Multivector Differential Calculus
Hitzer, Eckhard
2013-01-01
Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential calculus part of geometric calculus. The multivector differential is introduced, followed by the multivector derivative and the adjoint of multivector functions. The basic rules of multivector differentiation are derived explicitly, as well as a variety of basic m...
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.
On paragrassmann differential calculus
International Nuclear Information System (INIS)
The paper significantly extends and generalizes our previous paper. Here we discuss explicit general constructions for paragrassmann calculus with one and many variables. For one variable nondegenerate differentiation algebras are identified and shown to be equivalent to the algebra of (p+1)x(p+1) complex matrices. For many variables we give a general construction of the differentiation algebras. Some particular examples are related to the multiparametric quantum deformations of the harmonic oscillators. 18 refs
The absolute differential calculus (calculus of tensors)
Levi-Civita, Tullio
2013-01-01
Written by a towering figure of twentieth-century mathematics, this classic examines the mathematical background necessary for a grasp of relativity theory. Tullio Levi-Civita provides a thorough treatment of the introductory theories that form the basis for discussions of fundamental quadratic forms and absolute differential calculus, and he further explores physical applications.Part one opens with considerations of functional determinants and matrices, advancing to systems of total differential equations, linear partial differential equations, algebraic foundations, and a geometrical intro
The differential lambda-mu-calculus
Vaux, Lionel
2007-01-01
We define a differential lambda-mu-calculus which is an extension of both Parigot's lambda-mu-calculus and Ehrhard- Regnier's differential lambda-calculus. We prove some basic properties of the system: reduction enjoys Church-Rosser and simply typed terms are strongly normalizing.
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
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. .
Differential Calculus: Concepts and Notation.
Hobbs, David; Relf, Simon
1997-01-01
Suggests that many students with A-level mathematics, and even with a degree in mathematics or a related subject, do not have an understanding of the basic principles of calculus. Describes the approach used in three textbooks currently in use. Contains 14 references. (Author/ASK)
A few calculus rules for chain differentials
Clark, Daniel E.; Houssineau, Jeremie; Delande, Emmanuel D.
2015-01-01
This paper summarizes the core definitions and results regarding the chain differential for functions in locally convex topological vector spaces. In addition, it provides a few elementary calculus rules of practical interest, notably for the differentiation of characteristic functionals in various domains of physical science and engineering.
A Cone Pseudo-differential Calculus
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
@@ The calculus of pseudo-differential operators on singular spaces and theconcept of ellipti-city in operator algebras on manifolds with singularitieshave become an enormous challenge for analysists. The so-called cone algebras(with discrete and continuous asymptotics) are investigated by manymathematicians, especially by B. W. Schulze, who developed and enrichedcone and edge pseudo-differential calculus, see Schulze［4-7］, Rempel and Schulze ［2, 3］. In this note,we construct a cone pseudo-differentialcalculus for operators which respect conormal asymptotics of a prescribedasymptotic type.
Algorithmic Differentiation for Calculus-based Optimization
Walther, Andrea
2010-10-01
For numerous applications, the computation and provision of exact derivative information plays an important role for optimizing the considered system but quite often also for its simulation. This presentation introduces the technique of Algorithmic Differentiation (AD), a method to compute derivatives of arbitrary order within working precision. Quite often an additional structure exploitation is indispensable for a successful coupling of these derivatives with state-of-the-art optimization algorithms. The talk will discuss two important situations where the problem-inherent structure allows a calculus-based optimization. Examples from aerodynamics and nano optics illustrate these advanced optimization approaches.
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
DIFFERENTIAL AND INTEGRAL CALCULUS. A TENTATIVE CURRICULUM GUIDE.
BRANT, VINCENT; GERARDI, WILLIAM
A GUIDE FOR A 1-YEAR COURSE IN DIFFERENTIAL AND INTEGRAL CALCULUS PREREQUISITED KNOWLEDGE IN ALGEBRA, ANALYTIC TRIGONOMETRY, AND ELEMENTARY ANALYSIS. EACH ASSIGNMENT CONTAINED BOTH NEW AND REVIEW WORK TO REINFORCE THE NEW WORK. THERE WERE ELEVEN UNITS OF STUDY USING THE FOLLOWING FOUR BOOKS--"CALCULUS AND ANALYTIC GEOMETRY, THIRD EDITIONS," BY…
A Tasty Combination: Multivariable Calculus and Differential Forms
Goins, Edray Herber
2009-01-01
Differential Calculus is a staple of the college mathematics major's diet. Eventually one becomes tired of the same routine, and wishes for a more diverse meal. The college math major may seek to generalize applications of the derivative that involve functions of more than one variable, and thus enjoy a course on Multivariate Calculus. We serve this article as a culinary guide to differentiating and integrating functions of more than one variable -- using differential forms which are the basis for de Rham Cohomology.
Algebraic differential calculus for gauge theories
International Nuclear Information System (INIS)
The guiding idea in this paper is that, from the point of view of physics, functions and fields are more important than the (space time) manifold over which they are defined. The line pursued in these notes belongs to the general framework of ideas that replaces the space M by the ring of functions on it. Our essential observation, underlying this work, is that much of mathematical physics requires only a few differential operators (Lie derivative, d, δ) operating on modules of sections of suitable bundles. A connection (=gauge potential) can be described by a lift of vector fields from the base to the total space of a principal bundle. Much of the information can be encoded in the lift without reference to the bundle structures. In this manner, one arrives at an 'algebraic differential calculus' and its graded generalization that we are going to discuss. We are going to give an exposition of 'algebraic gauge theory' in both ungraded and graded versions. We show how to deal with the essential features of electromagnetism, Dirac, Kaluza-Klein and 't Hooft-Polyakov monopoles. We also show how to break the symmetry from SU(2) to U(1) without Higgs field. We briefly show how to deal with tests particles in external fields and with the Lagrangian formulation of field theories. (orig./HSI)
Poincare-Lie algebra and noncommutative differential calculus
International Nuclear Information System (INIS)
A realization of Poincare-Lie algebra in terms of noncommutative differential calculus is constructed. Corresponding relativistic quantum mechanics is considered. The important conclusion is that field equations appear in the integral form
Magnetic pseudo-differential Weyl calculus on nilpotent Lie groups
Beltita, Ingrid; Beltita, Daniel
2009-01-01
We develop a pseudo-differential Weyl calculus on nilpotent Lie groups which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie grup and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.
Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors
Bertram, Wolfgang
2011-01-01
We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the degree, and not exponentially as in the classical, ``cubic'' approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define We...
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.
Noncommutative Differential Calculus and Its Application on Discrete Spaces
Institute of Scientific and Technical Information of China (English)
WANG Ming-Liang; LIU Zhen; ZHANG Jin-Liang; BAI Yong-Qiang; LI Xiang-Zheng; WU Ke; GUO Han-Ying
2008-01-01
We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincar(e) lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.
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.
A TENTATIVE GUIDE, DIFFERENTIAL AND INTEGRAL CALCULUS.
BRANT, VINCENT; GERARDI, WILLIAM
THE COURSE IS INTENDED TO GO BEYOND THE REQUIREMENTS OF THE ADVANCED PLACEMENT PROGRAM IN MATHEMATICS AS DESIGNED BY THE COLLEGE ENTRANCE EXAMINATION BOARD. THE ADVANCED PLACEMENT PROGRAM CONSISTS OF A 1-YEAR COURSE COMBINING ANALYTIC GEOMETRY AND CALCULUS. PRESUPPOSED HERE ARE--A SEMESTER COURSE IN ANALYTIC GEOMETRY AND A THOROUGH KNOWLEDGE OF…
Relativistic differential-difference momentum operators and noncommutative differential calculus
International Nuclear Information System (INIS)
The relativistic kinetic momentum operators are introduced in the framework of the Quantum Mechanics (QM) in the Relativistic Configuration Space (RCS). These operators correspond to the half of the non-Euclidean distance in the Lobachevsky momentum space. In terms of kinetic momentum operators the relativistic kinetic energy is separated as the independent term of the total Hamiltonian. This relativistic kinetic energy term is not distinguishing in form from its nonrelativistic counterpart. The role of the plane wave (wave function of the motion with definite value of momentum and energy) plays the generating function for the matrix elements of the unitary irreps of Lorentz group (generalized Jacobi polynomials). The kinetic momentum operators are the interior derivatives in the framework of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions over the RCS
Covariant differential calculus on the quantum exterior vector space
International Nuclear Information System (INIS)
We formulate a differential calculus on the quantum exterior vector space spanned by the generators of a non-anticommutative algebra satisfying rij = θiθj+Bklijθkθl=0 i, j=1, 2, ..., n. and (θi)2=(θj)2=...=(θn)2=0, where Bklij 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 (2n)+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' conditions drij≅0 and the 'quadratic' condition rijxn≅0 where xn=dθn are the differentials of the variables. (orig.)
Ideas of Physical Forces and Differential Calculus in Ancient India
Girish, T E
2011-01-01
We have studied the context and development of the ideas of physical forces and differential calculus in ancient India by studying relevant literature related to both astrology and astronomy since pre-Greek periods. The concept of Naisargika Bala (natural force) discussed in Hora texts from India is defined to be proportional to planetary size and inversely related to planetary distance. This idea developed several centuries prior to Isaac Newton resembles fundamental physical forces in nature especially gravity. We show that the studies on retrograde motion and Chesta Bala of planets like Mars in the context of astrology lead to development of differential calculus and planetary dynamics in ancient India. The idea of instantaneous velocity was first developed during the 1st millennium BC and Indians could solve first order differential equations as early as 6th cent AD. Indian contributions to astrophysics and calculus during European dark ages can be considered as a land mark in the pre-renaissance history ...
The Malliavin calculus and hypoelliptic differential operators
Bell, Denis
2015-03-01
This article is intended as an introduction to Malliavin's stochastic calculus of variations and his probabilistic approach to hypoellipticity. Topics covered include an elementary derivation of the basic integration by parts formulae, a proof of the probabilistic version of Hörmander's theorem as envisioned by Malliavin and completed by Kusuoka and Stroock, and an extension of Hörmander's theorem valid for operators with degeneracy of exponential type due to the author and S. Mohammed.
Differential Calculus, Tensor Products and the Importance of Notation
Manton, Jonathan H.
2012-01-01
An efficient coordinate-free notation is elucidated for differentiating matrix expressions and other functions between higher-dimensional vector spaces. This method of differentiation is known, but not explained well, in the literature. Teaching it early in the curriculum would avoid the tedium of element-wise differentiation and provide a better footing for understanding more advanced applications of calculus. Additionally, it is shown to lead naturally to tensor products, a topic previously...
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
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.
Introduction to Differential Calculus Systematic Studies with Engineering Applications for Beginners
Rohde, Ulrich L; Poddar, Ajay K; Ghosh, A K
2011-01-01
Enables readers to apply the fundamentals of differential calculus to solve real-life problems in engineering and the physical sciences Introduction to Differential Calculus fully engages readers by presenting the fundamental theories and methods of differential calculus and then showcasing how the discussed concepts can be applied to real-world problems in engineering and the physical sciences. With its easy-to-follow style and accessible explanations, the book sets a solid foundation before advancing to specific calculus methods, demonstrating the connections between differential calculus
Ideas of Physical Forces and Differential Calculus in Ancient India
Girish, T. E.; Nair, C. Radhakrishnan
2010-01-01
We have studied the context and development of the ideas of physical forces and differential calculus in ancient India by studying relevant literature related to both astrology and astronomy since pre-Greek periods. The concept of Naisargika Bala (natural force) discussed in Hora texts from India is defined to be proportional to planetary size and inversely related to planetary distance. This idea developed several centuries prior to Isaac Newton resembles fundamental physical forces in natur...
Bicovariant differential calculus on quantum groups and wave mechanics
International Nuclear Information System (INIS)
The bicovariant differential calculus on quantum groups defined by Woronowicz and later worked out explicitly by Carow-Watamura et al. and Jurco for the real quantum groups SUq(N) and SOq(N) through a systematic construction of the bicovariant bimodules of these quantum groups, is reviewed for SUq(2) and SOq(N). The resulting vector fields build representations of the quantized universal enveloping algebras acting as covariant differential operators on the quantum groups and their associated quantum spaces. As an application, a free particle stationary wave equation on quantum space is formulated and solved in terms of a complete set of energy eigenfunctions. (author) 15 refs
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.
On the construction of unitary quantum group differential calculus
Pyatov, Pavel
2016-10-01
We develop a construction of the unitary type anti-involution for the quantized differential calculus over {{GL}}q(n) in the case | q| =1. To this end, we consider a joint associative algebra of quantized functions, differential forms and Lie derivatives over {{GL}}q(n)/{{SL}}q(n), which is bicovariant with respect to {{GL}}q(n)/{{SL}}q(n) coactions. We define a specific non-central spectral extension of this algebra by the spectral variables of three matrices of the algebra generators. In the spectrally expended algebra, we construct a three-parametric family of its inner automorphisms. These automorphisms are used for the construction of the unitary anti-involution for the (spectrally extended) calculus over {{GL}}q(n). This work has been funded by the Russian Academic Excellence Project ‘5-100’. The results of section 5 (propositions 5.2, 5.3 and theorem 5.5) have been obtained under support of the RSF grant No.16-11-10160.
A natural differential calculus on Lie bialgebras with dual of triangular type
Hijligenberg, N.W. van den; Martini, R.
1995-01-01
We prove that for a specific class of Lie bialgebras, there exists a natural differential calculus. This class consists of the Lie bialgebras for which the dual Lie bialgebra is of triangular type. The differential calculus is explicitly constructed with the help of the $R$-matrix from the dual. The
Bicovariant differential calculus on the superspace ${\\mathbb R}_q(1|2)$
Celik, Salih
2015-01-01
Super Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.
Restrictive metric regularity and generalized differential calculus in Banach spaces
Directory of Open Access Journals (Sweden)
Bingwu Wang
2004-10-01
Full Text Available We consider nonlinear mappings f:XÃ¢Â†Â’Y between Banach spaces and study the notion of restrictive metric regularity of f around some point xÃ‚Â¯, that is, metric regularity of f from X into the metric space E=f(X. Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case when f is strictly differentiable at xÃ‚Â¯ but its strict derivative Ã¢ÂˆÂ‡f(xÃ‚Â¯ is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions.
Henle, James M
2014-01-01
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.
Mathematics in the Classroom: Conceptual Cartography of Differential Calculus
Directory of Open Access Journals (Sweden)
María de Lourdes RODRÍGUEZ PERALTA
2015-12-01
Full Text Available This paper presents the results of a documentary investigation with the intention of substantiate how and why, and the level and depth of the topics used by the teacher in the classroom for the development of the mathematical knowledge on the part of higher level engineering students. The analysis of the mathematical object was made through the construction of conceptual cartography, being the core of the derivative concept. To construct the axes, the socio-formative theory of Sergio Tobón was used, together with the semiotic representation register of Raychmond Duval and Tall's mathematical advanced thought in the engineering context. The topic is a part of the Unit of learning: Differential and Integral Calculus. This corresponds to the first semester. The course lasts for a semester and is intended for students aged between 18 and 20 years. The research shows that by constructing a conceptual cartography involving at least 8 axes of analysis that the socio-formation orientates, and taking mathematics in the context of careers offered by the educational institution, the teacher is allowed to place the thematic content in the appropriate level and depth, guiding in a possible treatment of knowledge to be brought into the classroom.
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…
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.
Noncommutative Differential Calculus and Its Application on the Lattice%格点上的非交换微分运算及其应用
Institute of Scientific and Technical Information of China (English)
刘震; 白永强; 李起升
2007-01-01
By introducing the noncommutative differential calculus on the function space of the infinite/finite set and construct a homotopy operator, one prove the analogue of the Poincaré lemma for the difference complex. As an application of the differential calculus, a two dimensional integral model can be derived from the noncommutative differential calculus.
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…
Boehme, Thomas K
1987-01-01
Operational Calculus, Volume II is a methodical presentation of operational calculus. An outline of the general theory of linear differential equations with constant coefficients is presented. Integral operational calculus and advanced topics in operational calculus, including locally integrable functions and convergence in the space of operators, are also discussed. Formulas and tables are included.Comprised of four sections, this volume begins with a discussion on the general theory of linear differential equations with constant coefficients, focusing on such topics as homogeneous and non-ho
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
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.
BRST-operator for quantum Lie algebra and differential calculus on quantum groups
International Nuclear Information System (INIS)
For A Hopf algebra one determined structure of differential complex in two dual external Hopf algebras: A external expansion and in A* dual algebra external expansion. The Heisenberg double of these two Hopf algebras governs the differential algebra for the Cartan differential calculus on A algebra. The forst differential complex is the analog of the de Rame complex. The second complex coincide with the standard complex. Differential is realized as (anti)commutator with Q BRST-operator. Paper contains recursion relation that determines unequivocally Q operator. For Uq(gl(N)) Lie quantum algebra one constructed BRST- and anti-BRST-operators and formulated the theorem of the Hodge expansion
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.
n=3 differential calculus on a given reduced quantum plane and gauge theory
International Nuclear Information System (INIS)
We discuss the algebra of NxN matrices that seems to be as a reduced quantum plane. A new deformed differential calculus involving a complex parameter q is introduced. The two cases, q generic and q N-th root of unity are completely treated. As an application, we give connection with gauge field theory for the particular cases n=2 and n=3. (author)
The two-parameter deformation of GL(2), its differential calculus, and Lie algebra
International Nuclear Information System (INIS)
The Yang-Baxter equation is solved in two dimensions giving rise to a two-parameter deformation of GL(2). The transformation properties of quantum planes are briefly discussed. Non-central determinant and inverse are constructed. A right-invariant differential calculus is presented and the role of the different deformation parameters investigated. While the corresponding Lie algebra relations are simply deformed, the comultiplication exhibits both quantization parameters. (orig.)
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.
Conceptual Differential Calculus. I: First Order Local Linear Algebra
Bertram, Wolfgang
2015-01-01
We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina vector space or module, a locally linear map is defined to be a map respecting two canonical operationsliving ``over'' its domain of definition.These two operations are composition laws of a canonical groupoid and of a scaled action category, respectively,f...
Computer Graphics as an Instructional Aid in an Introductory Differential Calculus Course
Directory of Open Access Journals (Sweden)
Tapan Kumar Tiwari
2007-02-01
Full Text Available Mathematicians in general claim that the Computer Algebra Systems (CAS provide an excellent tool for illustrating calculus concepts. They caution, however, against heavy dependency on the CAS for all computational purposes without the mastery of the procedures involved. This study examined the effect of using the graphical and numerical capabilities of Mathematica as a supplemental instructional tool in enhancing the conceptual knowledge and problem solving abilities of students in a differential calculus course. Topics of differential calculus were introduced by the traditional lecture method to both the control and experimental groups comprised of students enrolled in two sections of the Business and Life Sciences I course. Mathematica was used only by the students of the experimental group to reinforce and illustrate the concepts developed by the traditional method. A content analysis was conducted using the qualitative data obtained from students’ explanations of the derivative of a function. The quantitative data, the students’ test scores, were analyzed using ANCOVA. The results showed that students in the experimental group scored higher than students in the control group on both the conceptual and the computational parts of the examination. The qualitative analysis results revealed that, compared to the control group, a higher percentage of students in the experimental group had a better understanding of the derivative.
Non-commutative Differential Calculus and the Axial Anomaly in Abelian Lattice Gauge Theories
Fujiwara, T; Wu, K; Fujiwara, Takanori; Suzuki, Hiroshi; Wu, Ke
2000-01-01
The axial anomaly in lattice gauge theories has topological nature when the Dirac operator satisfies the Ginsparg-Wilson relation. We study the axial anomaly in Abelian gauge theories on an infinite hypercubic lattice by utilizing cohomological techniques. The crucial tool in our approach is the non-commutative differential calculus (NCDC) which validates the Leibniz rule of exterior derivatives on the lattice. The topological nature of the ``Chern character'' on the lattice becomes manifest with NCDC. Our result provides an algebraic proof of Lüscher's theorem for a four-dimensional lattice and its generalization to arbitrary dimensions.
Nel's category theory based differential and integral Calculus, or did Newton know category theory ?
Rosinger, Elemer Elad
2005-01-01
In a series of publications in the early 1990s, L D Nel set up a study of non-normable topological vector spaces based on methods in category theory. One of the important results showed that the classical operations of derivative and integral in Calculus can in fact be obtained by a rather simple construction in categories. Here we present this result in a concise form. It is important to note that the respective differentiation does not lead to any so called generalized derivatives, for inst...
Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus
Cagnache, Eric; Wallet, Jean-Christophe
2008-01-01
Derivations of a (noncommutative) algebra can be used to construct various consistent differential calculi, the so-called derivation-based differential calculi. We apply this framework to the noncommutative Moyal algebras for which all the derivations are inner and analyse in detail the case where the derivation algebras generating the differential calculus are related to area preserving diffeomorphisms. The ordinary derivations corresponding to spatial dimensions are supplemented by additional derivations necessarely related to additional covariant coordinates. It is shown that these latter have a natural interpretation as Higgs fields when involved in gauge invariant actions built from the noncommutative curvature. The UV/IR mixing problem for (some of) the resulting Yang-Mills-Higgs models is discussed. A comparition to other noncommutative geometries already considered in the litterature is given.
So, Bing Kwan
2010-06-01
In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside the extended calculus; we construct the heat kernel of perturbed Laplacian operators; and prove an Atiyah-Singer type renormalized index formula for perturbed Dirac operators on the Bruhat sphere using the heat kernel method.
Matrix approach to discrete fractional calculus II: Partial fractional differential equations
Podlubny, Igor; Chechkin, Aleksei; Skovranek, Tomas; Chen, YangQuan; Vinagre Jara, Blas M.
2009-05-01
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359-386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Dirichlet boundary value problems for perturbed second-order differential equations on a half line are investigated in this paper. The methods mainly depend on the calculus of variations to the classical functionals. Sufficient conditions are obtained for the existence of the solutions.
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
International Nuclear Information System (INIS)
A n(n-1)/2+1 parameter solution of the Yang Baxter equation is presented giving rise to the quantum Group GLx;qij(n). Determinant and inverse are constructed. The group acts covariantly on a quantum vector space of non-commutative coordinates. The associated exterior space can be identified with the differentials exhibiting a multiparameter deformed differential calculus following the construction of Wess and Zumino. (orig.)
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
Commentary on Fundamental Theorem of Differential Calculus%微分学基本定理评注
Institute of Scientific and Technical Information of China (English)
张明会; 高婷婷
2015-01-01
微分学主要处理的是连续函数的变化率问题,它的基本定理、概念、原理和方法是高等数学的主干,也是高等数学尤其是数学分析的灵魂.文章从微分学基本理论出发,逐步分析并阐明了微分中值定理间内在联系,以及它们的特征和意义.%Differential calculus mainly deals with the change rate of continuous function, concept, principle and method, and is the backbone of higher mathematics and the soul of higher mathematics analysis. This paper, from the basic theory of differential calculus, has a step-by-step analysis and clarification of the relationship among differential mean value theorems, and their characteristic and the significance.
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. PMID:24892051
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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.
Calculus Demonstrations Using MATLAB
Dunn, Peter K.; Harman, Chris
2002-01-01
The note discusses ways in which technology can be used in the calculus learning process. In particular, five MATLAB programs are detailed for use by instructors or students that demonstrate important concepts in introductory calculus: Newton's method, differentiation and integration. Two of the programs are animated. The programs and the…
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
Institute of Scientific and Technical Information of China (English)
张宏伟; 张立卫; 等
2003-01-01
A Riesz space K1 whose elements are pairs of convex-set collections is presented for the study on the calculus of generalized quasi-differentiable functions.The space K1 is constructed by introducing a well-defined equivalence reation among pairs of collections of convex sets .Some important properties on the norm and operations is K1 are given.
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
Soergel calculus and Schubert calculus
He, Xuhua; Williamson, Geordie
2015-01-01
We reduce some key calculations of compositions of morphisms between Soergel bimodules ("Soergel calculus") to calculations in the nil Hecke ring ("Schubert calculus"). This formula has several applications in modular representation theory.
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
Desbrun, Mathieu; Hirani, Anil N.; Leok, Melvin; Marsden, Jerrold E.
2005-01-01
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous attempts at discrete exterior ca...
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
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
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 composition. We define a standard model and prove that CTC is relatively complete with respect to it. The core calculus is extended with operators for choice, information hiding, scalar multiplicatio...
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
Dirac operator, bicovariant differential calculus and gauge theory on κ-Minkowski space
International Nuclear Information System (INIS)
Connections between the κ-Poincare covariant space Γ of differential 1-forms on κ-Minkowski space, Dirac operator and Alain Connes formula are studied. The equations and Lagrangian of gauge theory are constructed. The appearance of an additional spin-0 gauge field according to the non-trivial structure of Γ is studied. (author)
On the full calculus of pseudo-differential operators on boundary groupoids with polynomial growth
So, Bing Kwan
2011-01-01
In this paper, we enlarge the space of uniformly supported pseudo-differential operators on some groupoids by considering kernels satisfying certain asymptotic estimates. We show that such enlarged space contains the compact parametrix, and the generalized inverse of uniformly supported operators with Fredholm vector representation.
Pseudodifferential calculus on manifolds with corners and groupoids
Monthubert, Bertrand
1997-01-01
We build a longitudinally smooth differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called b-calculus). We also define an algebra of rapidly decreasing functions on this groupoid; it contains the kernels of the smoothing operators of the (small) b-calculus.
Directory of Open Access Journals (Sweden)
J.A. Bergstra
2008-01-01
Full Text Available We introduce a calculus for tuplices, which are expressions that generalize matrices and vectors. Tuplices have an underlying data type for quantities that are taken from a zero-totalized field. We start with the core tuplix calculus CTC for entries and tests, which are combined using conjunctive composition. We define a standard model and prove that CTC is relatively complete with respect to it. The core calculus is extended with operators for choice, information hiding, scalar multiplication, clearing and encapsulation. We provide two examples of applications; one on incremental financial budgeting, and one on modular financial budget design.
Several problems of the application of differential and integral calculus%浅析微积分应用的若干问题
Institute of Scientific and Technical Information of China (English)
刘洪辰
2015-01-01
本文利用微积分方法解决力学、几何学、运动学中常见的问题，通过算例说明计算步骤、微元选取和建立模型的方法，探求用微积分方法解决工程实际问题的一般规律。%In this paper, using the method of differential and integral calculus to solve common problems in mechanics, geometry, kinematics, calculation steps, micro yuan selection is provided through calculating examples and the method of model, to explore the general rule of solving engineering problem with calculus method.
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.
Osserman, Robert
2011-01-01
The basic component of several-variable calculus, two-dimensional calculus is vital to mastery of the broader field. This extensive treatment of the subject offers the advantage of a thorough integration of linear algebra and materials, which aids readers in the development of geometric intuition. An introductory chapter presents background information on vectors in the plane, plane curves, and functions of two variables. Subsequent chapters address differentiation, transformations, and integration. Each chapter concludes with problem sets, and answers to selected exercises appear at the end o
Izadi, F A; Bagirov, G
2009-01-01
With its origins stretching back several centuries, discrete calculus is now an increasingly central methodology for many problems related to discrete systems and algorithms. The topics covered here usually arise in many branches of science and technology, especially in discrete mathematics, numerical analysis, statistics and probability theory as well as in electrical engineering, but our viewpoint here is that these topics belong to a much more general realm of mathematics; namely calculus and differential equations because of the remarkable analogy of the subject to this branch of mathemati
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.
Polynomial calculus: rethinking the role of calculus in high schools
Grant, Melva R.; Crombie, William; Enderson, Mary; Cobb, Nell
2016-08-01
Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.
ESeal Calculus: A Secure Mobile Calculus
Institute of Scientific and Technical Information of China (English)
Peng Rong; Chen Xin-meng; Liu Ping
2003-01-01
The ESeal Calculus is a secure mobile calculus based on Seal Calculus. By using open-channels, ESeal Calculus makes it possible to communicate between any two arbitrary seals with some secure restrictions. It improves the expression ability and efficiency of Seal calculus without losing security.
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...
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,
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
Foliated stochastic calculus: Harmonic measures
Catuogno, Pedro J; Ruffino, Paulo R
2010-01-01
In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Other results include a decomposition of the Laplacian in terms of the foliated and basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.
McGivney-Burelle, Jean; Xue, Fei
2013-01-01
In this paper we discuss flipping pedagogy and how it can transform the teaching and learning of calculus by applying pedagogical practices that are steeped in our understanding of how students learn most effectively. In particular, we describe the results of an exploratory study we conducted to examine the benefits and challenges of flipping a…
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.
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.
Introduction to Tensor Calculus
Sochi, Taha
2016-01-01
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is presumed.
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
Advanced Calculus An Introduction to Linear Analysis
Richardson, Leonard F
2008-01-01
Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra. Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis. Following an introdu
Cirstea, Horatiu; Kirchner, Claude
2000-01-01
The Rho-calculus is a new calculus that integrates in a uniform and simple setting first-order rewriting, lambda-calculus and non-deterministic computations. This paper describes the calculus from its syntax to its basic properties in the untyped case. We show how it embeds first-order conditional rewriting and lambda-calculus. Finally we use the Rho-calculus to give an operational semantics to the rewrite based language Elan.
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.
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
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...
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...
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, A
2011-01-01
Ideal for self-instruction as well as for classroom use, this text helps students improve their understanding and problem-solving skills in analysis, analytic geometry, and higher algebra. More than 1,200 problems appear in the text, with concise explanations of the basic notions and theorems to be used in their solution. Many are followed by complete answers; solutions for the others appear at the end of the book. Topics include sequences, functions of a single variable, limit of a function, differential calculus for functions of a single variable, fundamental theorems and applications of dif
Fractional Calculus and -Valently Starlike Functions
Directory of Open Access Journals (Sweden)
Özkan Öznur
2009-01-01
Full Text Available Abstract In this investigation, the authors prove coefficient bounds, distortion inequalities for fractional calculus of a family of multivalent functions with negative coefficients, which is defined by means of a certain nonhomogenous Cauchy-Euler differential equation.
Calculus with a quaternionic variable
Schwartz, Charles
2009-01-01
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn in trying to extend our reach to include quaternions. The noncommutativity of the quaternion algebra poses obstacles for the usual manipulations of calculus, but we show in this paper how many of those obstacles can be overcome. The surprising result is that the first order term in the expansion of F(x +δ) is a compact formula involving both F'(x) and [F(x )-F(x∗)]/(x -x∗). This advance in the differential calculus for quaternionic variables also leads us to some progress in studying integration.
Fluorescence detection of dental calculus
International Nuclear Information System (INIS)
This work is devoted to the optimization of fluorescence dental calculus diagnostics in optical spectrum. The optimal wavelengths for fluorescence excitation and registration are determined. Two spectral ranges 620 – 645 nm and 340 – 370 nm are the most convenient for supra- and subgingival calculus determination. The simple implementation of differential method free from the necessity of spectrometer using was investigated. Calculus detection reliability in the case of simple implementation is higher than in the case of spectra analysis at optimal wavelengths. The use of modulated excitation light and narrowband detection of informative signal allows us to decrease essentially its diagnostic intensity even in comparison with intensity of the low level laser dental therapy
Operational calculus and generalized functions
Erdelyi, Arthur
2013-01-01
This brief monograph by a distinguished professor is based on a mathematics course offered at the California Institute of Technology. The majority of students taking this course were advanced undergraduates and graduate students of engineering. A solid background in advanced calculus is a prerequisite.Topics include elementary and convergence theories of convolution quotients, differential equations involving operator functions, and exponential functions of operators. Tools developed in the preceding chapters are then applied to problems in partial differential equations. Solutions to selected
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.
Hatae, Ryusuke; Hata, Nobuhiro; Yoshimoto, Koji; Kuga, Daisuke; Akagi, Yojiro; Murata, Hideki; Suzuki, Satoshi O; Mizoguchi, Masahiro; Iihara, Koji
2016-01-01
High resolution melting (HRM) is a simple and rapid method for screening mutations. It offers various advantages for clinical diagnostic applications. Conventional HRM analysis often yields equivocal results, especially for surgically obtained tissues. We attempted to improve HRM analyses for more effective applications to clinical diagnostics. HRM analyses were performed for IDH1R132 and IDH2R172 mutations in 192 clinical glioma samples in duplicate and these results were compared with sequencing results. BRAFV600E mutations were analyzed in 52 additional brain tumor samples. The melting profiles were used for differential calculus analyses. Negative second derivative plots revealed additional peaks derived from heteroduplexes in PCR products that contained mutations; this enabled unequivocal visual discrimination of the mutations. We further developed a numerical expression, the HRM-mutation index (MI), to quantify the heteroduplex-derived peak of the mutational curves. Using this expression, all IDH1 mutation statuses matched those ascertained by sequencing, with the exception of three samples. These discordant results were all derived from the misinterpretation of sequencing data. The effectiveness of our approach was further validated by analyses of IDH2R172 and BRAFV600E mutations. The present analytical method enabled an unequivocal and objective HRM analysis and is suitable for reliable mutation scanning in surgically obtained glioma tissues. This approach could facilitate molecular diagnostics in clinical environments.
Hatae, Ryusuke; Hata, Nobuhiro; Yoshimoto, Koji; Kuga, Daisuke; Akagi, Yojiro; Murata, Hideki; Suzuki, Satoshi O; Mizoguchi, Masahiro; Iihara, Koji
2016-01-01
High resolution melting (HRM) is a simple and rapid method for screening mutations. It offers various advantages for clinical diagnostic applications. Conventional HRM analysis often yields equivocal results, especially for surgically obtained tissues. We attempted to improve HRM analyses for more effective applications to clinical diagnostics. HRM analyses were performed for IDH1R132 and IDH2R172 mutations in 192 clinical glioma samples in duplicate and these results were compared with sequencing results. BRAFV600E mutations were analyzed in 52 additional brain tumor samples. The melting profiles were used for differential calculus analyses. Negative second derivative plots revealed additional peaks derived from heteroduplexes in PCR products that contained mutations; this enabled unequivocal visual discrimination of the mutations. We further developed a numerical expression, the HRM-mutation index (MI), to quantify the heteroduplex-derived peak of the mutational curves. Using this expression, all IDH1 mutation statuses matched those ascertained by sequencing, with the exception of three samples. These discordant results were all derived from the misinterpretation of sequencing data. The effectiveness of our approach was further validated by analyses of IDH2R172 and BRAFV600E mutations. The present analytical method enabled an unequivocal and objective HRM analysis and is suitable for reliable mutation scanning in surgically obtained glioma tissues. This approach could facilitate molecular diagnostics in clinical environments. PMID:27529619
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.
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
Calculus ABCs: A Gateway for Freshman Calculus
Fulton, Scott R.
2003-01-01
This paper describes a gateway testing program designed to ensure that students acquire basic skills in freshman calculus. Students must demonstrate they have mastered standards for "Absolutely Basic Competency"--the Calculus ABCs--in order to pass the course with a grade of C or better. We describe the background, standards, and testing program.…
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.
Kotkar, Kunal; Thakkar, Ravi; Songra, MC
2011-01-01
Primary urethral calculus is rarely seen and is usually encountered in men with urethral stricture or diverticulum. We present a case of giant urethral calculus secondary to a urethral stricture in a man. The patient was treated with calculus extraction with end to end urethroplasty.
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...
Advanced Placement Mathematics Calculus, Grade 12 Curriculum Guide.
Scharf, John; And Others
This document is a guide to the advanced placement program in calculus for grade 12 in the city schools in Warren, Ohio. The program covers analytic geometry, differential and integral calculus of algebraic functions, elementary transcendental functions, and applications of differentiation and integration. The philosophy and aims of the program…
Using the Finite Difference Calculus to Sum Powers of Integers.
Zia, Lee
1991-01-01
Summing powers of integers is presented as an example of finite differences and antidifferences in discrete mathematics. The interrelation between these concepts and their analogues in differential calculus, the derivative and integral, is illustrated and can form the groundwork for students' understanding of differential and integral calculus.…
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
Fractional Vector Calculus and Fractional Maxwell's Equations
Vasily E. Tarasov
2009-01-01
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 i...
Cartan calculus on quantum Lie algebras
International Nuclear Information System (INIS)
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ''Cartan Calculus.''
Knight, Brian
1975-01-01
Each chapter in this book deals with a single mathematical topic, which ideally should form the basis of a single lecture. The chapter has been designed as a mixture of the following ingredients: -(i) Illustrative examples and notes for the student's pre-lecture reading. (ii) Class discussion exercises for study in a lecture or seminar. (iii) Graded problems for assignment work. Contents 1 Sets, functions page 11 2 Limits and continuity 17 3 The exponential and related functions 25 4 Inverse functions 30 5 Differentiation 35 6 Differentiation of implicit functions 44 7 Maxima and minima 50 8 Curve sketching 54 9 Expansion in series 61 10 Newton's method 67 11 Area and integration 72 12 Standard integrals 80 13 Applications of the fundamental theorem 87 14 Substitution in integrals 94 15 Use of partial fractions 100 16 Integration by parts 106 Answers to problems 110 Index 116 1 Sets, Functions A set is a collection of distinct objects. The objects be longing to a set are the elements (or members) of the set...
Blum, William
2009-01-01
Safety is a syntactic condition of higher-order grammars that constrains occurrences of variables in the production rules according to their type-theoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition to the setting of the simply-typed lambda calculus. In contrast to the original definition of safety, our calculus does not constrain types (to be homogeneous). We show that in the safe lambda calculus, there is no need to rename bound variables when performing substitution, as variable capture is guaranteed not to happen. We also propose an adequate notion of beta-reduction that preserves safety. In the same vein as Schwichtenberg's 1976 characterization of the simply-typed lambda calculus, we show that the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a characterization of representable word functions. We then study the ...
Fractional vector calculus and fractional Maxwell's equations
International Nuclear Information System (INIS)
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
Advanced calculus of several variables
Kumar, Devendra
2014-01-01
ADVANCED CALCULUS OF SEVERAL VARIABLES covers important topics of Transformations and topology on Euclidean in n-space Rn Functions of several variables, Differentiation in Rn, Multiple integrals and Integration in Rn. The topics have been presented in a simple clear and coherent style with a number of examples and exercises. Proofs have been made direct and simple. Unsolved problems just after relevant articles in the form of exercises and typical problems followed by suggestions have been given. This book will help the reader work on the problems of Numerical Analysis, Operations Research, Differential Equations and Engineering applications.
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.
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.)
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…
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…
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
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...
Cartan Calculus on Quantum Lie Algebras
Schupp, Peter; Watts, Paul; Zumino, Bruno
1993-01-01
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra, the ``Cartan Calculus''. (This is an extended version of a talk presented by P. Schupp at the XXII$^{th}$ International Conference on Differential Geo...
Baxter Algebras and Umbral Calculus
Guo, Li
2004-01-01
We apply recent constructions of free Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of Baxter algebra. This characterization leads to a natural generalization of the umbral calculus that include the classical umbral calculus in a family of $\\lambda$-umbral calculi parameterized by $\\lambda$ in the base ring.
Hermeneutic operative calculus
Ramakrishnan, Sivakumar; Isawasan, Pradeep; Mohanan, Vasuky
2014-07-01
The predicate calculus used currently by mathematical logic in computer science, philosophy and linguistic was found to be too restrictive and inadequate for describing the grammar of natural and artificial language. Therefore many higher order logics have been developed to overcome the limitation of predicate calculus. In this paper a new representation of logic using mathematical principles has been developed for the natural language called Hermeneutic Operative Calculus. This Hermeneutic Operative Calculus is a new language interpretive calculus developed to account for the syntactic, semantic and pragmatic features of natural language and allows removing the restrictions of any particular natural language in the semantic field its map out. The logic of Hermeneutic Operative Calculus capable of represent the syntactic and semantic of factual information of a natural language precisely in any language. The logic of this Hermeneutic Operative Calculus has two different forms of operations called object and meta-operations. The object operation allow for listing the various objects, picturing the various propositions and so forth. The meta-operation would specify what cannot be specified by the object operation like semantical stances of a proposition. The basic operative processes of linguistics and cognitive logic will be mathematically conceptualized and elaborated in this paper.
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
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…
Elsgolc, L E; Stark, M
1961-01-01
Calculus of Variations aims to provide an understanding of the basic notions and standard methods of the calculus of variations, including the direct methods of solution of the variational problems. The wide variety of applications of variational methods to different fields of mechanics and technology has made it essential for engineers to learn the fundamentals of the calculus of variations. The book begins with a discussion of the method of variation in problems with fixed boundaries. Subsequent chapters cover variational problems with movable boundaries and some other problems; sufficiency
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...
Introduction to the Rewriting Calculus
Cirstea, Horatiu; Kirchner, Claude
1999-01-01
The $\\rho$-calculus is a new calculus that integrates in a uniform and simple setting first-order rewriting, $\\lambda$-calculus and non-deterministic computations. This paper describes the calculus from its syntax to its basic properties in the untyped case. We show how it embeds first-order conditional rewriting and $\\lambda$-calculus. Finally we use the $\\rho$-calcul- us to give an operational semantics to the rewrite based language ELAN.
A transition calculus for Boolean functions. [logic circuit analysis
Tucker, J. H.; Bennett, A. W.
1974-01-01
A transition calculus is presented for analyzing the effect of input changes on the output of logic circuits. The method is closely related to the Boolean difference, but it is more powerful. Both differentiation and integration are considered.
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.
Complex Multiplicative Calculus
Bashirov, Agamirza; Riza, Mustafa
2011-01-01
In the present paper we extend the concepts of multiplicative de- rivative and integral to complex-valued functions of complex variable. Some drawbacks, arising with these concepts in the real case, are explained satis- factorily. Properties of complex multiplicative derivatives and integrals are studied. In particular, the fundamental theorem of complex multiplicative calculus, relating these concepts, is proved. It is shown that complex multi- plicative calculus is not just another realizat...
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...
More calculus of a single variable
Mercer, Peter R
2014-01-01
This book goes beyond the basics of a first course in calculus to reveal the power and richness of the subject. Standard topics from calculus — such as the real numbers, differentiation and integration, mean value theorems, the exponential function — are reviewed and elucidated before digging into a deeper exploration of theory and applications, such as the AGM inequality, convexity, the art of integration, and explicit formulas for π. Further topics and examples are introduced through a plethora of exercises that both challenge and delight the reader. While the reader is thereby exposed to the many threads of calculus, the coherence of the subject is preserved throughout by an emphasis on patterns of development, of proof and argumentation, and of generalization. More Calculus of a Single Variable is suitable as a text for a course in advanced calculus, as a supplementary text for courses in analysis, and for self-study by students, instructors, and, indeed, all connoisseurs of ingenious calculations.
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.
Institute of Scientific and Technical Information of China (English)
傅育熙
1998-01-01
An alternative presentation of the π－calculus is given.This version of the π-calculus is symmetric in the sense that communications are symmetric and there is no difference between input and output prefixes.The point of the symmetric π-calculus is that it has no abstract names.The set of closed names is therefore homogeneous.The π－calculus can be fully embedded into the symmetric π-calculus.The symmetry changes the emphasis of the communication mechanism of the π-calculus and opens up possibility for further variations.
Proof nets for the Displacement calculus
Moot, Richard
2016-01-01
We present a proof net calculus for the Displacement calculus and show its correctness. This is the first proof net calculus which models the Displacement calculus directly and not by some sort of translation into another formalism. The proof net calculus opens up new possibilities for parsing and proof search with the Displacement calculus.
Fractional Complex Transform for Fractional Differential Equations
Li, 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 applied to fractional calculus. Two examples are given.
Investigations on the dual calculus
Tzevelekos, Nikos
2006-01-01
The Dual Calculus, proposed recently by Wadler, is the outcome of two distinct lines of research in theoretical computer science: (A) Efforts to extend the Curry–Howard isomorphism, established between the simply-typed lambda calculus and intuitionistic logic, to classical logic. (B) Efforts to establish the tacit conjecture that call-by-value (CBV) reduction in lambda calculus is dual to call-by-name (CBN) reduction. This paper initially investigates relations of the Dual Calculus t...
A development calculus for specifications
Institute of Scientific and Technical Information of China (English)
李未
2003-01-01
A first order inference system, named R-calculus, is defined to develop the specifications.This system intends to eliminate the laws which are not consistent with users' requirements. TheR-calculus consists of the structural rules, an axiom, a cut rule, and the rules for logical connectives.Some examples are given to demonstrate the usage of the R-calculus. Furthermore, the propertiesregarding reachability and completeness of the R-calculus are formally defined and proved.
Malinowska, Agnieszka B
2014-01-01
This Brief puts together two subjects, quantum and variational calculi by considering variational problems involving Hahn quantum operators. The main advantage of its results is that they are able to deal with nondifferentiable (even discontinuous) functions, which are important in applications. Possible applications in economics are discussed. Economists model time as continuous or discrete. Although individual economic decisions are generally made at discrete time intervals, they may well be less than perfectly synchronized in ways discrete models postulate. On the other hand, the usual assumption that economic activity takes place continuously, is nothing else than a convenient abstraction that in many applications is far from reality. The Hahn quantum calculus helps to bridge the gap between the two families of models: continuous and discrete. Quantum Variational Calculus is self-contained and unified in presentation. It provides an opportunity for an introduction to the quantum calculus of variations fo...
The history of the calculus and its conceptual development (the concepts of the calculus)
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.
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
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...
Toward lattice fractional vector calculus
International Nuclear Information System (INIS)
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. (papers)
Gelfand, I M
2000-01-01
Based on a series of lectures given by I. M. Gelfand at Moscow State University, this book actually goes considerably beyond the material presented in the lectures. The aim is to give a treatment of the elements of the calculus of variations in a form both easily understandable and sufficiently modern. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws.The reader who merely wishes to become familiar with the most basic concepts and methods of the calculus of variations need on
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...
Pedersen, Steen
2015-01-01
This textbook features applications including a proof of the Fundamental Theorem of Algebra, space filling curves, and the theory of irrational numbers. In addition to the standard results of advanced calculus, the book contains several interesting applications of these results. The text is intended to form a bridge between calculus and analysis. It is based on the authors lecture notes used and revised nearly every year over the last decade. The book contains numerous illustrations and cross references throughout, as well as exercises with solutions at the end of each section
Calculus in physics classes at UFRGS: an exploratory study
Maria Cecilia Pereira Santarosa; Marco Antonio Moreira
2011-01-01
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 Calc...
Geometric constrained variational calculus. II: The second variation (Part I)
Massa, Enrico; Bruno, Danilo; Luria, Gianvittorio; Pagani, Enrico
2016-10-01
Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.
The simply typed rewriting calculus
Cirstea, Horatiu; Kirchner, Claude
2000-01-01
The rewriting calculus is a rule construction and application framework. As such it embeds in a uniform way term rewriting and lambda-calculus. Since rule application is an explicit object of the calculus, it allows us also to handle the set of results explicitly. We present a simply typed version of the rewriting calculus. With a good choice of the type system, we show that the calculus is type preserving and terminating, i.e. verifies the subject reduction and strong normalization properties.
On Multiplicative Fractional Calculus
Abdeljawad, Thabet
2015-01-01
We set the main concepts for multiplicative fractional calculus. We define Caputo, Riemann and Letnikov multiplicative fractional derivatives and multiplicative fractional integrals and study some of their properties. Finally, the multiplicative analogue of the local conformable fractional derivative and integral is studied.
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.
Steckroth, Jeffrey J.
2010-01-01
For nearly three decades, during which the author taught everything from basic algebra to advanced placement calculus, the author thought of himself as a secondary school mathematics teacher. The notion of teaching elementary school math never appealed to him because of its simplicity. The author stresses that anyone could teach children to count,…
Palmaccio, Richard J.
1982-01-01
A method of using vector analysis is presented that is an application of calculus that helps to find the best angle for tacking a boat into the wind. While the discussion is theoretical, it is seen as a good illustration of mathematical investigation of a given situation. (MP)
Jiang, Yuming
2009-01-01
Network calculus, a theory dealing with queuing systems found in computer networks, focuses on performance guarantees. This title presents a comprehensive treatment for the stochastic service-guarantee analysis research and provides basic introductory material on the subject, as well as discusses the various researches in the area.
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 encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing ca...
ESeal Calculus： A Secure Mobile Calculus
Institute of Scientific and Technical Information of China (English)
PengRong; UuPing
2003-01-01
The ESeal Calculus is a secure mobile calculus based on Seal Calculus. By using open-channels,ESeal Calculus makes it possible to communicate between any two arbitrary seals with some secure restrictions. It improves the expression ability and efficiency of Seal calculus without losing security.
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.
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....
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
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.
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...
Algebraic Properties of Propositional Calculus
Schuh, Bernd R.
2009-01-01
In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such they can be represented by uniquely defined elements of this algebra which we call "logical primes". The algebraic notations appear useful because they make it possible to derive well known properties of propositional calculus by simple calculations or to subs...
Lambda-mu-calculus and Bohm's theorem
David, René; Py, Walter
2001-01-01
The lambda mu-calculus is an extension of the lambda-calculus that has been introduced by M. Parigot to give an algorithmic content to classical proofs. We show that Bohm's theorem fails in this calculus.
On the origins of generalized fractional calculus
Kiryakova, Virginia
2015-11-01
In Fractional Calculus (FC), as in the (classical) Calculus, the notions of derivatives and integrals (of first, second, etc. or arbitrary, incl. non-integer order) are basic and co-related. One of the most frequent approach in FC is to define first the Riemann-Liouville (R-L) integral of fractional order, and then by means of suitable integer-order differentiation operation applied over it (or under its sign) a fractional derivative is defined - in the R-L sense (or in Caputo sense). The first mentioned (R-L type) is closer to the theoretical studies in analysis, but has some shortages - from the point of view of interpretation of the initial conditions for Cauchy problems for fractional differential equations (stated also by means of fractional order derivatives/ integrals), and also for the analysts' confusion that such a derivative of a constant is not zero in general. The Caputo (C-) derivative, arising first in geophysical studies, helps to overcome these problems and to describe models of applied problems with physically consistent initial conditions. The operators of the Generalized Fractional Calculus - GFC (integrals and derivatives) are based on commuting m-tuple (m = 1, 2, 3, …) compositions of operators of the classical FC with power weights (the so-called Erdélyi-Kober operators), but represented in compact and explicit form by means of integral, integro-differential (R-L type) or differential-integral (C-type) operators, where the kernels are special functions of most general hypergeometric kind. The foundations of this theory are given in Kiryakova 18. In this survey we present the genesis of the definitions of the GFC - the generalized fractional integrals and derivatives (of fractional multi-order) of R-L type and Caputo type, analyze their properties and applications. Their special cases are all the known operators of classical FC, their generalizations introduced by other authors, the hyper-Bessel differential operators of higher integer
Students' difficulties with vector calculus in electrodynamics
Bollen, Laurens; De Cock, Mieke
2015-01-01
Understanding Maxwell's equations in differential form is a prerequisite to study the electrodynamic phenomena that are discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell's equations in electrodynamics.
Some Applications of Fractional Calculus in Engineering
Directory of Open Access Journals (Sweden)
J. A. Tenreiro Machado
2010-01-01
Full Text Available Fractional Calculus (FC goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades, due to the progress in the area of chaos that revealed subtle relationships with the FC concepts. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses FC in the study of system dynamics and control. In this perspective, this paper investigates the use of FC in the fields of controller tuning, legged robots, redundant robots, heat diffusion, and digital circuit synthesis.
Necessary optimality conditions for the calculus of variations on time scales
Ferreira, Rui A. C.; Torres, Delfim F. M.
2007-01-01
We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems of the calculus of variations with delta-differential side conditions (Lagrange problem of the calculus of variations on time scales).
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.)
International Nuclear Information System (INIS)
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.
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.
Fluorescence spectroscopy of dental calculus
International Nuclear Information System (INIS)
The aim of the present study was to investigate the fluorescence properties of dental calculus in comparison with the properties of adjacent unaffected tooth structure using both lasers and LEDs in the UV-visible range for fluorescence excitation. The influence of calculus color on the informative signal is demonstrated. The optimal spectral bands of excitation and registration of the fluorescence are determined
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…
The Basic Principle of Calculus?
Hardy, Michael
2011-01-01
A simple partial version of the Fundamental Theorem of Calculus can be presented on the first day of the first-year calculus course, and then relied upon repeatedly in assigned problems throughout the course. With that experience behind them, students can use the partial version to understand the full-fledged Fundamental Theorem, with further…
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
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....
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
Barnes, David
2015-01-01
We show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the rational homology type of the input functor, whose layers are given by rational spectra with an action of $O(n)$. By work of Greenlees and Shipley, we see that these layers are classified by torsion $H^*(B SO(n))[O(n)/SO(n)]$-modules.
二元函数微分学两个定理的推广%Generalization of Two Theorem of Differential Calculus of Binary Function
Institute of Scientific and Technical Information of China (English)
孔祥凤
2011-01-01
This paper generalizes the differentiable binary function and the sufficient condition that advanced mixed partial derivative is not related to derivation order and then proves them.%本文对二元函数可微性及高阶混合偏导数与求导次序无关的充分条件进行了推广并加以证明.
A Simple Acronym for Doing Calculus: CAL
Hathaway, Richard J.
2008-01-01
An acronym is presented that provides students a potentially useful, unifying view of the major topics covered in an elementary calculus sequence. The acronym (CAL) is based on viewing the calculus procedure for solving a calculus problem P* in three steps: (1) recognizing that the problem cannot be solved using simple (non-calculus) techniques;…
Open Calculus: A Free Online Learning Environment
Korey, Jane; Rheinlander, Kim; Wallace, Dorothy
2007-01-01
Dartmouth College mathematicians have developed a free online calculus course called "Open Calculus." Open Calculus is an exportable distance-learning/self-study environment for learning calculus including written text, nearly 4000 online homework problems and instructional videos. The paper recounts the evaluation of course elements since 2000 in…
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.
Omega Model of Standard Calculus(续2)
Institute of Scientific and Technical Information of China (English)
Huang Cheng-gui
2004-01-01
Chapter Two. Construction of Omega Continuum and Special Rules of the Integral of Infinitesimals Purpose of the chapter Chapter one states the foundation of differential calculus. The task of the chapter is to construct Omega continuum,at the same time to interlude two axioms of the integral of infinitesimals.
Constrained variational calculus: the second variation (part I)
Massa, Enrico; Pagani, Enrico; Luria, Gianvittorio
2010-01-01
This paper is a direct continuation of arXiv:0705.2362 . The Hamiltonian aspects of the theory are further developed. Within the framework provided by the first paper, the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A necessary and sufficient condition for minimality is proved.
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…
Stochastic Calculus of Wrapped Compartments
Coppo, Mario; Drocco, Maurizio; Grassi, Elena; Troina, Angelo; 10.4204/EPTCS.28.6
2010-01-01
The Calculus of Wrapped Compartments (CWC) is a variant of the Calculus of Looping Sequences (CLS). While keeping the same expressiveness, CWC strongly simplifies the development of automatic tools for the analysis of biological systems. The main simplification consists in the removal of the sequencing operator, thus lightening the formal treatment of the patterns to be matched in a term (whose complexity in CLS is strongly affected by the variables matching in the sequences). We define a stochastic semantics for this new calculus. As an application we model the interaction between macrophages and apoptotic neutrophils and a mechanism of gene regulation in E.Coli.
DEFF Research Database (Denmark)
Clouston, Ranald; Bizjak, Aleš; Grathwohl, Hans Bugge;
2016-01-01
-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always......We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type...
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
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
The Initial Conditions of Fractional Calculus
International Nuclear Information System (INIS)
During the past fifty years , Fractional Calculus has become an original and renowned mathematical tool for the modelling of diffusion Partial Differential Equations and the design of robust control algorithms. However, in spite of these celebrated results, some theoretical problems have not yet received a satisfying solution. The mastery of initial conditions, either for Fractional Differential Equations (FDEs) or for the Caputo and Riemann-Liouville fractional derivatives, remains an open research domain. The solution of this fundamental problem, also related to the long range memory property, is certainly the necessary prerequisite for a satisfying approach to modelling and control applications. The fractional integrator and its continuously frequency distributed differential model is a valuable tool for the simulation of fractional systems and the solution of initial condition problems. Indeed, the infinite dimensional state vector of fractional integrators allows the direct generalization to fractional calculus of the theoretical results of integer order systems. After a reminder of definitions and properties related to fractional derivatives and systems, this presentation is intended to show, based on the results of two recent publications [1,2], how the fractional integrator provides the solution of the initial condition problem of FDEs and of Caputo and Riemann-Liouville fractional derivatives. Numerical simulation examples illustrate and validate these new theoretical concepts.
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...
The history of the calculus and its conceptual development
Boyer, Carl B
1959-01-01
This book, for the first time, provides laymen and mathematicians alike with a detailed picture of the historical development of one of the most momentous achievements of the human intellect ― the calculus. It describes with accuracy and perspective the long development of both the integral and the differential calculus from their early beginnings in antiquity to their final emancipation in the 19th century from both physical and metaphysical ideas alike and their final elaboration as mathematical abstractions, as we know them today, defined in terms of formal logic by means of the idea of a
The M-calculus: a Higher-Order Distributed Process Calculus
Schmitt, Alan; Stefani, Jean-Bernard
2002-01-01
This report presents a new distributed process calculus, called the -calculus. Key insights for the calculus are similar to those laid out by L. Cardelli for its calculus of ambients. Mobile Ambients and other recent distributed process calculi such as the Join calculus or the D-calculus introduce notions of distributed locations or localities, corresponding to a spatial partitioning of computations and embodying different features of distributed computations (e.g. failures, access control, p...
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.
Dynamic Visualizations of Calculus Ideas.
Embse, Charles Vonder
2001-01-01
Presents three fundamental ideas of calculus and explains using the coordinate plane geometrically. Uses Cabri Geometry II to show how computer geometry systems can facilitate student understanding of general conic objects and its dynamic algebraic equations. (KHR)
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....
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...
Decidability of Mean Value Calculus
Institute of Scientific and Technical Information of China (English)
LI Xiaoshan
1999-01-01
Mean Value Calculus (MVC)[1] is a real-time logicwhich can be used to specify and verify real-time systems[2]. As aconservative extension of Duration Calculus (DC)[3], MVC increasesthe expressive power but keeps the properties of DC. In this paper wepresent decidability results of MVC. An interesting result is that propositional MVC with chop star operator is still decidable, which develops the results of[4]and[5].
Hill, Greg
2011-01-01
Calculus is the basis of all advanced science and math. But it can be very intimidating, especially if you're learning it for the first time! If finding derivatives or understanding integrals has you stumped, this book can guide you through it. This indispensable resource offers hundreds of practice exercises and covers all the key concepts of calculus, including:- Limits of a function- Derivatives of a function- Monomials and polynomials- Calculating maxima and minima- Logarithmic differentials- Integrals- Finding the volume of irregularly shaped objectsBy breaking down challenging concepts a
Laguerre calculus and Paneitz operator on the Heisenberg group
Institute of Scientific and Technical Information of China (English)
CHANG; Der-Chen
2009-01-01
Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group.Many sub-elliptic partial differential operators can be inverted by Laguerre calculus.In this article,we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation.The Paneitz operator which plays an important role in CR geometry can be written as follows:Here{Zj}n j=1 is an orthonormal basis for the subbundle T(1,0)of the complex tangent bundle TC(Hn) and T is the"missing direction".The operator Lα is the sub-Laplacian on the Heisenberg group which is sub-elliptic ifαdoes not belong to an exceptional setΛα.We also construct projection operators and relative fundamental solution for the operator Lα whileα∈Λα.
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 ...
A Tutorial Review on Fractal Spacetime and Fractional Calculus
He, Ji-Huan
2014-11-01
This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.
Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus
Hirani, A. N.; Nakshatrala, K. B.; Chaudhry, J. H.
2015-05-01
We derive a numerical method for Darcy flow, and also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for a spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solutions in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. We also show numerical evidence of convergence of the flux and the pressure. A convergence experiment is included for Darcy flow on a surface. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this article. We also include a discussion of the boundary condition in terms of exterior calculus.
``Riemann equations'' in bidifferential calculus
Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.
2015-10-01
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.
Differential geometry and the calculus of variations
Hermann, Robert
1968-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
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
The untyped stack calculus and Bohm's theorem
Alberto Carraro
2013-01-01
The stack calculus is a functional language in which is in a Curry-Howard correspondence with classical logic. It enjoys confluence but, as well as Parigot's lambda-mu, does not admit the Bohm Theorem, typical of the lambda-calculus. We present a simple extension of stack calculus which is for the stack calculus what Saurin's Lambda-mu is for lambda-mu.
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....
Answers to selected problems in multivariable calculus with linear algebra and series
Trench, William F
1972-01-01
Answers to Selected Problems in Multivariable Calculus with Linear Algebra and Series contains the answers to selected problems in linear algebra, the calculus of several variables, and series. Topics covered range from vectors and vector spaces to linear matrices and analytic geometry, as well as differential calculus of real-valued functions. Theorems and definitions are included, most of which are followed by worked-out illustrative examples.The problems and corresponding solutions deal with linear equations and matrices, including determinants; vector spaces and linear transformations; eig
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.…
The Power of Investigative Calculus Projects
Perrin, John Robert; Quinn, Robert J.
2008-01-01
This article describes investigative calculus projects in which students explore a question or problem of their own construction. Three exemplary pieces of student work are showcased. Investigative calculus projects are an excellent way to foster student understanding and interest in calculus. (Contains 4 figures.)
Applying π-Calculus to Practice
DEFF Research Database (Denmark)
Abendroth, Jorg
2003-01-01
The π-Calculus has been developed to reason about behavioural equivalence. Different notations of equivalence are defined in terms of process interactions, as well as the context of processes. There are various extensions of the π-Calculus, such as the SPI calculus, which has primitives to facili...
Graphic lambda calculus and knot diagrams
Buliga, Marius
2012-01-01
In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.
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
Vaux, Lionel
2009-01-01
We introduce an extension of the pure lambda-calculus by endowing the set of terms with a structure of vector space, or more generally of module, over a fixed set of scalars. Terms are moreover subject to identities similar to usual point-wise definition of linear combinations of functions with values in a vector space. We then study a natural extension of beta-reduction in this setting: we prove it is confluent, then discuss consistency and conservativity over the ordinary lambda-calculus. W...
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...... barbed equivalences and study their possible applications....
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,…
The Algebra of Schubert Calculus
Gatto, Letterio
2004-01-01
A flexible unified framework for both classical and quantum Schubert calculus is proposed. It is based on a natural combinatorial approach relying on the Hasse-Schmidt extension of a certain family of pairwise commuting endomorphisms of an infinite free Z-module M to its exterior algebra.
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…
Calculus Students' Understanding of Volume
Dorko, Allison; Speer, Natasha M.
2013-01-01
Researchers have documented difficulties that elementary school students have in understanding volume. Despite its importance in higher mathematics, we know little about college students' understanding of volume. This study investigated calculus students' understanding of volume. Clinical interview transcripts and written responses to volume…
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.
ENERGY CALCULUS IN CHINESE LANGUAGESEGMENTATION
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
Based on cognitive science, the EnergyCalculus in Chinese language segmentation was presented to eliminate segmentation ambiguity. The notion of "EnergyCost" was advanced to denote the extent of the under-standability of a certain segmentation. EnergyCost function was defined with Z-notation. This approcah is effective to all natural language segmentation.
A "Model" Multivariable Calculus Course.
Beckmann, Charlene E.; Schlicker, Steven J.
1999-01-01
Describes a rich, investigative approach to multivariable calculus. Introduces a project in which students construct physical models of surfaces that represent real-life applications of their choice. The models, along with student-selected datasets, serve as vehicles to study most of the concepts of the course from both continuous and discrete…
Stochastic Pi-calculus Revisited
DEFF Research Database (Denmark)
Cardelli, Luca; Mardare, Radu Iulian
2013-01-01
We develop a version of stochastic Pi-calculus with a semantics based on measure theory. We dene the behaviour of a process in a rate environment using measures over the measurable space of processes induced by structural congruence. We extend the stochastic bisimulation to include the concept of...
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…
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…
Dyadic Modalities and Lambek Calculus
Roorda, Dirk; de Rijke, Maarten
1993-01-01
The Lambek calculus is a logic on the one hand, and a grammar on the other. The system is studied in different disciplines, having their own interests. The logician studies relations with other systems, models in general, cut elimination etc. The linguist is interested in parsing properties, express
Quantum stochastic calculus associated with quadratic quantum noises
Energy Technology Data Exchange (ETDEWEB)
Ji, Un Cig, E-mail: uncigji@chungbuk.ac.kr [Department of Mathematics, Research Institute of Mathematical Finance, Chungbuk National University, Cheongju, Chungbuk 28644 (Korea, Republic of); Sinha, Kalyan B., E-mail: kbs-jaya@yahoo.co.in [Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore-64, India and Department of Mathematics, Indian Institute of Science, Bangalore-12 (India)
2016-02-15
We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie †-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes, and the quantum Itô formula is revisited. The existence and uniqueness of solution of a quantum stochastic differential equation is proved. The unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined. The quantum stochastic calculus extends the Hudson-Parthasarathy quantum stochastic calculus.
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.
Quantum stochastic calculus associated with quadratic quantum noises
International Nuclear Information System (INIS)
We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie †-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes, and the quantum Itô formula is revisited. The existence and uniqueness of solution of a quantum stochastic differential equation is proved. The unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined. The quantum stochastic calculus extends the Hudson-Parthasarathy quantum stochastic calculus
A Calculus of Higher-Order Distributed Components
Stefani, Jean-Bernard
2003-01-01
This report presents a calculus for higher-order distributed components, the Kell calculus. The calculus can be understood as a direct extension of the higher-order -calculus with programmable locations. The report illustrates the expressive power of the Kell calculus by encoding several process calculi with explicit locations, including Mobile Ambients, the Distributed Join calculus and the . The latter encoding demonstrates that the Kell calculus retains the expressive power of the but in a...
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.
Professor Rudolf Gorenflo and his Contribution to Fractional Calculus
Luchko, Yury; Mainardi, Francesco; Rogosin, Sergei
2011-01-01
MSC 2010: 26A33 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary This paper presents a brief overview of the life story and professional career of Prof. R. Gorenflo - a well-known mathematician, an expert in the field of Differential and Integral Equations, Numerical Mathematics, Fractional Calculus and Applied Analysis, an interesting conversational partner, an experienced colleague, and a real friend. Especially his role in the modern Fraction...
Sampling, splitting and merging in coinductive stream calculus
Niqui, Milad; Rutten, Jan
2009-01-01
We study various operations for partitioning, projecting and merging streams of data. These operations are motivated by their use in dataflow programming and the stream processing languages. We use the framework of stream calculus and stream circuits for defining and proving properties of such operations using behavioural differential equations and coinduction proof principles. We study the invariance of certain well patterned classes of streams, namely rational and algebraic streams, under s...
λμ-calculus and Λμ-calculus: a Capital Difference
Herbelin, Hugo; Saurin, Alexis
2009-01-01
Since Parigot designed the λμ-calculus to algorithmically interpret classical natural deduction, several variants of λμ-calculus have been proposed. Some of these variants derived from an alteration of the original syntax due to de Groote, leading in particular to the Λμ-calculus of the second author, a calculus truly different from λμ-calculus since, in the untyped case, it provides a Böhm separation theorem that the original calculus does not satisfy. In addition to a survey of some aspects...
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 ...
A non-local vector calculus,non-local volume-constrained problems,and non-local balance laws
Du, Q.; Gunzburger, M.; Lehoucq, R. B.; Zhou, K.
2011-01-01
A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoints operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted int...
Malliavin differentiability of solutions of rough differential equations
Inahama, Yuzuru
2013-01-01
In this paper we study rough differential equations driven by Gaussian rough paths from the viewpoint of Malliavin calculus. Under mild assumptions on coefficient vector fields and underlying Gaussian processes, we prove that solutions at a fixed time is smooth in the sense of Malliavin calculus. Examples of Gaussian processes include fractional Brownian motion with Hurst parameter larger than $1/4$.
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…
Decidable Type Inference for the Polymorphic Rewriting Calculus
Cirstea, Horatiu; Kirchner, Claude; Liquori, Luigi; Wack, Benjamin
2006-01-01
The rewriting calculus is a minimal framework embedding lambda calculus and term rewriting systems that allows abstraction on variables and patterns. The rewriting calculus features higher-order functions (from the lambda calculus) and pattern matching (from term rewriting systems). In this paper, we study extensively the decidability of type inference in the second-order rewriting calculus à la Curry.
Cosmological modelling with Regge calculus
Liu, Rex G
2015-01-01
The late universe's matter distribution obeys the Copernican principle at only the coarsest of scales. The relative importance of such inhomogeneity is still not well understood. Because of the Einstein field equations' non-linear nature, some argue a non-perturbative approach is necessary to correctly model inhomogeneities and may even obviate any need for dark energy. We shall discuss an approach based on Regge calculus, a discrete approximation to general relativity: we shall discuss the Collins--Williams formulation of Regge calculus and its application to two toy universes. The first is a universe for which the continuum solution is well-established, the $\\Lambda$-FLRW universe. The second is an inhomogeneous universe, the `lattice universe' wherein matter consists solely of a lattice of point masses with pure vacuum in between, a distribution more similar to that of the actual universe compared to FLRW universes. We shall discuss both regular lattices and one where one mass gets perturbed.
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...
Fractal calculus involving gauge function
Golmankhaneh, Alireza K.; Baleanu, Dumitru
2016-08-01
Henstock-Kurzweil integral or gauge integral is the generalization of the Riemann integral. The functions which are not integrable because of singularity in the senses of Lebesgue or Riemann are gauge integrable. In this manuscript, we have generalized Fα-calculus using the gauge integral method for the integrating of the functions on fractal set subset of real-line where they have singularities. The suggested new method leads to the wider class of functions on the fractal subset of real-line that are *Fα-integrable. Using gauge function we define *Fα-derivative of functions their Fα-derivative is not exist. The reported results can be used for generalizing the fundamental theorem of Fα-calculus.
Schubert calculus and singularity theory
Gorbounov, Vassily; Petrov, Victor
2012-02-01
Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces, it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K-theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note, we show that there is, in some sense, the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of Lerche Vafa and Warner. The main conjecture these authors made was that the classical cohomology of a Hermitian symmetric homogeneous manifold is a Jacobi ring of an appropriate potential. We extend this conjecture and provide a simple proof. Namely, we show that the cohomology of the Hermitian symmetric space is a Jacobi ring of a certain potential and the equivariant and the quantum cohomology and the K-theory is a Jacobi ring of a particular deformation of this potential. This suggests to study the most general deformations of the Frobenius algebra of cohomology of these manifolds by considering the versal deformation of the appropriate potential. The structure of the Jacobi ring of such potential is a subject of well developed singularity theory. This gives a potentially new way to look at the classical, the equivariant, the quantum and other flavors of Schubert calculus.
Extended Report: The Implicit Calculus
Oliveira, Bruno C d S; Choi, Wontae; Lee, Wonchan; Yi, Kwangkeun
2012-01-01
Generic programming (GP) is an increasingly important trend in programming languages. Well-known GP mechanisms, such as type classes and the C++0x concepts proposal, usually combine two features: 1) a special type of interfaces; and 2) implicit instantiation of implementations of those interfaces. Scala implicits are a GP language mechanism, inspired by type classes, that break with the tradition of coupling implicit instantiation with a special type of interface. Instead, implicits provide only implicit instantiation, which is generalized to work for any types. This turns out to be quite powerful and useful to address many limitations that show up in other GP mechanisms. This paper synthesizes the key ideas of implicits formally in a minimal and general core calculus called the implicit calculus, and it shows how to build source languages supporting implicit instantiation on top of it. A novelty of the calculus is its support for partial resolution and higher-order rules (a feature that has been proposed bef...
A Calculus of Evolving Objects
Directory of Open Access Journals (Sweden)
M. Dezani-Ciancaglini
2008-01-01
Full Text Available The demands of developing modern, highly dynamic applications have led to an increasing interest in dynamic programming languages and mechanisms. Not only must applications evolve over time, but the object models themselves may need to be adapted to the requirements of different run-time contexts. Class-based models and prototype-based models, for example, may need to co-exist to meet the demands of dynamically evolving applications. Multi-dimensional dispatch, fine-grained and dynamic software composition, and run-time evolution of behaviour are further examples of diverse mechanisms which may need to co-exist in a dynamically evolving run-time environment. How can we model the semantics of these highly dynamic features, yet still offer some reasonable safety guarantees?To this end we present an original calculus in which objects can adapt their behaviour at run-time. Both objects and environments are represented by first-class mappings between variables and values. Message sends are dynamically resolved to method calls. Variables may be dynamically bound, making it possible to model a variety of dynamic mechanisms within the same calculus. Despite the highly dynamic nature of the calculus, safety properties are assured by a type assignment system.
A Process Calculus for Molecular Interaction Maps
Roberto Barbuti; Andrea Maggiolo-Schettini; Paolo Milazzo; Giovanni Pardini; Aureliano Rama
2009-01-01
We present the MIM calculus, a modeling formalism with a strong biological basis, which provides biologically-meaningful operators for representing the interaction capabilities of molecular species. The operators of the calculus are inspired by the reaction symbols used in Molecular Interaction Maps (MIMs), a diagrammatic notation used by biologists. Models of the calculus can be easily derived from MIM diagrams, for which an unambiguous and executable interpretation is thus obtained. We give...
Control Flow Analysis for SF Combinator Calculus
Lester, Martin
2015-01-01
Programs that transform other programs often require access to the internal structure of the program to be transformed. This is at odds with the usual extensional view of functional programming, as embodied by the lambda calculus and SK combinator calculus. The recently-developed SF combinator calculus offers an alternative, intensional model of computation that may serve as a foundation for developing principled languages in which to express intensional computation, including program transfo...
Monogenic Calculus as an Intertwining Operator
Kisil, Vladimir V.
2003-01-01
We revise a monogenic calculus for several non-commuting operators, which is defined through group representations. Instead of an algebraic homomorphism we use group covariance. The related notion of joint spectrum and spectral mapping theorem are discussed. The construction is illustrated by a simple example of calculus and joint spectrum of two non-commuting selfadjoint (n\\times n) matrices. Keywords: Functional calculus, spectrum, intertwining operator, spectral mapping theorem, jet spaces...
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.
The call-by-need lambda calculus (unabridged).
Maraist, John; Odersky, Martin; Wadler, Phil
2007-01-01
We present a calculus that captures the operational semantics of call-by-need.We demonstrate that the calculus is confluent and standardizable and entails the same observational equivalences as call-by-name lambda calculus.
Petri nets semantics ofπ-calculus
Institute of Scientific and Technical Information of China (English)
Zhenhua YU; Yuanli CAI; Haiping XU
2008-01-01
As π-calculus based on the interleaving semantics cannot depict the true concurrency and has few supporting tools,it is translated into Petri nets.π-calculus is divided into basic elements,sequence,concurrency,choice and recursive modules.These modules are translated into Petri nets to construct a complicated system.Petri nets semantics for π-calculus visualize system structure as well as system behaviors.The structural analysis techniques allow direct qualitative analysis of the system properties on the structure of the nets.Finally,Petri nets semantics for π-calculus are illustrated by applying them to mobile telephone systems.
CALCULUS AND THE RACE TRACK PRINCIPLE
Akritas, Alkiviadis
1999-01-01
Calculus and Mathematica (C&M) by Davis, Porta and Uhl ia a well thought-out method that, when used properly, gives students an intuitive understanding of, and a feeling for, all the major calculus concepts. It is comprised of the following four books: C&M / Derivatives, C&M / Integrals, C&M / Vector Calculus, and C&M / Approximation, known also as Books 1-4. In these books the authors advocate an explore-and-discover method for teaching the basic concepts of Calculus to u...
A Higher-Order Calculus for Categories
DEFF Research Database (Denmark)
Cáccamo, Mario José; Winskel, Glynn
2001-01-01
in their free variables. There are special binders for limits and more general ends. The rules for limits and ends support an algebraic manipulation of universal constructions as opposed to a more traditional diagrammatic approach. Duality within the calculus and applications in proving continuity are discussed......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...... with examples. The calculus gives a basis for mechanising a theory of categories in a generic theorem prover like Isabelle....
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...
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
Fractional calculus in hydrologic modeling: A numerical perspective.
Benson, David A; Meerschaert, Mark M; Revielle, Jordan
2013-01-01
Fractional derivatives can be viewed either as handy extensions 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 Lévy 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 Lévy) 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. PMID:23524449
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.
Fractional calculus with applications for nuclear reactor dynamics
Ray, Santanu Saha
2015-01-01
Introduces Novel Applications for Solving Neutron Transport EquationsWhile deemed nonessential in the past, fractional calculus is now gaining momentum in the science and engineering community. Various disciplines have discovered that realistic models of physical phenomenon can be achieved with fractional calculus and are using them in numerous ways. Since fractional calculus represents a reactor more closely than classical integer order calculus, Fractional Calculus with Applications for Nuclear Reactor Dynamics focuses on the application of fractional calculus to describe the physical behavi
The hidden structural rules of the discontinuous Lambek calculus
Valentín Fernández Gallart, José Oriol
2014-01-01
The sequent calculus sL for the Lambek calculus L (lambek 58) has no structural rules. Interestingly, sL is equivalent to a multimodal calculus mL, which consists of the nonassociative Lambek calculus with the structural rule of associativity. This paper proves that the sequent calculus or hypersequent calculus hD of the discontinuous Lambek calculus (Morrill and Valent\\'in), which like sL has no structural rules, is also equivalent to an omega-sorted multimodal calculus mD. More concretely, ...
Dental Calculus Arrest of Dental Caries
Keyes, Paul H.; Rams, Thomas E.
2016-01-01
Background An inverse relationship between dental calculus mineralization and dental caries demineralization on teeth has been noted in some studies. Dental calculus may even form superficial layers over existing dental caries and arrest their progression, but this phenomenon has been only rarely documented and infrequently considered in the field of Cariology. To further assess the occurrence of dental calculus arrest of dental caries, this study evaluated a large number of extracted human teeth for the presence and location of dental caries, dental calculus, and dental plaque biofilms. Materials and methods A total of 1,200 teeth were preserved in 10% buffered formal saline, and viewed while moist by a single experienced examiner using a research stereomicroscope at 15-25× magnification. Representative teeth were sectioned and photographed, and their dental plaque biofilms subjected to gram-stain examination with light microscopy at 100× magnification. Results Dental calculus was observed on 1,140 (95%) of the extracted human teeth, and no dental carious lesions were found underlying dental calculus-covered surfaces on 1,139 of these teeth. However, dental calculus arrest of dental caries was found on one (0.54%) of 187 evaluated teeth that presented with unrestored proximal enamel caries. On the distal surface of a maxillary premolar tooth, dental calculus mineralization filled the outer surface cavitation of an incipient dental caries lesion. The dental calculus-covered carious lesion extended only slightly into enamel, and exhibited a brown pigmentation characteristic of inactive or arrested dental caries. In contrast, the tooth's mesial surface, without a superficial layer of dental calculus, had a large carious lesion going through enamel and deep into dentin. Conclusions These observations further document the potential protective effects of dental calculus mineralization against dental caries. PMID:27446993
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, Maarten M.; Asveld, P.R.J.; Nijholt, A.
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: s
Raise Test Scores: Integrate Biology and Calculus.
Lukens, Jeffrey D.; Feinstein, Sheryl
This paper presents the results of research that compared the academic achievement of high school students enrolled in an integrated Advanced Placement Biology/Advanced Placement Calculus course with students enrolled in traditional Advanced Placement Biology and Advanced Placement Calculus courses. Study subjects included high school students…
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…
Heisenberg algebra and a graphical calculus
Khovanov, Mikhail
2010-01-01
A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of vector spaces of morphisms between products of generating objects in this category.
Aspects of Calculus for Preservice Teachers
Fothergill, Lee
2011-01-01
The purpose of this study was to compare the perspectives of faculty members who had experience teaching undergraduate calculus and preservice teachers who had recently completed student teaching in regards to a first semester undergraduate calculus course. An online survey was created and sent to recent student teachers and college mathematics…
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 si
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…
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…
Areas and Volumes in Pre-Calculus
Jarrett, Joscelyn A.
2008-01-01
This article suggests the introduction of the concepts of areas bounded by plane curves and the volumes of solids of revolution in Pre-calculus. It builds on the basic knowledge that students bring to a pre-calculus class, derives a few more formulas, and gives examples of some problems on plane areas and the volumes of solids of revolution that…
Attendance and Attainment in a Calculus Course
Meulenbroek, Bernard; van den Bogaard, Maartje
2013-01-01
In this paper the relationship between attendance and attainment in a standard calculus course is investigated. Calculus could in principle be studied without attending lectures due to the wealth of material available (in hardcopy and online). However, in this study we will show that the pass rate of students attending classes regularly (>75%…
Spikes in Quantum Regge Calculus
Ambjorn, J.; Nielsen, J.; Rolf, J.; Savvidy, G.
1997-01-01
We demonstrate by explicit calculation of the DeWitt-like measure in two-dimensional quantum Regge gravity that it is highly non-local and that the average values of link lengths $l, $, do not exist for sufficient high powers of $n$. Thus the concept of length has no natural definition in this formalism and a generic manifold degenerates into spikes. This might explain the failure of quantum Regge calculus to reproduce the continuum results of two-dimensional quantum gravity. It points to sev...
OVARIAN CALCIFICATION MIMICKING VESICLE CALCULUS
Directory of Open Access Journals (Sweden)
Pallavi
2013-04-01
Full Text Available INTRODUCTION: Calcification in ovary is usually dystrophic in natu re, forming secondary to degeneration of the epithelium or in association wit h areas of necrosis. It may occur in cases of endometriosis [1] or in some ovarian tumor eg. Fibro thecoma [2] , Brenner’s tumor [3] , cavernous hemangioma [4] etc. Benign unilateral densely calcified ovary wit hout any association with tumor or endometriosis has not been reported previously. We report a case of heavily calcified left ovary which mimicked as vesicle calculus on X- ray leading to confusion in diagnosis.
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
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
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
Cartan Calculus via Pauli Matrices
Mauro, D.
2002-01-01
In this paper we will provide a new operatorial counterpart of the path-integral formalism of classical mechanics developed in recent years. We call it new because the Jacobi fields and forms will be realized via finite dimensional matrices. As a byproduct of this we will prove that all the operations of the Cartan calculus, such as the exterior derivative, the interior contraction with a vector field, the Lie derivative and so on, can be realized by means of suitable tensor products of Pauli...
Cartan Calculus via Pauli Matrices
Mauro, D
2003-01-01
In this paper we will provide a new operatorial counterpart of the path-integral formalism of classical mechanics developed in recent years. We call it new because the Jacobi fields and forms will be realized via finite dimensional matrices. As a byproduct of this we will prove that all the operations of the Cartan calculus, such as the exterior derivative, the interior contraction with a vector field, the Lie derivative and so on, can be realized by means of suitable tensor products of Pauli and identity matrices.
A functional presentation of Pi calculus
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
From the very beginning process algebra introduced the dichotomy between channels and processes. This dichotomy prevails in all present process calculi.The situation is in contrast to that with lambda calculus which has only one class of entities——the lambda terms. We introduce in this paper a process calculus called Lamp in which channels are process names. The language is more uniform than existing process calculi in two aspects: First it has a unified treatment of channels and processes. There is only one class of syntactical entities——processes. Second it has a unified presentation of both first order and higher order process calculi. The language is functional in the sense that lambda calculus is functional.Two bisimulation equivalences, barbed and closed bisimilarities, are proved to coincide.A natural translation from Pi calculus to Lamp is shown to preserve both operational and algebraic semantics. The relationship between lazy lambda calculus and Lamp is discussed.
Laguerre calculus and Paneitz operator on the Heisenberg group
Institute of Scientific and Technical Information of China (English)
CHANG Der-Cheni; CHANG Shu-Cheng; TIE JingZhi
2009-01-01
Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article,we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation. The Paneitz operator which plays an important role in CR geometry can be written as follows: Ρ_α=(ν)_a(ν)_a=4/1[∑n/j=1(Z_jZ_j+Z_jZ_j]~2+a~2T~2.Here {Z~j}~n_j=1 is an orthonormal basis for the subbundle T~(1,0) of the complex tangent bundle T_c(H_n) and T is the "missing direction". The operator ν_a is the sub-Laplaeian on the Heisenberg group which is sub-elliptic if α does not belong to an exceptional set Aα. We also construct projection operators and relative fundamental solution for the operator (ν)_α while α∈ (A)_α.
Computer-Oriented Calculus Courses Using Finite Differences.
Gordon, Sheldon P.
The so-called discrete approach in calculus instruction involves introducing topics from the calculus of finite differences and finite sums, both for motivation and as useful tools for applications of the calculus. In particular, it provides an ideal setting in which to incorporate computers into calculus courses. This approach has been…
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…
New symbolic tools for differential geometry, gravitation, and field theory
Anderson, I. M.; Torre, C. G.
2012-01-01
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, spinor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors, algebraic classification of solutions of the Einstein equations, and symmetry reduction of field equations.
Distributed mean curvature on a discrete manifold for Regge calculus
Conboye, Rory; Ray, Shannon
2015-01-01
The integrated mean curvature of a simplicial manifold is well understood in both Regge Calculus and Discrete Differential Geometry. However, a well motivated pointwise definition of curvature requires a careful choice of volume over which to uniformly distribute the local integrated curvature. We show that hybrid cells formed using both the simplicial lattice and its circumcentric dual emerge as a remarkably natural structure for the distribution of this local integrated curvature. These hybrid cells form a complete tessellation of the simplicial manifold, contain a geometric orthonormal basis, and are also shown to give a pointwise mean curvature with a natural interpretation as a fractional rate of change of the normal vector.
Distributed mean curvature on a discrete manifold for Regge calculus
Conboye, Rory; Miller, Warner A.; Ray, Shannon
2015-09-01
The integrated mean curvature of a simplicial manifold is well understood in both Regge Calculus and Discrete Differential Geometry. However, a well motivated pointwise definition of curvature requires a careful choice of the volume over which to uniformly distribute the local integrated curvature. We show that hybrid cells formed using both the simplicial lattice and its circumcentric dual emerge as a remarkably natural structure for the distribution of this local integrated curvature. These hybrid cells form a complete tessellation of the simplicial manifold, contain a geometric orthonormal basis, and are also shown to give a pointwise mean curvature with a natural interpretation as the fractional rate of change of the normal vector.
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…
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…
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.
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
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 into ac...... to show the application of this calculus.......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...
Recursive sequences in first-year calculus
Krainer, Thomas
2016-02-01
This article provides ready-to-use supplementary material on recursive sequences for a second-semester calculus class. It equips first-year calculus students with a basic methodical procedure based on which they can conduct a rigorous convergence or divergence analysis of many simple recursive sequences on their own without the need to invoke inductive arguments as is typically required in calculus textbooks. The sequences that are accessible to this kind of analysis are predominantly (eventually) monotonic, but also certain recursive sequences that alternate around their limit point as they converge can be considered.
From X to Pi; Representing the Classical Sequent Calculus in the Pi-calculus
van Bakel, Steffen; Vigliotti, Maria Grazia
2011-01-01
We study the Pi-calculus, enriched with pairing and non-blocking input, and define a notion of type assignment that uses the type constructor "arrow". We encode the circuits of the calculus X into this variant of Pi, and show that all reduction (cut-elimination) and assignable types are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen's calculus LK, this implies that all proofs in LK have a representation in Pi.
Astrophysical Applications of Fractional Calculus
Stanislavsky, Aleksander A.
The paradigm of fractional calculus occupies an important place for the macroscopic description of subdiffusion. Its advance in theoretical astrophysics is expected to be very attractive too. In this report we discuss a recent development of the idea to some astrophysical problems. One of them is connected with a random migration of bright points associated with magnetic fields at the solar photosphere. The transport of the bright points has subdiffusive features that require the fractional generalization of the Leighton's model. Another problem is related to the angular distribution of radio beams, being propagated through a medium with random inhomogeneities. The peculiarity of this medium is that radio beams are trapped because of random wave localization. This idea can be useful for the diagnostics of interplanetary and interstellar turbulent media.
A CALCULUS FOR SERVICES INNOVATION
Institute of Scientific and Technical Information of China (English)
James M.TIEN; Daniel BERG
2007-01-01
Innovation in the services area - especially in the electronic services (e-services) domain - can be systematically developed by first considering the strategic drivers and foci, then the tactical principles and enablers, and finally the operational decision attributes, all of which constitute a process or calculus of services innovation. More specifically, there are four customer drivers (i.e., collaboration,customization, integration and adaptation), three business foci (i.e., creation-focused, solution-focused and competition-focused), six business principles (i.e., reconstruct market boundaries, focus on the big picture not numbers, reach beyond existing demand, get strategic sequence right, overcome organizational hurdles and build execution into strategy), eight technical enablers (i.e., software algorithms, automation, telecommunication, collaboration, standardization, customization,organization, and globalization), and six attributes of decision informatics (i.e., decision-driven,information-based, real-time, continuously-adaptive, customer-centric and computationally-intensive).It should be noted that the four customer drivers are all directed at empowering the individual - that is,at recognizing that the individual can, respectively, contribute in a collaborative situation, receive customized or personalized attention, access an integrated system or process, and obtain adaptive real-time or just-in-time input. The developed process or calculus serves to identify the potential white spaces or blue oceans for innovation. In addition to expanding on current innovations in services and related experiences, white spaces are identified for possible future innovations; they include those that can mitigate the unforeseen consequences or abuses of earlier innovations, safeguard our rights to privacy, protect us from the always-on, interconnected world, provide us with an authoritative search engine, and generate a GDP metric that can adequately measure the growing
The Calculus Concept Readiness (CCR) Instrument: Assessing Student Readiness for Calculus
Carlson, Marilyn; West, Richard
2010-01-01
The Calculus Concept Readiness (CCR) instrument is based on the broad body of mathematics education research that has revealed major understandings, representational abilities, and reasoning abilities students need to construct in precalculus level courses to be successful in calculus. The CCR is a 25-item multiple-choice instrument, and the CCR taxonomy articulates what the CCR assesses. The methodology used to develop and validate the CCR is described and illustrated. Results from administering the CCR as a readiness examination in calculus are provided along with data to guide others in using the CCR as a readiness examination for beginning calculus.
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
Newton Binomial Formulas in Schubert Calculus
Cordovez, Jorge; Gatto, Letterio; Santiago, Taise
2008-01-01
We prove Newton's binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points.
Umbral Calculus a Model from Convoloids
Kisil, V V
1997-01-01
We are working in the three-borders-point between combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures). The profit from the ``contraband'' in all directions is investigated.
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.
A Temporal Approach to Stochastic Network Calculus
Xie, Jing; Xie, Min
2011-01-01
Stochastic network calculus is a newly developed theory for stochastic service guarantee analysis of computer networks. In the current stochastic network calculus literature, its fundamental models are based on the cumulative amount of traffic or cumulative amount of service. However, there are network scenarios where direct application of such models is difficult. This paper presents a temporal approach to stochastic network calculus. The key idea is to develop models and derive results from the time perspective. Particularly, we define traffic models and service models based on the cumulative packet inter-arrival time and the cumulative packet service time, respectively. Relations among these models as well as with the existing models in the literature are established. In addition, we prove the basic properties of the proposed models, such as delay bound and backlog bound, output characterization, concatenation property and superposition property. These results form a temporal stochastic network calculus an...
Multi-instanton calculus in supersymmetric theories
International Nuclear Information System (INIS)
In this talk I review some recent results concerning multi-instanton calculus in supersymmetric field theories. More in detail, I will show how these computations can be efficiently performed using the formalism of topological field theories. (author)
Fractional Vector Calculus and Fractional Special Function
Li, Ming-Fan; Ren, Ji-Rong; Zhu, Tao
2010-01-01
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric functions.
Applying Change of Variable to Calculus Problems
Kachapova, Farida; Kachapov, Ilias
2011-01-01
This article describes the technique of introducing a new variable in some calculus problems to help students master the skills of integration and evaluation of limits. This technique is algorithmic and easy to apply.
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
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...
Tuplix Calculus Specifications of Financial Transfer Networks
Bergstra, J A; van der Zwaag, M B
2008-01-01
We study the application of Tuplix Calculus in modular financial budget design. We formalize organizational structure using financial transfer networks. We consider the notion of flux of money over a network, and a way to enforce the matching of influx and outflux for parts of a network. We exploit so-called signed attribute notation to make internal streams visible through encapsulations. Finally, we propose a Tuplix Calculus construct for the definition of data functions.
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...... of how well-established process calculi could be encoded into AbC is given by considering the translation into AbC of a proto-typical π-calculus process....
A Tableaux Calculus for Ambiguous Quantification
Monz, Christof; de Rijke, Maarten
2000-01-01
Coping with ambiguity has recently received a lot of attention in natural language processing. Most work focuses on the semantic representation of ambiguous expressions. In this paper we complement this work in two ways. First, we provide an entailment relation for a language with ambiguous expressions. Second, we give a sound and complete tableaux calculus for reasoning with statements involving ambiguous quantification. The calculus interleaves partial disambiguation steps with steps in a t...
A Superposition Calculus for Abductive Reasoning
Echenim, Mnacho; Peltier, Nicolas
2014-01-01
We present a modification of the superposition calculus that is meant to generate consequences of sets of first-order axioms. This approach is proven to be sound and deductive-complete in the presence of redundancy elimination rules, provided the considered consequences are built on a given finite set of ground terms, represented by constant symbols. In contrast to other approaches, most existing results about the termination of the superposition calculus can be carried over to our procedure....
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.
Variational time discretization of geodesic calculus
Rumpf, Martin; Wirth, Benedikt
2012-01-01
We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps, and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct methods in the calculus of variation, on $\\Gamma$-convergence, and on weighted finite ele...
Ordered Models of the Lambda Calculus
Salibra, Antonino; Carraro, Alberto
2013-01-01
Answering a question by Honsell and Plotkin, we show that there are two equations between lambda terms, the so-called subtractive equations, consistent with lambda calculus but not simultaneously satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the order-incompleteness of lambda calculus, by studying the connection between the notion of absolute unorderability in a specific point and a weaker notion of subtractivity ...
A phenomenological calculus of Wiener description space.
Richardson, I W; Louie, A H
2007-10-01
The phenomenological calculus is a categorical example of Robert Rosen's modeling relation. This paper is an alligation of the phenomenological calculus and generalized harmonic analysis, another categorical example. Our epistemological exploration continues into the realm of Wiener description space, in which constitutive parameters are extended from vectors to vector-valued functions of a real variable. Inherent in the phenomenology are fundamental representations of time and nearness to equilibrium. PMID:17955459
Superconformal tensor calculus in five dimensions
International Nuclear Information System (INIS)
We present a full superconformal tensor calculus in five spacetime dimensions in which the Weyl multiplet has 32 Bose plus 32 Fermi degrees of freedom. It is derived using dimensional reduction from the 6D superconformal tensor calculus. We present two types of 32+32 Weyl multiplets, a vector multiplet, linear multiplet, hypermultiplet and nonlinear multiplet. Their superconformal transformation laws and the embedding and invariant action formulas are given. (author)
Superconformal Tensor Calculus in Five Dimensions
Fujita, Tomoyuki; Ohashi, Keisuke
2001-01-01
We present a full superconformal tensor calculus in five spacetime dimensions in which the Weyl multiplet has 32 Bose plus 32 Fermi degrees of freedom. It is derived by the dimensional reduction from the 6D superconformal tensor calculus. We present two types of 32+32 Weyl multiplets, vector multiplet, linear multiplet, hypermultiplet and nonlinear multiplet. Their superconformal transformation laws and the embedding and invariant action formulas are given.
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...... ensures that errors do not occur outside sandboxes and that authentication and migration only happen when allowed. Futhermore a weak subject reduction result involving partial well sortedness is presented....
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.
Metaplectic Representation, Conley-Zehnder Index, and Weyl Calculus on Phase Space
de Gosson, Maurice
We define and study a metaplectically covariant class of pseudo-differential operators acting on functions on symplectic space and generalizing a modified form of the usual Weyl calculus. This construction requires a precise calculation of the twisted Weyl symbol of a class of generators of the metaplectic group and the use of a Conley-Zehnder type index for symplectic paths, defined without restrictions on the endpoint. Our calculus is related to the usual Weyl calculus using a family of isometries of L2(ℝn) on closed subspaces of L2(ℝ2n) and to an irreducible representation of the Heisenberg algebra distinct from the usual Schrödinger representation.
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.
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.; Hirani, Anil N.; Samtaney, Ravi
2016-05-01
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.
Barbed congruence of the asymmetric chi calculus
Institute of Scientific and Technical Information of China (English)
DONG Xiao-ju; FU Yu-xi
2006-01-01
The chi calculus is a model of mobile processes. It has evolved from the pi-calculus with motivations from simplification and communication-as-cut-elimination. This paper studies the chi calculus in the framework incorporating asymmetric communication. The major feature of the calculus is the identification of two actions:x/x and τ. The investigation on the barbed bisimilarity shows how the property affects the observational theory.Based on the definition of the barbed bisimilarity, the simulation properties of the barbed bisimilarity are studied. It shows that the algebraic properties of the barbed bisimilarity have changed greatly compared with the chi calculus. Although the definition of the barbed bisimilarity is very simple, the property of closeness under contexts makes it difficult to understand the barbed bisimilarity directly. Therefore an open style definition of the barbed bisimilarity is given, which is a context free description of barbed bisimilarity. Its definition is complex,but it is a well-behaved relation for it coincides with the barbed bisimilarity. It also helps to build an axiomatization system for the barbed congruence. Besides the axioms for the strong barbed bisimilarity, the paper proposes a new tau law and four new update laws for the barbed congruence. Both the operational and algebraic properties of the enriched calculus improve the understanding of the bisimulation behaviors of the model.
New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory (extended version)
Torre, Charles G.; Anderson, Ian M.
2011-01-01
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field...
The Sustainability of Dental Calculus for Archaeological Research
DEFF Research Database (Denmark)
Mackie, Meaghan Emma; Radini, Anita; Speller, Camilla
Dental calculus is a mineralized plaque biofilm formed by microbiota of the oral microbiome. Until recently, the information potential of dental calculus for archaeological study was not fully realised and it was often discarded. However, it is now recognized that dental calculus entombs and pres......Dental calculus is a mineralized plaque biofilm formed by microbiota of the oral microbiome. Until recently, the information potential of dental calculus for archaeological study was not fully realised and it was often discarded. However, it is now recognized that dental calculus entombs...... and preserve biomolecules related to diet, health and disease....
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.
Ancient DNA analysis of dental calculus.
Weyrich, Laura S; Dobney, Keith; Cooper, Alan
2015-02-01
Dental calculus (calcified tartar or plaque) is today widespread on modern human teeth around the world. A combination of soft starchy foods, changing acidity of the oral environment, genetic pre-disposition, and the absence of dental hygiene all lead to the build-up of microorganisms and food debris on the tooth crown, which eventually calcifies through a complex process of mineralisation. Millions of oral microbes are trapped and preserved within this mineralised matrix, including pathogens associated with the oral cavity and airways, masticated food debris, and other types of extraneous particles that enter the mouth. As a result, archaeologists and anthropologists are increasingly using ancient human dental calculus to explore broad aspects of past human diet and health. Most recently, high-throughput DNA sequencing of ancient dental calculus has provided valuable insights into the evolution of the oral microbiome and shed new light on the impacts of some of the major biocultural transitions on human health throughout history and prehistory. Here, we provide a brief historical overview of archaeological dental calculus research, and discuss the current approaches to ancient DNA sampling and sequencing. Novel applications of ancient DNA from dental calculus are discussed, highlighting the considerable scope of this new research field for evolutionary biology and modern medicine.
Standardization of a Call-By-Value Lambda-Calculus
Guerrieri, Giulio; Paolini, Luca; Ronchi Della Rocca, Simona
2015-01-01
We study an extension of Plotkin's call-by-value lambda-calculus by means of two commutation rules (sigma-reductions). Recently, it has been proved that this extended calculus provides elegant characterizations of many semantic properties, as for example solvability. We prove a standardization theorem for this calculus by generalizing Takahashi's approach of parallel reductions. The standardization property allows us to prove that our calculus is conservative with respect to the Plotkin's one...
Enhancing Students’ Understanding in Calculus Trough Writing
Directory of Open Access Journals (Sweden)
Noraini Idris
2009-02-01
Full Text Available The purpose of this study was to investigate the effects of using writing activities on students’ understanding and achievement in Calculus. The design of this study was quasi-experimental. The subjects of this study consisted of two secondary schools in one of the states in Malaysia. Each school was assigned one intact class of Form Four to be the experimental group and another one intact class as the control. The experimental group learned mathematics by using the writing activities for five weeks, while the control group learned mathematics by using traditional whole-class instruction. A 20-item Calculus Achievement test was designed with reliability .87. The findings showed that the experimental group exhibited significantly greater improvement on calculus achievement. The students showed positive reaction towards the use of writing. Findings of this study provide information to schools to take advantage of writing activities to promote understanding.
Formalizing BPEL-TC Through ?-Calculus
Directory of Open Access Journals (Sweden)
Preeti Marwaha
2013-07-01
Full Text Available WS-BPEL is way to define business processes that interact with external entities through webservice operations using WSDL. We have proposed BPEL-TC, an extension to existing WS-BPEL whichuses temporally customized Web Services (WSDL-TC as a model for process decomposition and assembly.WSDL-TC handles both backward compatible and incompatible changes and also maintains variousversions of the artifacts that results due to changes over time and customizations desired by the users. Inthis paper, we are using pi-calculus to formalize Business Process Execution Language- TemporalCustomization (BPEL-TC process. π -calculus is a model of computation for concurrent systems alongwith changing connectivity of interactive systems. Pi-calculus is an extension of the process algebra CCS,with added mobility to CCS while preserving its algebraic properties.
Quantum geometry in dynamical Regge calculus
International Nuclear Information System (INIS)
We study geometric properties of dynamical Regge calculus which is a hybridization of dynamical triangulation and quantum Regge calculus. Lattice diffeomorphisms are generated by certain elementary moves on a simplicial lattice in the hybrid model. At the semiclassical level, we discuss a possibility that the lattice diffeomorphisms give a simple explanation for the Bekenstein-Hawking entropy of a black hole. At the quantum level, numerical calculations of 3D pure gravity show that a fractal structure of the hybrid model is the same as that of dynamical triangulation in the strong-coupling phase. In the weak-coupling phase, on the other hand, space-time becomes a spiky configuration, which often occurs in quantum Regge calculus
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...... and Cardelli's denotational model is sound but not complete with respect to the operational semantics. We also construct a modal logic for the typed functional Sigma-calculus, provide a translation of types to a sub-logic and prove the translation is sound and complete. The amount work required to perform...... these investigations indicate, that although it is perfectly possible to use process calculus techniques on object oriented languages, such techniques will not come to widespread use, but only be limited to reasoning about critical parts of a language or program design....
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 SI....... 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.......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...
Differential Equations with Linear Algebra
Boelkins, Matthew R; Potter, Merle C
2009-01-01
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, t
Research of Semantic Comparison between χ-calculus and π-calculus%χ-演算与π-演算的语义比较研究
Institute of Scientific and Technical Information of China (English)
徐林; 傅育熙
2000-01-01
Through the comparison of syntactic structure,operational semantics and algebraic semantics between χ-calculus and π-calculus, this paper concludes that χ-calculus has more succinct syntactic structure,more explicit operational semantics,more intuitionistic algebraic semantics and more favorable algebraic property. And a translation from π-calculus to χ-calculus is presented.
Functional calculus for generators of analytic semigroups of operators
Lopushansky O.V.; Sharyn S.V.
2012-01-01
We construct a functional calculus for generators of one-parameter boundedanalytic semigroups of operators on a Banach space. The calculus symbol classconsist of the Laplace image of the convolution algebra $cal S'_+$ of tempereddistributions with supports in $[0, infty)$. Domain of constructed calculus isdense in the Banach space.
Functional calculus for generators of analytic semigroups of operators
Directory of Open Access Journals (Sweden)
Lopushansky O.V.
2012-06-01
Full Text Available We construct a functional calculus for generators of one-parameter boundedanalytic semigroups of operators on a Banach space. The calculus symbol classconsist of the Laplace image of the convolution algebra $cal S'_+$ of tempereddistributions with supports in $[0, infty$. Domain of constructed calculus isdense in the Banach space.
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…
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.
A BRUTUS Logic for a Spi-Calculus Dialect
Gnesi, S.; Latella, D.; Lenzini, G.
2000-01-01
In the field of process algebras, the spi-calculus, a modified version of the π-calculus with encryption primitives, is indicated as an expressive specification language for cryptographic protocols. In spi-calculus basic security properties, such as secrecy and integrity can be formalized as may-tes
Laparoscopic Ureterolithotomy for Giant Ureteric Calculus: A Case Report
Prasad V. Magdum; Rajendra B. Nerli; Shishir Devaraju; Hiremath, Murigendra B.
2015-01-01
We present a case of a 21 year old male who presented with symptomatic right upper ureteric calculus measuring 5 cm × 1.5 cm fulfilling the criteria to be named as giant ureteric calculus. Laparoscopic right ureterolithotomy was performed and the giant ureteric calculus was retrieved.
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…
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…
Some problems in fractal differential equations
Su, Weiyi
2016-06-01
Based upon the fractal calculus on local fields, or p-type calculus, or Gibbs-Butzer calculus ([1],[2]), we suggest a constructive idea for "fractal differential equations", beginning from some special examples to a general theory. However, this is just an original idea, it needs lots of later work to support. In [3], we show example "two dimension wave equations with fractal boundaries", and in this note, other examples, as well as an idea to construct fractal differential equations are shown.
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...... be exploited to reduce the "administrative overhead" of the analysis specification and thus simplify it. Finally, we use HyLoTab, a fully automated theorem prover for hybrid logic, both as a convenient platform for a prototype implementation as well as to formally prove the correctness of the analysis. (C...
Dynamical Regge calculus as lattice gravity
Hagura, Hiroyuki
2001-03-01
We propose a hybrid approach to lattice quantum gravity by combining simultaneously the dynamical triangulation with the Regge calculus, called the dynamical Regge calculus (DRC). In this approach lattice diffeomorphism is realized as an exact symmetry by some hybrid ( k, l) moves on the simplicial lattice. Numerical study of 3D pure gravity shows that an entropy of the DRC is not exponetially bounded if we adopt the uniform measure Π idli. On the other hand, using the scale-invariant measure Π idli/ li, we can calculate observables and observe a large hysteresis between two phases that indicates the first-order nature of the phase transition.
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...
Modular invariance and covariant loop calculus
International Nuclear Information System (INIS)
The covariant loop calculus provides and efficient technique for computing explicit expressions for the density on moduli space corresponding to arbitrary (bosonic string) loop diagrams. Since modular invariance is not manifest, however, we carry out a detailed comparison with known explicit 2- and 3- loop results derived using analytic geometry (1 loop is known to be ok). We establish identity to 'high' order in some moduli and exactly in others. Agreement is found as a result of various non-trivial cancellations, in part related to number theory. We feel our results provide very strong support for the correctness of the covariant loop calculus approach. (orig.)
Dynamical Regge calculus as lattice gravity
International Nuclear Information System (INIS)
We propose a hybrid approach to lattice quantum gravity by combining simultaneously the dynamical triangulation with the Regge calculus, called the dynamical Regge calculus (DRC). In this approach lattice diffeomorphism is realized as an exact symmetry by some hybrid (k, l) moves on the simplicial lattice. Numerical study of 3D pure gravity shows that an entropy of the DRC is not exponetially bounded if we adopt the uniform measure Πidli. On the other hand, using the scale-invariant measure Πidli/li, we can calculate observables and observe a large hysteresis between two phases that indicates the first-order nature of the phase transition
A residue calculus for root systems
Ban, E. P. van den; Schlichtkrull, H.
2001-01-01
Let V be a nite dimensional real vector space on which a root system is given. Consider a meromorphic function ' on VC = V +iV , the singular locus of which is a locally nite union of hyperplanes of the form f 2 VC j h; i = sg, 2 , s 2 R. Assume ' is of suitable decay in the imaginary directions, so that integrals of the form R +iV '() d make sense for generic 2 V . A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and Paley{Wiener t...
Variational calculus with constraints on general algebroids
International Nuclear Information System (INIS)
Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and geometrical settings. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers the majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in classical mechanics for E = TM
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
Mapping the Join Calculus to Heterogeneous Hardware
Directory of Open Access Journals (Sweden)
Peter Calvert
2013-02-01
Full Text Available As modern architectures introduce additional heterogeneity and parallelism, we look for ways to deal with this that do not involve specialising software to every platform. In this paper, we take the Join Calculus, an elegant model for concurrent computation, and show how it can be mapped to an architecture by a Cartesian-product-style construction, thereby making use of the calculus' inherent non-determinism to encode placement choices. This unifies the concepts of placement and scheduling into a single task.
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...
Lazo, Matheus J.; Delfim F. M. Torres
2012-01-01
Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is cur...
Calculus on manifolds a modern approach to classical theorems of advanced calculus
Spivak, Michael D
1965-01-01
This little book is especially concerned with those portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approa
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.
Detection, removal and prevention of calculus: Literature Review
Directory of Open Access Journals (Sweden)
Deepa G. Kamath
2014-01-01
Full Text Available Dental plaque is considered to be a major etiological factor in the development of periodontal disease. Accordingly, the elimination of supra- and sub-gingival plaque and calculus is the cornerstone of periodontal therapy. Dental calculus is mineralized plaque; because it is porous, it can absorb various toxic products that can damage the periodontal tissues. Hence, calculus should be accurately detected and thoroughly removed for adequate periodontal therapy. Many techniques have been used to identify and remove calculus deposits present on the root surface. The purpose of this review was to compile the various methods and their advantages for the detection and removal of calculus.
The Development of Newtonian Calculus in Britain, 1700-1800
Guicciardini, Niccoló
2003-11-01
Introduction; Overture: Newton's published work on the calculus of fluxions; Part I. The Early Period: 1. The diffusion of the calculus (1700-1730); 2. Developments in the calculus of fluxions (1714-1733); 3. The controversy on the foundations of the calculus (1734-1742); Part II. The Middle Period: 4. The textbooks on fluxions (1736-1758); 5. Some applications of the calculus (1740-1743); 6. The analytic art (1755-1785); Part III. The Reform: 7. Scotland (1785-1809); 8. The Military Schools (1773-1819); 9. Cambridge and Dublin (1790-1820); 10. Tables; Endnotes; Bibliography; Index.
About compositional analysis of pi-calculus processes
Martinelli, Fabio
2003-01-01
We set up a logical framework for the compositional analysis of finite pi-calculus processes. In particular, we extend the partial model checking techniques developed for value passing process algebras to a nominal calculus, i.e. the pi-calculus. The logic considered is an adaptation of the ambient logic to the pi-calculus. As one of the possible applications, we show that our techniques may be used to study interesting security properties as confidentiality for (finite) pi-calculus processes.
About compositional analysis of pi-calculus processes
Martinelli, Fabio
2001-01-01
We set up a logical framework for the compositional analysis of finite pi-calculus processes. In particular, we extend the partial model checking techniques developed for value passing process algebras to a nominal calculus, i.e. the pi-calculus. The logic considered is an adaptation of the ambient logic to the pi-calculus. As one of the possible applications, we show that our techniques may be used to study interesting security properties as confidentiality for (finite) pi-calculus processes.
The Calculus Concept Readiness (CCR) Instrument: Assessing Student Readiness for Calculus
Carlson, Marilyn; Madison, Bernard; West, Richard
2010-01-01
The Calculus Concept Readiness (CCR) instrument is based on the broad body of mathematics education research that has revealed major understandings, representational abilities, and reasoning abilities students need to construct in precalculus level courses to be successful in calculus. The CCR is a 25-item multiple-choice instrument, and the CCR taxonomy articulates what the CCR assesses. The methodology used to develop and validate the CCR is described and illustrated. Results from administe...
Flipping a Calculus Class: One Instructor's Experience
Palmer, Katrina
2015-01-01
This paper describes one instructor's experiences during a year of flipping four calculus classes. The first exploration attempts to understand student expectations of a math class and their preference towards a flipped classroom. The second examines success of students from a flipped classroom, and the last investigates relationships with student…
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.
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. PMID:26361555
A planar calculus for infinite index subfactors
Penneys, David
2011-01-01
We develop an analog of Jones' planar calculus for II_1-factor bimodules with arbitrary left and right von Neumann dimension. We generalize to bimodules Burns' results on rotations and extremality for infinite index subfactors. These results are obtained without Jones' basic construction and the resulting Jones projections.
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...
Advanced calculus of a single variable
Geveci, Tunc
2016-01-01
This advanced undergraduate textbook is based on a one-semester course on single variable calculus that the author has been teaching at San Diego State University for many years. The aim of this classroom-tested book is to deliver a rigorous discussion of the concepts and theorems that are dealt with informally in the first two semesters of a beginning calculus course. As such, students are expected to gain a deeper understanding of the fundamental concepts of calculus, such as limits (with an emphasis on ε-δ definitions), continuity (including an appreciation of the difference between mere pointwise and uniform continuity), the derivative (with rigorous proofs of various versions of L’Hôpital’s rule) and the Riemann integral (discussing improper integrals in-depth, including the comparison and Dirichlet tests). Success in this course is expected to prepare students for more advanced courses in real and complex analysis and this book will help to accomplish this. The first semester of advanced calculus...
Exposing Calculus Students to Advanced Mathematics
Griffiths, Barry J.; Haciomeroglu, Erhan Selcuk
2014-01-01
To ensure the competitiveness of the USA in the global economy, and its role as a leader in science and engineering, it is important to cultivate the next generation of home grown mathematicians. However, while universities across the USA offer calculus classes to thousands of undergraduate students each year, very few of them go on to major in…
Using Matlab in a Multivariable Calculus Course.
Schlatter, Mark D.
The benefits of high-level mathematics packages such as Matlab include both a computer algebra system and the ability to provide students with concrete visual examples. This paper discusses how both capabilities of Matlab were used in a multivariate calculus class. Graphical user interfaces which display three-dimensional surfaces, contour plots,…
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…
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,…
A Planar Calculus for Infinite Index Subfactors
Penneys, David
2013-05-01
We develop an analog of Jones' planar calculus for II 1-factor bimodules with arbitrary left and right von Neumann dimension. We generalize to bimodules Burns' results on rotations and extremality for infinite index subfactors. These results are obtained without Jones' basic construction and the resulting Jones projections.
DEFF Research Database (Denmark)
Hatcliff, John; Danvy, Olivier
1997-01-01
Thirty-five years ago, thunks were used to simulate call-by-name under call-by-value in Algol 60. Twenty years ago, Plotkin presented continuation-based simulations of call-by-name under call-by-value and vice versa in the λ-calculus. We connect all three of these classical simulations by...
DEFF Research Database (Denmark)
Hatcliff, John; Danvy, Olivier
1996-01-01
Thirty-five years ago, thunks were used to simulate call-by-name under call-by-value in Algol 60. Twenty years ago, Plotkin presented continuation-based simulations of call-by-name under call-by-value and vice versa in the λ-calculus. We connect all three of these classical simulations by...
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.
The Inductive Applications of Probability Calculus
Directory of Open Access Journals (Sweden)
Corrado Gini
2015-06-01
Full Text Available The Author goes back to Founders of Probability calculus to investigate their original interpretation of the probability measure in the applications of the probability theory to real problems. The Author puts in evidence some misunderstandings related to the inversion of deductions derived by the use of probability distributions for investigating the causes of events.
A Functional Calculus for Quotient Bounded Operators
Directory of Open Access Journals (Sweden)
Sorin Mirel Stoian
2006-12-01
Full Text Available If (X, P is a sequentially locally convex space, then a quotient bounded operator T beloging to QP is regular (in the sense of Waelbroeck if and only if it is a bounded element (in the sense of Allan of algebra QP. The classic functional calculus for bounded operators on Banach space is generalized for bounded elements of algebra QP.
Teaching Calculus with Wolfram|Alpha
Dimiceli, Vincent E.; Lang, Andrew S. I. D.; Locke, LeighAnne
2010-01-01
This article describes the benefits and drawbacks of using Wolfram|Alpha as the platform for teaching calculus concepts in the lab setting. It is a result of our experiences designing and creating an entirely new set of labs using Wolfram|Alpha. We present the reasoning behind our transition from using a standard computer algebra system (CAS) to…
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…
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…
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…
Expressing First-Order π-Calculus in Higher-Order Calculus of Communicating Systems
Institute of Scientific and Technical Information of China (English)
Xian Xu
2009-01-01
In the study of process calculi, encoding between different calculi is an effective way to compare the expressive power of calculi and can shed light on the essence of where the difference lies. Thomsen and Sangiorgi have worked on the higher-order calculi (higher-order Calculus of Communicating Systems (CCS) and higher-order It-calculus, respectively) and the encoding from and to first-order π-calculus. However a fully abstract encoding of first-order π-calculus with higher-order CCS is not available up-today. This is what we intend to settle in this paper. We follow the encoding strategy, first proposed by Thomsen, of translating first-order π-calculus into Plain CHOCS. We show that the encoding strategy is fully abstract with respect to early bisimilarity (first-order π-calculus) and wired bisimilarity (Plain CHOCS) (which is a bisimulation defined on wired processes only sending and receiving wires), that is the core of the encoding strategy. Moreover from the fact that the wired bisimilarity is contained by the well-established context bisimilarity, we secure the soundness of the encoding, with respect to early bisimilarity and context bisimilarity. We use index technique to get around all the technical details to reach these main results of this paper. Finally, we make some discussion on our work and suggest some future work.
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.
Geometric constrained variational calculus. III: The second variation (Part II)
Massa, Enrico; Luria, Gianvittorio; Pagani, Enrico
2016-03-01
The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.
Gouveia, Paulo D. F.; Delfim F. M. Torres
2004-01-01
English version of abstract: The dynamic optimization problems treated by the calculus of variations are usually solved with the help of the 2nd order Euler-Lagrange differential equations. These equations are, generally speaking, nonlinear, and very hard to solve. One way to address the problem is to obtain conservation laws of lower order than those of the corresponding Euler-Lagrange equations. While in Physics and Economics the question of existence of conservation laws is treated in a ra...
A General Delta-Nabla Calculus of Variations on Time Scales with Application to Economics
Dryl, Monika; Torres, Delfim F. M.
2014-01-01
We consider a general problem of the calculus of variations on time scales with a cost functional that is the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange delta-nabla differential equations are proved, which lead to important insights in the process of discretization. Application of the obtained results to a firm that wants to program its production and investment policies to reach a given production rate and to maximize its ...
Calculus, Biology and Medicine: A Case Study in Quantitative Literacy for Science Students
Kim Rheinlander; Dorothy Wallace
2011-01-01
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 a...
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.
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...
Differential topology an introduction
Gauld, David B
2006-01-01
Offering classroom-proven results, Differential Topology presents an introduction to point set topology via a naive version of nearness space. Its treatment encompasses a general study of surgery, laying a solid foundation for further study and greatly simplifying the classification of surfaces.This self-contained treatment features 88 helpful illustrations. Its subjects include topological spaces and properties, some advanced calculus, differentiable manifolds, orientability, submanifolds and an embedding theorem, and tangent spaces. Additional topics comprise vector fields and integral curv
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
GIANT VESICAL CALCULUS – A CASE REPORT
Directory of Open Access Journals (Sweden)
Hanumanthaiah
2014-06-01
Full Text Available Until 20th century, bladder stones were one of the most prevalent disorders among the poor class and the incidence was especially high in childhood and adolescent. 1 The decrease in incidence of bladder calculi is attributed mainly to dietary and nutritional progress especially in children. 2 A solitary bladder calculus is usual, although multiple stones are found in 25% of cases. 3 Bladder stones are rare, and they constitute about 5% of all urinary stones, 4, 5 it is classified as migrated from upper urinary tract, primary idiopathic, or secondary calculi. 6 Bladder stones are managed by Extracorporeal Shockwave Lithotripsy (ESWL, endourology procedures, or open surgery. We report an unusual case of giant vesical calculus weighing 600grams in a 55 year old female with no evidence of hematuria, urinary retention, and dysuria.
Attendance and attainment in a Calculus course
Meulenbroek, Bernard; van den Bogaard, Maartje
2013-10-01
In this paper the relationship between attendance and attainment in a standard calculus course is investigated. Calculus could in principle be studied without attending lectures due to the wealth of material available (in hardcopy and online). However, in this study we will show that the pass rate of students attending classes regularly (>75% of the classes) is much higher than the pass rate of students attending fewer classes. We use a logistic model to investigate whether this correlation is significant. We will argue why we believe that this correlation between attendance and attainment is causal, i.e. why it is necessary for most students to attend classes in order to (improve their chances to) pass the exam.
Intersection Logic in sequent calculus style
Della Rocca, Simona Ronchi; Stavrinos, Yiorgos; Veneti, Anastasia; 10.4204/EPTCS.45.2
2011-01-01
The intersection type assignment system has been designed directly as deductive system for assigning formulae of the implicative and conjunctive fragment of the intuitionistic logic to terms of lambda-calculus. But its relation with the logic is not standard. Between all the logics that have been proposed as its foundation, we consider ISL, which gives a logical interpretation of the intersection by splitting the intuitionistic conjunction into two connectives, with a local and global behaviour respectively, being the intersection the local one. We think ISL is a logic interesting by itself, and in order to support this claim we give a sequent calculus formulation of it, and we prove that it enjoys the cut elimination property.
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...... theorem prover for hybrid logic, both as a convenient platform for a prototype implementation as well as to formally prove the correctness of the analysis....
Affine connection form of Regge calculus
Khatsymovsky, V M
2015-01-01
Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein-Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the 3-simplices which play a role of a general discrete connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4,R) of the connection matrices. As a result, we have some action invariant w. r. t. arbitrary change of coordinates of the vertices (and related GL(4,R) transformations in...
Investigations on a Pedagogical Calculus of Constructions
Colson, Loïc
2012-01-01
In the last few years appeared pedagogical propositional natural deduction systems. In these systems, one must satisfy the pedagogical constraint: the user must give an example of any introduced notion. First we expose the reasons of such a constraint and properties of these "pedagogical" calculi: the absence of negation at logical side, and the "usefulness" feature of terms at computational side (through the Curry-Howard correspondence). Then we construct a simple pedagogical restriction of the calculus of constructions (CC) called CCr. We establish logical limitations of this system, and compare its computational expressiveness to Godel system T. Finally, guided by the logical limitations of CCr, we propose a formal and general definition of what a pedagogical calculus of constructions should be.
Semiclassical dynamics and magnetic Weyl calculus
International Nuclear Information System (INIS)
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.)
On the interpretation of Stratonovich calculus
International Nuclear Information System (INIS)
The Itô–Stratonovich dilemma is revisited from the perspective of the interpretation of Stratonovich calculus using shot noise. Over the long time scales of the displacement of an observable, the principal issue is how to deal with finite/zero autocorrelation of the stochastic noise. The former (non-zero) noise autocorrelation structure preserves the normal chain rule using a mid-point selection scheme, which is the basis Stratonovich calculus, whereas the instantaneous autocorrelation structure of Itô's approach does not. By considering the finite decay of the noise correlations on time scales very short relative to the overall displacement times of the observable, we suggest a generalization of the integral Taylor expansion criterion of Wong and Zakai (1965 Ann. Math. Stat. 36 1560–4) for the validity of the Stratonovich approach. (paper)
Two cosmological solutions of Regge calculus
International Nuclear Information System (INIS)
Two cosmological solutions of Regge calculus are presented which correspond to the flat Friedmann-Robertson-Walker and the Kasner solutions of general relativity. By taking advantage of the symmetries that are present, I am able to show explicitly that a limit of Regge calculus does yield Einstein's equations for these cases. The method of averaging these equations when taking limits is important, especially for the Kasner model. I display the leading error term that arises from keeping the Regge equations in discrete form rather than using their continuum limit. In particular, this work shows that for the ''Reggeized'' Friedmann model the minimum volume is a velocity-dominated singularity as in the continuum Friedmann model. However, unlike the latter, the Regge version has a nonzero minimum volume
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.)
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.
Predicateμ-Calculus for Mobile Ambients
Institute of Scientific and Technical Information of China (English)
Hui-Min Lin
2005-01-01
Ambient logics have been proposed to describe properties for mobile agents which may evolve over time as well as space. This paper takes a predicate-based approach to extending an ambient logic with recursion, yielding a predicate t-calculus in which fixpoint formulas are formed using predicate variables. An algorithm is developed for model checking finite-control mobile ambients against formulas of the logic, providing the first decidability result for model checking a spatial logic with recursion.
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, w....... In particular, the involution property amounts to the self-duality of the free (syntactic) model....
A Calculus for Higher Spin Interactions
Joung, Euihun; Waldron, Andrew
2013-01-01
Higher spin theories can be efficiently described in terms of auxiliary St\\"uckelberg or projective space field multiplets. By considering how higher spin models couple to scale, these approaches can be unified in a conformal geometry/tractor calculus framework. We review these methods and apply them to higher spin vertices to obtain a generating function for massless, massive and partially massless three-point interactions.
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.
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...
Variational calculus with constraints on general algebroids
Grabowska, Katarzyna; Grabowski, Janusz
2007-01-01
Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces t...
Mould Calculus for Hamiltonian Vector Fields
Cresson, Jacky; Morin, Guillaume
2008-01-01
We present the general framework of \\'Ecalle's moulds in the case of linearization of a formal vector field without and within resonances. We enlighten the power of moulds by their universality, and calculability. We modify then \\'Ecalle's technique to fit in the seek of a formal normal form of a Hamiltonian vector field in cartesian coordinates. We prove that mould calculus can also produce successive canonical transformations to bring a Hamiltonian vector field into a normal form. We then p...
VEST: abstract vector calculus simplification in Mathematica
Squire, Jonathan; Burby, Joshua; Qin, Hong
2013-01-01
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 l...
GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION
Institute of Scientific and Technical Information of China (English)
陈泽乾
2002-01-01
A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators (e)t and its dual, creation operators (e)*t.
Bigeometric Calculus and Runge Kutta Method
Riza, Mustafa; Eminağa, Buğçe
2014-01-01
The properties of the Bigeometric or proportional derivative are presented and discussed explicitly. Based on this derivative, the Bigeometric Taylor theorem is worked out. As an application of this calculus, the Bigeometric Runge-Kutta method is derived and is applied to academic examples, with known closed form solutions, and a sample problem from mathematical modelling in biology. The comparison of the results of the Bigeometric Runge-Kutta method with the ordinary Runge-Kutta method shows...
Calculus teaching and learning in South Korea
Karjanto, N.
2015-01-01
This article discusses an experience of teaching Calculus classes for the freshmen students enrolled at Sungkyunkwan University, one of the private universities in South Korea. The teaching and learning approach is a balance combination between the teacher-oriented traditional style of lecturing and other activities that encourage students for active learning and classroom participation. Based on the initial observation during several semesters, some anecdotal evidences show that students' le...
CLINICO-BACTERIOLOGICAL STUDY OF VESICAL CALCULUS
Directory of Open Access Journals (Sweden)
Pushpendra
2016-05-01
Full Text Available BACKGROUND Vesical calculi are the most common manifestation of lower urinary tract lithiasis. Urinary infections play an important role in aetiopathogenesis of vesical calculi. OBJECTIVE Aim of this study was proposed to establish the bacteriology of stone and urine in an attempt to evaluate the role of infection in the formation of stone. Associated factors like age, sex, site of infection, obstruction, diet were also evaluated. DESIGN Prospective cohort study. METHODS The patients were admitted in surgical ward as provisional diagnosed cases of vesical calculus, were subjected to investigations including CBC, RBS, urine analysis, renal function test, x-ray KUB region and ultrasonography. Patients who were fit for surgery, various surgical procedures were done. Gross examination and core culture of stone was done to establish their aetiology. RESULTS Ninety-four patients with vesical calculus were evaluated. Incidence of vesical calculus was 1.13%. Majority of cases were from rural areas (92.55%. Urinary tract infection was present in 37.2% of cases, majority of cases urine culture was positive (30.95%. Core culture of stone was positive in 18 cases (25.17%. E. coli was the predominant organism both in urine culture (19.04% and core culture of stone (25.71%. CONCLUSIONS There is significant association regarding the presence of vesical calculi and the development of urinary infections. E. coli was the predominant organism found both in urine and core culture of stone.
Institute of Scientific and Technical Information of China (English)
李明; 刘龙
2013-01-01
Calculus is an important public institution of all kinds of professional basic math lessons, and is the foundation of student learning and subsequent course of scientific and technological knowledge. Calculus teaching should run through a school year. In order to maintain students' enthusiasm for calculus, it is necessary to do some calculus teaching reform. Starting from a lesson, this article discussed how to introduce application instance into class teaching, enhance students' interest, and increase students ' learning motivation.% 微积分是高等院校各类专业一门重要的公共基础数学课，是学生学习后继课程和科技知识的基础。微积分的教学要贯穿一个学年，要想保持学生在整个微积分学习过程中的热情，就要对微积分教学做一些改革，本文主要从一节教学谈起如何在课堂教学中由应用实例引入教学内容，增强学生的好奇心，从而达到提高学生学习的积极性。
Hybrid Calculus of Wrapped Compartments
Coppo, Mario; Drocco, Maurizio; Grassi, Elena; Sciacca, Eva; Spinella, Salvatore; Troina, Angelo; 10.4204/EPTCS.40.8
2010-01-01
The modelling and analysis of biological systems has deep roots in Mathematics, specifically in the field of ordinary differential equations (ODEs). Alternative approaches based on formal calculi, often derived from process algebras or term rewriting systems, provide a quite complementary way to analyze the behaviour of biological systems. These calculi allow to cope in a natural way with notions like compartments and membranes, which are not easy (sometimes impossible) to handle with purely numerical approaches, and are often based on stochastic simulation methods. Recently, it has also become evident that stochastic effects in regulatory networks play a crucial role in the analysis of such systems. Actually, in many situations it is necessary to use stochastic models. For example when the system to be described is based on the interaction of few molecules, when we are at the presence of a chemical instability, or when we want to simulate the functioning of a pool of entities whose compartmentalised structur...
微积分让数据说话%Calculus:Data’s Apology
Institute of Scientific and Technical Information of China (English)
林群
2013-01-01
Calculus can be unified by a philosophical formula with a quantitative ratio:0.9 truthAbsolute truthRelative =0.9. Why? Because calculus is attributed to computing the heights of curves (See the figure 4 in the paper), where a small triangle (Relative truth) replaces a small curved triangle (Absolute truth), and the ratio, Small Height alDifferenti , is used to measure the local error. This procedure goes ahead repeatedly until the sum of differentials replaces the full-height;and then, the ratio, the full-height alsthe sum of differenti , is used to measure the global error. We can observe whether the ratio is closer and closer to 1 (It just means whether Relative truth coincides with Absolute truth). From the experimental data, we found:Small Height alDifferenti =0.999..., the full-height alsthe sum of differenti =0.999... When the partition is refined, there are more 9 in the end of the figure. The repetition leads to an infinite decimal, 0.9, and a unified ratio formula:truthAbsolute truthRelative=0.999...... What an ingenious calculation! The two ratios can be rewritten as Scant Slope SlopeTangent=0.999..., the integral alsthe sum of differenti =0.999... .%微积分统一到一个哲学公式，但有定量的比例：绝对真理/相对真理=0.9.
Extending the Lambda Calculus to Express Randomized and Quantumized Algorithms
Maymin, Philip
1996-01-01
This paper introduces a formal metalanguage called the lambda-q calculus for the specification of quantum programming languages. This metalanguage is an extension of the lambda calculus, which provides a formal setting for the specification of classical programming languages. As an intermediary step, we introduce a formal metalanguage called the lambda-p calculus for the specification of programming languages that allow true random number generation. We demonstrate how selected randomized alg...
Ecological Modelling with the Calculus of Wrapped Compartments
Pablo, de, P.J.; Angelo
2015-01-01
The Calculus of Wrapped Compartments is a framework based on stochastic multiset rewriting in a compartmentalised setting originally developed for the modelling and analysis of biological interactions. In this paper, we propose to use this calculus for the description of ecological systems and we provide the modelling guidelines to encode within the calculus some of the main interactions leading ecosystems evolution. As a case study, we model the distribution of height of Croton wagneri, a sh...
Detection, removal and prevention of calculus: Literature Review
Kamath, Deepa G.; Sangeeta Umesh Nayak
2013-01-01
Dental plaque is considered to be a major etiological factor in the development of periodontal disease. Accordingly, the elimination of supra- and sub-gingival plaque and calculus is the cornerstone of periodontal therapy. Dental calculus is mineralized plaque; because it is porous, it can absorb various toxic products that can damage the periodontal tissues. Hence, calculus should be accurately detected and thoroughly removed for adequate periodontal therapy. Many techniques have been used t...
Numerical method for Darcy flow derived using Discrete Exterior Calculus
Hirani, Anil N.; Nakshatrala, Kalyana B.; Chaudhry, Jehanzeb H.
2008-01-01
We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equa...
The Complex Gradient Operator and the CR-Calculus
Kreutz-Delgado, Ken
2009-01-01
A thorough discussion and development of the calculus of real-valued functions of complex-valued vectors is given using the framework of the Wirtinger Calculus. The presented material is suitable for exposition in an introductory Electrical Engineering graduate level course on the use of complex gradients and complex Hessian matrices, and has been successfully used in teaching at UC San Diego. Going beyond the commonly encountered treatments of the first-order complex vector calculus, second-...
Non-mathematics Students’ Reasoning in Calculus Tasks
Matić, Ljerka Jukić
2014-01-01
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 the students’ calculus course. We examined the reasoning of six students in the middle of the calculus course and at the end of the course. The ana...
Mikusi\\'nski's Operational Calculus with Algebraic Foundations and Applications to Bessel Functions
Bengochea, Gabriel; G, Gabriel López
2013-01-01
We construct an operational calculus supported on the algebraic operational calculus introduced by Bengochea and Verde. With this operational calculus we study the solution of certain Bessel type equations.
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
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
Categorical Models for a Semantically Linear Lambda-calculus
Marco Gaboardi; Mauro Piccolo
2010-01-01
This paper is about a categorical approach to model a very simple Semantically Linear lambda calculus, named Sll-calculus. This is a core calculus underlying the programming language SlPCF. In particular, in this work, we introduce the notion of Sll-Category, which is able to describe a very large class of sound models of Sll-calculus. Sll-Category extends in the natural way Benton, Bierman, Hyland and de Paiva's Linear Category, in order to soundly interpret all the constructs of Sll-calculu...
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...
Topology,randomness and noise in process calculus
Institute of Scientific and Technical Information of China (English)
YING Mingsheng
2007-01-01
Formal models of communicating and concurrent systems are one of the most important topics in formal methods,and process calculus is one of the most successful formal models of communicating and concurrent systems.In the previous works,the author systematically studied topology in process calculus,probabilistic process calculus and pi-calculus with noisy channels in order to describe approximate behaviors of communicating and concurrent systems as well as randonmess and noise in them.This article is a brief survey of these works.
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....
Maximum range of a projectile launched from a height h: a non-calculus treatment
International Nuclear Information System (INIS)
The classical example of problem solving, maximizing the range of a projectile launched from height h with velocity v over the ground level, has received various solutions. In some of these, one can find the maximization of the range R by differentiating R as a function of an independent variable or through the implicit differentiation in Cartesian or polar coordinates. In other papers, various elegant non-calculus solutions can be found. In this paper, this problem is revisited on the basis of the elementary analytical geometry and the trigonometry only. (papers)
Depolarizing differential Mueller matrices.
Ortega-Quijano, Noé; Arce-Diego, José Luis
2011-07-01
The evolution of a polarized beam can be described by the differential formulation of Mueller calculus. The nondepolarizing differential Mueller matrices are well known. However, they only account for 7 out of the 16 independent parameters that are necessary to model a general anisotropic depolarizing medium. In this work we present the nine differential Mueller matrices for general depolarizing media, highlighting the physical implications of each of them. Group theory is applied to establish the relationship between the differential matrix and the set of transformation generators in the Minkowski space, of which Lorentz generators constitute a particular subgroup. PMID:21725434
Mueller matrix differential decomposition.
Ortega-Quijano, Noé; Arce-Diego, José Luis
2011-05-15
We present a Mueller matrix decomposition based on the differential formulation of the Mueller calculus. The differential Mueller matrix is obtained from the macroscopic matrix through an eigenanalysis. It is subsequently resolved into the complete set of 16 differential matrices that correspond to the basic types of optical behavior for depolarizing anisotropic media. The method is successfully applied to the polarimetric analysis of several samples. The differential parameters enable one to perform an exhaustive characterization of anisotropy and depolarization. This decomposition is particularly appropriate for studying media in which several polarization effects take place simultaneously. PMID:21593943
Thompson, John; Christensen, Warren; Mountcastle, Donald
2010-03-01
In work on student understanding of concepts in advanced thermal physics, we are exploring student understanding of the mathematics required for productive reasoning about the physics. By analysis of student use of mathematics in responses to conceptual physics questions, as well as analogous math questions stripped of physical meaning, we find evidence that students often enter upper-level physics courses lacking the assumed prerequisite mathematics knowledge and/or the ability to apply it productively in a physics context. Our focus is in two main areas: interpretation of P-V diagrams, requiring an understanding of integration, and material properties and the Maxwell relations, involving partial differentiation. We have also assessed these mathematical concepts among students in multivariable calculus. Calculus results support the findings among physics students: some observed difficulties are not just with transfer of math knowledge to physics contexts, but seem to have origins in the understanding of the math concepts themselves.
Stochastic differential games with inside information
Draouil, Olfa; Øksendal, Bernt
2016-08-01
We study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida-Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.
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.
Extended finite operator calculus as an example of algebraization of analysis
Kwasniewski, A. K.
2008-01-01
A calculus of sequences started by professor morgan ward constitutes the general scheme for extensions of classical operator calculus of the distinguished gian carlo rota considered by many afterwards and after ward morgan. Because of the historically now established notation we call the wardian calculus of sequences in its afterwards elaborated form a psi calculus. The psi calculus in parts appears to be almost automatic, natural extension of classical operator calculus or equivalently of um...
Institute of Scientific and Technical Information of China (English)
王娜
2011-01-01
For a long time,calculus is always the one of active fields in mathematics education reform.The reform of calculus in American makes the calculus teaching full of the energy.It pays attention to the essence,and funds the innovative method to help students achieving deep understand on calculus idea.By analyzing the main textbook of calculus in USA,Calculus written by James Stewart,and comparing the limit theory,differential calculus and integral calculus in our country＇s textbook,puts forward some suggestions for our country＇s calculus teaching.%长期以来,微积分始终是数学教育改革中最活跃的领域之一。美国的微积分改革使得美国的微积分教学充满了活力,它注重本质,发现能帮助学生获得对微积分思想深刻理解的创造性方法。通过分析美国微积分的主流教材,即James Stewart所著的《微积分》教材,并与我国微积分教材中的极限理论、微分学和积分学部分进行了比较研究,根据比较研究的结论,对我国微积分教学提出了若干建议。
Differential subordination for meromorphic multivalent quasi-convex functions
Directory of Open Access Journals (Sweden)
R. W. Ibrahim
2010-02-01
Full Text Available We introduce new classes of meromorphic multivalent quasi-convex functions and find some sufficient differential subordination theorems for such classes in punctured unit disk with applications in fractional calculus.
A physically based connection between fractional calculus and fractal geometry
Butera, Salvatore
2015-01-01
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation i...
Geometric constrained variational calculus I: Piecewise smooth extremals
Massa, Enrico; Bruno, Danilo; Luria, Gianvittorio; Pagani, Enrico
2015-05-01
A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in Hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints.
A physically based connection between fractional calculus and fractal geometry
International Nuclear Information System (INIS)
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry
A physically based connection between fractional calculus and fractal geometry
Energy Technology Data Exchange (ETDEWEB)
Butera, Salvatore, E-mail: sg.butera@gmail.com [SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, EH14 4AS (United Kingdom); Di Paola, Mario, E-mail: mario.dipaola@unipa.it [Dipartimento di Ingegneria Civile Ambientale, Aerospaziale e dei Materiali (DICAM), Università degli Studi di Palermo, Viale delle Scienze, Ed.8, 90128 - Palermo (Italy)
2014-11-15
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.
An Evaluative Calculus Project: Applying Bloom's Taxonomy to the Calculus Classroom
Karaali, Gizem
2011-01-01
In education theory, Bloom's taxonomy is a well-known paradigm to describe domains of learning and levels of competency. In this article I propose a calculus capstone project that is meant to utilize the sixth and arguably the highest level in the cognitive domain, according to Bloom et al.: evaluation. Although one may assume that mathematics is…
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…
Experimental Design: 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…
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…
Type Inference for Session Types in the Pi-Calculus
DEFF Research Database (Denmark)
Huttel, Hans; Graversen, Eva Fajstrup; Wahl, Sebastian;
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...
A compositional proof system for the modal μ-calculus
DEFF Research Database (Denmark)
Andersen, Henrik Reif; Stirling, C.; Winskel,, G.
1994-01-01
We present a proof system for determining satisfaction between processes in a fairly general process algebra and assertions of the modal μ-calculus. The proof system is compositional in the structure of processes. It extends earlier work on compositional reasoning within the modal μ-calculus and ...
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…
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…
Calculus of Variations with Fractional and Classical Derivatives
Odzijewicz, Tatiana
2010-01-01
We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both fractional and classical derivatives are proved. The fundamental problem of the calculus of variations with mixed integer and fractional order derivatives as well as isoperimetric problems are considered.
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.
Evaluating the Use of Learning Objects for Improving Calculus Readiness
Kay, Robin; Kletskin, Ilona
2010-01-01
Pre-calculus concepts such as working with functions and solving equations are essential for students to explore limits, rates of change, and integrals. Yet many students have a weak understanding of these key concepts which impedes performance in their first year university Calculus course. A series of online learning objects was developed to…
Improving Student Success in Calculus at Seattle University
Carter, J. D.; Helliwell, D.; Henrich, Allison; Principe, M.; Sloughter, J. M.
2016-01-01
Finding ways to improve student success in calculus is a critically important step on the path to supporting students who are pursuing degrees in STEM fields. Far too many students fail calculus 1 and are pushed to drop their majors in technical fields. One way of addressing this issue is by following a program that was pioneered at University of…
The Arrow Calculus as a Quantum Programming Language
Vizzotto, Juliana Kaizer; Bois, Andre Rauber Du; Sabry, Amr
2009-01-01
We express quantum computations (with measurements) using the arrow calculus extended with monadic constructions. This framework expresses quantum programming using well-understood and familiar classical patterns for programming in the presence of computational effects. In addition, the five laws of the arrow calculus provide a convenient framework for equational reasoning about quantum computations that include measurements.
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...
Calculus Reform and Graphing Calculators: A University View.
Stick, Marvin E.
1997-01-01
Describes the results of a teacher's exploration of the effects of using graphing calculators in calculus instruction in sections other than those that are experimental. Two experimental and two traditional sections of Calculus I and II participated in the study. (DDR)
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…
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…
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 pl...
Towards Model Checking a Spi-Calculus Dialect
Gnesi, S.; Latella, D.; Lenzini, G.
2002-01-01
We present a model checking framework for a spi-calculus dialect which uses a linear time temporal logic for expressing security properties. We have provided our spi-calculus dialect, called SPID, with a semantics based on labeled transition systems (LTS), where the intruder is modeled in the Dolev-
Coordinating Multiple Representations in a Reform Calculus Textbook
Chang, Briana L.; Cromley, Jennifer G.; Tran, Nhi
2015-01-01
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…
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…
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...
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…
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 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...
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...
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.