors calculus or for an introduction to analysis. Beginning with the real number axioms, and involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate math majors. This fourth edition includes an additional chapter on the fundamental theorems in their full Lebesgue generality, based on the Sunrise Lemma. Key features of this text include: • Applications from several parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more; • A heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero; • Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals; • A self-contained t...
Functional analysis and the Feynman operator calculus
Gill, Tepper L
2016-01-01
This book provides the mathematical foundations for Feynman's operator calculus and for the Feynman path integral formulation of quantum mechanics as a natural extension of analysis and functional analysis to the infinite-dimensional setting. In one application, the results are used to prove the last two remaining conjectures of Freeman Dyson for quantum electrodynamics. In another application, the results are used to unify methods and weaken domain requirements for non-autonomous evolution equations. Other applications include a general theory of Lebesgue measure on Banach spaces with a Schauder basis and a new approach to the structure theory of operators on uniformly convex Banach spaces. This book is intended for advanced graduate students and researchers.
Correlators in tensor models from character calculus
Mironov, A.; Morozov, A.
2017-11-01
We explain how the calculations of [20], which provided the first evidence for non-trivial structures of Gaussian correlators in tensor models, are efficiently performed with the help of the (Hurwitz) character calculus. This emphasizes a close similarity between technical methods in matrix and tensor models and supports a hope to understand the emerging structures in very similar terms. We claim that the 2m-fold Gaussian correlators of rank r tensors are given by r-linear combinations of dimensions with the Young diagrams of size m. The coefficients are made from the characters of the symmetric group Sm and their exact form depends on the choice of the correlator and on the symmetries of the model. As the simplest application of this new knowledge, we provide simple expressions for correlators in the Aristotelian tensor model as tri-linear combinations of dimensions.
Correlators in tensor models from character calculus
Directory of Open Access Journals (Sweden)
A. Mironov
2017-11-01
Full Text Available We explain how the calculations of [20], which provided the first evidence for non-trivial structures of Gaussian correlators in tensor models, are efficiently performed with the help of the (Hurwitz character calculus. This emphasizes a close similarity between technical methods in matrix and tensor models and supports a hope to understand the emerging structures in very similar terms. We claim that the 2m-fold Gaussian correlators of rank r tensors are given by r-linear combinations of dimensions with the Young diagrams of size m. The coefficients are made from the characters of the symmetric group Sm and their exact form depends on the choice of the correlator and on the symmetries of the model. As the simplest application of this new knowledge, we provide simple expressions for correlators in the Aristotelian tensor model as tri-linear combinations of dimensions.
A Fractional Probability Calculus View of Allometry
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Bruce J. West
2014-04-01
Full Text Available The scaling of respiratory metabolism with body size in animals is considered by many to be a fundamental law of nature. An apparent corollary of this law is the scaling of physiologic time with body size, implying that physiologic time is separate and distinct from clock time. However, these are only two of the many allometry relations that emerge from empirical studies in the physical, social and life sciences. Herein, we present a theory of allometry that provides a foundation for the allometry relation between a network function and the size that is entailed by the hypothesis that the fluctuations in the two measures are described by a scaling of the joint probability density. The dynamics of such networks are described by the fractional calculus, whose scaling solutions entail the empirically observed allometry relations.
VEST: Abstract vector calculus simplification in Mathematica
Squire, J.; Burby, J.; Qin, H.
2014-01-01
We present a new package, VEST (Vector Einstein Summation Tools), that performs abstract vector calculus computations in Mathematica. Through the use of index notation, VEST is able to reduce three-dimensional scalar and vector expressions of a very general type to a well defined standard form. In addition, utilizing properties of the Levi-Civita symbol, the program can derive types of multi-term vector identities that are not recognized by reduction, subsequently applying these to simplify large expressions. In a companion paper Burby et al. (2013) [12], we employ VEST in the automation of the calculation of high-order Lagrangians for the single particle guiding center system in plasma physics, a computation which illustrates its ability to handle very large expressions. VEST has been designed to be simple and intuitive to use, both for basic checking of work and more involved computations.
Improving Student Success in Calculus: A Comparison of Four College Calculus Classes
Bagley, Spencer Franklin
The quality of education in science, technology, engineering, and mathematics (STEM) fields is an issue of particular educational and economic importance, and Calculus I is a linchpin course in STEM major tracks. A national study is currently being conducted examining the characteristics of successful programs in college calculus (CSPCC, 2012). In work related to the CSPCC program, this study examines the effects on student outcomes of four different teaching strategies used at a single institution. The four classes were a traditional lecture, a lecture with discussion, a lecture incorporating both discussion and technology, and an inverted model. This dissertation was guided by three questions: (1) What impact do these four instructional approaches have on students' persistence, beliefs about mathematics, and conceptual and procedural achievement in calculus? (2) How do students at the local institution compare to students in the national database? And (3) How do the similarities and differences in opportunities for learning presented in the four classes contribute to the similarities and differences in student outcomes? Quantitative analysis of surveys and exams revealed few statistically significant differences in outcomes, and students in the inverted classroom often had poorer outcomes than those in other classes. Students in the technology-enhanced class scored higher on conceptual items on the final exam than those in other classes. Comparing to the national database, local students had similar switching rates but less expert-like attitudes and beliefs about mathematics than the national average. Qualitative analysis of focus group interviews, classroom observations, and student course evaluations showed that several implementation issues, some the result of pragmatic constraints, others the result of design choice, weakened affordances provided by innovative features and shrunk the differences between classes. There were substantial differences between the
The use of concept maps as an indicator of significant learning in Calculus
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Naíma Soltau Ferrão
2014-03-01
Full Text Available This paper contains reflections and results of a research that aimed to apply and analyze the use of concept maps in Higher Education as an indicator of significant learning concerning derivative as mathematical object with students that finished Differential and Integral Calculus. This is a qualitative approach, situated in the area of mathematics education, based on Ausubel's Theory of Meaningful Learning and on technique of Novak's Concept Mapping. As data acquisition instruments, use of classroom observations, questionnaire, brainstorming and digital conceptual mapping, made by an undergraduate physics course. To analyze we defined four aspects to be observed in the maps constructed by students: (i validity of propositions formed with concepts, (ii hierarchization, (iii cross-links between the propositions, and (vi the presence of applications. The identification of these elements, taken as reference to analyze the maps, allowed the collection of information about how each student has structured and correlated the set of concepts learned on the derivative of a function along their course. Based on the results, we have identified in the digital conceptual maps effective tools to evaluate the students in terms of meaningful learning about specific contents of Differential and Integral Calculus by the hierarchy of concepts, progressive differentiation and integrative reconciliation as defined in the Theory of Meaningful Learning.
Anisotropic fractal media by vector calculus in non-integer dimensional space
Tarasov, Vasily E.
2014-08-01
A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.
Discrete calculus applied analysis on graphs for computational science
Grady, Leo J
2010-01-01
This unique text brings together into a single framework current research in the three areas of discrete calculus, complex networks, and algorithmic content extraction. Many example applications from several fields of computational science are provided.
From Calculus to Wavelets: A New Mathematical Technique Wavelet ...
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 2; Issue 4. From Calculus to Wavelets: A New Mathematical Technique Wavelet Analysis Physical Properties. Gerald B Folland. General Article Volume 2 Issue 4 April 1997 pp 25-37 ...
Forest Carbon Uptake and the Fundamental Theorem of Calculus
Zobitz, John
2013-01-01
Using the fundamental theorem of calculus and numerical integration, we investigate carbon absorption of ecosystems with measurements from a global database. The results illustrate the dynamic nature of ecosystems and their ability to absorb atmospheric carbon.
Model-checking dense-time Duration Calculus
DEFF Research Database (Denmark)
Fränzle, Martin
2004-01-01
Since the seminal work of Zhou Chaochen, M. R. Hansen, and P. Sestoft on decidability of dense-time Duration Calculus [Zhou, Hansen, Sestoft, 1993] it is well-known that decidable fragments of Duration Calculus can only be obtained through withdrawal of much of the interesting vocabulary...... of this logic. While this was formerly taken as an indication that key-press verification of implementations with respect to elaborate Duration Calculus specifications were also impossible, we show that the model property is well decidable for realistic designs which feature natural constraints...... suitably sparser model classes we obtain model-checking procedures for rich subsets of Duration Calculus. Together with undecidability results also obtained, this sheds light upon the exact borderline between decidability and undecidability of Duration Calculi and related logics....
A Fluent Calculus Approach to Automatic Web Service Composition
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CHIFU, V.
2009-10-01
Full Text Available Web service composition is mandatory when complex functional requirements cannot be satisfied by a single Web service. Because of the exponential growth of available Web services, their automatic discovery and composition are highly desirable tasks. This paper presents a new approach for automatic Web service composition based on the formalism of Fluent Calculus using semantic service descriptions. In our approach, the Web service composition process is viewed as an AI planning problem in the Fluent Calculus formalism. To semantically describe Web services, we have used a Web service domain ontology which is then translated into a Fluent Calculus knowledge base, necessary for the composition planning phase. For verifying the composed services, the Label Transition System Analyzer (LTSA formalism is used. The paper also presents an experimental prototype for the Fluent Calculus based Web service composition and demonstrates its effectiveness with the help of an application scenario from the social event planning domain.
Model checking biological systems described using ambient calculus
DEFF Research Database (Denmark)
Mardare, Radu Iulian; Priami, Corrado; Qualia, Paola
2005-01-01
Model checking biological systems described using ambient calculus. In Proc. of the second International Workshop on Computational Methods in Systems Biology (CMSB04), Lecture Notes in Bioinformatics 3082:85-103, Springer, 2005.......Model checking biological systems described using ambient calculus. In Proc. of the second International Workshop on Computational Methods in Systems Biology (CMSB04), Lecture Notes in Bioinformatics 3082:85-103, Springer, 2005....
Calculus introductory theory and applications in physical and life science
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
Dynamic Compartments in the Imperative π-Calculus
John, Mathias; Lhoussaine, Cédric; Niehren, Joachim
Dynamic compartments with mutable configurations and variable volumes are of basic interest for the stochastic modeling of biochemistry in cells. We propose a new language to express dynamic compartments that we call the imperative π -calculus. It is obtained from the attributed π -calculus by adding imperative assignment operations to a global store. Previous approaches to dynamic compartments are improved in flexibility or efficiency. This is illustrated by an appropriate model of osmosis and a correct encoding of bioambBioAmbients.
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...... of a language increases its expressive power by means of a separation result in the style of Palamidessi's result for mixed choice....
UTILIZING MICROSOFT MATHEMATICS IN TEACHING AND LEARNING CALCULUS
Rina Oktaviyanthi; Yani Supriani
2015-01-01
The experimental design was conducted to investigate the use of Microsoft Mathematics, free software made by Microsoft Corporation, in teaching and learning Calculus. This paper reports results from experimental study details on implementation of Microsoft Mathematics in Calculus, students’ achievement and the effects of the use of Microsoft Mathematics on students’ attitudes in relation to such experience. Two classes of the students from the first year student in Universitas Serang Raya wer...
Applications of Malliavin calculus to Monte Carlo methods in finance
Eric Fournié; Jean-Michel Lasry; Pierre-Louis Lions; Jérôme Lebuchoux; Nizar Touzi
1999-01-01
This paper presents an original probabilistic method for the numerical computations of Greeks (i.e. price sensitivities) in finance. Our approach is based on the {\\it integration-by-parts} formula, which lies at the core of the theory of variational stochastic calculus, as developed in the Malliavin calculus. The Greeks formulae, both with respect to initial conditions and for smooth perturbations of the local volatility, are provided for general discontinuous path-dependent payoff functional...
Fractional calculus transmutation for the Airy WKB solutions and Stokes phenomenon
Kiryakova, Virginia
2016-12-01
We apply the transmutation method to give a new explanation of the Stokes phenomenon for the Airy differential equation and of the change of the coeffcients in its asymptotic solutions for large values of argument in different parts of the complex plane. As a transmutation operator, a Weyl type fractional order integral is used. But this scheme is a special case of the so-called Poisson- Sonine-Dimovski transmutation operators related to the hyper-Bessel differential equations of arbitrary integer order, and of the generalized fractional calculus operators related to differential equations of fractional multi-order and their solutions, including a number of special functions. We analyze also the previous results of other authors and suggest some perspectives to use the same method in more general cases.
Hall, Angela Renee
2011-01-01
This investigative research focuses on the level of readiness of Science, Technology, Engineering, and Mathematics (STEM) students entering Historically Black Colleges and Universities (HBCU) in the college Calculus sequence. Calculus is a fundamental course for STEM courses. The level of readiness of the students for Calculus can very well play a…
Rewriting Modulo β in the λΠ-Calculus Modulo
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Ronan Saillard
2015-07-01
Full Text Available The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with dependent types where beta-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type systems in a shallow way. Basic properties such as subject reduction or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo. However, they hold if the rewrite system generated by the rewrite rules together with beta-reduction is confluent. But this is too restrictive. To handle the case where non confluence comes from the interference between the beta-reduction and rewrite rules with lambda-abstraction on their left-hand side, we introduce a notion of rewriting modulo beta for the lambda-Pi-calculus Modulo. We prove that confluence of rewriting modulo beta is enough to ensure subject reduction and uniqueness of types. We achieve our goal by encoding the lambda-Pi-calculus Modulo into Higher-Order Rewrite System (HRS. As a consequence, we also make the confluence results for HRSs available for the lambda-Pi-calculus Modulo.
Water content contribution in calculus phantom ablation during Q-switched Tm:YAG laser lithotripsy.
Zhang, Jian J; Rajabhandharaks, Danop; Xuan, Jason Rongwei; Wang, Hui; Chia, Ray W J; Hasenberg, Tom; Kang, Hyun Wook
2015-01-01
Q-switched (QS) Tm:YAG laser ablation mechanisms on urinary calculi are still unclear to researchers. Here, dependence of water content in calculus phantom on calculus ablation performance was investigated. White gypsum cement was used as a calculus phantom model. The calculus phantoms were ablated by a total 3-J laser pulse exposure (20 mJ, 100 Hz, 1.5 s) and contact mode with N=15 sample size. Ablation volume was obtained on average 0.079, 0.122, and 0.391 mm3 in dry calculus in air, wet calculus in air, and wet calculus in-water groups, respectively. There were three proposed ablation mechanisms that could explain the effect of water content in calculus phantom on calculus ablation performance, including shock wave due to laser pulse injection and bubble collapse, spallation, and microexplosion. Increased absorption coefficient of wet calculus can cause stronger spallation process compared with that caused by dry calculus; as a result, higher calculus ablation was observed in both wet calculus in air and wet calculus in water. The test result also indicates that the shock waves generated by short laser pulse under the in-water condition have great impact on the ablation volume by Tm:YAG QS laser.
q-fractional calculus and equations
Annaby, Mahmoud H
2012-01-01
This nine-chapter monograph introduces a rigorous investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson’s type before turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular q-Sturm–Liouville theory is also introduced; Green’s function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi. Hence fractional q-calculi of the types Riemann–Liouville; Grünwald–Letnikov; Caputo; Erdélyi–Kober and Weyl are defined analytically. Fractional q-Leibniz rules with applications in q-series are also obtained with rigorous proofs of the formal results of Al-Salam-Verma, which remained unproved for decades. In working ...
Meshes optimized for discrete exterior calculus (DEC).
Energy Technology Data Exchange (ETDEWEB)
Mousley, Sarah C. [Univ. of Illinois, Urbana-Champaign, IL (United States); Deakin, Michael [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Knupp, Patrick [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Mitchell, Scott A. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2017-12-01
We study the optimization of an energy function used by the meshing community to measure and improve mesh quality. This energy is non-traditional because it is dependent on both the primal triangulation and its dual Voronoi (power) diagram. The energy is a measure of the mesh's quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. The energy of a mesh gives an upper bound on the error of the discrete diagonal approximation of the Hodge star operator. In practice, one begins with an initial mesh and then makes adjustments to produce a mesh of lower energy. However, we have discovered several shortcomings in directly optimizing this energy, e.g. its non-convexity, and we show that the search for an optimized mesh may lead to mesh inversion (malformed triangles). We propose a new energy function to address some of these issues.
Şen, Mürsel; Çalık, Abdullah Engin; Ertik, Hüseyin
2014-09-01
Reduction of beta-ray intensity with respect to thickness of absorber material exhibits a non-exponential behavior due to the different types of the energy loss processes and many different fractal-like paths followed by beta particles in material. According to Caputo formalism of fractional calculus, the reduction process of beta-ray intensity is governed by using a simple fractional differential equation of order α ≈ 0.31. The solution is obtained in terms of Mittag-Leffler function which depends on a mass attenuation coefficient μm and a fractional order α that can be considered as a measure of fractality of absorbing material. In the experimental part of the study, 99Tc, 36Cl, 14C, 210Pb and 147Pm radioisotopes have been used as beta sources. In the framework of fractional calculus approach, the experimental and calculated half-value thicknesses of all samples have been obtained in agreement with each other.
Obstacles to Mathematization in Physics: The Case of the Differential
López-Gay, R.; Martinez Sáez, J.; Martinez Torregrosa, J.
2015-01-01
The process of the mathematization of physical situations through differential calculus requires an understanding of the justification for and the meaning of the differential in the context of physics. In this work, four different conceptions about the differential in physics are identified and assessed according to their utility for the…
On flipping first-semester calculus: a case study
Petrillo, Joseph
2016-05-01
High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are 'flipping' (or inverting) their classrooms. By flipping, we mean administering course content outside of the classroom and replacing the traditional in-class lectures with discussion, practice, group work, and other elements of active learning. This paper presents the major results from a three-year study of a flipped, first-semester calculus course at a small, comprehensive, American university with a well-known engineering programme. The data we have collected help quantify the positive and substantial effects of our flipped calculus course on failure rates, scores on the common final exam, student opinion of calculus, teacher impact on measurable outcomes, and success in second-semester calculus. While flipping may not be suitable for every teacher, every student, and in every situation, this report provides some evidence that it may be a viable option for those seeking an alternative to the traditional lecture model.
Backpropagation and ordered derivatives in the time scales calculus.
Seiffertt, John; Wunsch, Donald C
2010-08-01
Backpropagation is the most widely used neural network learning technique. It is based on the mathematical notion of an ordered derivative. In this paper, we present a formulation of ordered derivatives and the backpropagation training algorithm using the important emerging area of mathematics known as the time scales calculus. This calculus, with its potential for application to a wide variety of inter-disciplinary problems, is becoming a key area of mathematics. It is capable of unifying continuous and discrete analysis within one coherent theoretical framework. Using this calculus, we present here a generalization of backpropagation which is appropriate for cases beyond the specifically continuous or discrete. We develop a new multivariate chain rule of this calculus, define ordered derivatives on time scales, prove a key theorem about them, and derive the backpropagation weight update equations for a feedforward multilayer neural network architecture. By drawing together the time scales calculus and the area of neural network learning, we present the first connection of two major fields of research.
VOLMER, M; WOLTHERS, BG; METTING, HJ; DEHAAN, THY; COENEGRACHT, PMJ; VANDERSLIK, W
Infrared (IR) spectroscopy is used to analyze urinary calculus (renal stone) constituents. However, interpretation of IR spectra for quantifying urinary calculus constituents in mixtures is difficult, requiring expert knowledge by trained technicians. In our laboratory IR spectra of unknown calculi
Stochastic differential equations, backward SDEs, partial differential equations
Pardoux, Etienne
2014-01-01
This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter. Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has...
Differential form representation of stochastic electromagnetic fields
Directory of Open Access Journals (Sweden)
M. Haider
2017-09-01
Full Text Available In this work, we revisit the theory of stochastic electromagnetic fields using exterior differential forms. We present a short overview as well as a brief introduction to the application of differential forms in electromagnetic theory. Within the framework of exterior calculus we derive equations for the second order moments, describing stochastic electromagnetic fields. Since the resulting objects are continuous quantities in space, a discretization scheme based on the Method of Moments (MoM is introduced for numerical treatment. The MoM is applied in such a way, that the notation of exterior calculus is maintained while we still arrive at the same set of algebraic equations as obtained for the case of formulating the theory using the traditional notation of vector calculus. We conclude with an analytic calculation of the radiated electric field of two Hertzian dipole, excited by uncorrelated random currents.
Differential form representation of stochastic electromagnetic fields
Haider, Michael; Russer, Johannes A.
2017-09-01
In this work, we revisit the theory of stochastic electromagnetic fields using exterior differential forms. We present a short overview as well as a brief introduction to the application of differential forms in electromagnetic theory. Within the framework of exterior calculus we derive equations for the second order moments, describing stochastic electromagnetic fields. Since the resulting objects are continuous quantities in space, a discretization scheme based on the Method of Moments (MoM) is introduced for numerical treatment. The MoM is applied in such a way, that the notation of exterior calculus is maintained while we still arrive at the same set of algebraic equations as obtained for the case of formulating the theory using the traditional notation of vector calculus. We conclude with an analytic calculation of the radiated electric field of two Hertzian dipole, excited by uncorrelated random currents.
Multivariable calculus with Matlab with applications to geometry and physics
Lipsman, Ronald L
2017-01-01
This comprehensive treatment of multivariable calculus focuses on the numerous tools that MATLAB® brings to the subject, as it presents introductions to geometry, mathematical physics, and kinematics. Covering simple calculations with MATLAB®, relevant plots, integration, and optimization, the numerous problem sets encourage practice with newly learned skills that cultivate the reader’s understanding of the material. Significant examples illustrate each topic, and fundamental physical applications such as Kepler’s Law, electromagnetism, fluid flow, and energy estimation are brought to prominent position. Perfect for use as a supplement to any standard multivariable calculus text, a “mathematical methods in physics or engineering” class, for independent study, or even as the class text in an “honors” multivariable calculus course, this textbook will appeal to mathematics, engineering, and physical science students. MATLAB® is tightly integrated into every portion of this book, and its graphical ...
Bacteria and archaea paleomicrobiology of the dental calculus: a review.
Huynh, H T T; Verneau, J; Levasseur, A; Drancourt, M; Aboudharam, G
2016-06-01
Dental calculus, a material observed in the majority of adults worldwide, emerged as a source for correlating paleomicrobiology with human health and diet. This mini review of 48 articles on the paleomicrobiology of dental calculus over 7550 years discloses a secular core microbiota comprising nine bacterial phyla - Firmicutes, Actinobacteria, Proteobacteria, Bacteroidetes, TM7, Synergistetes, Chloroflexi, Fusobacteria, Spirochetes - and one archaeal phylum Euryarchaeota; and some accessory microbiota that appear and disappear according to time frame. The diet residues and oral microbes, including bacteria, archaea, viruses and fungi, consisting of harmless organisms and pathogens associated with local and systemic infections have been found trapped in ancient dental calculus by morphological approaches, immunolabeling techniques, isotope analyses, fluorescent in situ hybridization, DNA-based approaches, and protein-based approaches. These observations led to correlation of paleomicrobiology, particularly Streptococcus mutans and archaea, with past human health and diet. © 2015 John Wiley & Sons A/S. Published by John Wiley & Sons Ltd.
Golden quantum oscillator and Binet-Fibonacci calculus
Energy Technology Data Exchange (ETDEWEB)
Pashaev, Oktay K; Nalci, Sengul, E-mail: oktaypashaev@iyte.edu.tr [Department of Mathematics, Izmir Institute of Technology, Urla-Izmir 35430 (Turkey)
2012-01-13
The Binet formula for Fibonacci numbers is treated as a q-number and a q-operator with Golden ratio bases q = {phi} and Q = -1/{phi}, and the corresponding Fibonacci or Golden calculus is developed. A quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given only by Fibonacci numbers. The ratio of successive energy levels is found to be the Golden sequence, and for asymptotic states in the limit n {yields} {infinity} it appears as the Golden ratio. We call this oscillator the Golden oscillator. Using double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived. Relations of Fibonacci calculus with a q-deformed fermion oscillator and entangled N-qubit states are indicated. (paper)
Understanding and Interpreting Calculus Graphs: Refining an Instrument
Perez-Goytia, Nadia; Dominguez, Angeles; Zavala, Genaro
2010-10-01
The objective of this ongoing study is to refine an instrument to evaluate conceptual understanding and graphical interpretation of a function and its derivative. The instrument is based on a modified version of the Test of Understanding Graphs in Kinematics (TUG-K) which consists of 26 items (7 dimensions). In the new instrument, Test of Understanding Graphs in Calculus (TUG-C), the kinematics context has been removed from the items creating a new context-free version. To favor the translation from kinematics to Calculus, the focus is on 5 out of the 7 original dimensions of the test, giving a 16-item test. A total of 526 students from a university level Introductory Physics course participated in the study. Half of the students were administered the kinematics test and the other half took the calculus test. This work will present data showing preliminary results of the instrument and new directions on improving the instrument.
Dental wax decreases calculus accumulation in small dogs.
Smith, Mark M; Smithson, Christopher W
2014-01-01
A dental wax was evaluated after unilateral application in 20 client-owned, mixed and purebred small dogs using a clean, split-mouth study model. All dogs had clinical signs of periodontal disease including plaque, calculus, and/or gingivitis. The wax was randomly applied to the teeth of one side of the mouth daily for 30-days while the contralateral side received no treatment. Owner parameters evaluated included compliance and a subjective assessment of ease of wax application. Gingivitis, plaque and calculus accumulation were scored at the end of the study period. Owners considered the wax easy to apply in all dogs. Compliance with no missed application days was achieved in 8 dogs. The number of missed application days had no effect on wax efficacy. There was no significant difference in gingivitis or plaque accumulation scores when comparing treated and untreated sides. Calculus accumulation scores were significantly less (22.1 %) for teeth receiving the dental wax.
The early period of the calculus of variations
Freguglia, Paolo
2016-01-01
This monograph explores the early development of the calculus of variations in continental Europe during the Eighteenth Century by illustrating the mathematics of its founders. Closely following the original papers and correspondences of Euler, Lagrange, the Bernoullis, and others, the reader is immersed in the challenge of theory building. We see what the founders were doing, the difficulties they faced, the mistakes they made, and their triumphs. The authors guide the reader through these works with instructive commentaries and complements to the original proofs, as well as offering a modern perspective where useful. The authors begin in 1697 with Johann Bernoulli’s work on the brachystochrone problem and the events leading up to it, marking the dawn of the calculus of variations. From there, they cover key advances in the theory up to the development of Lagrange’s δ-calculus, including: • The isoperimetrical problems • Shortest lines and geodesics • Euler’s Methodus Inveniendi and the two Addi...
Integral calculus problem solving: an fMRI investigation.
Krueger, Frank; Spampinato, Maria Vittoria; Pardini, Matteo; Pajevic, Sinisa; Wood, Jacqueline N; Weiss, George H; Landgraf, Steffen; Grafman, Jordan
2008-07-16
Only a subset of adults acquires specific advanced mathematical skills, such as integral calculus. The representation of more sophisticated mathematical concepts probably evolved from basic number systems; however its neuroanatomical basis is still unknown. Using fMRI, we investigated the neural basis of integral calculus while healthy participants were engaged in an integration verification task. Solving integrals activated a left-lateralized cortical network including the horizontal intraparietal sulcus, posterior superior parietal lobe, posterior cingulate gyrus, and dorsolateral prefrontal cortex. Our results indicate that solving of more abstract and sophisticated mathematical facts, such as calculus integrals, elicits a pattern of brain activation similar to the cortical network engaged in basic numeric comparison, quantity manipulation, and arithmetic problem solving.
UTILIZING MICROSOFT MATHEMATICS IN TEACHING AND LEARNING CALCULUS
Directory of Open Access Journals (Sweden)
Rina Oktaviyanthi
2015-01-01
Full Text Available The experimental design was conducted to investigate the use of Microsoft Mathematics, free software made by Microsoft Corporation, in teaching and learning Calculus. This paper reports results from experimental study details on implementation of Microsoft Mathematics in Calculus, students’ achievement and the effects of the use of Microsoft Mathematics on students’ attitudes in relation to such experience. Two classes of the students from the first year student in Universitas Serang Raya were participated in the study. This study found that students who taught by using Microsoft Mathematics had higher achievement and has a positive effect on students’ confidence of mathematics.Keywords: Calculus, Educational Technology, Mathematics Education, Microsoft Mathematics, Technology Application DOI: dx.doi.org/10.22342/jme.61.63
A study of ∇-discrete fractional calculus operator on the radial ...
African Journals Online (AJOL)
The fractional calculus includes concepts of integrals and derivatives of any complex or real order. The fractional calculus is as old as the usual calculus. Recently, many scientists have been studying on this eld to provide the development and applicability to various areas of mathematics, physics, engineering and other ...
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…
Teaching and learning of pre-calculus: an insights of educators ...
African Journals Online (AJOL)
The high-failure of Calculus courses has bring dilemma to the educators worldwide. This study has focus on teaching and learning of Pre-Calculus, a fundamental course for higher level Calculus courses. A well-designed questionnaire was distributed to all respondents to find the difficulties faced by both lecturers and ...
The Development and Nature of Problem-Solving among First-Semester Calculus Students
Dawkins, Paul Christian; Epperson, James A. Mendoza
2014-01-01
This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students' problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We investigate…
Recalling Prerequisite Material in a Calculus II Course to Improve Student Success
Mokry, Jeanette
2016-01-01
This article discusses preparation assignments used in a Calculus II course that cover material from prerequisite courses. Prior to learning new material, students work on problems outside of class involving concepts from algebra, trigonometry, and Calculus I. These problems are directly built upon in order to answer Calculus II questions,…
Linear Uniform Receptiveness in a Pi-Calculus with Location Failures
DEFF Research Database (Denmark)
Hüttel, Hans; Garm Nielsen, Johannes; Finderup, Peter
2010-01-01
The notion of receptiveness arises in the pi-calculus as a guarantee of determinacy in the behaviour of callable entities and was first investigated by Sangiorgi. The DpiF process calculus, introduced by Francalanza and Hennessy, extends the pi-calculus with located processes and location and link...
Control Flow Analysis for the Pi-calculus
DEFF Research Database (Denmark)
Bodei, C.; Degano, P.; Nielson, Flemming
1998-01-01
Control Flow Analysis is a static technique for predicting safe and computable approximations to the set of values that the objects of a program may assume during its execution. We present an analysis for the pi-calculus that shows how names will be bound to actual channels at run time....... The formulation of the analysis requires no extensions to the pi-calculus, except for assigning ``channels'' to the occurrences of names within restrictions, and assigning ``binders'' to the occurrences of names within input prefixes. The result of our analysis establishes a super-set of the set of names to which...
Three qubit entanglement within graphical Z/X-calculus
Directory of Open Access Journals (Sweden)
Bob Coecke
2011-03-01
Full Text Available The compositional techniques of categorical quantum mechanics are applied to analyse 3-qubit quantum entanglement. In particular the graphical calculus of complementary observables and corresponding phases due to Duncan and one of the authors is used to construct representative members of the two genuinely tripartite SLOCC classes of 3-qubit entangled states, GHZ and W. This nicely illustrates the respectively pairwise and global tripartite entanglement found in the W- and GHZ-class states. A new concept of supplementarity allows us to characterise inhabitants of the W class within the abstract diagrammatic calculus; these method extends to more general multipartite qubit states.
A Calculus for Control Flow Analysis of Security Protocols
DEFF Research Database (Denmark)
Buchholtz, Mikael; Nielson, Hanne Riis; Nielson, Flemming
2004-01-01
analysis methodology. We pursue an analysis methodology based on control flow analysis in flow logic style and we have previously shown its ability to analyse a variety of security protocols. This paper develops a calculus, LysaNS that allows for much greater control and clarity in the description......The design of a process calculus for anaysing security protocols is governed by three factors: how to express the security protocol in a precise and faithful manner, how to accommodate the variety of attack scenarios, and how to utilise the strengths (and limit the weaknesses) of the underlying...
Laser modeling a numerical approach with algebra and calculus
Csele, Mark Steven
2014-01-01
Offering a fresh take on laser engineering, Laser Modeling: A Numerical Approach with Algebra and Calculus presents algebraic models and traditional calculus-based methods in tandem to make concepts easier to digest and apply in the real world. Each technique is introduced alongside a practical, solved example based on a commercial laser. Assuming some knowledge of the nature of light, emission of radiation, and basic atomic physics, the text:Explains how to formulate an accurate gain threshold equation as well as determine small-signal gainDiscusses gain saturation and introduces a novel pass
The GHZ/W-calculus contains rational arithmetic
Directory of Open Access Journals (Sweden)
Bob Coecke
2011-03-01
Full Text Available Graphical calculi for representing interacting quantum systems serve a number of purposes: compositionally, intuitive graphical reasoning, and a logical underpinning for automation. The power of these calculi stems from the fact that they embody generalized symmetries of the structure of quantum operations, which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One such calculus takes the GHZ and W states as its basic generators. Here we show that this language allows one to encode standard rational calculus, with the GHZ state as multiplication, the W state as addition, the Pauli X gate as multiplicative inversion, and the Pauli Z gate as additive inversion.
Utilizing Microsoft Mathematics In Teaching And Learning Calculus
Directory of Open Access Journals (Sweden)
Rina Oktaviyanthi
2015-10-01
Full Text Available The experimental design was conducted to investigate the use of Microsoft Mathematics, free software made by Microsoft Corporation, in teaching and learning Calculus. This paper reports results from experimental study details on implementation of Microsoft Mathematics in Calculus, students’ achievement and the effects of the use of Microsoft Mathematics on students’ attitudes in relation to such experience. Two classes of the students from the first year student in Universitas Serang Raya were participated in the study. This study found that students who taught by using Microsoft Mathematics had higher achievement and has a positive effect on students’ confidence of mathematics.
Improved diffusion Monte Carlo propagators for bosonic systems using Itô calculus
Hâkansson, P.; Mella, M.; Bressanini, Dario; Morosi, Gabriele; Patrone, Marta
2006-11-01
The construction of importance sampled diffusion Monte Carlo (DMC) schemes accurate to second order in the time step is discussed. A central aspect in obtaining efficient second order schemes is the numerical solution of the stochastic differential equation (SDE) associated with the Fokker-Plank equation responsible for the importance sampling procedure. In this work, stochastic predictor-corrector schemes solving the SDE and consistent with Itô calculus are used in DMC simulations of helium clusters. These schemes are numerically compared with alternative algorithms obtained by splitting the Fokker-Plank operator, an approach that we analyze using the analytical tools provided by Itô calculus. The numerical results show that predictor-corrector methods are indeed accurate to second order in the time step and that they present a smaller time step bias and a better efficiency than second order split-operator derived schemes when computing ensemble averages for bosonic systems. The possible extension of the predictor-corrector methods to higher orders is also discussed.
Differential equations for dummies
Holzner, Steven
2008-01-01
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Calculus an intuitive and physical approach
Kline, Morris
1998-01-01
Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, and theorems on differentiation and antidifferentiation lead to a definition of the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Clear-cut explanations, numerous drills, illustrative examples. 1967 edition. Solution guide available upon request.
Mahalingam, Harshavardhan; Lal, Anupam; Mandal, Arup K; Singh, Shrawan Kumar; Bhattacharyya, Shalmoli; Khandelwal, Niranjan
2015-08-01
This study aimed to assess the accuracy of low-dose dual-energy computed tomography (DECT) in predicting the composition of urinary calculi. A total of 52 patients with urinary calculi were scanned with a 128-slice dual-source DECT scanner by use of a low-dose protocol. Dual-energy (DE) ratio, weighted average Hounsfield unit (HU) of calculi, radiation dose, and image noise levels were recorded. Two radiologists independently rated study quality. Stone composition was assessed after extraction by Fourier transform infrared spectroscopy (FTIRS). Analysis of variance was used to determine if the differences in HU values and DE ratios between the various calculus groups were significant. Threshold cutoff values to classify the calculi into separate groups were identified by receiver operating characteristic curve analysis. A total of 137 calculi were detected. FTIRS analysis differentiated the calculi into five groups: uric acid (n=17), struvite (n=3), calcium oxalate monohydrate and dihydrate (COM-COD, n=84), calcium oxalate monohydrate (COM, n=28), and carbonate apatite (n=5). The HU value could differentiate only uric acid calculi from calcified calculi (p80% sensitivity and specificity to differentiate them. The DE ratio could not differentiate COM from COM-COD calculi. No study was rated poor in quality by either of the observers. The mean radiation dose was 1.8 mSv. Low-dose DECT accurately predicts urinary calculus composition in vivo while simultaneously reducing radiation exposure without compromising study quality.
Advances in fractional calculus theoretical developments and applications in physics and engineering
Sabatier, J; Machado, J A Teneiro
2007-01-01
Fractional Calculus is a new growing field. Up to this point, researchers, scientists, and engineers have been reluctant to accept the fact that Fractional Calculus can be used in the analysis and design of many systems of practical interests, whereas in similar applications the traditional calculus either fails or provides poor solutionsMany engineers, scientists, and applied mathematicians are looking for books that can provide many applications of Fractional Calculus. This book will provide a partial solution to this problem. Since it covers recent applications of Fractional Calculus, it will be attractive to many engineers, scientists, and applied mathematicians.
Peer Conferences in Calculus: The Impact of Systematic Training
Reinholz, Daniel
2017-01-01
This paper describes an intervention for improving the quality of peer assessment conferences in calculus. Although a body of work highlights the learning benefits of peer assessment, few papers have described the nature of student conversations during peer conferences/assessment in detail. This paper provides deeper insight into what those…
Spontaneous calyceal rupture caused by a ureteral calculus
African Journals Online (AJOL)
A. Chaabouni
Ureteroscopy revealed a little calculus in distal ureter. The patient successfully underwent endoscopic treatment. The clinical manifestation of spontaneous urinary extravasation is diverse, ranging from mild flank discomfort to unremitting abdom- inal pain such as acute abdomen. Gastrointestinal symptoms like nausea and ...
Algorithms, The Calculus and Programming-An Intuitive Approach
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 18; Issue 4. Algorithms, The Calculus and Programming - An Intuitive Approach. Abhijat Vichare. General Article Volume 18 Issue 4 April 2013 pp 345-367. Fulltext. Click here to view fulltext PDF. Permanent link:
Mathematica empowered applets for learning school algebra and calculus
Heck, A.; Boon, P.; van Velthoven, W.
2008-01-01
We describe the development and use of Mathematica empowered applets for learning and practising algebra and calculus at secondary school level. We discuss how Mathematica can be used as back engine to evaluate students' answers and provide appropriate feedback. These applets provide scoring
Airfoil Design in Multivariable Calculus: Tying It All Together
Laverty, Rich; Povich, Timothy; Williams, Tasha
2005-01-01
Near the conclusion of their final term in the calculus sequence at The United States Military Academy, cadets are given a week long group project. At the end of the week, the project is briefed to their instructors, classmates, and superior officers. From a teaching perspective, the goal is to encapsulate as much of the course as possible in one…
Assessing Online Homework in First-Semester Calculus
Callahan, Jason T.
2016-01-01
This paper describes and assesses the implementation of online homework in a first-semester calculus course. Comparing sections of the course before implementation to those after, we find statistically significant improvements in retention rates, measures of student engagement, and participation on homework. We do not, however, find statistically…
Towards a calculus for expressive timing in music
Desain, P.; Honing, H.
1991-01-01
This paper presents a calculus that enables expressive timing to be transformed on the basis of the structural aspects of the music. Expression within a unit is defined as the deviations of its parts with respect to the norm set by the unit itself. The behaviour of musical material under expressive
Relation of Spatial Skills to Calculus Proficiency: A Brief Report
Cromley, Jennifer G.; Booth, Julie L.; Wills, Theodore W.; Chang, Briana L.; Tran, Nhi; Madeja, Michael; Shipley, Thomas F.; Zahner, William
2017-01-01
Spatial skills have been shown in various longitudinal studies to be related to multiple science, technology, engineering, and math (STEM) achievement and retention. The specific nature of this relation has been probed in only a few domains, and has rarely been investigated for calculus, a critical topic in preparing students for and in STEM…
Quantum calculus new concepts, impulsive IVPs and BVPs, inequalities
Ahmad, Bashir; Tariboon, Jessada
2016-01-01
The main objective of this book is to extend the scope of the q-calculus based on the definition of q-derivative [Jackson (1910)] to make it applicable to dense domains. As a matter of fact, Jackson's definition of q-derivative fails to work for impulse points while this situation does not arise for impulsive equations on q-time scales as the domains consist of isolated points covering the case of consecutive points. In precise terms, we study quantum calculus on finite intervals.In the first part, we discuss the concepts of qk-derivative and qk-integral, and establish their basic properties. As applications, we study initial and boundary value problems of impulsive qk-difference equations and inclusions equipped with different kinds of boundary conditions. We also transform some classical integral inequalities and develop some new integral inequalities for convex functions in the context of qk-calculus. In the second part, we develop fractional quantum calculus in relation to a new qk-shifting operator and e...
Calculus Students' Understanding of Area and Volume Units
Dorko, Allison; Speer, Natasha
2015-01-01
Units of measure are critical in many scientific fields. While instructors often note that students struggle with units, little research has been conducted about the nature and extent of these difficulties or why they exist. We investigated calculus students' unit use in area and volume computations. Seventy-three percent of students gave…
Using Origami Boxes to Explore Concepts of Geometry and Calculus
Wares, Arsalan
2011-01-01
The purpose of this classroom note is to provide an example of how a simple origami box can be used to explore important concepts of geometry and calculus. This article describes how an origami box can be folded, then it goes on to describe how its volume and surface area can be calculated. Finally, it describes how the box could be folded to…
Executable Behaviour and the π-Calculus (extended abstract
Directory of Open Access Journals (Sweden)
Bas Luttik
2015-08-01
Full Text Available Reactive Turing machines extend classical Turing machines with a facility to model observable interactive behaviour. We call a behaviour executable if, and only if, it is behaviourally equivalent to the behaviour of a reactive Turing machine. In this paper, we study the relationship between executable behaviour and behaviour that can be specified in the pi-calculus. We establish that all executable behaviour can be specified in the pi-calculus up to divergence-preserving branching bisimilarity. The converse, however, is not true due to (intended limitations of the model of reactive Turing machines. That is, the pi-calculus allows the specification of behaviour that is not executable up to divergence-preserving branching bisimilarity. Motivated by an intuitive understanding of executability, we then consider a restriction on the operational semantics of the pi-calculus that does associate with every pi-term executable behaviour, at least up to the version of branching bisimilarity that does not require the preservation of divergence.
A Team Taught Interdisciplinary Approach To Physics and Calculus Education.
Johnson, David B.
The Special Intensive Program for Scientists and Engineers (SIPSE) at Diablo Valley College in California replaces the traditional engineering calculus and physics sequences with a single sequence that combines the two subjects into an integrated whole. The project report provides an overview of SIPSE, a section that traces the project from…
SOME CONSIDERATIONS REGARDING THE STRENGTH CALCULUS OF A LATHE TOOL
Directory of Open Access Journals (Sweden)
Catălin ROŞU
2013-05-01
Full Text Available In this paper a strength calculus of a lathe tool is made. The main purpose of this study is to determineequivalent stress relations that can be useful for an engineer in the design situation. The simplifying assumptionsand the equivalent stress relations are presented from an original point of view.
Formalization of the Resolution Calculus for First-Order Logic
DEFF Research Database (Denmark)
Schlichtkrull, Anders
2016-01-01
A formalization in Isabelle/HOL of the resolution calculus for first-order logic is presented. Its soundness and completeness are formally proven using the substitution lemma, semantic trees, Herbrand’s theorem, and the lifting lemma. In contrast to previous formalizations of resolution...
Formalization of the Integral Calculus in the PVS Theorem Prover
Directory of Open Access Journals (Sweden)
Ricky Wayne Butler
2009-04-01
Full Text Available The PVS Theorem prover is a widely used formal verification tool used for the analysis of safetycritical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higher-order logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and Computer Science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed.The theory and proofs were based on Rosenlicht’s classic text on real analysis and follow the traditional epsilon-delta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.
Using the Pottery Wheel to Explore Topics in Calculus
Farnell, Elin; Snipes, Marie A.
2015-01-01
Students sometimes struggle with visualizing the three-dimensional solids encountered in certain integral problems in a calculus class. We present a project in which students create solids of revolution with clay on a pottery wheel and estimate the volumes of these objects using Riemann sums. In addition to giving students an opportunity for…
Formalization of the Integral Calculus in the PVS Theorem Prover
Butler, Ricky W.
2004-01-01
The PVS Theorem prover is a widely used formal verification tool used for the analysis of safety-critical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higher-order logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and Computer Science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed.The theory and proofs were based on Rosenlicht's classic text on real analysis and follow the traditional epsilon-delta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.
Inertial Navigation: A Bridge between Kinematics and Calculus
Sadler, Philip M.; Garfield, Eliza N.; Tremblay, Alex; Sadler, Daniel J.
2012-01-01
Those who come to Cambridge soon learn that the fastest route between Harvard and MIT is by the subway. For many students, this short ride is a quick and easy way to link physics and calculus. A simple, homemade accelerometer provides all the instrumentation necessary to produce accurate graphs of acceleration, velocity, and displacement position…
Non-Mathematics Students' Reasoning in Calculus Tasks
Jukic Matic, Ljerka
2015-01-01
This paper investigates the reasoning of first year non-mathematics students in non-routine calculus tasks. The students in this study were accustomed to imitative reasoning from their primary and secondary education. In order to move from imitative reasoning toward more creative reasoning, non-routine tasks were implemented as an explicit part of…
Unusual giant prostatic urethral calculus | Bello | Journal of Surgical ...
African Journals Online (AJOL)
Giant vesico-prostatic urethral calculus is uncommon. Urethral stones rarely form primarily in the urethra, and they are usually associated with urethral strictures, posterior urethral valve or diverticula. We report a case of a 32-year-old man with giant vesico-prostatic (collar-stud) urethral stone presenting with sepsis and ...
The Resolution Calculus for First-Order Logic
DEFF Research Database (Denmark)
Schlichtkrull, Anders
2016-01-01
This theory is a formalization of the resolution calculus for first-order logic. It is proven sound and complete. The soundness proof uses the substitution lemma, which shows a correspondence between substitutions and updates to an environment. The completeness proof uses semantic trees, i.e. trees...
Flipping the Calculus Classroom: A Cost-Effective Approach
Young, Andrea
2015-01-01
This article discusses a cost-effective approach to flipping the calculus classroom. In particular, the emphasis is on low-cost choices, both monetarily and with regards to faculty time, that make the daunting task of flipping a course manageable for a single instructor. Student feedback and overall impressions are also presented.
Bladder calculus resulting from an intravesical translocation of ...
African Journals Online (AJOL)
Although perforation of the uterus by an intrauterine contraceptive device (IUCD) is commonly encountered, intravesical translocation and secondary calculus formation is a very rare complication.We report a case of a 60-year old multiparous woman in whom an intrauterine contraceptive Copper-T device inserted 12 years ...
Simple Trigonometric Identities and Basic Calculus Leading to ...
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 12; Issue 6. Simple Trigonometric Identities and Basic Calculus Leading to Interesting Series. R Govindaraj. Classroom Volume 12 Issue 6 June 2007 pp 67-70. Fulltext. Click here to view fulltext PDF. Permanent link:
Calculus of multivariate functions: it's application in business | Awen ...
African Journals Online (AJOL)
Calculus of multivariate functions is a mathematical concept that has to do with two sets of variables, the dependent variables and the independent variables. Multivariate functions can be applied to situations in business organizations like a manufacturer‟s profit (which is a dependent variable) depending on sales, ...
Inquiry-Based Learning of Transcendental Functions in Calculus
Ekici, Celil; Gard, Andrew
2017-01-01
In a series of group activities supplemented with independent explorations and assignments, calculus students investigate functions similar to their own derivatives. Graphical, numerical, and algebraic perspectives are suggested, leading students to develop deep intuition into elementary transcendental functions even as they lay the foundation for…
Bladder Calculus Following an Unusual Vesical Foreign Body ...
African Journals Online (AJOL)
Bladder calculus following vesical foreign bodies is uncommon. The usual presentation is presence of lower urinary symptoms. Most of these foreign bodies are either left inadvertently after open bladder operations or migrate from adjacent structures. This is a case report of an unusual self inserted foreign body in a female ...
On Flipping First-Semester Calculus: A Case Study
Petrillo, Joseph
2016-01-01
High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are "flipping" (or inverting) their classrooms. By flipping, we…
Contrasting Cases of Calculus Students' Understanding of Derivative Graphs
Haciomeroglu, Erhan Selcuk; Aspinwall, Leslie; Presmeg, Norma C.
2010-01-01
This study adds momentum to the ongoing discussion clarifying the merits of visualization and analysis in mathematical thinking. Our goal was to gain understanding of three calculus students' mental processes and images used to create meaning for derivative graphs. We contrast the thinking processes of these three students as they attempted to…
Calculus of One and More Variables with Maple
Samkova, Libuse
2012-01-01
This is a guide to using Maple in teaching fundamental calculus of one, two and three variables (limits, derivatives, integrals, etc.), also suitable for Maple beginners. It outlines one of the ways to effective use of computers in the teaching process. It scans advantages and disadvantages of using Maple in relation to students and teacher. The…
Calculus Problem Solution And Simulation Using GUI Of Matlab
Directory of Open Access Journals (Sweden)
Syaharuddin
2017-09-01
Full Text Available This research aims to develop the project application in teaching and learning Calculus using Grapichal User Interface GUI of Matlab. The scope of the development result includes 1 PreCalculus 2 Function Limit 3 Derivative and its application and 4 Integral and its application. Every aspect of development is presented according to the standards of material competence learning achievement indicators and solutions or settlement steps on each issue presented. This project application can be used by lecturers and students in universities to improve motivation mastery of material and student learning outcomes in the Calculus course. Development of this project application is done through 4 stages called 4-D namely Define Design Development and Dissemination. Based on the data analysis the results obtained at the validation stage of expert validity level average of 3.575 which means quite valid. Then on a limited trial the average student gave a response of 92.00 which means very good. While in the first field trial the average student gave a response of 89.30 which means very good and in the second field trial the average student gave a responded of 90.15 which means very good. Besides that we also solve computational calculus problems in Edwin J. Purcell Dale Varbergs Book of Volume 1 of Edition 5 of 48 questions that are divided into 64 cases.
A metric model of lambda calculus with guarded recursion
DEFF Research Database (Denmark)
Birkedal, Lars; Schwinghammer, Jan; Støvring, Kristian
2010-01-01
We give a model for Nakano’s typed lambda calculus with guarded recursive definitions in a category of metric spaces. By proving a computational adequacy result that relates the interpretation with the operational semantics, we show that the model can be used to reason about contextual equivalence....
Elements of stream calculus : an extensive exercise in coinduction
J.J.M.M. Rutten (Jan)
2001-01-01
textabstractBased on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers), a coinductive calculus of streams is developed. The main ingredient is the notion of stream derivative, with which both coinductive proofs and definitions can be formulated.
Integrating Precalculus Review with the First Course in Calculus.
Sevilla, Alicia; Somers, Kay
1993-01-01
Describes a course designed by Moravian College, Pennsylvania, to integrate precalculus topics as needed into a first calculus course. The textbook developed for the course covers the concepts of functions, Cartesian coordinates, limits, continuity, infinity, and the derivative. Examples are discussed. (MDH)
A Calculus for Modelling, Simulating and Analysing Compartmentalized Biological Systems
DEFF Research Database (Denmark)
Mardare, Radu Iulian; Ihekwaba, Adoha
2007-01-01
A. Ihekwaba, R. Mardare. A Calculus for Modelling, Simulating and Analysing Compartmentalized Biological Systems. Case study: NFkB system. In Proc. of International Conference of Computational Methods in Sciences and Engineering (ICCMSE), American Institute of Physics, AIP Proceedings, N 2...
Functional Calculus via Transference, Double Operator Integrals and Applications
Rozendaal, J.
2015-01-01
This thesis is dedicated to the study of several aspects of the theory of functional calculus. This theory considers the combination of an operator A and a function f(z) of a variable z, resulting in an operator f(A). One then attempts to study properties of the operator f(A) in terms of properties
The Vector Calculus Gap: Mathematics (Does Not Equal) Physics.
Dray, Tevian; Manogue, Corinne A.
1999-01-01
Discusses some of the differences between the ways mathematicians and physicists view vector calculus and the gap between the way this material is traditionally taught by mathematicians and the way physicists use it. Suggests some ways to narrow the gap. (Author/ASK)
Use of WIRIS Quizzes in an Online Calculus Course
Calm, Remei; Masià, Ramon; Olivé, Carme; Parés, Núria; Pozo, Francesc; Ripoll, Jordi; Sancho-Vinuesa, Teresa
2017-01-01
Calculus courses often present a large number of difficulties to undergraduate students of scientific studies, especially in engineering degrees. These difficulties are sometimes related to teaching and assessment strategies. In this paper, a teaching innovation experience is presented within the framework of the Universitat Oberta de Catalunya.…
Liu, Ping; He, Xinrong; Guo, Mei
2010-04-01
To investigate the correlation effects between single or combined administration of Calculus Bovis or zolpidem and changes of inhibitive neurotransmitter in rat striatum corpora. Sampling from rat striatum corpora was carried out through microdialysis. The content of two inhibitive neurotransmitters in rat corpus striatum- glycine (Gly) and gama aminobutyric acid (GABA), was determined by HPLC, which involved pre-column derivation with orthophthaladehyde, reversed-phase gradient elution and fluorescence detection. GABA content of rat striatum corpora in Calculus Bovis group was significantly increased compared with saline group (P Calculus Boris plus zolpidem group were increased largely compared with saline group as well (P Calculus Bovis group was higher than combination group (P Calculus Bovis or zolpidem group was markedly increased compared with saline group or combination group (P Calculus Bovis group, zolpidem group and combination group. The magnitude of increase was lower in combination group than in Calculus Bovis group and Zolpidem group, suggesting that Calculus Bovis promoted encephalon inhibition is more powerful than zolpidem. The increase in two inhibitive neurotransmitters did not show reinforcing effect in combination group, suggesting that Calculus Bovis and zolpidem may compete the same receptors. Therefore, combination of Calculus Bovis containing drugs and zolpidem has no clinical significance. Calculus Bovis shouldn't as an aperture-opening drugs be used for resuscitation therapy.
Assessing college students' retention and transfer from calculus to physics
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
Fractional vector calculus and fluid mechanics
Lazopoulos, Konstantinos A.; Lazopoulos, Anastasios K.
2017-04-01
Basic fluid mechanics equations are studied and revised under the prism of fractional continuum mechanics (FCM), a very promising research field that satisfies both experimental and theoretical demands. The geometry of the fractional differential has been clarified corrected and the geometry of the fractional tangent spaces of a manifold has been studied in Lazopoulos and Lazopoulos (Lazopoulos KA, Lazopoulos AK. Progr. Fract. Differ. Appl. 2016, 2, 85-104), providing the bases of the missing fractional differential geometry. Therefore, a lot can be contributed to fractional hydrodynamics: the basic fractional fluid equations (Navier Stokes, Euler and Bernoulli) are derived and fractional Darcy's flow in porous media is studied.
Differential geometry and mathematical physics
Rudolph, Gerd
Starting from an undergraduate level, this book systematically develops the basics of • Calculus on manifolds, vector bundles, vector fields and differential forms, • Lie groups and Lie group actions, • Linear symplectic algebra and symplectic geometry, • Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory. The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics. The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous d...
CALCULUS FROM THE PAST: MULTIPLE DELAY SYSTEMS ARISING IN CANCER CELL MODELLING
WAKE, G. C.
2013-01-01
Nonlocal calculus is often overlooked in the mathematics curriculum. In this paper we present an interesting new class of nonlocal problems that arise from modelling the growth and division of cells, especially cancer cells, as they progress through the cell cycle. The cellular biomass is assumed to be unstructured in size or position, and its evolution governed by a time-dependent system of ordinary differential equations with multiple time delays. The system is linear and taken to be autonomous. As a result, it is possible to reduce its solution to that of a nonlinear matrix eigenvalue problem. This method is illustrated by considering case studies, including a model of the cell cycle developed recently by Simms, Bean and Koeber. The paper concludes by explaining how asymptotic expressions for the distribution of cells across the compartments can be determined and used to assess the impact of different chemotherapeutic agents. Copyright © 2013 Australian Mathematical Society.
Directory of Open Access Journals (Sweden)
Aristophanes Dimakis
2013-02-01
Full Text Available We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D−2 commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski-Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato and five (single and double Myers-Perry black holes, black saturn, bicycling black rings.
Primary Vaginal Calculus in a Woman with Disability: Case Report and Literature Review.
Castellan, Pietro; Nicolai, Michele; De Francesco, Piergustavo; Di Tizio, Luciano; Castellucci, Roberto; Bada, Maida; Marchioni, Michele; Cindolo, Luca; Schips, Luigi
2017-01-01
Background: Vaginal stones are rare and often unknown entities. Most urologists may never see a case in their careers. Case Presentation: We present the case of a 34-year-old bedridden Caucasian woman with mental and physical disabilities who presented with a large primary vaginal calculus, which, surprisingly, had remained undiagnosed until the patient suffered a right renal colic caused by a ureteral stone. The vagina was completely filled and a digital examination was not possible. For this reason, the stone was removed using surgical pliers with some maneuvering. A vesicovaginal fistula was excluded, as well as foreign bodies or other nidi of infection. After, urethral lithotripsy was performed as planned. The postoperative course and follow-up were uneventful. Conclusion: Although vaginal calculi are extremely rare in literature, their differential diagnosis should be considered in women with incontinence and associated disabilities, paraplegia, or prolonged immobilization in recumbent position.
SOME ASPECTS OF THE MANAGEMENT OF BILATERAL CALCULUS DISEASE OF THE KIDNEY
Flocks, R. H.
1950-01-01
In the diagnosis of bilateral calculus disease of the kidneys, it is important to differentiate between cystine, uric acid, calcium oxalate and phosphate renal lithiasis. Methods for distinguishing one from another are described. Dietary therapy is the method of choice for cystine and uric acid lithiasis. In calcium and phosphate urolithiasis, dietary therapy is a very useful adjunct. It must be regulated by careful studies of its effect on urinary calcium precipitability, a new test for which is described based upon the demonstration of the existence of two forms of calcium in the urine. Irrigation therapy for calcium phosphate and phosphate lithiasis is briefly discussed. Surgical therapy for large renal phosphatic calculi is discussed to show how considerations of renal counterbalance and urinary calcium, magnesium and phosphate excretion through damaged kidney substances influence the surgical plan in each case. ImagesFigure 1.Figure 5.Figure 4.Figure 4. PMID:15408493
The symmetric Mellin transform in quantum calculus
Directory of Open Access Journals (Sweden)
Brahim Kamel Brahim
2015-12-01
Full Text Available In this paper, we define the q-analogue of Mellin Transform symmetric under interchange of q and 1/q, and present some of its main properties and explore the possibility of using the integral transform to solve a class of differential equations q-differences.
Composition formulas in the Weyl calculus
DEFF Research Database (Denmark)
Kobayashi, Toshiyuki; Ørsted, Bent; Pevzner, Michael
2009-01-01
In pseudodifferential analysis, the usual composition formula, which has asymptotic value, extends that valid for differential operators. The one developed here is based instead on the decomposition of symbols (functions in Rn×Rn ) as integral superpositions of homogeneous ones, of degrees lying ...
Stability analysis of a class of fractional delay differential equations
Indian Academy of Sciences (India)
[14] A A Kilbas, H M Srivastava and J J Trujillo, Theory and applications of fractional differential equations (Elsevier, Amsterdam, 2006). [15] R L Magin, Fractional calculus in bioengineering (Begll House Publishers, USA, 2006). [16] H Smith, An introduction to delay differential equations with applications to the life sciences.
The impacts of gingivitis and calculus on Thai children's quality of life.
Krisdapong, Sudaduang; Prasertsom, Piyada; Rattanarangsima, Khanit; Sheiham, Aubrey; Tsakos, Georgios
2012-09-01
To assess associations of socio-demographic, behavioural and the extent of gingivitis and calculus with oral health-related quality of life (OHRQoL) in nationally representative samples of 12- and 15-year-old Thai children. In the Thailand National Oral Health Survey, 1,063 twelve-year olds and 811 fifteen-year olds were clinically examined and interviewed for OHRQoL using the Child-OIDP and OIDP indices, respectively, and completed a behavioural questionnaire. We assessed associations of condition-specific impacts (CS-impacts) with gingivitis and calculus, adjusted for socio-demographic and behavioural factors. Gingivitis and calculus were highly prevalent: 79.3% in 12-year and 81.5% in 15-year olds. CS-impacts relating to calculus and/or gingivitis were reported by 26.0% of 12-year and 29.6% of 15-year olds. Except for calculus without gingivitis, calculus and/or gingivitis in any form was significantly related to any level of CS-impacts. At a moderate or higher level of CS-impacts, there were significant relationships with extensive calculus and/or gingivitis in 12-year olds and for extensive gingivitis and gingivitis without calculus in 15-year olds. Gingivitis was generally associated with any level of CS-impacts attributed to calculus and/or gingivitis. CS-impacts were related more to gingivitis than to calculus. © 2012 John Wiley & Sons A/S.
Intelligent numerical methods applications to fractional calculus
Anastassiou, George A
2016-01-01
In this monograph the authors present Newton-type, Newton-like and other numerical methods, which involve fractional derivatives and fractional integral operators, for the first time studied in the literature. All for the purpose to solve numerically equations whose associated functions can be also non-differentiable in the ordinary sense. That is among others extending the classical Newton method theory which requires usual differentiability of function. Chapters are self-contained and can be read independently and several advanced courses can be taught out of this book. An extensive list of references is given per chapter. The book’s results are expected to find applications in many areas of applied mathematics, stochastics, computer science and engineering. As such this monograph is suitable for researchers, graduate students, and seminars of the above subjects, also to be in all science and engineering libraries.
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Boolean differential equations
Steinbach, Bernd
2013-01-01
The Boolean Differential Calculus (BDC) is a very powerful theory that extends the structure of a Boolean Algebra significantly. Based on a small number of definitions, many theorems have been proven. The available operations have been efficiently implemented in several software packages. There is a very wide field of applications. While a Boolean Algebra is focused on values of logic functions, the BDC allows the evaluation of changes of function values. Such changes can be explored for pairs of function values as well as for whole subspaces. Due to the same basic data structures, the BDC can
Ordinary differential equations
Cox, William
1995-01-01
Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further
Time-division multiplexing vs network calculus: A comparison
DEFF Research Database (Denmark)
Puffitsch, Wolfgang; Sørensen, Rasmus Bo; Schoeberl, Martin
2015-01-01
Networks-on-chip are increasingly common in modern multicore architectures. However, general-purpose networks-on-chip are not always well suited for real-time applications that require bandwidth and latency guarantees. Two approaches to provide real-time guarantees have emerged: time...... that time-division multiplexing leads to better worst-case latencies, while network calculus supports higher bandwidths. Furthermore, time-division multiplexing leads to a simpler hardware implementation, while dynamically scheduled networks-on-chip allow the integration of best-effort traffic in the on-chip......-division multiplexing, where traffic is scheduled according to a precalculated static schedule, and network calculus, a mathematical framework to reason about dynamically scheduled networks. This paper compares the two approaches to provide insight into their relative advantages and disadvantages. The results show...
Female urethral diverticulum associated with a large urinary calculus
Directory of Open Access Journals (Sweden)
Alexandre Oliveira Rodrigues
2009-12-01
Full Text Available The diverticula of female urethra are very uncommon, but more frequently found between the third and fifth decade of life. Diverticula area mostly relate to repeated urinary infections of the periurethral glands or urethra’s trauma. The diverticula may cause infection, calculus formation and rarely endometriosis or cancer. A case of a 65-year old Caucasian female with vaginal mass over six months is herein reported. There was no urinary loss, urethral secretion or urinary symptoms. A cystourethrography showed diverticula with calculus inside. The patient was submitted to surgery and dismissed from the hospital on the first postoperative day. The pathologic examination revealed no malignancy. In six months of follow-up, the patient was continent with no complaints.
Geometric calculus according to the Ausdehnungslehre of H. Grassmann
Peano, Giuseppe
2000-01-01
Calcolo Geometrico, G. Peano's first publication in mathematical logic, is a model of expository writing, with a significant impact on 20th century mathematics. Kannenberg's lucid and crisp translation, Geometric Calculus, will appeal to historians of mathematics, researchers, graduate students, and general readers interested in the foundations of mathematics and the development of a formal logical language. In Chapter IX, with the innocent-sounding title "Transformations of a linear system," one finds the crown jewel of the book: Peano's axiom system for a vector space, the first-ever presentation of a set of such axioms. The very wording of the axioms (which Peano calls "definitions") has a remarkably modern ring, almost like a modern introduction to linear algebra. Peano also presents the basic calculus of set operation, introducing the notation for 'intersection,' 'union,' and 'element of,' many years before it was accepted. Despite its uniqueness, Calcolo Geometrico has been strangely neglected by histor...
k-Schur functions and affine Schubert calculus
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 ...
Model Checking Algorithms for the mu-Calculus,
1996-09-23
Model Checking Algorithms for the -Calculus Sergey Berezin Edmund Clarke Somesh Jha Will Marrero September 23, 1996 CMU-CS-96-180 School of Computer...authorized to reproduce and distribute reprints for Governmental purposes. Keywords: automatic veri cation, temporal logic, model checking , binary decision di...comes from the existence of ecient model checking algorithms for this formalism. As a consequence, veri cation procedures for many temporal and modal
On fractal space-time and fractional calculus
Directory of Open Access Journals (Sweden)
Hu Yue
2016-01-01
Full Text Available This paper gives an explanation of fractional calculus in fractal space-time. On observable scales, continuum models can be used, however, when the scale tends to a smaller threshold, a fractional model has to be adopted to describe phenomena in micro/nano structure. A time-fractional Fornberg-Whitham equation is used as an example to elucidate the physical meaning of the fractional order, and its solution process is given by the fractional complex transform.