The quantum probability calculus
Jauch, J.M.
1976-01-01
The Wigner anomaly (1932) for the joint distribution of noncompatible observables is an indication that the classical probability calculus is not applicable for quantum probabilities. It should, therefore, be replaced by another, more general calculus, which is specifically adapted to quantal systems. In this article this calculus is exhibited and its mathematical axioms and the definitions of the basic concepts such as probability field, random variable, and expectation values are given. (B.R.H)
General quantum variational calculus
Artur M. C. Brito da Cruz
2018-02-01
Full Text Available We develop a new variational calculus based in the general quantum difference operator recently introduced by Hamza et al. In particular, we obtain optimality conditions for generalized variational problems where the Lagrangian may depend on the endpoints conditions and a real parameter, for the basic and isoperimetric problems, with and without fixed boundary conditions. Our results provide a generalization to previous results obtained for the $q$- and Hahn-calculus.
Generalized Gaussian Error Calculus
Grabe, Michael
2010-01-01
For the first time in 200 years Generalized Gaussian Error Calculus addresses a rigorous, complete and self-consistent revision of the Gaussian error calculus. Since experimentalists realized that measurements in general are burdened by unknown systematic errors, the classical, widespread used evaluation procedures scrutinizing the consequences of random errors alone turned out to be obsolete. As a matter of course, the error calculus to-be, treating random and unknown systematic errors side by side, should ensure the consistency and traceability of physical units, physical constants and physical quantities at large. The generalized Gaussian error calculus considers unknown systematic errors to spawn biased estimators. Beyond, random errors are asked to conform to the idea of what the author calls well-defined measuring conditions. The approach features the properties of a building kit: any overall uncertainty turns out to be the sum of a contribution due to random errors, to be taken from a confidence inter...
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.
Generalized Cartan Calculus in general dimension
Wang, Yi-Nan
2015-07-01
We develop the generalized Cartan Calculus for the groups and SO(5 , 5). They are the underlying algebraic structures of d = 9 , 7 , 6 exceptional field theory, respectively. These algebraic identities are needed for the "tensor hierarchy" structure in exceptional field theory. The validity of Poincaré lemmas in this new differential geometry is also discussed. Finally we explore some possible extension of the generalized Cartan calculus beyond the exceptional series.
Generalized Probability-Probability Plots
Mushkudiani, N.A.; Einmahl, J.H.J.
2004-01-01
We introduce generalized Probability-Probability (P-P) plots in order to study the one-sample goodness-of-fit problem and the two-sample problem, for real valued data.These plots, that are constructed by indexing with the class of closed intervals, globally preserve the properties of classical P-P
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.
Generalized Multiparameters Fractional Variational Calculus
Om Prakash Agrawal
2012-01-01
Full Text Available This paper builds upon our recent paper on generalized fractional variational calculus (FVC. Here, we briefly review some of the fractional derivatives (FDs that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives (GFDs which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases of the proposed GFDs. We develop several parts of FVC in terms of one parameter GFDs. We point out how many other parts could be developed using the properties of the one-parameter GFDs. Subsequently, we introduce two new two- and three-parameter GFDs. We introduce some of their properties, and discuss how they can be used to develop FVC. In addition, we indicate how these formulations could be used in various fields, and how the generalizations presented here can be further extended.
Class dependency of fuzzy relational database using relational calculus and conditional probability
Deni Akbar, Mohammad; Mizoguchi, Yoshihiro; Adiwijaya
2018-03-01
In this paper, we propose a design of fuzzy relational database to deal with a conditional probability relation using fuzzy relational calculus. In the previous, there are several researches about equivalence class in fuzzy database using similarity or approximate relation. It is an interesting topic to investigate the fuzzy dependency using equivalence classes. Our goal is to introduce a formulation of a fuzzy relational database model using the relational calculus on the category of fuzzy relations. We also introduce general formulas of the relational calculus for the notion of database operations such as ’projection’, ’selection’, ’injection’ and ’natural join’. Using the fuzzy relational calculus and conditional probabilities, we introduce notions of equivalence class, redundant, and dependency in the theory fuzzy relational database.
Variational calculus with constraints on general algebroids
Grabowska, Katarzyna [Physics Department, Division of Mathematical Methods in Physics, University of Warsaw, Hoza 69, 00-681 Warszawa (Poland); Grabowski, Janusz [Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, PO Box 21, 00-956 Warszawa (Poland)], E-mail: konieczn@fuw.edu.pl, E-mail: jagrab@impan.gov.pl
2008-05-02
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.
Variational calculus with constraints on general algebroids
Grabowska, Katarzyna; Grabowski, Janusz
2008-01-01
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 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
Generalized Probability Functions
Alexandre Souto Martinez
2009-01-01
Full Text Available From the integration of nonsymmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions (pdfs. A very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. We obtain, as particular cases, the generalized error function, the Zipf-Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.
Enabling quaternion derivatives: the generalized HR calculus
Xu, Dongpo; Jahanchahi, Cyrus; Took, Clive C.; Mandic, Danilo P.
2015-01-01
Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis. PMID:26361555
Toward a generalized probability theory: conditional probabilities
Cassinelli, G.
1979-01-01
The main mathematical object of interest in the quantum logic approach to the foundations of quantum mechanics is the orthomodular lattice and a set of probability measures, or states, defined by the lattice. This mathematical structure is studied per se, independently from the intuitive or physical motivation of its definition, as a generalized probability theory. It is thought that the building-up of such a probability theory could eventually throw light on the mathematical structure of Hilbert-space quantum mechanics as a particular concrete model of the generalized theory. (Auth.)
Intitialization, Conceptualization, and Application in the Generalized Fractional Calculus
Lorenzo, Carl F.; Hartley, Tom T.
1998-01-01
This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.
Initialization, conceptualization, and application in the generalized (fractional) calculus.
Lorenzo, Carl F; Hartley, Tom T
2007-01-01
This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.
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.
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
A generalization of tensor calculus and its application to physics
Ashtekar, A.
1982-01-01
Penrose's abstract index notation and axiomatic introduction of covariant derivatives in tensor calculus is generalized to fields with internal degrees of freedom. The result provides, in particular, an intrinsic formulation of gauge theories without the use of bundles. (author)
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 ...
Larson, Ron
2014-01-01
The Larson CALCULUS program has a long history of innovation in the calculus market. It has been widely praised by a generation of students and professors for its solid and effective pedagogy that addresses the needs of a broad range of teaching and learning styles and environments. Each title is just one component in a comprehensive calculus course program that carefully integrates and coordinates print, media, and technology products for successful teaching and learning.
Grossman, Stanley I
1981-01-01
Calculus, Second Edition discusses the techniques and theorems of calculus. This edition introduces the sine and cosine functions, distributes ?-? material over several chapters, and includes a detailed account of analytic geometry and vector analysis.This book also discusses the equation of a straight line, trigonometric limit, derivative of a power function, mean value theorem, and fundamental theorems of calculus. The exponential and logarithmic functions, inverse trigonometric functions, linear and quadratic denominators, and centroid of a plane region are likewise elaborated. Other topics
Integration on supermanifolds and a generalized Cartan calculus
Picken, R.F.; Sundermeyer, K.
1986-01-01
A suggestion by Berezin for a method of integration on supermanifolds is given a precise differential geometric meaning by assuming that a supermanifold is the total space of a fibre bundle with connection. The relevant objects for integration are identified as suitable horizontal/vertical projections of hyperforms. The latter are generalizations of differential forms having both covariant and contravariant indices. The exterior calculus of these projected hyperforms is developed, analogously to the Cartan calculus, by introducing appropriate derivations and determining their commutators, respectively anticommutators. (orig.)
A Generalized Nonlocal Calculus with Application to the Peridynamics Model for Solid Mechanics
Alali, Bacim; Liu, Kuo; Gunzburger, Max
2014-01-01
A nonlocal vector calculus was introduced in [2] that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A generalization 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...
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.
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
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.
On certain fractional calculus operators involving generalized Mittag-Leffler function
Dinesh Kumar
2016-01-01
The object of this paper is to establish certain generalized fractional integration and differentiation involving generalized Mittag-Leffler function defined by Salim and Faraj [25]. The considered generalized fractional calculus operators contain the Appell's function $F_3$ [2, p.224] as kernel and are introduced by Saigo and Maeda [23]. The Marichev-Saigo-Maeda fractional calculus operators are the generalization of the Saigo fractional calculus operators. The established results provide ex...
Jumarie, Guy
2009-01-01
A probability distribution of fractional (or fractal) order is defined by the measure μ{dx} = p(x)(dx) α , 0 α (D x α h α )f(x) provided by the modified Riemann Liouville definition, one can expand a probability calculus parallel to the standard one. A Fourier's transform of fractional order using the Mittag-Leffler function is introduced, together with its inversion formula; and it provides a suitable generalization of the characteristic function of fractal random variables. It appears that the state moments of fractional order are more especially relevant. The main properties of this fractional probability calculus are outlined, it is shown that it provides a sound approach to Fokker-Planck equation which are fractional in both space and time, and it provides new results in the information theory of non-random functions.
Swenson, Daniel
2015-01-01
We walk through a module intended for undergraduates in mathematics, with the focus of finding the best strategies for competing in the Showcase Showdown on the game show "The Price Is Right." Students should have completed one semester of calculus, as well as some probability. We also give numerous suggestions for further questions that…
Particular case of operator calculus for generalized functions with supports in cone
A. V. Solomko
2009-06-01
Full Text Available In this work the construction of functional calculus for strongly continuous semigroups of operators in Schwartz distribution algebra on some cone is generalized. The partial case of vector valued calculus on the base of modification operator Fourier transformation is researched.
Restrictive metric regularity and generalized differential calculus in Banach spaces
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.
Agarkov S.A.
2015-03-01
Full Text Available A new approach to the identification of parameters of the Nomoto generalized vessel model has been proposed. The apparatus of classical calculus and the method of least squares have been used
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
Bergstra, J. A.; Ponse, A.; van der Zwaag, M. B.
2007-01-01
We introduce a calculus for tuplices, which are expressions that generalize matrices and vectors. Tuplices have an underlying data type for quantities that are taken from a zero-totalized field. We start with the core tuplix calculus CTC for entries and tests, which are combined using conjunctive composition. We define a standard model and prove that CTC is relatively complete with respect to it. The core calculus is extended with operators for choice, information hiding, scalar multiplicatio...
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
Mori, Ryuhei
2015-01-01
The holographic transformation, belief propagation and loop calculus are generalized to problems in generalized probabilistic theories including quantum mechanics. In this work, the partition function of classical factor graph is represented by an inner product of two high-dimensional vectors both of which can be decomposed to tensor products of low-dimensional vectors. On the representation, the holographic transformation is clearly understood by using adjoint linear maps. Furthermore, on th...
Benakli, Nadia; Kostadinov, Boyan; Satyanarayana, Ashwin; Singh, Satyanand
2017-04-01
The goal of this paper is to promote computational thinking among mathematics, engineering, science and technology students, through hands-on computer experiments. These activities have the potential to empower students to learn, create and invent with technology, and they engage computational thinking through simulations, visualizations and data analysis. We present nine computer experiments and suggest a few more, with applications to calculus, probability and data analysis, which engage computational thinking through simulations, visualizations and data analysis. We are using the free (open-source) statistical programming language R. Our goal is to give a taste of what R offers rather than to present a comprehensive tutorial on the R language. In our experience, these kinds of interactive computer activities can be easily integrated into a smart classroom. Furthermore, these activities do tend to keep students motivated and actively engaged in the process of learning, problem solving and developing a better intuition for understanding complex mathematical concepts.
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
Lunsford, M. Leigh; Rowell, Ginger Holmes; Goodson-Espy, Tracy
2006-01-01
We applied a classroom research model to investigate student understanding of sampling distributions of sample means and the Central Limit Theorem in post-calculus introductory probability and statistics courses. Using a quantitative assessment tool developed by previous researchers and a qualitative assessment tool developed by the authors, we…
Calculus on Surfaces with General Closest Point Functions
Mä rz, Thomas; Macdonald, Colin B.
2012-01-01
The closest point method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys., 227 (2008), pp. 1943- 1961] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization of this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs. © 2012 Society for Industrial and Applied Mathematics.
Certain Properties of Some Families of Generalized Starlike Functions with respect to q-Calculus
Ben Wongsaijai
2016-01-01
Full Text Available By making use of the concept of q-calculus, various types of generalized starlike functions of order α were introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types of q-starlike functions of order α. We also introduce and study a new subclass of q-starlike functions of order α. Moreover, we give some properties of those q-starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class.
Guoqing Tang
2004-02-01
Full Text Available In this paper we present an approach of incorporating interactive and media-enhanced lectures to promote active learning in Calculus and General Physics courses. The pedagogical practice of using interactive techniques in lectures to require "heads-on" and "hands-on" learning, and involve students more as active participants than passive receivers is a part of academic curricular reform efforts undertaken currently by the mathematics, physics and chemistry departments at North Carolina A&T State University under the NSF funded project "Talent-21: Gateway for Advancing Science and Mathematics Talents."
Schubert, Martin
2012-01-01
This book develops a mathematical framework for modeling and optimizing interference-coupled multiuser systems. At the core of this framework is the concept of general interference functions, which provides a simple means of characterizing interdependencies between users. The entire analysis builds on the two core axioms scale-invariance and monotonicity. The proposed network calculus has its roots in power control theory and wireless communications. It adds theoretical tools for analyzing the typical behavior of interference-coupled networks. In this way it complements existing game-theoretic approaches. The framework should also be viewed in conjunction with optimization theory. There is a fruitful interplay between the theory of interference functions and convex optimization theory. By jointly exploiting the properties of interference functions, it is possible to design algorithms that outperform general-purpose techniques that only exploit convexity. The title “network calculus” refers to the fact tha...
Tensor Calculus: Unlearning Vector Calculus
Lee, Wha-Suck; Engelbrecht, Johann; Moller, Rita
2018-01-01
Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can…
Multiple-event probability in general-relativistic quantum mechanics
Hellmann, Frank; Mondragon, Mauricio; Perez, Alejandro; Rovelli, Carlo
2007-01-01
We discuss the definition of quantum probability in the context of 'timeless' general-relativistic quantum mechanics. In particular, we study the probability of sequences of events, or multievent probability. In conventional quantum mechanics this can be obtained by means of the 'wave function collapse' algorithm. We first point out certain difficulties of some natural definitions of multievent probability, including the conditional probability widely considered in the literature. We then observe that multievent probability can be reduced to single-event probability, by taking into account the quantum nature of the measuring apparatus. In fact, by exploiting the von-Neumann freedom of moving the quantum/classical boundary, one can always trade a sequence of noncommuting quantum measurements at different times, with an ensemble of simultaneous commuting measurements on the joint system+apparatus system. This observation permits a formulation of quantum theory based only on single-event probability, where the results of the wave function collapse algorithm can nevertheless be recovered. The discussion also bears on the nature of the quantum collapse
Generalizing Automath by means of a lambda-typed lambda calculus
Bruijn, de N.G.; Nederpelt, R.P.; Geuvers, J.H.; Vrijer, de R.C.
1994-01-01
The calculus delta conjunction developed in this paper is claimed to be able to embrace all the essential aspects of Automath, apart from the feature of type inclusion which will not be considered in this paper. The calculus deal with a correctness notion for lambda-typed lambda formulas (which are
Decision making generalized by a cumulative probability weighting function
dos Santos, Lindomar Soares; Destefano, Natália; Martinez, Alexandre Souto
2018-01-01
Typical examples of intertemporal decision making involve situations in which individuals must choose between a smaller reward, but more immediate, and a larger one, delivered later. Analogously, probabilistic decision making involves choices between options whose consequences differ in relation to their probability of receiving. In Economics, the expected utility theory (EUT) and the discounted utility theory (DUT) are traditionally accepted normative models for describing, respectively, probabilistic and intertemporal decision making. A large number of experiments confirmed that the linearity assumed by the EUT does not explain some observed behaviors, as nonlinear preference, risk-seeking and loss aversion. That observation led to the development of new theoretical models, called non-expected utility theories (NEUT), which include a nonlinear transformation of the probability scale. An essential feature of the so-called preference function of these theories is that the probabilities are transformed by decision weights by means of a (cumulative) probability weighting function, w(p) . We obtain in this article a generalized function for the probabilistic discount process. This function has as particular cases mathematical forms already consecrated in the literature, including discount models that consider effects of psychophysical perception. We also propose a new generalized function for the functional form of w. The limiting cases of this function encompass some parametric forms already proposed in the literature. Far beyond a mere generalization, our function allows the interpretation of probabilistic decision making theories based on the assumption that individuals behave similarly in the face of probabilities and delays and is supported by phenomenological models.
Paragrassmann differential calculus
Filippov, A.T.; Isaev, A.P.; Kurdikov, A.V.
1993-01-01
This paper significantly extends and generalizes the paragrassmann calculus previous paper. Explicit general constructions for paragrassmann calculus with one and many vaiables are discussed. A general construction of many-variable differentiation algebras is given. Some particular examples are related to multi-parametric quantum deformation of the harmonic oscillators
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...
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Ogawa, Shigeyoshi
2017-01-01
This book presents an elementary introduction to the theory of noncausal stochastic calculus that arises as a natural alternative to the standard theory of stochastic calculus founded in 1944 by Professor Kiyoshi Itô. As is generally known, Itô Calculus is essentially based on the "hypothesis of causality", asking random functions to be adapted to a natural filtration generated by Brownian motion or more generally by square integrable martingale. The intention in this book is to establish a stochastic calculus that is free from this "hypothesis of causality". To be more precise, a noncausal theory of stochastic calculus is developed in this book, based on the noncausal integral introduced by the author in 1979. After studying basic properties of the noncausal stochastic integral, various concrete problems of noncausal nature are considered, mostly concerning stochastic functional equations such as SDE, SIE, SPDE, and others, to show not only the necessity of such theory of noncausal stochastic calculus but ...
Continuation of probability density functions using a generalized Lyapunov approach
Baars, S., E-mail: s.baars@rug.nl [Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen (Netherlands); Viebahn, J.P., E-mail: viebahn@cwi.nl [Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB, Amsterdam (Netherlands); Mulder, T.E., E-mail: t.e.mulder@uu.nl [Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht (Netherlands); Kuehn, C., E-mail: ckuehn@ma.tum.de [Technical University of Munich, Faculty of Mathematics, Boltzmannstr. 3, 85748 Garching bei München (Germany); Wubs, F.W., E-mail: f.w.wubs@rug.nl [Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen (Netherlands); Dijkstra, H.A., E-mail: h.a.dijkstra@uu.nl [Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht (Netherlands); School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY (United States)
2017-05-01
Techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved. Here, we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points, under a small noise approximation. Key innovation is the efficient solution of a generalized Lyapunov equation using an iterative method involving low-rank approximations. We apply and illustrate the capabilities of the method using a problem in physical oceanography, i.e. the occurrence of multiple steady states of the Atlantic Ocean circulation.
Ritow, Ira
2003-01-01
This brief introductory text presents the basic principles of calculus from the engineering viewpoint. Excellent either as a refresher or as an introductory course, it focuses on developing familiarity with the basic principles rather than presenting detailed proofs.Topics include differential calculus, in terms of differentiation and elementary differential equations; integral calculus, in simple and multiple integration forms; time calculus; equations of motion and their solution; complex variables; complex algebra; complex functions; complex and operational calculus; and simple and inverse
Wick calculus on spaces of generalized functions of compound poisson white noise
Lytvynov, Eugene W.; Rebenko, Alexei L.; Shchepan'ur, Gennadi V.
1997-04-01
We derive white noise calculus for a compound Poisson process. Namely, we consider, on the Schwartz space of tempered distributions, S', a measure of compound Poisson white noise, μcp, and construct a whole scale of standard nuclear triples ( Scp) - x ⊃ L2cp) ≡ L2( S', dμcp) ⊃( Scpx, x≥ 0, which are obtained as images under some isomorphism of the corresponding triples centred at a Fock space. It turns out that the most interesting case is x = 1, when our triple coincides with the triple that is constructed by using a system of Appell polynomials in the framework of non-Gaussian biorthogonal analysis. Our special attention is paid to the Wick calculus of the Poisson field, or the quantum compound Poisson white noise process in other terms, which is the family of operators acting from ( Scp) 1 into ( Scp) 1 as multiplication by the compound Poisson white noise ω( t).
Shiryaev, A N
1996-01-01
This book contains a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, ergodic theory, weak convergence of probability measures, stationary stochastic processes, and the Kalman-Bucy filter Many examples are discussed in detail, and there are a large number of exercises The book is accessible to advanced undergraduates and can be used as a text for self-study This new edition contains substantial revisions and updated references The reader will find a deeper study of topics such as the distance between probability measures, metrization of weak convergence, and contiguity of probability measures Proofs for a number of some important results which were merely stated in the first edition have been added The author included new material on the probability of large deviations, and on the central limit theorem for sums of dependent random variables
Geron, B.; Geuvers, J.H.; de'Liguoro, U.; Saurin, A.
2013-01-01
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
A General Schema for Constructing One-Point Bases in the Lambda Calculus
Goldberg, Mayer
2001-01-01
In this paper, we present a schema for constructing one-point bases for recursively enumerable sets of lambda terms. The novelty of the approach is that we make no assumptions about the terms for which the one-point basis is constructed: They need not be combinators and they may contain constants...... and free variables. The significance of the construction is twofold: In the context of the lambda calculus, it characterises one-point bases as ways of ``packaging'' sets of terms into a single term; And in the context of realistic programming languages, it implies that we can define a single procedure...
The stochastic quality calculus
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...
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
Continuation of probability density functions using a generalized Lyapunov approach
Baars, S.; Viebahn, J. P.; Mulder, T. E.; Kuehn, C.; Wubs, F. W.; Dijkstra, H. A.
2017-01-01
Techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved. Here, we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial
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
Modified Stieltjes Transform and Generalized Convolutions of Probability Distributions
Lev B. Klebanov
2018-01-01
Full Text Available The classical Stieltjes transform is modified in such a way as to generalize both Stieltjes and Fourier transforms. This transform allows the introduction of new classes of commutative and non-commutative generalized convolutions. A particular case of such a convolution for degenerate distributions appears to be the Wigner semicircle distribution.
Polynomial Calculus: Rethinking the Role of Calculus in High Schools
Grant, Melva R.; Crombie, William; Enderson, Mary; Cobb, Nell
2016-01-01
Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in…
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.
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Means and Variances without Calculus
Kinney, John J.
2005-01-01
This article gives a method of finding discrete approximations to continuous probability density functions and shows examples of its use, allowing students without calculus access to the calculation of means and variances.
Eerkens, Jelmer W; Nichols, Ruth V; Murray, Gemma G R; Perez, Katherine; Murga, Engel; Kaijankoski, Phil; Rosenthal, Jeffrey S; Engbring, Laurel; Shapiro, Beth
2018-05-25
Next Generation Sequencing (NGS) of ancient dental calculus samples from a prehistoric site in San Francisco Bay, CA-SCL-919, reveals a wide range of potentially pathogenic bacteria. One older adult woman, in particular, had high levels of Neisseria meningitidis and low levels of Haemophilus influenzae, species that were not observed in the calculus from three other individuals. Combined with the presence of incipient endocranial lesions and pronounced meningeal grooves, we interpret this as an ancient case of meningococcal disease. This disease afflicts millions around the globe today, but little is known about its (pre)history. With additional sampling, we suggest NGS of calculus offers an exciting new window into the evolutionary history of these bacterial species and their interactions with humans. Copyright © 2018 Elsevier Inc. All rights reserved.
Genefer: Programs for Finding Large Probable Generalized Fermat Primes
Iain Arthur Bethune
2015-11-01
Full Text Available Genefer is a suite of programs for performing Probable Primality (PRP tests of Generalised Fermat numbers 'b'2'n'+1 (GFNs using a Fermat test. Optimised implementations are available for modern CPUs using single instruction, multiple data (SIMD instructions, as well as for GPUs using CUDA or OpenCL. Genefer has been extensively used by PrimeGrid – a volunteer computing project searching for large prime numbers of various kinds, including GFNs. Genefer’s architecture separates the high level logic such as checkpointing and user interface from the architecture-specific performance-critical parts of the implementation, which are suitable for re-use. Genefer is released under the MIT license. Source and binaries are available from www.assembla.com/spaces/genefer.
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.
Calculus of variations in rate of reactions tax using the general pertubation theory
Silva, F.C. da.
1981-02-01
A perturbation expression to calculate the variations in the rates of integral parameters (such as reaction rates) of a reactor using a Time-Independent Generalized Perturbation Theory, was developed. This theory makes use of the concepts of neutron generation and neutron importance with respect to a given process occurring in a system. The application of Time-Dependent Generalized Perturbation Theory to the calculation of Burnup, by using the expressions derived by A. Gandini, along with the perturbation expression derived in the Time Independent Generalized Perturbation Theory, is done. (Author) [pt
Generalization of Poisson distribution for the case of changing probability of consequential events
Kushnirenko, E.
1995-01-01
The generalization of the Poisson distribution for the case of changing probabilities of the consequential events is done. It is shown that the classical Poisson distribution is the special case of this generalized distribution when the probabilities of the consequential events are constant. The using of the generalized Poisson distribution gives the possibility in some cases to obtain analytical result instead of making Monte-Carlo calculation
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...
Noncommutative operational calculus
Henry E. Heatherly
1999-12-01
Full Text Available Oliver Heaviside's operational calculus was placed on a rigorous mathematical basis by Jan Mikusinski, who constructed an algebraic setting for the operational methods. In this paper, we generalize Mikusi'{n}ski's methods to solve linear ordinary differential equations in which the unknown is a matrix- or linear operator-valued function. Because these functions can be zero-divisors and do not necessarily commute, Mikusi'{n}ski's one-dimensional calculus cannot be used. The noncommuative operational calculus developed here,however, is used to solve a wide class of such equations. In addition, we provide new proofs of existence and uniqueness theorems for certain matrix- and operator valued Volterra integral and integro-differential equations. Several examples are given which demonstrate these new methods.
Topology, calculus and approximation
Komornik, Vilmos
2017-01-01
Presenting basic results of topology, calculus of several variables, and approximation theory which are rarely treated in a single volume, this textbook includes several beautiful, but almost forgotten, classical theorems of Descartes, Erdős, Fejér, Stieltjes, and Turán. The exposition style of Topology, Calculus and Approximation follows the Hungarian mathematical tradition of Paul Erdős and others. In the first part, the classical results of Alexandroff, Cantor, Hausdorff, Helly, Peano, Radon, Tietze and Urysohn illustrate the theories of metric, topological and normed spaces. Following this, the general framework of normed spaces and Carathéodory's definition of the derivative are shown to simplify the statement and proof of various theorems in calculus and ordinary differential equations. The third and final part is devoted to interpolation, orthogonal polynomials, numerical integration, asymptotic expansions and the numerical solution of algebraic and differential equations. Students of both pure an...
Geometric calculus: a new computational tool for Riemannian geometry
Moussiaux, A.; Tombal, P.
1988-01-01
We compare geometric calculus applied to Riemannian geometry with Cartan's exterior calculus method. The correspondence between the two methods is clearly established. The results obtained by a package written in an algebraic language and doing general manipulations on multivectors are compared. We see that the geometric calculus is as powerful as exterior calculus
Jet calculus beyond leading logarithms
Kalinowski, J.; Konishi, K.; Taylor, T.R.
1981-01-01
It is shown that the evolution of hadronic jets produced in hard processes can be studied in terms of a simple parton branching picture, beyond the leading log approximation of QCD. The jet calculus is generalized to any given order of logs (but always to all orders of αsub(s)). We discuss the general structure of the formalism. Universality of jet evolution is discussed. We consider also a jet calorimetry measure and the multiplicity distribution of final states in a form which allows a systematic improvement of approximation. To the next-to-leading order, we prove the finiteness and elucidate the scheme dependence of parton subprocess probabilities. The physical inclusive cross section is shown to be scheme independent: next-to-leading results for e + e - → q (nonsinglet) + X agree with those of Curci and others. (orig.)
On Functional Calculus Estimates
Schwenninger, F.L.
2015-01-01
This thesis presents various results within the field of operator theory that are formulated in estimates for functional calculi. Functional calculus is the general concept of defining operators of the form $f(A)$, where f is a function and $A$ is an operator, typically on a Banach space. Norm
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
On paragrassmann differential calculus
Filippov, A.T.; Isaev, A.P.; Kurdikov, A.B.
1992-01-01
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
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...
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...
The malliavin calculus and related topics
Nualart, David
1995-01-01
The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on the Wiener space Originally, it was developed to prove a probabilistic proof to Hörmander's "sum of squares" theorem, but more recently it has found application in a variety of stochastic differential equation problems This monograph presents the main features of the Malliavin calculus and discusses in detail its connection with the anticipating stochastic calculus The author begins by developing analysis on the Wiener space, and then uses this to analyze the regularity of probability laws and to prove Hörmander's theorem Subsequent chapters apply the Malliavin calculus to anticipating stochastic differential equations and to studying the Markov property of solutions to stochastic differential equations with boundary conditions
Height probabilities in the Abelian sandpile model on the generalized finite Bethe lattice
Chen, Haiyan; Zhang, Fuji
2013-08-01
In this paper, we study the sandpile model on the generalized finite Bethe lattice with a particular boundary condition. Using a combinatorial method, we give the exact expressions for all single-site probabilities and some two-site joint probabilities. As a by-product, we prove that the height probabilities of bulk vertices are all the same for the Bethe lattice with certain given boundary condition, which was found from numerical evidence by Grassberger and Manna ["Some more sandpiles," J. Phys. (France) 51, 1077-1098 (1990)], 10.1051/jphys:0199000510110107700 but without a proof.
Scheunert, M.
1982-10-01
We develop a graded tensor calculus corresponding to arbitrary Abelian groups of degrees and arbitrary commutation factors. The standard basic constructions and definitions like tensor products, spaces of multilinear mappings, contractions, symmetrization, symmetric algebra, as well as the transpose, adjoint, and trace of a linear mapping, are generalized to the graded case and a multitude of canonical isomorphisms is presented. Moreover, the graded versions of the classical Lie algebras are introduced and some of their basic properties are described. (orig.)
Using Copulas in Data Mining Based on the Observational Calculus
Holeňa, Martin; Bajer, L.; Ščavnický, M.
2015-01-01
Roč. 27, č. 10 (2015), s. 2851-2864 ISSN 1041-4347 R&D Projects: GA ČR GA13-17187S Grant - others:SLU(CZ) SGS/21/2014 Institutional support: RVO:67985807 Keywords : data mining * observational calculus * generalized quantifiers * joint probability distribution * copulas * hierarchical Archimedean copulas Subject RIV: IN - Informatics, Computer Science Impact factor: 2.476, year: 2015
Olivieri, E.; Scoppola, E.
1996-01-01
In this paper we consider aperiodic ergodic Markov chains with transition probabilities exponentially small in a large parameter β. We extend to the general, not necessarily reversible case the analysis, started in part I of this work, of the first exit problem from a general domain Q containing many stable equilibria (attracting equilibrium points for the β = ∞ dynamics). In particular we describe the tube of typical trajectories during the first excursion outside Q
Synthesizing controllers from duration calculus
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....
Testicular calculus: A rare case.
Sen, Volkan; Bozkurt, Ozan; Demır, Omer; Tuna, Burcin; Yorukoglu, Kutsal; Esen, Adil
2015-01-01
Testicular calculus is an extremely rare case with unknown etiology and pathogenesis. To our knowledge, here we report the third case of testicular calculus. A 31-year-old man was admitted to our clinic with painful solid mass in left testis. After diagnostic work-up for a possible testicular tumour, he underwent inguinal orchiectomy and histopathologic examination showed a testicular calculus. Case hypothesis: Solid testicular lesions in young adults generally correspond to testicular cancer. Differential diagnosis should be done carefully. Future implications: In young adults with painful and solid testicular mass with hyperechogenic appearance on scrotal ultrasonography, testicular calculus must be kept in mind in differential diagnosis. Further reports on this topic may let us do more clear recommendations about the etiology and treatment of this rare disease.
Bodewig, E
1959-01-01
Matrix Calculus, Second Revised and Enlarged Edition focuses on systematic calculation with the building blocks of a matrix and rows and columns, shunning the use of individual elements. The publication first offers information on vectors, matrices, further applications, measures of the magnitude of a matrix, and forms. The text then examines eigenvalues and exact solutions, including the characteristic equation, eigenrows, extremum properties of the eigenvalues, bounds for the eigenvalues, elementary divisors, and bounds for the determinant. The text ponders on approximate solutions, as well
Ouellette,, Jennifer
2011-01-01
Jennifer Ouellette never took maths in the sixth form, mostly because she like most of us assumed she wouldn't need it much in real life. But then the English graduate, now an award-winning science-writer, had a change of heart and decided to revisit the equations and formulas that had haunted her youth. The Calculus Diaries is the fun and fascinating account of a year spent confronting her numbers-phobia head on. With wit and verve, Ouellette explains how she discovered that maths could apply to everything from petrol mileages to dieting, rollercoaster rides to winning in Las Vegas.
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
Fitzpatrick, Patrick M
2009-01-01
Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is self-contained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclide
Time of Arrival Estimation in Probability-Controlled Generalized CDMA Systems
Hagit Messer
2007-11-01
Full Text Available In recent years, more and more wireless communications systems are required to provide also a positioning measurement. In code division multiple access (CDMA communication systems, the positioning accuracy is significantly degraded by the multiple access interference (MAI caused by other users in the system. This MAI is commonly managed by a power control mechanism, and yet, MAI has a major effect on positioning accuracy. Probability control is a recently introduced interference management mechanism. In this mechanism, a user with excess power chooses not to transmit some of its symbols. The information in the nontransmitted symbols is recovered by an error-correcting code (ECC, while all other users receive a more reliable data during these quiet periods. Previous research had shown that the implementation of a probability control mechanism can significantly reduce the MAI. In this paper, we show that probability control also improves the positioning accuracy. We focus on time-of-arrival (TOA based positioning systems. We analyze the TOA estimation performance in a generalized CDMA system, in which the probability control mechanism is employed, where the transmitted signal is noncontinuous with a symbol transmission probability smaller than 1. The accuracy of the TOA estimation is determined using appropriate modifications of the Cramer-Rao bound on the delay estimation. Keeping the average transmission power constant, we show that the TOA accuracy of each user does not depend on its transmission probability, while being a nondecreasing function of the transmission probability of any other user. Therefore, a generalized, noncontinuous CDMA system with a probability control mechanism can always achieve better positioning performance, for all users in the network, than a conventional, continuous, CDMA system.
Larsen, Kim Guldstrand; Mardare, Radu Iulian; Xue, Bingtian
2016-01-01
We introduce a version of the probabilistic µ-calculus (PMC) built on top of a probabilistic modal logic that allows encoding n-ary inequational conditions on transition probabilities. PMC extends previously studied calculi and we prove that, despite its expressiveness, it enjoys a series of good...... metaproperties. Firstly, we prove the decidability of satisﬁability checking by establishing the small model property. An algorithm for deciding the satisﬁability problem is developed. As a second major result, we provide a complete axiomatization for the alternation-free fragment of PMC. The completeness proof...
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...
Soury, Hamza
2012-06-01
This letter considers the average bit error probability of binary coherent signaling over flat fading channels subject to additive generalized Gaussian noise. More specifically, a generic closed form expression in terms of the Fox\\'s H function is offered for the extended generalized-K fading case. Simplifications for some special fading distributions such as generalized-K fading and Nakagami-m fading and special additive noise distributions such as Gaussian and Laplacian noise are then presented. Finally, the mathematical formalism is illustrated by some numerical examples verified by computer based simulations for a variety of fading and additive noise parameters. © 2012 IEEE.
Soury, Hamza
2013-07-01
This paper considers the average symbol error probability of square Quadrature Amplitude Modulation (QAM) coherent signaling over flat fading channels subject to additive generalized Gaussian noise. More specifically, a generic closedform expression in terms of the Fox H function and the bivariate Fox H function is offered for the extended generalized-K fading case. Simplifications for some special fading distributions such as generalized-K fading, Nakagami-m fading, and Rayleigh fading and special additive noise distributions such as Gaussian and Laplacian noise are then presented. Finally, the mathematical formalism is illustrated by some numerical examples verified by computer based simulations for a variety of fading and additive noise parameters.
Crawford, Forrest W.; Suchard, Marc A.
2011-01-01
A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with n current particles, a new particle is born with instantaneous rate λn and a particle dies with instantaneous rate μn. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics. PMID:21984359
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.
Feynman quasi probability distribution for spin-(1/2), and its generalizations
Colucci, M.
1999-01-01
It has been examined the Feynman's paper Negative probability, in which, after a discussion about the possibility of attributing a real physical meaning to quasi probability distributions, he introduces a new kind of distribution for spin-(1/2), with a possible method of generalization to systems with arbitrary number of states. The principal aim of this article is to shed light upon the method of construction of these distributions, taking into consideration their application to some experiments, and discussing their positive and negative aspects
A Calculus for Context-Awareness
Zimmer, Pascal
2005-01-01
In order to answer the challenge of pervasive computing, we propose a new process calculus, whose aim is to describe dynamic systems composed of agents able to move and react differently depending on their location. This Context-Aware Calculus features a hierarchical structure similar to mobile...... ambients, and a generic multi-agent synchronization mechanism, inspired from the join-calculus. After general ideas and introduction, we review the full calculus' syntax and semantics, as well as some motivating examples, study its expressiveness, and show how the notion of computation itself can be made...
A Higher-Order Calculus for Categories
Cáccamo, Mario José; Winskel, Glynn
2001-01-01
A calculus for a fragment of category theory is presented. The types in the language denote categories and the expressions functors. The judgements of the calculus systematise categorical arguments such as: an expression is functorial in its free variables; two expressions are naturally isomorphic...... in their free variables. There are special binders for limits and more general ends. The rules for limits and ends support an algebraic manipulation of universal constructions as opposed to a more traditional diagrammatic approach. Duality within the calculus and applications in proving continuity are discussed...... with examples. The calculus gives a basis for mechanising a theory of categories in a generic theorem prover like Isabelle....
Baronti, Marco; van der Putten, Robertus; Venturi, Irene
2016-01-01
This book, intended as a practical working guide for students in Engineering, Mathematics, Physics, or any other field where rigorous calculus is needed, includes 450 exercises. Each chapter starts with a summary of the main definitions and results, which is followed by a selection of solved exercises accompanied by brief, illustrative comments. A selection of problems with indicated solutions rounds out each chapter. A final chapter explores problems that are not designed with a single issue in mind but instead call for the combination of a variety of techniques, rounding out the book’s coverage. Though the book’s primary focus is on functions of one real variable, basic ordinary differential equations (separation of variables, linear first order and constant coefficients ODEs) are also discussed. The material is taken from actual written tests that have been delivered at the Engineering School of the University of Genoa. Literally thousands of students have worked on these problems, ensuring their real-...
Malliavin Calculus With Applications to Stochastic Partial Differential Equations
Sanz-Solé, Marta
2005-01-01
Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics.This book presents applications of Malliavin calculus to the analysis of probability laws of solutions to stochastic partial differential equations driven by Gaussian noises that are white in time and coloured in space. The first five chapters introduce the calculus itself
Covariant differential calculus on the quantum hyperplane
Wess, J.
1991-01-01
We develop a differential calculus on the quantum hyperplane covariant with respect to the action of the quantum group GL q (n). This is a concrete example of noncommutative differential geometry. We describe the general constraints for a noncommutative differential calculus and verify that the example given here satisfies all these constraints. We also discuss briefly the integration over the quantum plane. (orig.)
Multiple-event probability in general-relativistic quantum mechanics. II. A discrete model
Mondragon, Mauricio; Perez, Alejandro; Rovelli, Carlo
2007-01-01
We introduce a simple quantum mechanical model in which time and space are discrete and periodic. These features avoid the complications related to continuous-spectrum operators and infinite-norm states. The model provides a tool for discussing the probabilistic interpretation of generally covariant quantum systems, without the confusion generated by spurious infinities. We use the model to illustrate the formalism of general-relativistic quantum mechanics, and to test the definition of multiple-event probability introduced in a companion paper [Phys. Rev. D 75, 084033 (2007)]. We consider a version of the model with unitary time evolution and a version without unitary time evolution
Efficient Outage Probability Evaluation of Diversity Receivers Over Generalized Gamma Channels
Ben Issaid, Chaouki; Alouini, Mohamed-Slim; Tempone, Raul
2016-01-01
In this paper, we are interested in determining the cumulative distribution function of the sum of generalized Gamma in the setting of rare event simulations. To this end, we present an efficient importance sampling estimator. The main result of this work is the bounded relative property of the proposed estimator. This result is used to accurately estimate the outage probability of multibranch maximum ratio combining and equal gain combining diversity receivers over generalized Gamma fading channels. Selected numerical simulations are discussed to show the robustness of our estimator compared to naive Monte Carlo.
Efficient Outage Probability Evaluation of Diversity Receivers Over Generalized Gamma Channels
Ben Issaid, Chaouki
2016-10-01
In this paper, we are interested in determining the cumulative distribution function of the sum of generalized Gamma in the setting of rare event simulations. To this end, we present an efficient importance sampling estimator. The main result of this work is the bounded relative property of the proposed estimator. This result is used to accurately estimate the outage probability of multibranch maximum ratio combining and equal gain combining diversity receivers over generalized Gamma fading channels. Selected numerical simulations are discussed to show the robustness of our estimator compared to naive Monte Carlo.
Series expansion in fractional calculus and fractional differential equations
Li, Ming-Fan; Ren, Ji-Rong; Zhu, Tao
2009-01-01
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this theorem, in this paper we introduce fractional series expansion method to fractional calculus. We define a kind of fractional Taylor series of an infinitely fractionally-differentiable function. Further, based on our definition we generalize hypergeometric functio...
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is Part of a series of 41 Calculus Based Physics (CBP) modules totaling about 1,000 Pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized courses in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Fuller, Robert G., Ed.; And Others
This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules indlude study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…
Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process
Chuancun Yin
2015-01-01
Full Text Available We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy.
Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process
Yuen, Kam Chuen; Shen, Ying
2015-01-01
We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy. PMID:26351655
Fast Outage Probability Simulation for FSO Links with a Generalized Pointing Error Model
Ben Issaid, Chaouki
2017-02-07
Over the past few years, free-space optical (FSO) communication has gained significant attention. In fact, FSO can provide cost-effective and unlicensed links, with high-bandwidth capacity and low error rate, making it an exciting alternative to traditional wireless radio-frequency communication systems. However, the system performance is affected not only by the presence of atmospheric turbulences, which occur due to random fluctuations in the air refractive index but also by the existence of pointing errors. Metrics, such as the outage probability which quantifies the probability that the instantaneous signal-to-noise ratio is smaller than a given threshold, can be used to analyze the performance of this system. In this work, we consider weak and strong turbulence regimes, and we study the outage probability of an FSO communication system under a generalized pointing error model with both a nonzero boresight component and different horizontal and vertical jitter effects. More specifically, we use an importance sampling approach which is based on the exponential twisting technique to offer fast and accurate results.
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.
HITZER, Eckhard MS
2002-01-01
This paper treats the fundamentals of the vector differential calculus part of universal geometric calculus. Geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. In order to make the treatment self-contained, I first compile all important geometric algebra relationships,which are necesssary for vector differential calculus. Then differentiation by vectors is introduced and a host of major ve...
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....
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.
Multivariable calculus and differential geometry
Walschap, Gerard
2015-01-01
This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.
Hreniuc, V.; Hreniuc, A.; Pescaru, A.
2017-08-01
Solving a general strength problem of a ship hull may be done using analytical approaches which are useful to deduce the buoyancy forces distribution, the weighting forces distribution along the hull and the geometrical characteristics of the sections. These data are used to draw the free body diagrams and to compute the stresses. The general strength problems require a large amount of calculi, therefore it is interesting how a computer may be used to solve such problems. Using computer programming an engineer may conceive software instruments based on analytical approaches. However, before developing the computer code the research topic must be thoroughly analysed, in this way being reached a meta-level of understanding of the problem. The following stage is to conceive an appropriate development strategy of the original software instruments useful for the rapid development of computer aided analytical models. The geometrical characteristics of the sections may be computed using a bool algebra that operates with ‘simple’ geometrical shapes. By ‘simple’ we mean that for the according shapes we have direct calculus relations. In the set of ‘simple’ shapes we also have geometrical entities bounded by curves approximated as spline functions or as polygons. To conclude, computer programming offers the necessary support to solve general strength ship hull problems using analytical methods.
Calculus and analysis in Euclidean space
Shurman, Jerry
2016-01-01
The graceful role of analysis in underpinning calculus is often lost to their separation in the curriculum. This book entwines the two subjects, providing a conceptual approach to multivariable calculus closely supported by the structure and reasoning of analysis. The setting is Euclidean space, with the material on differentiation culminating in the inverse and implicit function theorems, and the material on integration culminating in the general fundamental theorem of integral calculus. More in-depth than most calculus books but less technical than a typical analysis introduction, Calculus and Analysis in Euclidean Space offers a rich blend of content to students outside the traditional mathematics major, while also providing transitional preparation for those who will continue on in the subject. The writing in this book aims to convey the intent of ideas early in discussion. The narrative proceeds through figures, formulas, and text, guiding the reader to do mathematics resourcefully by marshaling the skil...
Area Regge calculus and continuum limit
Khatsymovsky, V.M.
2002-01-01
Encountered in the literature generalisations of general relativity to independent area variables are considered, the discrete (generalised Regge calculus) and continuum ones. The generalised Regge calculus can be either with purely area variables or, as we suggest, with area tensor-connection variables. Just for the latter, in particular, we prove that in analogy with corresponding statement in ordinary Regge calculus (by Feinberg, Friedberg, Lee and Ren), passing to the (appropriately defined) continuum limit yields the generalised continuum area tensor-connection general relativity
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
Zegarelli, Mark
2012-01-01
An easy-to-understand primer on advanced calculus topics Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. Calculus II For Dummies offers expert instruction, advice, and tips to help second semester calculus students get a handle on the subject and ace their exams. It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals. This hands-on guide also covers sequences and series, wit
Dimakis, N.; Terzis, Petros A.; Zampeli, Adamantia; Christodoulakis, T.
2016-09-01
The high degree of symmetry renders the dynamics of cosmological as well as some black hole spacetimes describable by a system of finite degrees of freedom. These systems are generally known as minisuperspace models. One of their important key features is the invariance of the corresponding reduced actions under reparametrizations of the independent variable, a fact that can be seen as the remnant of the general covariance of the full theory. In the case of a system of n degrees of freedom, described by a Lagrangian quadratic in velocities, one can use the lapse by either gauge fixing it or letting it be defined by the constraint and subsequently substitute into the rest of the equations. In the first case, the system of the second-order equations of motion is solvable for all n accelerations and the constraint becomes a restriction among constants of integration. In the second case, the system can be solved for only n -1 accelerations and the "gauge" freedom is transferred to the choice of one of the scalar degrees of freedom. In this paper, we take the second path and express all n -1 scalar degrees of freedom in terms of the remaining one, say q . By considering these n -1 degrees of freedom as arbitrary but given functions of q , we manage to extract a two-dimensional pure gauge system consisting of the lapse N and the arbitrary q : in a way, we decouple the reparametrization invariance from the rest of the equations of motion, which are thus describing the "true" dynamics. The solution of the corresponding quantum two-dimensional system is used for the definition of a generalized probability for every configuration fi(q ), be it classical or not. The main result is that, interestingly enough, this probability attains its extrema on the classical solution of the initial n -dimensional system.
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.
From Calculus to Wavelets: ANew Mathematical Technique
Home; Journals; Resonance – Journal of Science Education; Volume 2; Issue 4. From Calculus to Wavelets: A New Mathematical Technique Wavelet Analysis Physical Properties. Gerald B Folland. General Article Volume 2 Issue 4 April 1997 pp 25-37 ...
Paul B. Slater
2015-01-01
Full Text Available Previously, a formula, incorporating a 5F4 hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments ρPTnρk/ρk of 4×4 density-matrices (ρ and their partial transposes (|ρPT|, was applied with k=0 to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over “induced measures in the space of mixed quantum states.” The associated induced-measure separability probabilities (k=1,2,… are found—via a high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit (α=1/2, (standard two-qubit (α=1, and two-quater[nionic]bit (α=2 cases. We deduce rather simple companion (rebit, qubit, quaterbit, … formulas that successfully reproduce the rational values assumed for general k. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.
Relativistic collapse using Regge calculus: Pt. 1
Dubal, M.R.; Leicester Univ.
1989-01-01
Regge calculus is used to simulate the dynamical collapse of model stars. In this paper we describe the general methodology of including a perfect fluid in dynamical Regge calculus spacetimes. The Regge-Einstein equations for spherical collapse are obtained and are then specialised to mimic a particular continuum gauge. The equivalent continuum problem is also set up. This is to be solved using standard numerical techniques (i.e. the method of finite difference). A subsequent paper will consider the solution of the equations presented here and will use the continuum problem for comparison purposes in order to check the Regge calculus results. (author)
Applications of fractional calculus in physics
2000-01-01
Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to noninteger orders. While these results have been accumulated over centuries in various branches of mathematics, they have until recently found little appreciation or application in physics and other mathematically oriented sciences. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus.This volume provides an introduction to fractional calculus for physicists, and co
Brownian motion and stochastic calculus
Karatzas, Ioannis
1998-01-01
This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization). This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large num...
Grinstead, Charles M; Snell, J Laurie
2011-01-01
This book explores four real-world topics through the lens of probability theory. It can be used to supplement a standard text in probability or statistics. Most elementary textbooks present the basic theory and then illustrate the ideas with some neatly packaged examples. Here the authors assume that the reader has seen, or is learning, the basic theory from another book and concentrate in some depth on the following topics: streaks, the stock market, lotteries, and fingerprints. This extended format allows the authors to present multiple approaches to problems and to pursue promising side discussions in ways that would not be possible in a book constrained to cover a fixed set of topics. To keep the main narrative accessible, the authors have placed the more technical mathematical details in appendices. The appendices can be understood by someone who has taken one or two semesters of calculus.
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...
Introduction to stochastic calculus
Karandikar, Rajeeva L
2018-01-01
This book sheds new light on stochastic calculus, the branch of mathematics that is most widely applied in financial engineering and mathematical finance. The first book to introduce pathwise formulae for the stochastic integral, it provides a simple but rigorous treatment of the subject, including a range of advanced topics. The book discusses in-depth topics such as quadratic variation, Ito formula, and Emery topology. The authors briefly address continuous semi-martingales to obtain growth estimates and study solution of a stochastic differential equation (SDE) by using the technique of random time change. Later, by using Metivier–Pellumail inequality, the solutions to SDEs driven by general semi-martingales are discussed. The connection of the theory with mathematical finance is briefly discussed and the book has extensive treatment on the representation of martingales as stochastic integrals and a second fundamental theorem of asset pricing. Intended for undergraduate- and beginning graduate-level stud...
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.
Space complexity in polynomial calculus
Filmus, Y.; Lauria, M.; Nordström, J.; Ron-Zewi, N.; Thapen, Neil
2015-01-01
Roč. 44, č. 4 (2015), s. 1119-1153 ISSN 0097-5397 R&D Projects: GA AV ČR IAA100190902; GA ČR GBP202/12/G061 Institutional support: RVO:67985840 Keywords : proof complexity * polynomial calculus * lower bounds Subject RIV: BA - General Mathematics Impact factor: 0.841, year: 2015 http://epubs.siam.org/doi/10.1137/120895950
Dixmier, Marc.
1980-10-01
A general expression of the diffusion coefficient (d.c.) of neutrons was given, with stress being put on symmetries. A system of first-flight collision probabilities for the case of a random stack of any number of types of one- and two-zoned spherical pebbles, with an albedo at the frontiers of the elements or (either) consideration of the interstital medium, was built; to that end, the bases of collision probability theory were reviewed, and a wide generalisation of the reciprocity theorem for those probabilities was demonstrated. The migration area of neutrons was expressed for any random stack of convex, 'simple' and 'regular-contact' elements, taking into account the correlations between free-paths; the average cosinus of re-emission of neutrons by an element, in the case of a homogeneous spherical pebble and the transport approximation, was expressed; the superiority of the so-found result over Behrens' theory, for the type of media under consideration, was established. The 'fine structure current term' of the d.c. was also expressed, and it was shown that its 'polarisation term' is negligible. Numerical applications showed that the global heterogeneity effect on the d.c. of pebble-bed reactors is comparable with that for Graphite-moderated, Carbon gas-cooled, natural Uranium reactors. The code CARACOLE, which integrates all the results here obtained, was introduced [fr
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
Mouchtouris, S.; Kokkoris, G.
2018-01-01
A generalized equation for the electron energy probability function (EEPF) of inductively coupled Ar plasmas is proposed under conditions of nonlocal electron kinetics and diffusive cooling. The proposed equation describes the local EEPF in a discharge and the independent variable is the kinetic energy of electrons. The EEPF consists of a bulk and a depleted tail part and incorporates the effect of the plasma potential, Vp, and pressure. Due to diffusive cooling, the break point of the EEPF is eVp. The pressure alters the shape of the bulk and the slope of the tail part. The parameters of the proposed EEPF are extracted by fitting to measure EEPFs (at one point in the reactor) at different pressures. By coupling the proposed EEPF with a hybrid plasma model, measurements in the gaseous electronics conference reference reactor concerning (a) the electron density and temperature and the plasma potential, either spatially resolved or at different pressure (10-50 mTorr) and power, and (b) the ion current density of the electrode, are well reproduced. The effect of the choice of the EEPF on the results is investigated by a comparison to an EEPF coming from the Boltzmann equation (local electron kinetics approach) and to a Maxwellian EEPF. The accuracy of the results and the fact that the proposed EEPF is predefined renders its use a reliable alternative with a low computational cost compared to stochastic electron kinetic models at low pressure conditions, which can be extended to other gases and/or different electron heating mechanisms.
Cui, Helen; Thomas, Johanna; Kumar, Sunil
2013-04-10
We present a case of a renal calculus treated solely with antibiotics which has not been previously reported in the literature. A man with a 17 mm lower pole renal calculus and concurrent Escherichia coli urine infection was being worked up to undergo percutaneous nephrolithotomy. However, after a course of preoperative antibiotics the stone was no longer seen on retrograde pyelography or CT imaging.
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.
Sutherland, Melissa
2006-01-01
In this paper we discuss manipulatives and hands-on investigations for Calculus involving volume, arc length, and surface area to motivate and develop formulae which can then be verified using techniques of integration. Pre-service teachers in calculus courses using these activities experience a classroom in which active learning is encouraged and…
On exterior variational calculus
Aldrovandi, R.; Kraenkel, R.A.
1987-01-01
Exterior variational calculus is introduced through examples in field theory. It provides a very simple technique to decide on the existence of Lagrangians for given equations of motions and, in the case, to find them. Only local aspects are discussed but the analogy to exterior calculus on finite dimensional manifolds is complete, strongly suggesting its suitability to the study of topological aspects. (Author) [pt
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…
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.
Ross, Sheldon
2014-01-01
A First Course in Probability, Ninth Edition, features clear and intuitive explanations of the mathematics of probability theory, outstanding problem sets, and a variety of diverse examples and applications. This book is ideal for an upper-level undergraduate or graduate level introduction to probability for math, science, engineering and business students. It assumes a background in elementary calculus.
Multivariable calculus with applications
Lax, Peter D
2017-01-01
This text in multivariable calculus fosters comprehension through meaningful explanations. Written with students in mathematics, the physical sciences, and engineering in mind, it extends concepts from single variable calculus such as derivative, integral, and important theorems to partial derivatives, multiple integrals, Stokes’ and divergence theorems. Students with a background in single variable calculus are guided through a variety of problem solving techniques and practice problems. Examples from the physical sciences are utilized to highlight the essential relationship between calculus and modern science. The symbiotic relationship between science and mathematics is shown by deriving and discussing several conservation laws, and vector calculus is utilized to describe a number of physical theories via partial differential equations. Students will learn that mathematics is the language that enables scientific ideas to be precisely formulated and that science is a source for the development of mathemat...
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...
Lax, Peter D
2014-01-01
This new edition of Lax, Burstein, and Lax's Calculus with Applications and Computing offers meaningful explanations of the important theorems of single variable calculus. Written with students in mathematics, the physical sciences, and engineering in mind, and revised with their help, it shows that the themes of calculation, approximation, and modeling are central to mathematics and the main ideas of single variable calculus. This edition brings the innovation of the first edition to a new generation of students. New sections in this book use simple, elementary examples to show that when applying calculus concepts to approximations of functions, uniform convergence is more natural and easier to use than point-wise convergence. As in the original, this edition includes material that is essential for students in science and engineering, including an elementary introduction to complex numbers and complex-valued functions, applications of calculus to modeling vibrations and population dynamics, and an introduc...
Dan Ye
2013-01-01
Full Text Available This paper is concerned with delay-dependent stochastic stability for time-delay Markovian jump systems (MJSs with sector-bounded nonlinearities and more general transition probabilities. Different from the previous results where the transition probability matrix is completely known, a more general transition probability matrix is considered which includes completely known elements, boundary known elements, and completely unknown ones. In order to get less conservative criterion, the state and transition probability information is used as much as possible to construct the Lyapunov-Krasovskii functional and deal with stability analysis. The delay-dependent sufficient conditions are derived in terms of linear matrix inequalities to guarantee the stability of systems. Finally, numerical examples are exploited to demonstrate the effectiveness of the proposed method.
Supercalculators and University Entrance Calculus Examinations.
Hong, Ye Yoon; Thomas, Mike; Kiernan, Christine
2000-01-01
Investigates whether the use of computer algebra systems could provide a significant advantage to students taking standard university entrance calculus examinations. Indicates that supercalculators would probably provide a significant advantage, particularly for lower-achieving students. Demonstrates that it is possible to write questions in which…
Supergravity tensor calculus in 5D from 6D
Kugo, Taichiro; Ohashi, Keisuke
2000-01-01
Supergravity tensor calculus in five spacetime dimensions is derived by dimensional reduction from the d=6 superconformal tensor calculus. In particular, we obtain an off-shell hypermultiplet in 5D from the on-shell hypermultiplet in 6D. Our tensor calculus retains the dilatation gauge symmetry, so that it is a trivial gauge fixing to make the Einstein term canonical in a general matter-Yang-Mills-supergravity coupled system. (author)
Impact of Calculus Reform in a Liberal Arts Calculus Course.
Brosnan, Patricia A.; Ralley, Thomas G.
This report describes the changes in a freshman-level calculus course that occurred as a consequence of adopting the Harvard Consortium Calculus text. The perspective is that of the lecturer. The course is intended as an introduction to calculus for liberal arts students, that is, students who will not be expected to use calculus as a mathematical…
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.
Probabilistic Analysis of the Quality Calculus
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...
Feynman's operational calculus and beyond noncommutativity and time-ordering
Johnson, George W; Nielsen, Lance
2015-01-01
This book is aimed at providing a coherent, essentially self-contained, rigorous and comprehensive abstract theory of Feynman's operational calculus for noncommuting operators. Although it is inspired by Feynman's original heuristic suggestions and time-ordering rules in his seminal 1951 paper An operator calculus having applications in quantum electrodynamics, as will be made abundantly clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman's work. Hence, the second part of the main title of this book. The basic properties of the operational calculus are developed and certain algebraic and analytic properties of the operational calculus are explored. Also, the operational calculus will be seen to possess some pleasant stability properties. Furthermore, an evolution equation and a generalized integral equation obeyed by the operational calculus are discussed and connections wi...
Command injection attacks, continuations, and the Lambek calculus
Hayo Thielecke
2016-06-01
Full Text Available This paper shows connections between command injection attacks, continuations, and the Lambek calculus: certain command injections, such as the tautology attack on SQL, are shown to be a form of control effect that can be typed using the Lambek calculus, generalizing the double-negation typing of continuations. Lambek's syntactic calculus is a logic with two implicational connectives taking their arguments from the left and right, respectively. These connectives describe how strings interact with their left and right contexts when building up syntactic structures. The calculus is a form of propositional logic without structural rules, and so a forerunner of substructural logics like Linear Logic and Separation Logic.
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
Stochastic calculus in physics
Fox, R.F.
1987-01-01
The relationship of Ito-Stratonovich stochastic calculus to studies of weakly colored noise is explained. A functional calculus approach is used to obtain an effective Fokker-Planck equation for the weakly colored noise regime. In a smooth limit, this representation produces the Stratonovich version of the Ito-Stratonovich calculus for white noise. It also provides an approach to steady state behavior for strongly colored noise. Numerical simulation algorithms are explored, and a novel suggestion is made for efficient and accurate simulation of white noise equations
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
Regge calculus from discontinuous metrics
Khatsymovsky, V.M.
2003-01-01
Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface defined by continuity conditions. Quantum theory of the discontinuous metric system is assumed to be fixed somehow in the form of quantum measure on (the space of functionals on) the superspace. The problem of reducing this measure to the Regge hypersurface is addressed. The quantum Regge calculus measure is defined from a discontinuous metric measure by inserting the δ-function-like phase factor. The requirement that continuity conditions be imposed in a 'face-independent' way fixes this factor uniquely. The term 'face-independent' means that this factor depends only on the (hyper)plane spanned by the face, not on it's form and size. This requirement seems to be natural from the viewpoint of existence of the well-defined continuum limit maximally free of lattice artefacts
Alberto Carraro
2013-03-01
Full Text Available We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry-Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without any restriction. Its type system enforces strong normalization of expressions and it is a sound and complete system for full implicational Classical Logic. We give a very simple denotational semantics which allows easy calculations of the interpretation of expressions.
Ody, Heinrich; Fränzle, Martin; Hansen, Michael Reichhardt
2016-01-01
To formally reason about the temporal quality of systems discounting was introduced to CTL and LTL. However, these logic are discrete and they cannot express duration properties. In this work we introduce discounting for a variant of Duration Calculus. We prove decidability of model checking...... for a useful fragment of discounted Duration Calculus formulas on timed automata under mild assumptions. Further, we provide an extensive example to show the usefulness of the fragment....
Calculus of bivariant function
PTÁČNÍK, Jan
2011-01-01
This thesis deals with the introduction of function of two variables and differential calculus of this function. This work should serve as a textbook for students of elementary school's teacher. Each chapter contains a summary of basic concepts and explanations of relationships, then solved model exercises of the topic and finally the exercises, which should solve the student himself. Thesis have transmit to students basic knowledges of differential calculus of functions of two variables, inc...
Malinowska , Agnieszka B.; Torres , Delfim
2014-01-01
International audience; Introduces readers to the treatment of the calculus of variations with q-differences and Hahn difference operators Provides the reader with the first extended treatment of quantum variational calculus Shows how the techniques described can be applied to economic models as well as other mathematical systems This Brief puts together two subjects, quantum and variational calculi by considering variational problems involving Hahn quantum operators. The main advantage of it...
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
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…
A compositional proof system for the modal μ-calculus
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 ...
Probability functions in the context of signed involutive meadows
Bergstra, J.A.; Ponse, A.
2016-01-01
The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability theory are restated in this framework.
Yi-Hua Zhu; Ding-Hua Shi; Yong Xiong; Ji Gao; He-Zhi Luo
2004-01-01
Mobility management is a challenging topic in mobile computing environment. Studying the situation of mobiles crossing the boundaries of location areas is significant for evaluating the costs and performances of various location management strategies. Hitherto, several formulae were derived to describe the probability of the number of location areas' boundaries crossed by a mobile. Some of them were widely used in analyzing the costs and performances of mobility management strategies. Utilizing the density evolution method of vector Markov processes, we propose a general probability formula of the number of location areas' boundaries crossed by a mobile between two successive calls. Fortunately, several widely-used formulae are special cases of the proposed formula.
Dental calculus formation in children and adolescents undergoing hemodialysis.
Martins, Carla; Siqueira, Walter Luiz; Oliveira, Elizabeth; Nicolau, José; Primo, Laura Guimarães
2012-10-01
This study aimed to determine whether dental calculus formation is really higher among patients with chronic kidney disease undergoing hemodialysis than among controls. Furthermore, the study evaluated correlations between dental calculus formation and dental plaque, variables that are related to renal disease and/or saliva composition. The Renal Group was composed of 30 patients undergoing hemodialysis, whereas the Healthy Group had 30 clinically healthy patients. Stimulated whole saliva and parotid saliva were collected. Salivary flow rate and calcium and phosphate concentrations were determined. In the Renal Group the saliva collection was carried out before and after a hemodialysis session. Patients from both groups received intraoral exams, oral hygiene instructions, and dental scaling. Three months later, the dental calculus was measured by the Volpe-Manhold method to determine the rate of dental calculus formation. The Renal Group presented a higher rate of dental calculus formation (p dental calculus formation and whole saliva flow rate in the Renal Group after a hemodialysis session (r = 0.44, p dental calculus was associated with phosphate concentration in whole saliva from the Renal Group (p dental calculus formation, probably due to salivary variables.
The absolute differential calculus calculus of tensors
Levi-Cività, Tullio
1926-01-01
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
Fractional vector calculus and fractional Maxwell's equations
Tarasov, Vasily E.
2008-01-01
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 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
General Exact Solution to the Problem of the Probability Density for Sums of Random Variables
Tribelsky, Michael I.
2002-07-01
The exact explicit expression for the probability density pN(x) for a sum of N random, arbitrary correlated summands is obtained. The expression is valid for any number N and any distribution of the random summands. Most attention is paid to application of the developed approach to the case of independent and identically distributed summands. The obtained results reproduce all known exact solutions valid for the, so called, stable distributions of the summands. It is also shown that if the distribution is not stable, the profile of pN(x) may be divided into three parts, namely a core (small x), a tail (large x), and a crossover from the core to the tail (moderate x). The quantitative description of all three parts as well as that for the entire profile is obtained. A number of particular examples are considered in detail.
Yanbo Li
2014-01-01
Full Text Available This paper is devoted to the investigation of the design of robust guaranteed cost observer for a class of linear singular Markovian jump time-delay systems with generally incomplete transition probability. In this singular model, each transition rate can be completely unknown or only its estimate value is known. Based on stability theory of stochastic differential equations and linear matrix inequality (LMI technique, we design an observer to ensure that, for all uncertainties, the resulting augmented system is regular, impulse free, and robust stochastically stable with the proposed guaranteed cost performance. Finally, a convex optimization problem with LMI constraints is formulated to design the suboptimal guaranteed cost filters for linear singular Markovian jump time-delay systems with generally incomplete transition probability.
Tolhoek, H.A.; Groot, S.R. de
1949-01-01
In the general case of a quantum mechanical system with a Hamiltonian that is invariant for rotations spatial degeneracy will exist. So the initial state must be characterized except by the energy also by e.g. the magnetic quantum number. Both for emission of light and electrons plus neutrinos
Ludeke, Steven G; DeYoung, Colin G
2014-06-01
Many of the characteristics cited in Hibbing et al.'s account are ineffective predictors of economic conservatism. However, these same characteristics are often associated with differences not only in social conservatism but also in religiousness and authoritarianism. Hibbing et al. may have offered a useful explanation of traditionalism and attitudes toward change across domains rather than of general political attitudes.
Qingwu Gao
2012-01-01
Full Text Available We discuss the uniformly asymptotic estimate of the finite-time ruin probability for all times in a generalized compound renewal risk model, where the interarrival times of successive accidents and all the claim sizes caused by an accident are two sequences of random variables following a wide dependence structure. This wide dependence structure allows random variables to be either negatively dependent or positively dependent.
Discrete Quantum Gravity in the Regge Calculus Formalism
Khatsymovsky, V.M.
2005-01-01
We discuss an approach to the discrete quantum gravity in the Regge calculus formalism that was developed in a number of our papers. The Regge calculus is general relativity for a subclass of general Riemannian manifolds called piecewise flat manifolds. The Regge calculus deals with a discrete set of variables, triangulation lengths, and contains continuous general relativity as a special limiting case where the lengths tend to zero. In our approach, the quantum length expectations are nonzero and of the order of the Plank scale, 10 -33 cm, implying a discrete spacetime structure on these scales
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.
Null-strut calculus. I. Kinematics
Kheyfets, A.; LaFave, N.J.; Miller, W.A.
1990-01-01
This paper describes the kinematics of null-strut calculus---a 3+1 Regge calculus approach to general relativity. We show how to model the geometry of spacetime with simplicial spacelike three-geometries (TET's) linked to ''earlier'' and ''later'' momentumlike lattice surfaces (TET * ) entirely by light rays or ''null struts.'' These three-layered lattice spacetime geometries are defined and analyzed using combinatorial formulas for the structure of polytopes. The following paper in this series describes how these three-layered spacetime lattices are used to model spacetimes in full conformity with Einstein's theory of gravity
Calculus of tensors and differential forms
Sinha, Rajnikant
2014-01-01
Calculus of tensors and differential forms is an introductory-level textbook. Through this book, students will familiarize themselves with tools they need in order to use for further study on general relativity and research, such as affine tensors, tensor calculus on manifolds, relative tensors, Lie derivatives, wedge products, differential forms, and Stokes' theorem. The treatment is concrete and in detail, so that abstract concepts do not deter even physics and engineering students. This self contained book requires undergraduate-level calculus of several variables and linear algebra as prerequisite. Fubini's theorem in real analysis, to be used in Stokes' theorem, has been proved earlier than Stokes' theorem so that students don't have to search elsewhere.
White noise calculus and Fock space
Obata, Nobuaki
1994-01-01
White Noise Calculus is a distribution theory on Gaussian space, proposed by T. Hida in 1975. This approach enables us to use pointwise defined creation and annihilation operators as well as the well-established theory of nuclear space.This self-contained monograph presents, for the first time, a systematic introduction to operator theory on fock space by means of white noise calculus. The goal is a comprehensive account of general expansion theory of Fock space operators and its applications. In particular,first order differential operators, Laplacians, rotation group, Fourier transform and their interrelations are discussed in detail w.r.t. harmonic analysis on Gaussian space. The mathematical formalism used here is based on distribution theory and functional analysis , prior knowledge of white noise calculus is not required.
Maxima and Minima Without Calculus.
Birnbaum, Ian
1982-01-01
Approaches to extrema that do not require calculus are presented to help free maxima/minima problems from the confines of calculus. Many students falsely suppose that these types of problems can only be dealt with through calculus, since few, if any, noncalculus examples are usually presented. (MP)
Cartan calculus on quantum Lie algebras
Schupp, P.; Watts, P.; Zumino, B.
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 au into one big algebra, the ''Cartan Calculus.''
A new approach to the Regge calculus
Porter, J.
1987-01-01
The paper develops a new approach to Regge calculus, a numerical technique used for the calculation of general relativistic spacetimes. The method is developed in an original '3 + 1' form in such a way that it can be applied to inhomogeneous spacetimes. (author)
Using Discovery in the Calculus Class
Shilgalis, Thomas W.
1975-01-01
This article shows how two discoverable theorems from elementary calculus can be presented to students in a manner that assists them in making the generalizations themselves. The theorems are the mean value theorems for derivatives and for integrals. A conjecture is suggested by pictures and then refined. (Author/KM)
Is Calculus a Failure in Cryptography?
Home; Journals; Resonance – Journal of Science Education; Volume 21; Issue 3. Is Calculus a Failure in Cryptography? P Vanchinathan. General Article Volume 21 Issue 3 March 2016 pp 239-245. Fulltext. Click here to view fulltext PDF. Permanent link: https://www.ias.ac.in/article/fulltext/reso/021/03/0239-0245. Keywords.
Full Lambek Calculus with Contraction is Undecidable
Chvalovský, Karel; Horčík, Rostislav
2016-01-01
Roč. 81, č. 2 (2016), s. 524-540 ISSN 0022-4812 R&D Projects: GA ČR GAP202/11/1632 Institutional support: RVO:67985807 Keywords : substructural logic * full Lambek calculus * contraction rule * square-increasing residuated lattice * equational theory * decidability Subject RIV: BA - General Mathematics Impact factor: 0.511, year: 2016
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 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.
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
Nielson, Hanne Riis; Nielson, Flemming; Vigo, Roberto
2013-01-01
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......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...
Functional Fractional Calculus
Das, Shantanu
2011-01-01
When a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its application to real life problems. It is extraordinary because it does not deal with 'ordinary' differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, with physical mathematic
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
Sigdel, G; Agarwal, A; Keshaw, B W
2014-01-01
Urethral calculi are rare forms of urolithiasis. Majority of the calculi are migratory from urinary bladder or upper urinary tract. Primary urethral calculi usually occur in presence of urethral stricture or diverticulum. In this article we report a case of a giant posterior urethral calculus measuring 7x3x2 cm in a 47 years old male. Patient presented with acute retention of urine which was preceded by burning micturition and dribbling of urine for one week. The calculus was pushed in to the bladder through the cystoscope and was removed by suprapubic cystolithotomy.
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.
Philip Atzemoglou
2014-12-01
Full Text Available We present a novel lambda calculus that casts the categorical approach to the study of quantum protocols into the rich and well established tradition of type theory. Our construction extends the linear typed lambda calculus with a linear negation of "trivialised" De Morgan duality. Reduction is realised through explicit substitution, based on a symmetric notion of binding of global scope, with rules acting on the entire typing judgement instead of on a specific subterm. Proofs of subject reduction, confluence, strong normalisation and consistency are provided, and the language is shown to be an internal language for dagger compact categories.
A proposition calculus in quantum mechanisms
Omnes, R.
1987-01-01
In quantum mechanics, to a set of n+1 observables A 0 , A 1 ...A n and a set of time instants, one can associate a probabilized space (X, B, P) where X is the direct product of the spectra of A 1 ...A n . The sigma-field B has a basis that is not a direct product but constructed in a well-defined order using sets in the spectra or equivalently projectors in the Hilbert space. The probability measure P on B satisfies the axioms of probability theory if some compatibility conditions, first found by R. Griffiths, are satisfied. To do so, one must use some quasi-classical Fefferman approximants for A 1 ...A n . Sets in B can be used as predicates in a proposition calculus using P, such that a proposition π 1 implies a proposition π-2, also gives probability 1 for π 2 . This is consistent with the logical axioms about implication. Here E 0 , projector on a set on the spectrum of A 0 is a common first predicate.This formalism is used to analyze the Einstein-Podolsky-Rosen gedankenexperiment and turns out not to contradict the finite-velocity propogation axiom of special relativity. Since only propositions and no measuring apparatus nor external observer have to be introduced, this theory generalizing quantum mechanics satisfies the criteria of objectivity, and remains non-separable. It turns out that, when an actual measuring apparatus is used, wave-packet reduction is the logico-mathematical operation that takes care of the measurement result as a proposition [fr
Discrete quantum gravitation in formalism of Regge calculus
Khatsimovskij, V.M.
2005-01-01
One deals with approach to the discrete quantum gravitation in terms of the Regge calculus formalism. The Regge calculus represents the general relativity theory for the Riemann varieties - the piecewise planar varieties. The Regge calculus makes use of the discrete set of variables, triangulation lengths, and contains the continuous general relativity theory serving as a limiting special case when lengths tend to zero. In terms of our approach the quantum mean values of the mentioned lengths differ from zero and 10 -33 cm Planck length and it implies the discrete structure of space-time at the mentioned scales [ru
Dai, Huanping; Micheyl, Christophe
2015-05-01
Proportion correct (Pc) is a fundamental measure of task performance in psychophysics. The maximum Pc score that can be achieved by an optimal (maximum-likelihood) observer in a given task is of both theoretical and practical importance, because it sets an upper limit on human performance. Within the framework of signal detection theory, analytical solutions for computing the maximum Pc score have been established for several common experimental paradigms under the assumption of Gaussian additive internal noise. However, as the scope of applications of psychophysical signal detection theory expands, the need is growing for psychophysicists to compute maximum Pc scores for situations involving non-Gaussian (internal or stimulus-induced) noise. In this article, we provide a general formula for computing the maximum Pc in various psychophysical experimental paradigms for arbitrary probability distributions of sensory activity. Moreover, easy-to-use MATLAB code implementing the formula is provided. Practical applications of the formula are illustrated, and its accuracy is evaluated, for two paradigms and two types of probability distributions (uniform and Gaussian). The results demonstrate that Pc scores computed using the formula remain accurate even for continuous probability distributions, as long as the conversion from continuous probability density functions to discrete probability mass functions is supported by a sufficiently high sampling resolution. We hope that the exposition in this article, and the freely available MATLAB code, facilitates calculations of maximum performance for a wider range of experimental situations, as well as explorations of the impact of different assumptions concerning internal-noise distributions on maximum performance in psychophysical experiments.
Dunne, Jacklyn; Kimergård, Andreas; Brown, Jamie; Beard, Emma; Buykx, Penny; Michie, Susan; Drummond, Colin
2018-03-25
To compare the proportion of people in England with probable alcohol dependence [Alcohol Use Disorders Identification Test (AUDIT) score ≥ 20] with those with other drinking patterns (categorized by AUDIT scores) in terms of motivation to reduce drinking and use of alcohol support resources. A combination of random probability and simple quota sampling to conduct monthly cross-sectional household computer-assisted interviews between March 2014 and August 2017. The general population in all nine regions of England. Participants in the Alcohol Toolkit Study (ATS), a monthly household survey of alcohol consumption among people aged 16 years and over in England (n = 69 826). The mean age was 47 years [standard deviation (SD) = 18.78; 95% confidence interval (CI) = 46.8-47] and 51% (n = 35 560) were female. χ 2 tests were used to investigate associations with demographic variables, motivation to quit drinking, attempts to quit drinking, general practitioner (GP) engagement and types of support accessed in the last 12 months across AUDIT risk zones. A total of 0.6% were classified as people with probable alcohol dependence (95% CI = 0.5-0.7). Motivation to quit (χ 2 = 1692.27, P AUDIT risk zone. People with probable dependence were more likely than other ATS participants to have a past-year attempt to cut down or quit (51.8%) and have received a specialist referral from their GP about drinking (13.7%), and less likely to report no motivation to reduce their drinking (26.2%). Those with probable dependence had higher use of self-help books and mobile applications (apps) than other ATS participants; however, 27.7% did not access any resources during their most recent attempt to cut down. Adults in England with probable alcohol dependence, measured through the Alcohol Use Disorders Identification Test, demonstrate higher motivation to quit drinking and greater use of both specialist treatment and self-driven support compared with those in other
Quantum Probability Zero-One Law for Sequential Terminal Events
Rehder, Wulf
1980-07-01
On the basis of the Jauch-Piron quantum probability calculus a zero-one law for sequential terminal events is proven, and the significance of certain crucial axioms in the quantum probability calculus is discussed. The result shows that the Jauch-Piron set of axioms is appropriate for the non-Boolean algebra of sequential events.
IAS Admin
Sphere–Cylinder Theorem, vol- ume and surface area of the torus, volume and surface area of a slice of a solid sphere. The author earned his PhD degree in mathematics. (topology), in 2000, from. Panjab University,. Chandigarh and since then he has been teaching analysis, algebra, calculus and discrete mathematics at.
Provability Calculus of Constructions
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...
Calculus Courses' Assessment Data
Pauna, Matti
2017-01-01
In this paper we describe computer-aided assessment methods used in online Calculus courses and the data they produce. The online learning environment collects a lot of time-stamped data about every action a student makes. Assessment data can be harnessed into use as a feedback, predictor, and recommendation facility for students and instructors.…
Duration Calculus: Logical Foundations
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...
Giant vesical calculus. A case report. H. H. LAUBSCHER. Summary. An exceptional case of bladder stone is presented. The case is unusual as regards the size of the stone and the fact that the patient did··not seek medical assistance much earlier, as this was readily avail- able. Furthermore, recovery after removal of the.
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...
Yilmaz, Ferkan
2012-07-01
Analysis of the average binary error probabilities (ABEP) and average capacity (AC) of wireless communications systems over generalized fading channels have been considered separately in past years. This paper introduces a novel moment generating function (MGF)-based unified expression for the ABEP and AC of single and multiple link communications with maximal ratio combining. In addition, this paper proposes the hyper-Fox\\'s H fading model as a unified fading distribution of a majority of the well-known generalized fading environments. As such, the authors offer a generic unified performance expression that can be easily calculated, and that is applicable to a wide variety of fading scenarios. The mathematical formulism is illustrated with some selected numerical examples that validate the correctness of the authors\\' newly derived results. © 1972-2012 IEEE.
Yilmaz, Ferkan
2014-04-01
The main idea in the moment generating function (MGF) approach is to alternatively express the conditional bit error probability (BEP) in a desired exponential form so that possibly multi-fold performance averaging is readily converted into a computationally efficient single-fold averaging - sometimes into a closed-form - by means of using the MGF of the signal-to-noise ratio. However, as presented in [1] and specifically indicated in [2] and also to the best of our knowledge, there does not exist an MGF-based approach in the literature to represent Wojnar\\'s generic BEP expression in a desired exponential form. This paper presents novel MGF-based expressions for calculating the average BEP of binary signalling over generalized fading channels, specifically by expressing Wojnar\\'s generic BEP expression in a desirable exponential form. We also propose MGF-based expressions to explore the amount of dispersion in the BEP for binary signalling over generalized fading channels.
Toward lattice fractional vector calculus
Tarasov, Vasily E
2014-01-01
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. (papers)
Zhou Sumin; Das, Shiva; Wang Zhiheng; Marks, Lawrence B.
2004-01-01
The generalized equivalent uniform dose (GEUD) model uses a power-law formalism, where the outcome is related to the dose via a power law. We herein investigate the mathematical compatibility between this GEUD model and the Poisson statistics based tumor control probability (TCP) model. The GEUD and TCP formulations are combined and subjected to a compatibility constraint equation. This compatibility constraint equates tumor control probability from the original heterogeneous target dose distribution to that from the homogeneous dose from the GEUD formalism. It is shown that this constraint equation possesses a unique, analytical closed-form solution which relates radiation dose to the tumor cell survival fraction. It is further demonstrated that, when there is no positive threshold or finite critical dose in the tumor response to radiation, this relationship is not bounded within the realistic cell survival limits of 0%-100%. Thus, the GEUD and TCP formalisms are, in general, mathematically inconsistent. However, when a threshold dose or finite critical dose exists in the tumor response to radiation, there is a unique mathematical solution for the tumor cell survival fraction that allows the GEUD and TCP formalisms to coexist, provided that all portions of the tumor are confined within certain specific dose ranges
Nuclear data uncertainties: I, Basic concepts of probability
Smith, D.L.
1988-12-01
Some basic concepts of probability theory are presented from a nuclear-data perspective, in order to provide a foundation for thorough understanding of the role of uncertainties in nuclear data research. Topics included in this report are: events, event spaces, calculus of events, randomness, random variables, random-variable distributions, intuitive and axiomatic probability, calculus of probability, conditional probability and independence, probability distributions, binomial and multinomial probability, Poisson and interval probability, normal probability, the relationships existing between these probability laws, and Bayes' theorem. This treatment emphasizes the practical application of basic mathematical concepts to nuclear data research, and it includes numerous simple examples. 34 refs.
Nuclear data uncertainties: I, Basic concepts of probability
Smith, D.L.
1988-12-01
Some basic concepts of probability theory are presented from a nuclear-data perspective, in order to provide a foundation for thorough understanding of the role of uncertainties in nuclear data research. Topics included in this report are: events, event spaces, calculus of events, randomness, random variables, random-variable distributions, intuitive and axiomatic probability, calculus of probability, conditional probability and independence, probability distributions, binomial and multinomial probability, Poisson and interval probability, normal probability, the relationships existing between these probability laws, and Bayes' theorem. This treatment emphasizes the practical application of basic mathematical concepts to nuclear data research, and it includes numerous simple examples. 34 refs
Feynman path integral in area tensor Regge calculus and positivity
Khatsymovsky, V.M.
2004-01-01
The versions of quantum measure in the area tensor Regge calculus constructed in the previous paper are studied on the simplest configurations of the system. These are found to be positively defined in the Euclidean case on physical surface corresponding to the ordinary Regge calculus (but not outside this surface), that is, adopt probabilistic interpretation. (Since Euclidean measure is defined via analytical continuation, positivity is not evident property.) An argument for positivity on physical surface on general configurations of area tensor Regge calculus is given
Conditional Independence in Applied Probability.
Pfeiffer, Paul E.
This material assumes the user has the background provided by a good undergraduate course in applied probability. It is felt that introductory courses in calculus, linear algebra, and perhaps some differential equations should provide the requisite experience and proficiency with mathematical concepts, notation, and argument. The document is…
Feinsilver, Philip; Schott, Rene
2009-01-01
We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated Heisenberg-Weyl algebra is called a TAA algebra after Tekin, Aydin and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the Heisenberg-Weyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, including a review of Krawtchouk polynomials that illustrates some curious properties of the Heisenberg-Weyl algebra, as well as presenting an approach to computing Krawtchouk expansions. From a mathematical perspective, we are providing indications as to how to implement infinite terms Rota's 'finite operator calculus'.
Multivariate calculus and geometry
Dineen, Seán
2014-01-01
Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations. In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions. Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.
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.
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.
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
Tall, David
1985-01-01
A number of significant changes have have occurred recently that give us a golden opportunity to review the teaching of calculus. The most obvious is the arrival of the microcomputer in the mathematics classroom, allowing graphic demonstrations and individual investigations into the mathematical ideas. But equally potent are new\\ud insights into mathematics and mathematics education that suggest new ways of approaching the subject.\\ud In this article I shall consider some of the difficulties ...
Roman, Steven
2005-01-01
Geared toward upper-level undergraduates and graduate students, this elementary introduction to classical umbral calculus requires only an acquaintance with the basic notions of algebra and a bit of applied mathematics (such as differential equations) to help put the theory in mathematical perspective. Subjects include Sheffer sequences and operators and their adjoints, with numerous examples of associated and other sequences. Related topics encompass the connection constants problem and duplication formulas, the Lagrange inversion formula, operational formulas, inverse relations, and binomial
Scale calculus and the Schroedinger equation
Cresson, Jacky
2003-01-01
This paper is twofold. In a first part, we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus. The scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f(t,ε) such that f(t,ε)→f(t) when ε goes to zero. This led to several new notions as representations: fractal functions and ε-differentiability. The basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated with them. Finally, we discuss a convenient geometric object associated with our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativity theory developed by Nottale. We obtain a nonlinear Schroedinger equation via the classical Newton's equation of dynamics using the scale operator. Under special assumptions we recover the classical Schroedinger equation and we discuss the relevance of these assumptions
Length expectation values in quantum Regge calculus
Khatsymovsky, V.M.
2004-01-01
Regge calculus configuration superspace can be embedded into a more general superspace where the length of any edge is defined ambiguously depending on the 4-tetrahedron containing the edge. Moreover, the latter superspace can be extended further so that even edge lengths in each the 4-tetrahedron are not defined, only area tensors of the 2-faces in it are. We make use of our previous result concerning quantization of the area tensor Regge calculus which gives finite expectation values for areas. Also our result is used showing that quantum measure in the Regge calculus can be uniquely fixed once we know quantum measure on (the space of the functionals on) the superspace of the theory with ambiguously defined edge lengths. We find that in this framework quantization of the usual Regge calculus is defined up to a parameter. The theory may possess nonzero (of the order of Planck scale) or zero length expectation values depending on whether this parameter is larger or smaller than a certain value. Vanishing length expectation values means that the theory is becoming continuous, here dynamically in the originally discrete framework
Contributions to quantum probability
Fritz, Tobias
2010-01-01
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels ''possible to occur'' or ''impossible to occur'' to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome
Contributions to quantum probability
Fritz, Tobias
2010-06-25
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels ''possible to occur'' or ''impossible to occur'' to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a
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...
Differential Calculus on Quantum Spheres
Welk, Martin
1998-01-01
We study covariant differential calculus on the quantum spheres S_q^2N-1. Two classification results for covariant first order differential calculi are proved. As an important step towards a description of the noncommutative geometry of the quantum spheres, a framework of covariant differential calculus is established, including a particular first order calculus obtained by factorization, higher order calculi and a symmetry concept.
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
Ivar Alberto Martins Hartmann
2016-01-01
Full Text Available The Brazilian equivalent of the Attorney General, as a brief historical analysis shows, plays an essential role in concentrated constitutional review. However, empirical studies of its performance remain scarce. Our goal is to test the AGs track record in Direct Actions of Unconstitutionality (ADIs at the Brazilian Supreme Court. We use a data set obtained from the Supreme Court in Numbers project’s database in order to run regressions with different models. The hypothesis we tested is that there is no statistically significant relationship between the performance of the AG as plaintiff in concentrated constitutional review and the outcome of the cases. This hypothesis was found to be disproved: ADIs started by the AG, as well as those proposed by representatives of the Executive branch, have, in fact, greater statistical probability of success.
Parvaneh Mohammadkhani
2015-03-01
Full Text Available The aim of this study was to evaluate suicide probability in Iranian males with substance abuse or dependence disorder and to investigate the predictors of suicide probability based on trait mindfulness, reasons for living and severity of general psychiatric symptoms.Participants were 324 individuals with substance abuse or dependence in an outpatient setting and prison. Reasons for living questionnaire, Mindfulness Attention Awareness Scale and Suicide probability Scale were used as instruments. Sample was selected based on convenience sampling method. Data were analyzed using SPSS and AMOS.The life-time prevalence of suicide attempt in the outpatient setting was35% and it was 42% in the prison setting. Suicide probability in the prison setting was significantly higher than in the outpatient setting (p<0.001. The severity of general symptom strongly correlated with suicide probability. Trait mindfulness, not reasons for living beliefs, had a mediating effect in the relationship between the severity of general symptoms and suicide probability. Fear of social disapproval, survival and coping beliefs and child-related concerns significantly predicted suicide probability (p<0.001.It could be suggested that trait mindfulness was more effective in preventing suicide probability than beliefs about reasons for living in individuals with substance abuse or dependence disorders. The severity of general symptom should be regarded as an important risk factor of suicide probability.
Three-plus-one formulation of Regge calculus
Piran, T.; Williams, R.M.
1986-01-01
Following the work of Lund and Regge for homogeneous spaces, we construct the action for Regge calculus in its three-plus-one form for general space-times. This is achieved in two ways: a first-order formalism and a second-order formalism. We describe the Regge-calculus analogue of solving the initial-value equations using conformal transformations. The second-order formalism is used to study the time development of two simple model universes
Fisher information, Borges operators, and q-calculus
Pennini, F.; Plastino, A.; Ferri, G. L.
2008-10-01
We discuss applying the increasingly popular q-calculus, or deformed calculus, so as to suitably generalize Fisher’s information measure and the Cramer-Rao inequality. A q-deformation can be attained in multiple ways, and we show that most of them do not constitute legitimate procedures. Within such a context, the only completely acceptable q-deformation is that ensuing from using the so-called Borges derivative [E.P. Borges, Physica A 340 (2004) 95].
DiBonaventura MD
2017-11-01
Full Text Available Marco D DiBonaventura,1 Alesia Sadosky,2 Kristen Concialdi,1 Markay Hopps,2 Ian Kudel,1 Bruce Parsons,2 Joseph C Cappelleri,3 Patrick Hlavacek,2 Andrea H Alexander,2 Brett R Stacey,4 John D Markman,5 John T Farrar6 1Health Outcomes Practice, Kantar Health, 2Pfizer Inc, New York, NY, 3Pfizer Inc, Groton, CT, 4University of Washington, Seattle, WA, 5University of Rochester School of Medicine and Dentistry, Rochester, NY, 6University of Pennsylvania Perelman School of Medicine, Philadelphia, PA, US Background: The prevalence of neuropathic pain (NeP has been estimated within specific health conditions; however, there are no published data on its broad prevalence in the US. The current exploratory study addresses this gap using the validated PainDetect questionnaire as a screener for probable NeP in a general-population health survey conducted with a multimodal recruitment strategy to maximize demographic representativeness. Materials and methods: Adult respondents were recruited from a combination of Internet panels, telephone lists, address lists, mall-based interviews, and store-receipt invitations using a random stratified-sampling framework, with strata defined by age, sex, and race/ethnicity. Older persons and minorities were oversampled to improve prevalence estimates. Results were weighted to match the total adult US population using US Census data. Demographic information was collected, and respondents who experienced physical pain in the past 12 months completed the PainDetect and provided additional pain history. A cutoff score of 19 or greater on the PainDetect was used to define probable NeP. Results: A total of 24,925 respondents (average response rate 2.5% provided demographic data (52.2% female, mean age 51.5 years; 15,751 respondents reported pain (63.7%, of which 2,548 (15.7%, 95% confidence interval 14.9%–16.5% had probable NeP based on the PainDetect, which was 10% (95% confidence interval 9.5%–10.5% of all respondents. Among
Introduction to probability theory with contemporary applications
Helms, Lester L
2010-01-01
This introduction to probability theory transforms a highly abstract subject into a series of coherent concepts. Its extensive discussions and clear examples, written in plain language, expose students to the rules and methods of probability. Suitable for an introductory probability course, this volume requires abstract and conceptual thinking skills and a background in calculus.Topics include classical probability, set theory, axioms, probability functions, random and independent random variables, expected values, and covariance and correlations. Additional subjects include stochastic process
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
The Probabilities of Unique Events
2012-08-30
Washington, DC USA Max Lotstein and Phil Johnson-Laird Department of Psychology Princeton University Princeton, NJ USA August 30th 2012...social justice and also participated in antinuclear demonstrations. The participants ranked the probability that Linda is a feminist bank teller as...retorted that such a flagrant violation of the probability calculus was a result of a psychological experiment that obscured the rationality of the
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.)
Independent variables in 3 + 1 Regge calculus
Tuckey, P.A.
1989-01-01
The space of metrics in 3+1 Regge calculus is discussed, and the problems of counting its dimensions, and of finding independent variables to parametrise the space, are addressed. The most general natural class of metrics is considered first, and bounds on its dimension are obtained, although no good parametrisations are found. The relationship between these metrics and those used in canonical Regge calculus is shown, and this leads to an interesting result via the Bianchi identities. A restricted class of metrics is then considered and independent variables, which parametrise these metrics and which may be computationally convenient, are given. The dimension of this space of metrics gives an improved lower bound for the dimension of the general space. (author)
Differential calculus on quantized simple Lie groups
Jurco, B.
1991-01-01
Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q ε R are also discussed. (orig.)
Time-evolution problem in Regge calculus
Sorkin, R.
1975-01-01
The simplectic approximation to Einstein's equations (''Regge calculus'') is derived by considering the net to be actually a (singular) Riemannian manifold. Specific nets for open and closed spaces are introduced in terms of which one can formulate the general time-evolution problem, which thereby reduces to the repeated solution of finite sets of coupled nonlinear (algebraic) equations. The initial-value problem is also formulated in simplectic terms
Neutrix calculus and finite quantum field theory
Ng, Y Jack; Dam, H van
2005-01-01
In general, quantum field theories (QFT) require regularizations and infinite renormalizations due to ultraviolet divergences in their loop calculations. Furthermore, perturbation series in theories like quantum electrodynamics are not convergent series, but are asymptotic series. We apply neutrix calculus, developed in connection with asymptotic series and divergent integrals, to QFT, obtaining finite renormalizations. While none of the physically measurable results in renormalizable QFT is changed, quantum gravity is rendered more manageable in the neutrix framework. (letter to the editor)
Geometrical foundations of tensor calculus and relativity
Schuller , Frédéric; Lorent , Vincent
2006-01-01
Manifolds, particularly space curves: basic notions 1 The first groundform, the covariant metric tensor 11 The second groundform, Meusnier's theorem 19 Transformation groups in the plane 28 Co- and contravariant components for a special affine transformation in the plane 29 Surface vectors 32 Elements of tensor calculus 36 Generalization of the first groundform to the space 46 The covariant (absolute) derivation 57 Examples from elasticity theory 61 Geodesic lines 63 Main curvatur...
Proof Nets for Lambek Calculus
Roorda, Dirk
1992-01-01
The proof nets of linear logic are adapted to the non-commutative Lambek calculus. A different criterion for soundness of proof nets is given, which gives rise to new algorithms for proof search. The order sensitiveness of the Lambek calculus is reflected by the planarity condition on proof nets;
Fluorescence spectroscopy of dental calculus
Bakhmutov, D; Gonchukov, S; Sukhinina, A
2010-01-01
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
Fluorescence spectroscopy of dental calculus
Bakhmutov, D.; Gonchukov, S.; Sukhinina, A.
2010-05-01
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…
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
Ernst, Erik; Ostermann, Klaus; Cook, William Randall
2006-01-01
Virtual classes are class-valued attributes of objects. Like virtual methods, virtual classes are defined in an object's class and may be redefined within subclasses. They resemble inner classes, which are also defined within a class, but virtual classes are accessed through object instances...... model for virtual classes has been a long-standing open question. This paper presents a virtual class calculus, vc, that captures the essence of virtual classes in these full-fledged programming languages. The key contributions of the paper are a formalization of the dynamic and static semantics of vc...
Calculus problems and solutions
Ginzburg, Abraham
2011-01-01
Ideal for self-instruction as well as for classroom use, this text helps students improve their understanding and problem-solving skills in analysis, analytic geometry, and higher algebra. More than 1,200 problems appear in the text, with concise explanations of the basic notions and theorems to be used in their solution. Many are followed by complete answers; solutions for the others appear at the end of the book. Topics include sequences, functions of a single variable, limit of a function, differential calculus for functions of a single variable, fundamental theorems and applications of dif
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
Elsgolc, Lev D
2007-01-01
This concise text offers both professionals and students an introduction to the fundamentals and standard methods of the calculus of variations. In addition to surveys of problems with fixed and movable boundaries, it explores highly practical direct methods for the solution of variational problems.Topics include the method of variation in problems with fixed boundaries; variational problems with movable boundaries and other problems; sufficiency conditions for an extremum; variational problems of constrained extrema; and direct methods of solving variational problems. Each chapter features nu
Extended finite operator calculus as an example of algebraization of analysis
Kwasniewski, A. K.; Borak, E.
2004-01-01
A wardian calculus of sequences started almost seventy years ago constitutes the general scheme for extensions of the classical umbral operator calculus considered by many afterwards . At the same time this calculus is an example of the algebraization of the analysis here restricted to the algebra of formal series. This is a review article based on the recent first author contributions. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by p...
Rethinking the Privacy Calculus: On the Role of Dispositional Factors and Affect
Kehr, Flavius; Wentzel, Daniel; Mayer, Peter
2013-01-01
Existing research on information privacy has mostly relied on the privacy calculus model which views privacy-related decision making as a rational process where individuals weigh the anticipated risks of disclosing personal data against the potential benefits. However, scholars have recently challenged two basic propositions of the privacy calculus model. First, some authors have distinguished between general and situational factors in the context of privacy calculus and have argued that ...
The impacts of gingivitis and calculus on Thai children's quality of life.
Krisdapong, Sudaduang; Prasertsom, Piyada; Rattanarangsima, Khanit; Sheiham, Aubrey; Tsakos, Georgios
2012-09-01
To assess associations of socio-demographic, behavioural and the extent of gingivitis and calculus with oral health-related quality of life (OHRQoL) in nationally representative samples of 12- and 15-year-old Thai children. In the Thailand National Oral Health Survey, 1,063 twelve-year olds and 811 fifteen-year olds were clinically examined and interviewed for OHRQoL using the Child-OIDP and OIDP indices, respectively, and completed a behavioural questionnaire. We assessed associations of condition-specific impacts (CS-impacts) with gingivitis and calculus, adjusted for socio-demographic and behavioural factors. Gingivitis and calculus were highly prevalent: 79.3% in 12-year and 81.5% in 15-year olds. CS-impacts relating to calculus and/or gingivitis were reported by 26.0% of 12-year and 29.6% of 15-year olds. Except for calculus without gingivitis, calculus and/or gingivitis in any form was significantly related to any level of CS-impacts. At a moderate or higher level of CS-impacts, there were significant relationships with extensive calculus and/or gingivitis in 12-year olds and for extensive gingivitis and gingivitis without calculus in 15-year olds. Gingivitis was generally associated with any level of CS-impacts attributed to calculus and/or gingivitis. CS-impacts were related more to gingivitis than to calculus. © 2012 John Wiley & Sons A/S.
Rached, Nadhir B.
2016-12-24
In this paper, we develop a generalized momentbased approach for the evaluation of the outage probability (OP) in the presence of co-channel interference and additive white Gaussian noise. The proposed method allows the evaluation of the OP of the signal-to-interference-plus-noise ratio by a power series expansion in the threshold value. Its main advantage is that it does not require a particular distribution for the interference channels. The only necessary ingredients are a power series expansion for the cumulative distribution function of the desired user power and the cross-moments of the interferers\\' powers. These requirements are easily met in many practical fading models, for which the OP might not be obtained in closed-form expression. For a sake of illustration, we consider the application of our method to the Rician fading environment. Under this setting, we carry out a convergence study of the proposed power series and corroborate the validity of our method for different values of fading parameters and various numbers of co-channel interferers.
On the interpretation of Stratonovich calculus
Moon, W; Wettlaufer, J S
2014-01-01
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
Lewis, S.M.
1982-01-01
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
Probability elements of the mathematical theory
Heathcote, C R
2000-01-01
Designed for students studying mathematical statistics and probability after completing a course in calculus and real variables, this text deals with basic notions of probability spaces, random variables, distribution functions and generating functions, as well as joint distributions and the convergence properties of sequences of random variables. Includes worked examples and over 250 exercises with solutions.
Leveraging Prior Calculus Study with Embedded Review
Nikolov, Margaret C.; Withers, Wm. Douglas
2016-01-01
We propose a new course structure to address the needs of college students with previous calculus study but no course validations as an alternative to repeating the first year of calculus. Students are introduced directly to topics from Calculus III unpreceded by a formal review of topics from Calculus I or II, but with additional syllabus time…
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.
Kok, de A.G.
2003-01-01
In this paper we present fast and accurate approximations for the probability of ruin over a finite number of periods, assuming inhomogeneous independent claim size distributions and arbitrary premium income in subsequent periods. We develop exact recursive expressions for the non-ruin probabilities
Fractional Order Generalized Information
José Tenreiro Machado
2014-04-01
Full Text Available This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization.
The calculus a genetic approach
Toeplitz, Otto
2007-01-01
When first published posthumously in 1963, this book presented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique a
Fluorescence detection of dental calculus
Gonchukov, S.; Biryukova, T.; Sukhinina, A.; Vdovin, Yu
2010-11-01
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.
The Vectorial $\\lambda$-Calculus
Arrighi, Pablo; Díaz-Caro, Alejandro; Valiron, Benoît
2013-01-01
We describe a type system for the linear-algebraic $\\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\\lambda$-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We prove that the resulting typed $\\lambda$-calculus is strongly normalising and fea...
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
Stochastic calculus an introduction through theory and exercises
Baldi, Paolo
2017-01-01
This book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications. It is the only textbook on the subject to include more than two hundred exercises with complete solutions. After explaining the basic elements of probability, the author introduces more advanced topics such as Brownian motion, martingales and Markov processes. The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance. The final chapter provides detailed solutions to all exercises, in some cases presenting various solution techniques together with a discussion of advantages and drawbacks of the methods used. Stochastic Calculus will be particularly useful to advanced undergraduate and graduate students wishing to acquire a solid understanding of the subject through the theory and exercises. Including full mathematical ...
Waste pipe calculus extensions
O'Connell, W.J.
1979-01-01
The waste pipe calculus provides a rapid method, using Laplace transforms, to calculate the transport of a pollutant such as nuclear waste, by a network of one-dimensional flow paths. The present note extends previous work as follows: (1) It provides an alternate approximation to the time-domain function (inverse Laplace transform) for the resulting transport. This algebraic approximation may be viewed as a simpler and more approximate model of the transport process. (2) It identifies two scalar quantities which may be used as summary consequence measures of the waste transport (or inversely, waste retention) system, and provides algebraic expressions for them. (3) It includes the effects of radioactive decay on the scalar quantity results, and further provides simplifying approximations for the cases of medium and long half-lives. This algebraic method can be used for quick approximate analyses of expected results, uncertainty and sensitivity, in evaluating selection and design choices for nuclear waste disposal systems
Lei, Qian
2017-01-01
This book offers a comprehensive and systematic review of the latest research findings in the area of intuitionistic fuzzy calculus. After introducing the intuitionistic fuzzy numbers’ operational laws and their geometrical and algebraic properties, the book defines the concept of intuitionistic fuzzy functions and presents the research on the derivative, differential, indefinite integral and definite integral of intuitionistic fuzzy functions. It also discusses some of the methods that have been successfully used to deal with continuous intuitionistic fuzzy information or data, which are different from the previous aggregation operators focusing on discrete information or data. Mainly intended for engineers and researchers in the fields of fuzzy mathematics, operations research, information science and management science, this book is also a valuable textbook for postgraduate and advanced undergraduate students alike.
Smirnov, Vladimir A
2006-01-01
The problem of evaluating Feynman integrals over loop momenta has existed from the early days of perturbative quantum field theory. The goal of the book is to summarize those methods for evaluating Feynman integrals that have been developed over a span of more than fifty years. `Feynman Integral Calculus' characterizes the most powerful methods in a systematic way. It concentrates on the methods that have been employed recently for most sophisticated calculations and illustrates them with numerous examples, starting from very simple ones and progressing to nontrivial examples. It also shows how to choose adequate methods and combine them in a non-trivial way. This is a textbook version of the previous book (Evaluating Feynman integrals, STMP 211) of the author. Problems and solutions have been included, Appendix G has been added, more details have been presented, recent publications on evaluating Feynman integrals have been taken into account and the bibliography has been updated.
Mestres-Missé, Anna; Trampel, Robert; Turner, Robert; Kotz, Sonja A
2016-04-01
A key aspect of optimal behavior is the ability to predict what will come next. To achieve this, we must have a fairly good idea of the probability of occurrence of possible outcomes. This is based both on prior knowledge about a particular or similar situation and on immediately relevant new information. One question that arises is: when considering converging prior probability and external evidence, is the most probable outcome selected or does the brain represent degrees of uncertainty, even highly improbable ones? Using functional magnetic resonance imaging, the current study explored these possibilities by contrasting words that differ in their probability of occurrence, namely, unbalanced ambiguous words and unambiguous words. Unbalanced ambiguous words have a strong frequency-based bias towards one meaning, while unambiguous words have only one meaning. The current results reveal larger activation in lateral prefrontal and insular cortices in response to dominant ambiguous compared to unambiguous words even when prior and contextual information biases one interpretation only. These results suggest a probability distribution, whereby all outcomes and their associated probabilities of occurrence--even if very low--are represented and maintained.
Regge calculus and observations. II. Further applications
Williams, R.M.; Ellis, G.F.R.
1983-03-01
The method, developed in an earlier paper, for tracing geodesics of particles and light rays through Regge calculus space-times, is applied to a number of problems in the Schwarschild geometry. It is possible to obtain accurate predictions of light-bending by taking sufficiently small Regge blocks. Calculations of perihelion precession, Thomas precession and the distortion of a ball of fluid moving on a geodesic can also show good agreement with the analytic solution. However difficulties arise in obtaining accurate predictions for general orbits in these space-times. Applications to other problems in general relativity are discussed briefly. (author)
Regge calculus and observations. II. Further applications.
Williams, Ruth M.; Ellis, G. F. R.
1984-11-01
The method, developed in an earlier paper, for tracing geodesies of particles and light rays through Regge calculus space-times, is applied to a number of problems in the Schwarzschild geometry. It is possible to obtain accurate predictions of light bending by taking sufficiently small Regge blocks. Calculations of perihelion precession, Thomas precession, and the distortion of a ball of fluid moving on a geodesic can also show good agreement with the analytic solution. However difficulties arise in obtaining accurate predictions for general orbits in these space-times. Applications to other problems in general relativity are discussed briefly.
A Formal Calculus for Categories
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...
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.
Differential calculus and its applications
Field, Michael J
2013-01-01
Based on undergraduate courses in advanced calculus, the treatment covers a wide range of topics, from soft functional analysis and finite-dimensional linear algebra to differential equations on submanifolds of Euclidean space. 1976 edition.
Catwalk: First-Semester Calculus.
Speiser, Bob; Walter, Chuck
1994-01-01
Describes the use of time-lapse photographs of a running cat as a model to investigate the concepts of function and derivative in a college calculus course. Discusses student difficulties and implications for teachers. (MKR)
Making an Interactive Calculus Textbook.
Larson, Timothy R.
1995-01-01
Presents a case study of the design and production of "Interactive Calculus," an interactive multimedia textbook. Discusses reasons for using multimedia textbooks; what an interactive textbook is; content, organization, graphic design, authoring and composition; and work flow. (AEF)
Quantum Regge calculus in the Lorentzian domain and its Hamiltonian formulation
Williams, R.M.; Cambridge Univ.
1986-01-01
A formalism is set up for quantum Regge calculus in the Lorentzian domain, calculating the inverse propagator in the free field case. The variables in the Arnowitt-Deser-Misner [1962, Gravitation, an Introduction to Current Research, ed. L. Witten (New York: Wiley) p 227] 3 + 1 formulation of general relativity are related to the Regge calculus variables. (author)
Affine connection form of Regge calculus
Khatsymovsky, V. M.
2016-12-01
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 three-simplices which play the 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 the four-simplices). Excluding GL(4, R) connection from this action via the equations of motion we have exactly the Regge action for the considered spacetime.
Covariant differential calculus on the quantum exterior vector space
Parashar, P.; Soni, S.K.
1992-01-01
We formulate a differential calculus on the quantum exterior vector space spanned by the generators of a non-anticommutative algebra satisfying r ij = θ i θ j +B kl ij θ k θ l =0 i, j=1, 2, ..., n. and (θ i ) 2 =(θ j ) 2 =...=(θ n ) 2 =0, where B kl ij is the most general matrix defined in terms of complex deformation parameters. Following considerations analogous to those of Wess and Zumino, we are able to exhibit covariance of our calculus under ( 2 n )+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 dr ij ≅0 and the 'quadratic' condition r ij x n ≅0 where x n =dθ n are the differentials of the variables. (orig.)
Equations involving Malliavin calculus operators applications and numerical approximation
Levajković, Tijana
2017-01-01
This book provides a comprehensive and unified introduction to stochastic differential equations and related optimal control problems. The material is new and the presentation is reader-friendly. A major contribution of the book is the development of generalized Malliavin calculus in the framework of white noise analysis, based on chaos expansion representation of stochastic processes and its application for solving several classes of stochastic differential equations with singular data involving the main operators of Malliavin calculus. In addition, applications in optimal control and numerical approximations are discussed. The book is divided into four chapters. The first, entitled White Noise Analysis and Chaos Expansions, includes notation and provides the reader with the theoretical background needed to understand the subsequent chapters. In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator, the Skorokhod integral and the Ornstein-Uhlenbeck operator are introdu...
Wilson, Jason; Lawman, Joshua; Murphy, Rachael; Nelson, Marissa
2011-01-01
This article describes a probability project used in an upper division, one-semester probability course with third-semester calculus and linear algebra prerequisites. The student learning outcome focused on developing the skills necessary for approaching project-sized math/stat application problems. These skills include appropriately defining…
Kok, de A.G.
2003-01-01
In this paper, we present fast and accurate approximations for the probability of ruin over a finite number of periods, assuming inhomogeneous independent claim size distributions and arbitrary premium income in subsequent periods. We develop exact recursive expressions for the non-ruin
Ben Issaid, Chaouki; Park, Kihong; Alouini, Mohamed-Slim
2017-01-01
When assessing the performance of the free space optical (FSO) communication systems, the outage probability encountered is generally very small, and thereby the use of nave Monte Carlo simulations becomes prohibitively expensive. To estimate these rare event probabilities, we propose in this work an importance sampling approach which is based on the exponential twisting technique to offer fast and accurate results. In fact, we consider a variety of turbulence regimes, and we investigate the outage probability of FSO communication systems, under a generalized pointing error model based on the Beckmann distribution, for both single and multihop scenarios. Selected numerical simulations are presented to show the accuracy and the efficiency of our approach compared to naive Monte Carlo.
Ben Issaid, Chaouki
2017-07-28
When assessing the performance of the free space optical (FSO) communication systems, the outage probability encountered is generally very small, and thereby the use of nave Monte Carlo simulations becomes prohibitively expensive. To estimate these rare event probabilities, we propose in this work an importance sampling approach which is based on the exponential twisting technique to offer fast and accurate results. In fact, we consider a variety of turbulence regimes, and we investigate the outage probability of FSO communication systems, under a generalized pointing error model based on the Beckmann distribution, for both single and multihop scenarios. Selected numerical simulations are presented to show the accuracy and the efficiency of our approach compared to naive Monte Carlo.
Yang Yang
2011-01-01
Full Text Available We propose a general continuous-time risk model with a constant interest rate. In this model, claims arrive according to an arbitrary counting process, while their sizes have dominantly varying tails and fulfill an extended negative dependence structure. We obtain an asymptotic formula for the finite-time ruin probability, which extends a corresponding result of Wang (2008.
Differential calculus on quantized simple Lie groups
Jurco, B. (Dept. of Optics, Palacky Univ., Olomouc (Czechoslovakia))
1991-07-01
Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU{sub q}(2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q {epsilon} R are also discussed. (orig.).
Sequent Calculus Representations for Quantum Circuits
Cameron Beebe
2016-06-01
Full Text Available When considering a sequent-style proof system for quantum programs, there are certain elements of quantum mechanics that we may wish to capture, such as phase, dynamics of unitary transformations, and measurement probabilities. Traditional quantum logics which focus primarily on the abstract orthomodular lattice theory and structures of Hilbert spaces have not satisfactorily captured some of these elements. We can start from 'scratch' in an attempt to conceptually characterize the types of proof rules which should be in a system that represents elements necessary for quantum algorithms. This present work attempts to do this from the perspective of the quantum circuit model of quantum computation. A sequent calculus based on single quantum circuits is suggested, and its ability to incorporate important conceptual and dynamic aspects of quantum computing is discussed. In particular, preserving the representation of phase helps illustrate the role of interference as a resource in quantum computation. Interference also provides an intuitive basis for a non-monotonic calculus.
Backpropagation and ordered derivatives in the time scales calculus.
Seiffertt, John; Wunsch, Donald C
2010-08-01
Backpropagation is the most widely used neural network learning technique. It is based on the mathematical notion of an ordered derivative. In this paper, we present a formulation of ordered derivatives and the backpropagation training algorithm using the important emerging area of mathematics known as the time scales calculus. This calculus, with its potential for application to a wide variety of inter-disciplinary problems, is becoming a key area of mathematics. It is capable of unifying continuous and discrete analysis within one coherent theoretical framework. Using this calculus, we present here a generalization of backpropagation which is appropriate for cases beyond the specifically continuous or discrete. We develop a new multivariate chain rule of this calculus, define ordered derivatives on time scales, prove a key theorem about them, and derive the backpropagation weight update equations for a feedforward multilayer neural network architecture. By drawing together the time scales calculus and the area of neural network learning, we present the first connection of two major fields of research.
Differential calculus on deformed E(2) group
Giller, S.; Gonera, C.; Kosinski, P.; Maslanka, P.
1997-01-01
Four dimensional bi-covariant differential *-calculus on quantum E(2) group is constructed. The relevant Lie algebra is obtained and covariant differential calculus on quantum plane is found. (author)
Stochastic Model Checking of the Stochastic Quality Calculus
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....... This gives rise to Generalised Semi-Markov Decision Processes for which few analytical techniques are available. We restrict delays on output actions to be exponentially distributed while still admitting real-time constraints on the quality binders. This facilitates developing analytical techniques based...
Three qubit entanglement within graphical Z/X-calculus
Bob Coecke
2011-03-01
Full Text Available The compositional techniques of categorical quantum mechanics are applied to analyse 3-qubit quantum entanglement. In particular the graphical calculus of complementary observables and corresponding phases due to Duncan and one of the authors is used to construct representative members of the two genuinely tripartite SLOCC classes of 3-qubit entangled states, GHZ and W. This nicely illustrates the respectively pairwise and global tripartite entanglement found in the W- and GHZ-class states. A new concept of supplementarity allows us to characterise inhabitants of the W class within the abstract diagrammatic calculus; these method extends to more general multipartite qubit states.
Probability, Statistics, and Stochastic Processes
Olofsson, Peter
2011-01-01
A mathematical and intuitive approach to probability, statistics, and stochastic processes This textbook provides a unique, balanced approach to probability, statistics, and stochastic processes. Readers gain a solid foundation in all three fields that serves as a stepping stone to more advanced investigations into each area. This text combines a rigorous, calculus-based development of theory with a more intuitive approach that appeals to readers' sense of reason and logic, an approach developed through the author's many years of classroom experience. The text begins with three chapters that d
The untyped stack calculus and Bohm's theorem
Alberto Carraro
2013-03-01
Full Text Available 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.
The development and nature of problem-solving among first-semester calculus students
Dawkins, Paul Christian; Mendoza Epperson, James A.
2014-08-01
This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students' problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We investigate this blended instruction as a local representative of the US calculus reform movements that helped foster it. These reform movements tended to emphasize problem-solving as well as multiple mathematical registers and quantitative modelling. Our statistical analysis reveals the influence of the blended traditional/reform calculus instruction on students' ability to solve calculus-related, non-routine problems through repeated measures over the semester. The calculus instruction in this study significantly improved students' performance on non-routine problems, though performance improved more regarding strategies and accuracy than it did for drawing conclusions and providing justifications. We identified problem-solving behaviours that characterized top performance or attrition in the course. Top-performing students displayed greater algebraic proficiency, calculus skills, and more general heuristics than their peers, but overused algebraic techniques even when they proved cumbersome or inappropriate. Students who subsequently withdrew from calculus often lacked algebraic fluency and understanding of the graphical register. The majority of participants, when given a choice, relied upon less sophisticated trial-and-error approaches in the numerical register and rarely used the graphical register, contrary to the goals of US calculus reform. We provide explanations for these patterns in students' problem-solving performance in view of both their preparation for university calculus and the courses' assessment structure, which preferentially rewarded algebraic reasoning. While instruction improved students' problem
Fractional dynamic calculus and fractional dynamic equations on time scales
Georgiev, Svetlin G
2018-01-01
Pedagogically organized, this monograph introduces fractional calculus and fractional dynamic equations on time scales in relation to mathematical physics applications and problems. Beginning with the definitions of forward and backward jump operators, the book builds from Stefan Hilger’s basic theories on time scales and examines recent developments within the field of fractional calculus and fractional equations. Useful tools are provided for solving differential and integral equations as well as various problems involving special functions of mathematical physics and their extensions and generalizations in one and more variables. Much discussion is devoted to Riemann-Liouville fractional dynamic equations and Caputo fractional dynamic equations. Intended for use in the field and designed for students without an extensive mathematical background, this book is suitable for graduate courses and researchers looking for an introduction to fractional dynamic calculus and equations on time scales. .
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.
Fluorescence detection of dental calculus
Gonchukov, S; Sukhinina, A; Vdovin, Yu; Biryukova, T
2010-01-01
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
On the fractional calculus of Besicovitch function
Liang Yongshun
2009-01-01
Relationship between fractional calculus and fractal functions has been explored. Based on prior investigations dealing with certain fractal functions, fractal dimensions including Hausdorff dimension, Box dimension, K-dimension and Packing dimension is shown to be a linear function of order of fractional calculus. Both Riemann-Liouville fractional calculus and Weyl-Marchaud fractional derivative of Besicovitch function have been discussed.
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.…
Exterior calculus and two-dimensional supersymmetric models
Sciuto, S.
1980-01-01
An important property of the calculus of differential forms on superspace is pointed out, and an economical way to treat the linear problem associated with certain supersymmetric two-dimensional models is discussed. A generalization of the super sine-Gordon model is proposed; its bosonic limit is a new model whose associate linear set has an SU(3) structure. (orig.)
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…
Superconformal tensor calculus and matter couplings in six dimensions
Bergshoeff, E.; Sezgin, E.; Proeyen, A. Van
1986-01-01
Using superconformal tensor calculus we construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. We start from the superconformal algebra which we realize on a 40+40 Weyl multiplet and on several matter multiplets. A
Quantum mechanics and umbral calculus
Lopez-Sendino, J E; Negro, J; Olmo, M A del; Salgado, E
2008-01-01
In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schroedinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schroedinger equation with a potential depending only in the space variable is discussed.
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
Sequent Calculus and Equational Programming
Nicolas Guenot
2015-07-01
Full Text Available Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums.
Ball, G.
1990-01-01
The development and analysis of methods for generating first-flight collision probabilities in two-dimensional geometries consistent with Light Water Moderated (LWR) fuel assemblies are examined. A new ray-tracing algorithm is discussed. A number of numerical results are given demonstrating the feasibility of this algorithm and the effects of the moderator (and fuel) sectorizations on the resulting flux distributions. The collision probabilties have been introduced and their subsequent utilization in the flux calculation procedures illustrated. A brief description of the Coxy-1 and Coxy-2 programs (which were developed in the Reactor Theory Division of the Atomic Energy Agency of South Africa Ltd) has also been added. 41 figs., 9 tabs., 18 refs
Quantum interference of probabilities and hidden variable theories
Srinivas, M.D.
1984-01-01
One of the fundamental contributions of Louis de Broglie, which does not get cited often, has been his analysis of the basic difference between the calculus of the probabilities as predicted by quantum theory and the usual calculus of probabilities - the one employed by most mathematicians, in its standard axiomatised version due to Kolmogorov. This paper is basically devoted to a discussion of the 'quantum interference of probabilities', discovered by de Broglie. In particular, it is shown that it is this feature of the quantum theoretic probabilities which leads to some serious constraints on the possible 'hidden-variable formulations' of quantum mechanics, including the celebrated theorem of Bell. (Auth.)
Calculus in physics classes at UFRGS: an exploratory study
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.
Applying π-Calculus to Practice
Abendroth, Jorg
2003-01-01
The π-Calculus has been developed to reason about behavioural equivalence. Different notations of equivalence are defined in terms of process interactions, as well as the context of processes. There are various extensions of the π-Calculus, such as the SPI calculus, which has primitives...... modles are instantiated correctly. In this paper we will utilize the to π-Calculus reason about access control policies and mechanism. An equivalence of different policy implementations, as well as access control mechanism will be shown. Finally some experiences regarding the use of π-Calculus...
A Calculus for Trust Management
Carbone, Marco; Nielsen, Mogens; Sassone, Vladimiro
2004-01-01
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...
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.
POGIL in the Calculus Classroom
Bénéteau, Catherine; Guadarrama, Zdenka; Guerra, Jill E.; Lenz, Laurie; Lewis, Jennifer E.; Straumanis, Andrei
2017-01-01
In this paper, we will describe the experience of the authors in using process-oriented guided inquiry learning (POGIL) in calculus at four institutions across the USA. We will briefly examine how POGIL compares to and fits in with other kinds of inquiry-based learning approaches. In particular, we will first discuss the unique structure of a…
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…
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…
Constructivized Calculus in College Mathematics
Lawrence, Barbara Ann
2012-01-01
The purpose of this study is to present some of the classical concepts, definitions, and theorems of calculus from the constructivists' point of view in the spirit of the philosophies of L.E.J. Brouwer and Errett Bishop. This presentation will compare the classical statements to the constructivized statements. The method focuses on giving…
Stochastic Pi-calculus Revisited
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...
The fermion stochastic calculus I
Streater, R.F.
1984-01-01
The author describes the stochastic calculus of quantum processes with fermions. After a description of the Clifford algebra as the csup(*)-algebra generated by spinor fields the damped harmonic oscillator with quantum noise is considered as example. Then the Clifford process is described. Finally the Ito-Clifford integral and the Ito-Clifford isometry are presented. (HSI)
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…
Calculus Student Descending a Staircase.
Mueller, William
1999-01-01
Common student attitudes toward reform methods are conveyed through the thoughts of a student leaving a multivariable calculus exam and musings range over textbooks, homework, workload, group work, writing, noncomputational problems, instructional problems, instructional styles, and classroom activities. (Author/ASK)
The Pendulum and the Calculus.
Sworder, Steven C.
A pair of experiments, appropriate for the lower division fourth semester calculus or differential equations course, are presented. The second order differential equation representing the equation of motion of a simple pendulum is derived. The period of oscillation for a particular pendulum can be predicted from the solution to this equation. As a…
Reasoning about objects using process calculus techniques
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......-calculus turns out to be insufficient. Based on our experiences, we present a translation of a typed imperative Sigma-calculus, which looks promising. We are able to provide simple proofs of the equivalence of different Sigma-calculus objects using this translation. We use a labelled transition system adapted...... to the Sigma-calculus to investigate the use of process calculi techniques directly on the Sigma-calculus. The results obtained are of a fairly theoretical nature. We investigate the connection between the operational and denotaional semantics for a typed functional Sigma-calculus. The result is that Abadi...
The dental calculus metabolome in modern and historic samples.
Velsko, Irina M; Overmyer, Katherine A; Speller, Camilla; Klaus, Lauren; Collins, Matthew J; Loe, Louise; Frantz, Laurent A F; Sankaranarayanan, Krithivasan; Lewis, Cecil M; Martinez, Juan Bautista Rodriguez; Chaves, Eros; Coon, Joshua J; Larson, Greger; Warinner, Christina
2017-01-01
Dental calculus is a mineralized microbial dental plaque biofilm that forms throughout life by precipitation of salivary calcium salts. Successive cycles of dental plaque growth and calcification make it an unusually well-preserved, long-term record of host-microbial interaction in the archaeological record. Recent studies have confirmed the survival of authentic ancient DNA and proteins within historic and prehistoric dental calculus, making it a promising substrate for investigating oral microbiome evolution via direct measurement and comparison of modern and ancient specimens. We present the first comprehensive characterization of the human dental calculus metabolome using a multi-platform approach. Ultra performance liquid chromatography-tandem mass spectrometry (UPLC-MS/MS) quantified 285 metabolites in modern and historic (200 years old) dental calculus, including metabolites of drug and dietary origin. A subset of historic samples was additionally analyzed by high-resolution gas chromatography-MS (GC-MS) and UPLC-MS/MS for further characterization of metabolites and lipids. Metabolite profiles of modern and historic calculus were compared to identify patterns of persistence and loss. Dipeptides, free amino acids, free nucleotides, and carbohydrates substantially decrease in abundance and ubiquity in archaeological samples, with some exceptions. Lipids generally persist, and saturated and mono-unsaturated medium and long chain fatty acids appear to be well-preserved, while metabolic derivatives related to oxidation and chemical degradation are found at higher levels in archaeological dental calculus than fresh samples. The results of this study indicate that certain metabolite classes have higher potential for recovery over long time scales and may serve as appropriate targets for oral microbiome evolutionary studies.
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.
The impact of taking a college pre-calculus course on students' college calculus performance
Sonnert, Gerhard; Sadler, Philip M.
2014-11-01
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 four-year colleges continues to grow, and these courses are well-populated with students who already took pre-calculus in high school. We examine student performance in college calculus, using regression discontinuity to estimate the effects of taking college pre-calculus or not, in a national US sample of 5507 students at 132 institutions. We find that students who take college pre-calculus do not earn higher calculus grades.
Probability and statistics for computer science
Johnson, James L
2011-01-01
Comprehensive and thorough development of both probability and statistics for serious computer scientists; goal-oriented: ""to present the mathematical analysis underlying probability results"" Special emphases on simulation and discrete decision theory Mathematically-rich, but self-contained text, at a gentle pace Review of calculus and linear algebra in an appendix Mathematical interludes (in each chapter) which examine mathematical techniques in the context of probabilistic or statistical importance Numerous section exercises, summaries, historical notes, and Further Readings for reinforcem
Quantum probability measures and tomographic probability densities
Amosov, GG; Man'ko, [No Value
2004-01-01
Using a simple relation of the Dirac delta-function to generalized the theta-function, the relationship between the tomographic probability approach and the quantum probability measure approach with the description of quantum states is discussed. The quantum state tomogram expressed in terms of the
Regge calculus: applications to classical and quantum gravity
Lewis, S.M.
1983-01-01
Regge calculus is a simplicial approximation to general relativity which preserves many topological and geometrical properties of the exact theory. After discussing the foundations of this technique and deriving some basic identities, specific solutions to Regge calculus are analyzed. In particular, the flat Friedmann-Robertson-Walker (FRW) model is shown. This particular model is used in the discussion of the initial value problem for Regge calculus. An Arnowitt-Deser-Misner type of 3 + 1 decomposition is possible only under very special circumstances; solutions with a non-spatially constant lapse can not generally be decomposed. The flat FRW model is also used to compute the accuracy of this approximation method developed by Regge. A three-dimensional toy model of quantum gravity is discussed that was originally formulated by Ponzano and Regge. A more thorough calculation is performed that takes into account additional terms. The renormalization properties of this model are shown. Finally, speculations are made on the interaction of the geometry, topology and quantum effects using Regge calculus, which, because of its simplicial nature, makes these effects more amenable to calculation and intuition
The Initial Conditions of Fractional Calculus
Trigeassou, J. C.; Maamri, N.
2011-01-01
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.
Factors Associated with Success in College Calculus II
Rosasco, Margaret E.
2013-01-01
Students are entering college having earned credit for college Calculus 1 based on their scores on the College Board's Advanced Placement (AP) Calculus AB exam. Despite being granted credit for college Calculus 1, it is unclear whether these students are adequately prepared for college Calculus 2. College calculus classes are often taught from a…
Sabzikar, Farzad, E-mail: sabzika2@stt.msu.edu [Department of Statistics and Probability, Michigan State University, East Lansing, MI 48823 (United States); Meerschaert, Mark M., E-mail: mcubed@stt.msu.edu [Department of Statistics and Probability, Michigan State University, East Lansing, MI 48823 (United States); Chen, Jinghua, E-mail: cjhdzdz@163.com [School of Sciences, Jimei University, Xiamen, Fujian, 361021 (China)
2015-07-15
Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.
Sabzikar, Farzad; Meerschaert, Mark M.; Chen, Jinghua
2015-07-01
Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.
Sabzikar, Farzad; Meerschaert, Mark M.; Chen, Jinghua
2015-01-01
Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series
Reggeon calculus at collider energies
Pajares, C.; Varias, A.; Yepes, P.
1983-01-01
The phenomenology of the perturbative reggeon calculus at collider energies is studied. It is found that the graphs which were neglected at ISR energies are still negligeable at √s=540 GeV. The perturbative series for the total cross section still converges reasonably fast. The values of the different parameters which describe rightly the data up to ISR energies give rise to a total cross section of around 60 mb at √s=540 GeV. For these values, the corresponding low mass and high mass eikonal series converges much more slowly. The non perturbative reggeon calculus gives rise to a total cross section less than 60 mb. (orig.)
Metric regularity and subdifferential calculus
Ioffe, A D
2000-01-01
The theory of metric regularity is an extension of two classical results: the Lyusternik tangent space theorem and the Graves surjection theorem. Developments in non-smooth analysis in the 1980s and 1990s paved the way for a number of far-reaching extensions of these results. It was also well understood that the phenomena behind the results are of metric origin, not connected with any linear structure. At the same time it became clear that some basic hypotheses of the subdifferential calculus are closely connected with the metric regularity of certain set-valued maps. The survey is devoted to the metric theory of metric regularity and its connection with subdifferential calculus in Banach spaces
Discrete calculus methods for counting
Mariconda, Carlo
2016-01-01
This book provides an introduction to combinatorics, finite calculus, formal series, recurrences, and approximations of sums. Readers will find not only coverage of the basic elements of the subjects but also deep insights into a range of less common topics rarely considered within a single book, such as counting with occupancy constraints, a clear distinction between algebraic and analytical properties of formal power series, an introduction to discrete dynamical systems with a thorough description of Sarkovskii’s theorem, symbolic calculus, and a complete description of the Euler-Maclaurin formulas and their applications. Although several books touch on one or more of these aspects, precious few cover all of them. The authors, both pure mathematicians, have attempted to develop methods that will allow the student to formulate a given problem in a precise mathematical framework. The aim is to equip readers with a sound strategy for classifying and solving problems by pursuing a mathematically rigorous yet ...
Formalization of the Integral Calculus in the PVS Theorem Prover
Ricky Wayne Butler
2009-04-01
Full Text Available The PVS Theorem prover is a widely used formal verification tool used for the analysis of safetycritical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higher-order logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and Computer Science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed.The theory and proofs were based on Rosenlicht’s classic text on real analysis and follow the traditional epsilon-delta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.
Formalization of the Integral Calculus in the PVS Theorem Prover
Butler, Ricky W.
2004-01-01
The PVS Theorem prover is a widely used formal verification tool used for the analysis of safety-critical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higher-order logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and Computer Science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed.The theory and proofs were based on Rosenlicht's classic text on real analysis and follow the traditional epsilon-delta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.
[Fluorescence control of dental calculus removal].
Bakhmutov, D N; Gonchukov, S A; Lonkina, T V; Sukhinina, A V
2012-01-01
The main condition of periodontitis prevention is the full calculus removal from the teeth surface. This procedure should be fulfilled without harming adjacent unaffected tooth tissues. Nevertheless the problem of sensitive and precise estimating of tooth-calculus interface exists and potential risk of hard tissue damage remains. In the frames of this work it was shown that fluorescence diagnostics during calculus removal can be successfully used for precise detection of tooth-calculus interface. In so doing the simple implementation of this method free from the necessity of spectrometer using can be employed. Such a simple implementation of calculus detection set-up can be aggregated with the devices of calculus removing (as ultrasonic or laser devices).
Complete staghorn calculus in polycystic kidney disease: infection is still the cause.
Mao, Zhiguo; Xu, Jing; Ye, Chaoyang; Chen, Dongping; Mei, Changlin
2013-08-01
Kidney stones in patients with autosomal dominant polycystic kidney disease are common, regarded as the consequence of the combination of anatomic abnormality and metabolic risk factors. However, complete staghorn calculus is rare in polycystic kidney disease and predicts a gloomy prognosis of kidney. For general population, recent data showed metabolic factors were the dominant causes for staghorn calculus, but for polycystic kidney disease patients, the cause for staghorn calculus remained elusive. We report a case of complete staghorm calculus in a polycystic kidney disease patient induced by repeatedly urinary tract infections. This 37-year-old autosomal dominant polycystic kidney disease female with positive family history was admitted in this hospital for repeatedly upper urinary tract infection for 3 years. CT scan revealed the existence of a complete staghorn calculus in her right kidney, while there was no kidney stone 3 years before, and the urinary stone component analysis showed the composition of calculus was magnesium ammonium phosphate. UTI is an important complication for polycystic kidney disease and will facilitate the formation of staghorn calculi. As staghorn calculi are associated with kidney fibrosis and high long-term renal deterioration rate, prompt control of urinary tract infection in polycystic kidney disease patient will be beneficial in preventing staghorn calculus formation.
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 t...
Jet-calculus approach including coherence effects
Jones, L.M.; Migneron, R.; Narayanan, K.S.S.
1987-01-01
We show how integrodifferential equations typical of jet calculus can be combined with an averaging procedure to obtain jet-calculus-based results including the Mueller interference graphs. Results in longitudinal-momentum fraction x for physical quantities are higher at intermediate x and lower at large x than with the conventional ''incoherent'' jet calculus. These results resemble those of Marchesini and Webber, who used a Monte Carlo approach based on the same dynamics
How Can Histograms Be Useful for Introducing Continuous Probability Distributions?
Derouet, Charlotte; Parzysz, Bernard
2016-01-01
The teaching of probability has changed a great deal since the end of the last century. The development of technologies is indeed part of this evolution. In France, continuous probability distributions began to be studied in 2002 by scientific 12th graders, but this subject was marginal and appeared only as an application of integral calculus.…
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.
A κ-symmetry calculus for superparticles
Gauntlett, J.P.
1991-01-01
We develop a κ-symmetry calculus for the d=2 and d=3, N=2 massive superparticles, which enables us to construct higher order κ-invariant actions. The method relies on a reformulation of these models as supersymmetric sigma models that are invariant under local worldline superconformal transformations. We show that the κ-symmetry is embedded in the superconformal symmetry so that a calculus for the κ-symmetry is equivalent to a tensor calculus for the latter. We develop such a calculus without the introduction of a wordline supergravity multiplet. (orig.)
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
Introduction to calculus and classical analysis
Hijab, Omar
2016-01-01
This completely self-contained text is intended either for a course in honors calculus or for an introduction to analysis. Beginning with the real number axioms, and involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate math majors. This fourth edition includes an additional chapter on the fundamental theorems in their full Lebesgue generality, based on the Sunrise Lemma. Key features of this text include: • Applications from several parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more; • A heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero; • Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals; • A self-contained t...
VEST: Abstract vector calculus simplification in Mathematica
Squire, J.; Burby, J.; Qin, H.
2014-01-01
We present a new package, VEST (Vector Einstein Summation Tools), that performs abstract vector calculus computations in Mathematica. Through the use of index notation, VEST is able to reduce three-dimensional scalar and vector expressions of a very general type to a well defined standard form. In addition, utilizing properties of the Levi-Civita symbol, the program can derive types of multi-term vector identities that are not recognized by reduction, subsequently applying these to simplify large expressions. In a companion paper Burby et al. (2013) [12], we employ VEST in the automation of the calculation of high-order Lagrangians for the single particle guiding center system in plasma physics, a computation which illustrates its ability to handle very large expressions. VEST has been designed to be simple and intuitive to use, both for basic checking of work and more involved computations.
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.
Dental Calculus Arrest of Dental Caries.
Keyes, Paul H; Rams, Thomas E
An inverse relationship between dental calculus mineralization and dental caries demineralization on teeth has been noted in some studies. Dental calculus may even form superficial layers over existing dental caries and arrest their progression, but this phenomenon has been only rarely documented and infrequently considered in the field of Cariology. To further assess the occurrence of dental calculus arrest of dental caries, this study evaluated a large number of extracted human teeth for the presence and location of dental caries, dental calculus, and dental plaque biofilms. A total of 1,200 teeth were preserved in 10% buffered formal saline, and viewed while moist by a single experienced examiner using a research stereomicroscope at 15-25× magnification. Representative teeth were sectioned and photographed, and their dental plaque biofilms subjected to gram-stain examination with light microscopy at 100× magnification. Dental calculus was observed on 1,140 (95%) of the extracted human teeth, and no dental carious lesions were found underlying dental calculus-covered surfaces on 1,139 of these teeth. However, dental calculus arrest of dental caries was found on one (0.54%) of 187 evaluated teeth that presented with unrestored proximal enamel caries. On the distal surface of a maxillary premolar tooth, dental calculus mineralization filled the outer surface cavitation of an incipient dental caries lesion. The dental calculus-covered carious lesion extended only slightly into enamel, and exhibited a brown pigmentation characteristic of inactive or arrested dental caries. In contrast, the tooth's mesial surface, without a superficial layer of dental calculus, had a large carious lesion going through enamel and deep into dentin. These observations further document the potential protective effects of dental calculus mineralization against dental caries.
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
Asmussen, Søren; Albrecher, Hansjörg
The book gives a comprehensive treatment of the classical and modern ruin probability theory. Some of the topics are Lundberg's inequality, the Cramér-Lundberg approximation, exact solutions, other approximations (e.g., for heavy-tailed claim size distributions), finite horizon ruin probabilities......, extensions of the classical compound Poisson model to allow for reserve-dependent premiums, Markov-modulation, periodicity, change of measure techniques, phase-type distributions as a computational vehicle and the connection to other applied probability areas, like queueing theory. In this substantially...... updated and extended second version, new topics include stochastic control, fluctuation theory for Levy processes, Gerber–Shiu functions and dependence....
Calculation of parton fragmentation functions from jet calculus: gluon applications
Lassila, K.E.; Ng, A.
1985-01-01
A method is presented for calculation of general parton fragmentation functions based on jet calculus plus meson and baryon wave functions. Results for gluon fragmentation into mesons and baryons are discussed and related to recent information on upsilon decay into gluons. The expressions derived can be used directly in e + e - cross section predictions and will need to be folded in with baryon parton distribution functions when used in p-barp collisions. (author)
Simulating non-prenex cuts in quantified propositional calculus
Jeřábek, Emil; Nguyen, P.
2011-01-01
Roč. 57, č. 5 (2011), s. 524-532 ISSN 0942-5616 R&D Projects: GA AV ČR IAA100190902; GA MŠk(CZ) 1M0545 Institutional research plan: CEZ:AV0Z10190503 Keywords : proof complexity * prenex cuts * quantified propositional calculus Subject RIV: BA - General Mathematics Impact factor: 0.496, year: 2011 http://onlinelibrary.wiley.com/doi/10.1002/malq.201020093/abstract
Study on bioactive compounds of in vitro cultured Calculus Suis and natural Calculus Bovis.
Wan, Tien-Chun; Cheng, Fu-Yuan; Liu, Yu-Tse; Lin, Liang-Chuan; Sakata, Ryoichi
2009-12-01
The purpose of the study was to investigate bioactive compounds of in vitro cultured Calculus Suis and natural Calculus Bovis obtained as valuable by-products from animals used for meat production. The results showed that the components of natural Calculus Bovis were rich in bilirubin and biliverdin and had higher content of essential amino acids. The major amino acids of in vitro cultured Calculus Suis were identified as glycine, alanine, glutamic acid and aspartic acid, and those for natural Calculus Bovis were found to be glutamic acid, aspartic acid, proline, and arginine. The methionine and cysteine contents of precursors for glutathione in natural Calculus Bovis were significantly higher than those of in vitro cultured Calculus Suis. The mineral contents of zinc, iron and manganese of natural Calculus Bovis were significantly higher than those of in vitro cultured Calculus Suis. The major bile acids in both products were cholic acid and dehydrocholic acid, respectively. The chenodeoxycholic and ursodeoxycholic acid content of in vitro cultured Calculus Suis was significantly higher than that of natural Calculus Bovis.
Shiryaev, Albert N
2016-01-01
This book contains a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, the measure-theoretic foundations of probability theory, weak convergence of probability measures, and the central limit theorem. Many examples are discussed in detail, and there are a large number of exercises. The book is accessible to advanced undergraduates and can be used as a text for independent study. To accommodate the greatly expanded material in the third edition of Probability, the book is now divided into two volumes. This first volume contains updated references and substantial revisions of the first three chapters of the second edition. In particular, new material has been added on generating functions, the inclusion-exclusion principle, theorems on monotonic classes (relying on a detailed treatment of “π-λ” systems), and the fundamental theorems of mathematical statistics.
Earth Data Analysis Center, University of New Mexico — USFS, State Forestry, BLM, and DOI fire occurrence point locations from 1987 to 2008 were combined and converted into a fire occurrence probability or density grid...
Thematization of the Calculus Graphing Schema
Cooley, Laurel; Baker, Bernadette; Trigueros, Maria
2003-01-01
This article is the result of an investigation of students' conceptualizations of calculus graphing techniques after they had completed at least two semesters of calculus. The work and responses of 27 students to a series of questions that solicit information about the graphical implications of the first derivative, second derivative, continuity,…
RAMAN-SPECTRA OF HUMAN DENTAL CALCULUS
TSUDA, H; ARENDS, J
1993-01-01
Raman spectra of human dental calculus have been observed for the first time by use of micro-Raman spectroscopy. The spectral features of calculus were influenced easily by heating caused by laser irradiation. Therefore, the measurements were carried out at relatively low power (5 mW, 1-mu m spot
Educating about Sustainability while Enhancing Calculus
Pfaff, Thomas J.
2011-01-01
We give an overview of why it is important to include sustainability in mathematics classes and provide specific examples of how to do this for a calculus class. We illustrate that when students use "Excel" to fit curves to real data, fundamentally important questions about sustainability become calculus questions about those curves. (Contains 5…
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…
Hybrid Logical Analyses of the Ambient Calculus
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...
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…
An Introductory Calculus-Based Mechanics Investigation
Allen, Bradley
2017-01-01
One challenge for the introductory physics teacher is incorporating calculus techniques into the laboratory setting. It can be difficult to strike a balance between presenting an experimental task for which calculus is essential and making the mathematics accessible to learners who may be apprehensive about applying it. One-dimensional kinematics…
Multiplicative calculus in biomedical image analysis
Florack, L.M.J.; Assen, van H.C.
2011-01-01
We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. It provides a natural framework in problems in which positive images or positive definite matrix fields and positivity preserving operators are of interest. Indeed,
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…
Improving student learning in calculus through applications
Young, C. Y.; Georgiopoulos, M.; Hagen, S. C.; Geiger, C. L.; Dagley-Falls, M. A.; Islas, A. L.; Ramsey, P. J.; Lancey, P. M.; Straney, R. A.; Forde, D. S.; Bradbury, E. E.
2011-07-01
Nationally only 40% of the incoming freshmen Science, Technology, Engineering and Mathematics (STEM) majors are successful in earning a STEM degree. The University of Central Florida (UCF) EXCEL programme is a National Science Foundation funded STEM Talent Expansion Programme whose goal is to increase the number of UCF STEM graduates. One of the key requirements for STEM majors is a strong foundation in Calculus. To improve student learning in calculus, the EXCEL programme developed two special courses at the freshman level called Applications of Calculus I (Apps I) and Applications of Calculus II (Apps II). Apps I and II are one-credit classes that are co-requisites for Calculus I and II. These classes are teams taught by science and engineering professors whose goal is to demonstrate to students where the calculus topics they are learning appear in upper level science and engineering classes as well as how faculty use calculus in their STEM research programmes. This article outlines the process used in producing the educational materials for the Apps I and II courses, and it also discusses the assessment results pertaining to this specific EXCEL activity. Pre- and post-tests conducted with experimental and control groups indicate significant improvement in student learning in Calculus II as a direct result of the application courses.
A direct extension of Meller's calculus
E. L. Koh
1982-01-01
Full Text Available This paper extends the operational calculus of Meller for the operator Bα=t−αddttα+1ddt to the case where α∈(0,∞. The development is àla Mikusinski calculus and uses Meller's convolution process with a fractional derivative operator.
A Snapshot of the Calculus Classroom
Weathers, Tony D.; Latterell, Carmen M.
2003-01-01
Essentially a focus group to discuss textbook related issues, a meeting of calculus instructors from a wide variety of environments was convened and sponsored by McGraw Hill to provide feedback on the current state of the calculus classroom. This paper provides a description of the group's discussions.
Programming Language Concepts - The Lambda Calculus Approach
Fokkinga, M.M.; Asveld, P.R.J.; Nijholt, Antinus
1987-01-01
The Lambda Calculus is a formal system, originally intended as a tool in the foundation of mathematics, but mainly used to study the concepts of algorithm and effective computability. Recently, the Lambda Calculus and related systems acquire attention from Computer Science for another reason too:
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…
Laparoscopic Cholecystectomy in Chronic Calculus Cholecystitis
Prakash Sapkota
2013-12-01
Full Text Available Introduction: Laparoscopic cholecystectomy has clearly become the choice over open cholecystectomy in the treatment of hepatobiliary disease since its introduction by Mouret in 1987. This study evaluates a series of patients with chronic calculus cholecystitis who were treated with laparoscopic and open cholecystectomy and assesses the outcomes of both techniques. Objective: To evaluate the efficacy of laparoscopic vs open cholecystectomy in chronic calculus cholecystitis and establish the out-comes of this treatment modality at Lumbini Medical College and Teaching Hospital. Methods: This was a retrospective analysis over a one-year period (January 1, 2012 to December 31, 2012, per-formed by single surgeon at Lumbini Medical College and Teaching Hospital located midwest of Nepal. 166 patients underwent surgical treatment for chronic calculus cholecystitis. Patients included were only chronic calculus cholecystitis proven histopathologocally and the rest were excluded. Data was collected which included patients demographics, medical history, presentation, complications, conversion rates from laparoscopic. cholecystectomy to open cholecystectomy, operative and postoperative time. Results: Patients treated with laparoscopic cholecystectomy for chronic calculus cholecystitis had shorter operating times and length of stay compared to patients treated with open cholecystectomy for chronic calculus cholecystitis. Conversion rates were 3.54% in chronic calculus cholecystitis during the study period. Complications were also lower in patients who underwent laparoscopic cholecystectomy versus open cholecystectomy for cholelithiasis. Conclusions: Laparoscopic cholecystectomy appears to be a reliable, safe, and cost-effective treatment modality for chronic calculus cholecystitis.
A type system for continuation calculus
Geuvers, J.H.; Geraedts, W.; Geron, B.; Stegeren, van J.; Oliva, P.
2014-01-01
Continuation Calculus (CC), introduced by Geron and Geuvers, is a simple foundational model for functional computation. It is closely related to lambda calculus and term rewriting, but it has no variable binding and no pattern matching. It is Turing complete and evaluation is deterministic. Notions
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
Technical calculus with analytic geometry
Gersting, Judith L
2010-01-01
This well-thought-out text, filled with many special features, is designed for a two-semester course in calculus for technology students with a background in college algebra and trigonometry. The author has taken special care to make the book appealing to students by providing motivating examples, facilitating an intuitive understanding of the underlying concepts involved, and by providing much opportunity to gain proficiency in techniques and skills.Initial chapters cover functions and graphs, straight lines and conic sections, new coordinate systems, the derivative, using the derivative, in
Borden, Robert S
1997-01-01
This remarkable undergraduate-level text offers a study in calculus that simultaneously unifies the concepts of integration in Euclidean space while at the same time giving students an overview of other areas intimately related to mathematical analysis. The author achieves this ambitious undertaking by shifting easily from one related subject to another. Thus, discussions of topology, linear algebra, and inequalities yield to examinations of innerproduct spaces, Fourier series, and the secret of Pythagoras. Beginning with a look at sets and structures, the text advances to such topics as lim
Bridging the Vector Calculus Gap
Dray, Tevian; Manogue, Corinne
2003-05-01
As with Britain and America, mathematicians and physicists are separated from each other by a common language. In a nutshell, mathematics is about functions, but physics is about things. For the last several years, we have led an NSF-supported effort to "bridge the vector calculus gap" between mathematics and physics. The unifying theme we have discovered is to emphasize geometric reasoning, not (just) algebraic computation. In this talk, we will illustrate the language differences between mathematicians and physicists, and how we are trying reconcile them in the classroom. For further information about the project go to: http://www.physics.orst.edu/bridge
Quantum chemistry and scientific calculus
Gervais, H.P.
1988-01-01
The 1988 progress report of the Polytechnic School research team, concerning the quantum chemistry and the scientific calculus. The research program involves the following topics: the transition metals - carbon monoxide systems, which are a suitable model for the chemisorption phenomena; the introduction of the vibronic perturbations in the magnetic screen constants; the gauge invariance method (used in the calculation of the magnetic perturbations), extended to the case of the static or dynamic electrical polarizabilities. The published papers, the congress communications and the thesis are listed [fr
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
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
An Overview of Generalized Gamma Mittag–Leffler Model and Its Applications
Seema S. Nair
2015-08-01
Full Text Available Recently, probability models with thicker or thinner tails have gained more importance among statisticians and physicists because of their vast applications in random walks, Lévi flights, financial modeling, etc. In this connection, we introduce here a new family of generalized probability distributions associated with the Mittag–Leffler function. This family gives an extension to the generalized gamma family, opens up a vast area of potential applications and establishes connections to the topics of fractional calculus, nonextensive statistical mechanics, Tsallis statistics, superstatistics, the Mittag–Leffler stochastic process, the Lévi process and time series. Apart from examining the properties, the matrix-variate analogue and the connection to fractional calculus are also explained. By using the pathway model of Mathai, the model is further generalized. Connections to Mittag–Leffler distributions and corresponding autoregressive processes are also discussed.
Enriching an effect calculus with linear types
Egger, Jeff; Møgelberg, Rasmus Ejlers; Simpson, Alex
2009-01-01
We define an ``enriched effect calculus'' by conservatively extending a type theory for computational effects with primitives from linear logic. By doing so, we obtain a generalisation of linear type theory, intended as a formalism for expressing linear aspects of effects. As a worked example, we...... formulate linearly-used continuations in the enriched effect calculus. These are captured by a fundamental translation of the enriched effect calculus into itself, which extends existing call-by-value and call-by-name linearly-used CPS translations. We show that our translation is involutive. Full...... completeness results for the various linearly-used CPS translations follow. Our main results, the conservativity of enriching the effect calculus with linear primitives, and the involution property of the fundamental translation, are proved using a category-theoretic semantics for the enriched effect calculus...
Brémaud, Pierre
2017-01-01
The emphasis in this book is placed on general models (Markov chains, random fields, random graphs), universal methods (the probabilistic method, the coupling method, the Stein-Chen method, martingale methods, the method of types) and versatile tools (Chernoff's bound, Hoeffding's inequality, Holley's inequality) whose domain of application extends far beyond the present text. Although the examples treated in the book relate to the possible applications, in the communication and computing sciences, in operations research and in physics, this book is in the first instance concerned with theory. The level of the book is that of a beginning graduate course. It is self-contained, the prerequisites consisting merely of basic calculus (series) and basic linear algebra (matrices). The reader is not assumed to be trained in probability since the first chapters give in considerable detail the background necessary to understand the rest of the book. .
Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus
Aguirre, Alejandro; Barthe, Gilles; Birkedal, Lars
2018-01-01
We extend the simply-typed guarded $\\lambda$-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like...
Vock, David M; Wolfson, Julian; Bandyopadhyay, Sunayan; Adomavicius, Gediminas; Johnson, Paul E; Vazquez-Benitez, Gabriela; O'Connor, Patrick J
2016-06-01
Models for predicting the probability of experiencing various health outcomes or adverse events over a certain time frame (e.g., having a heart attack in the next 5years) based on individual patient characteristics are important tools for managing patient care. Electronic health data (EHD) are appealing sources of training data because they provide access to large amounts of rich individual-level data from present-day patient populations. However, because EHD are derived by extracting information from administrative and clinical databases, some fraction of subjects will not be under observation for the entire time frame over which one wants to make predictions; this loss to follow-up is often due to disenrollment from the health system. For subjects without complete follow-up, whether or not they experienced the adverse event is unknown, and in statistical terms the event time is said to be right-censored. Most machine learning approaches to the problem have been relatively ad hoc; for example, common approaches for handling observations in which the event status is unknown include (1) discarding those observations, (2) treating them as non-events, (3) splitting those observations into two observations: one where the event occurs and one where the event does not. In this paper, we present a general-purpose approach to account for right-censored outcomes using inverse probability of censoring weighting (IPCW). We illustrate how IPCW can easily be incorporated into a number of existing machine learning algorithms used to mine big health care data including Bayesian networks, k-nearest neighbors, decision trees, and generalized additive models. We then show that our approach leads to better calibrated predictions than the three ad hoc approaches when applied to predicting the 5-year risk of experiencing a cardiovascular adverse event, using EHD from a large U.S. Midwestern healthcare system. Copyright © 2016 Elsevier Inc. All rights reserved.
A Calculus of Located Entities
Adriana Compagnoni
2014-03-01
Full Text Available We define BioScapeL, a stochastic pi-calculus in 3D-space. A novel aspect of BioScapeL is that entities have programmable locations. The programmer can specify a particular location where to place an entity, or a location relative to the current location of the entity. The motivation for the extension comes from the need to describe the evolution of populations of biochemical species in space, while keeping a sufficiently high level description, so that phenomena like diffusion, collision, and confinement can remain part of the semantics of the calculus. Combined with the random diffusion movement inherited from BioScape, programmable locations allow us to capture the assemblies of configurations of polymers, oligomers, and complexes such as microtubules or actin filaments. Further new aspects of BioScapeL include random translation and scaling. Random translation is instrumental in describing the location of new entities relative to the old ones. For example, when a cell secretes a hydronium ion, the ion should be placed at a given distance from the originating cell, but in a random direction. Additionally, scaling allows us to capture at a high level events such as division and growth; for example, daughter cells after mitosis have half the size of the mother cell.
Quantum Probabilities as Behavioral Probabilities
Vyacheslav I. Yukalov
2017-03-01
Full Text Available We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable to the description of human decision making. The applicability of quantum rules for describing decision making is connected with the nontrivial process of making decisions in the case of composite prospects under uncertainty. Such a process involves deliberations of a decision maker when making a choice. In addition to the evaluation of the utilities of considered prospects, real decision makers also appreciate their respective attractiveness. Therefore, human choice is not based solely on the utility of prospects, but includes the necessity of resolving the utility-attraction duality. In order to justify that human consciousness really functions similarly to the rules of quantum theory, we develop an approach defining human behavioral probabilities as the probabilities determined by quantum rules. We show that quantum behavioral probabilities of humans do not merely explain qualitatively how human decisions are made, but they predict quantitative values of the behavioral probabilities. Analyzing a large set of empirical data, we find good quantitative agreement between theoretical predictions and observed experimental data.
Rojas-Nandayapa, Leonardo
Tail probabilities of sums of heavy-tailed random variables are of a major importance in various branches of Applied Probability, such as Risk Theory, Queueing Theory, Financial Management, and are subject to intense research nowadays. To understand their relevance one just needs to think...... analytic expression for the distribution function of a sum of random variables. The presence of heavy-tailed random variables complicates the problem even more. The objective of this dissertation is to provide better approximations by means of sharp asymptotic expressions and Monte Carlo estimators...
Generalized quantal equation of motion
Morsy, M.W.; Embaby, M.
1986-07-01
In the present paper, an attempt is made for establishing a generalized equation of motion for quantal objects, in which intrinsic self adjointness is naturally built in, independently of any prescribed representation. This is accomplished by adopting Hamilton's principle of least action, after incorporating, properly, the quantal features and employing the generalized calculus of variations, without being restricted to fixed end points representation. It turns out that our proposed equation of motion is an intrinsically self-adjoint Euler-Lagrange's differential equation that ensures extremization of the quantal action as required by Hamilton's principle. Time dependence is introduced and the corresponding equation of motion is derived, in which intrinsic self adjointness is also achieved. Reducibility of the proposed equation of motion to the conventional Schroedinger equation is examined. The corresponding continuity equation is established, and both of the probability density and the probability current density are identified. (author)
Geometric calculus according to the Ausdehnungslehre of H. Grassmann
Peano, Giuseppe
2000-01-01
Calcolo Geometrico, G. Peano's first publication in mathematical logic, is a model of expository writing, with a significant impact on 20th century mathematics. Kannenberg's lucid and crisp translation, Geometric Calculus, will appeal to historians of mathematics, researchers, graduate students, and general readers interested in the foundations of mathematics and the development of a formal logical language. In Chapter IX, with the innocent-sounding title "Transformations of a linear system," one finds the crown jewel of the book: Peano's axiom system for a vector space, the first-ever presentation of a set of such axioms. The very wording of the axioms (which Peano calls "definitions") has a remarkably modern ring, almost like a modern introduction to linear algebra. Peano also presents the basic calculus of set operation, introducing the notation for 'intersection,' 'union,' and 'element of,' many years before it was accepted. Despite its uniqueness, Calcolo Geometrico has been strangely neglected by histor...
Quantum stochastic calculus associated with quadratic quantum noises
Ji, Un Cig; Sinha, Kalyan B.
2016-01-01
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
Quantum stochastic calculus associated with quadratic quantum noises
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.
k-Schur functions and affine Schubert calculus
Lam, Thomas; Morse, Jennifer; Schilling, Anne; Shimozono, Mark; Zabrocki, Mike
2014-01-01
This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry. This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with ...
Semiclassical regime of Regge calculus and spin foams
Bianchi, Eugenio; Satz, Alejandro
2009-01-01
Recent attempts to recover the graviton propagator from spin foam models involve the use of a boundary quantum state peaked on a classical geometry. The question arises whether beyond the case of a single simplex this suffices for peaking the interior geometry in a semiclassical configuration. In this paper we explore this issue in the context of quantum Regge calculus with a general triangulation. Via a stationary phase approximation, we show that the boundary state succeeds in peaking the interior in the appropriate configuration, and that boundary correlations can be computed order by order in an asymptotic expansion. Further, we show that if we replace at each simplex the exponential of the Regge action by its cosine-as expected from the semiclassical limit of spin foam models-then the contribution from the sign-reversed terms is suppressed in the semiclassical regime and the results match those of conventional Regge calculus
Quantum stochastic calculus in Fock space: A review
Hudson, R.L.
1986-01-01
This paper presents a survey of the recently developed theory of quantum stochastic calculus in Boson Fock space, together with its applications. The work focuses on a non-commutative generalization of the classical Ito stochastic calculus of Brownian motion, which exploits to the full the Wiener-Segal duality transformation identifying the L 2 space of Wiener measure with a Boson Fock space. This Fock space emerges as the natural home of not only Brownian motion but also classical Poisson processes, and even of Fermionic processes of the type developed by Barnett et al. The principle physical application of the theory to the construction and characterization of unitary dilations of quantum dynamical semigroups is also described
Fractional Calculus in Hydrologic Modeling: A Numerical Perspective
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.
Giant calculus: review and report of a case.
Woodmansey, Karl; Severine, Anthony; Lembariti, Bakari S
2013-01-01
Dental calculus is a common oral finding. The term giant calculus is used to describe unusually large deposits of dental calculus. Several extreme cases have been reported in the dental literature. The specific etiology of these cases remains uncertain. This paper reviews previously reported cases, and presents another extreme example of giant calculus.
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…
Fraassen, B.C. van
1979-01-01
The interpretation of probabilities in physical theories are considered, whether quantum or classical. The following points are discussed 1) the functions P(μ, Q) in terms of which states and propositions can be represented, are classical (Kolmogoroff) probabilities, formally speaking, 2) these probabilities are generally interpreted as themselves conditional, and the conditions are mutually incompatible where the observables are maximal and 3) testing of the theory typically takes the form of confronting the expectation values of observable Q calculated with probability measures P(μ, Q) for states μ; hence, of comparing the probabilities P(μ, Q)(E) with the frequencies of occurrence of the corresponding events. It seems that even the interpretation of quantum mechanics, in so far as it concerns what the theory says about the empirical (i.e. actual, observable) phenomena, deals with the confrontation of classical probability measures with observable frequencies. This confrontation is studied. (Auth./C.F.)
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…
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…
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.
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
Restricted diversity of dental calculus methanogens over five centuries, France
Hong T. T. Huynh; Vanessa D. Nkamga; Michel Signoli; Stéfan Tzortzis; Romuald Pinguet; Gilles Audoly; Gérard Aboudharam; Michel Drancourt
2016-01-01
Methanogens are acknowledged archaeal members of modern dental calculus microbiota and dental pathogen complexes. Their repertoire in ancient dental calculus is poorly known. We therefore investigated archaea in one hundred dental calculus specimens collected from individuals recovered from six archaeological sites in France dated from the 14th to 19th centuries AD. Dental calculus was demonstrated by macroscopic and cone-beam observations. In 56 calculus specimens free of PCR inhibition, PCR...
On Some Syntactic Properties of the Modalized Heyting Calculus
Muravitsky, Alexei
2016-01-01
We show that the modalized Heyting calculus introduced by Leo Esakia admits a normal axiomatization. Then, we prove that the inference rules $\\square\\alpha/\\alpha$ and $\\square\\alpha\\rightarrow\\alpha/\\alpha$ are admissible in this calculus. Finally, we show that this calculus and intuitionistic propositional calculus are assertorically equipollent, which leads to a variant of limited separation property for the modalized Heyting calculus.
Dorogovtsev, A Ya; Skorokhod, A V; Silvestrov, D S; Skorokhod, A V
1997-01-01
This book of problems is intended for students in pure and applied mathematics. There are problems in traditional areas of probability theory and problems in the theory of stochastic processes, which has wide applications in the theory of automatic control, queuing and reliability theories, and in many other modern science and engineering fields. Answers to most of the problems are given, and the book provides hints and solutions for more complicated problems.
Algebraic differential calculus for gauge theories
Landi, G.; Marmo, G.
1990-01-01
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)
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
Data Quality Indicators Composition and Calculus: Engineering and Information Systems Approaches
Leon REZNIK
2015-02-01
Full Text Available Big Data phenomenon is a result of novel technological developments in sensor, computer and communication technologies. Nowadays more and more data are produced by nanoscale photonic, optoelectronic and electronic devices. However, their quality characteristics could be very low. The paper proposes new methods of the data management with huge data amounts that is based on associating of data quality indicators with each data entity. To achieve this goal, one needs to define the composition of the data quality indicators and to develop their integration calculus. As data quality evaluation involves multi-disciplinary research, various metrics have been investigated. The paper describes two major approaches in assigning the data quality indicators and developing their integration calculus. The information systems approach employs traditional high-level metrics like data accuracy, consistency and completeness. The engineering approach utilizes signal characteristics processed with the probability based calculus. The data quality metrics composition and calculus are discussed. The tools developed to automate the metrics selection and calculus procedures are presented. The user- friendly interface examples are provided.
SRXRF analysis with spatial resolution of dental calculus
Sanchez, Hector Jorge; Perez, Carlos Alberto; Grenon, Miriam
2000-01-01
This work presents elemental-composition studies of dental calculus by X-ray fluorescence analysis using synchrotron radiation. The intrinsic characteristics of synchrotron light allow for a semi-quantitative analysis with spatial resolution. The experiments were carried out in the high-vacuum station of the XRF beamline at the Synchrotron Light National Laboratory (Campinas, Brazil). All the measurements were performed in conventional geometry (45 deg. + 45 deg.) and the micro-collimation was attained via a pair of orthogonal slits mounted in the beamline. In this way, pixels of 50 μmx50 μm were obtained keeping a high flux of photons on the sample. Samples of human dental calculus were measured in different positions along their growing axis, in order to determine variations of the compositions in the pattern of deposit. Intensity ratios of minor elements and traces were obtained, and linear profiles and surface distributions were determined. As a general summary, we can conclude that μXRF experiments with spatial resolution on dental calculus are feasible with simple collimation and adequate positioning systems, keeping a high flux of photon. These results open interesting perspectives for the future station of the line, devoted to μXRF, which will reach resolutions of the order of 10 μm
SRXRF analysis with spatial resolution of dental calculus
Sánchez, Héctor Jorge; Pérez, Carlos Alberto; Grenón, Miriam
2000-09-01
This work presents elemental-composition studies of dental calculus by X-ray fluorescence analysis using synchrotron radiation. The intrinsic characteristics of synchrotron light allow for a semi-quantitative analysis with spatial resolution. The experiments were carried out in the high-vacuum station of the XRF beamline at the Synchrotron Light National Laboratory (Campinas, Brazil). All the measurements were performed in conventional geometry (45°+45°) and the micro-collimation was attained via a pair of orthogonal slits mounted in the beamline. In this way, pixels of 50 μm×50 μm were obtained keeping a high flux of photons on the sample. Samples of human dental calculus were measured in different positions along their growing axis, in order to determine variations of the compositions in the pattern of deposit. Intensity ratios of minor elements and traces were obtained, and linear profiles and surface distributions were determined. As a general summary, we can conclude that μXRF experiments with spatial resolution on dental calculus are feasible with simple collimation and adequate positioning systems, keeping a high flux of photon. These results open interesting perspectives for the future station of the line, devoted to μXRF, which will reach resolutions of the order of 10 μm.
Fractional variational calculus in terms of Riesz fractional derivatives
Agrawal, O P
2007-01-01
This paper presents extensions of traditional calculus of variations for systems containing Riesz fractional derivatives (RFDs). Specifically, we present generalized Euler-Lagrange equations and the transversality conditions for fractional variational problems (FVPs) defined in terms of RFDs. We consider two problems, a simple FVP and an FVP of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives, functions and parameters, and to unspecified boundary conditions. For the second problem, we present Lagrange-type multiplier rules. For both problems, we develop the Euler-Lagrange-type necessary conditions which must be satisfied for the given functional to be extremum. Problems are considered to demonstrate applications of the formulations. Explicitly, we introduce fractional momenta, fractional Hamiltonian, fractional Hamilton equations of motion, fractional field theory and fractional optimal control. The formulations presented and the resulting equations are similar to the formulations for FVPs given in Agrawal (2002 J. Math. Anal. Appl. 272 368, 2006 J. Phys. A: Math. Gen. 39 10375) and to those that appear in the field of classical calculus of variations. These formulations are simple and can be extended to other problems in the field of fractional calculus of variations
Vector calculus in non-integer dimensional space and its applications to fractal media
Tarasov, Vasily E.
2015-02-01
We suggest a generalization of vector calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are defined. For simplification we consider scalar and vector fields that are independent of angles. We formulate a generalization of vector calculus for rotationally covariant scalar and vector functions. This generalization allows us to describe fractal media and materials in the framework of continuum models with non-integer dimensional space. As examples of application of the suggested calculus, we consider elasticity of fractal materials (fractal hollow ball and fractal cylindrical pipe with pressure inside and outside), steady distribution of heat in fractal media, electric field of fractal charged cylinder. We solve the correspondent equations for non-integer dimensional space models.
Real-Time Exponential Curve Fits Using Discrete Calculus
Rowe, Geoffrey
2010-01-01
An improved solution for curve fitting data to an exponential equation (y = Ae(exp Bt) + C) has been developed. This improvement is in four areas -- speed, stability, determinant processing time, and the removal of limits. The solution presented avoids iterative techniques and their stability errors by using three mathematical ideas: discrete calculus, a special relationship (be tween exponential curves and the Mean Value Theorem for Derivatives), and a simple linear curve fit algorithm. This method can also be applied to fitting data to the general power law equation y = Ax(exp B) + C and the general geometric growth equation y = Ak(exp Bt) + C.
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.
Extending Stochastic Network Calculus to Loss Analysis
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.
Differential calculus for q-deformed twistors
Akulov, V.P.; Duplij, S.A.; Chitov, V.V.
1998-01-01
Brief type of q-deformed differential calculus at light cone with using of twistor representation is suggested. Commutative relations between coordinates and moments are obtained. Considered quasiclassical limit gives exact form of vanish from mass shell
Null-strut calculus. II. Dynamics
Kheyfets, A.; LaFave, N.J.; Miller, W.A.
1990-01-01
In this paper, we continue from the preceding paper to develop a fully functional Regge calculus geometrodynamic algorithm from the null-strut-calculus construction. The developments discussed include (a) the identification of the Regge calculus analogue of the constraint and evolution equations on the null-strut lattice, (b) a description of the Minkowski solid geometry for the simplicial blocks of the null-strut lattice, (c) a description of the evolution algorithm for the geometrodynamic scheme and an analysis of its consistency, and (d) a presentation of the dynamical degrees of freedom for a simplicial hypersurface and the description of an initial-value prescription. To demonstrate qualitatively this new approach to geometrodynamics, we present the most simple application of null-strut calculus that we know of---the Friedmann cosmology using the three-boundary of a 600-cell simplicial polytope to model the simplicial hypersurface
One Answer to "What Is Calculus?"
Shilgalis, Thomas W.
1979-01-01
A number of questions are posed that can be answered with the aid of calculus. These include best value problems, best shape problems, problems involving integration, and growth and decay problems. (MP)
The contracted Bianchi identities in Regge calculus
Williams, Ruth M
2012-01-01
In this note, we show explicitly how the linearized contracted Bianchi identities at a vertex in four-dimensional Regge calculus are related to a sum of the equations of motion for all the edges meeting at that vertex. (note)
A primer on exterior differential calculus
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. .
The origins of Cauchy's rigorous calculus
Grabiner, Judith V
2005-01-01
This text examines the reinterpretation of calculus by Augustin-Louis Cauchy and his peers in the 19th century. These intellectuals created a collection of well-defined theorems about limits, continuity, series, derivatives, and integrals. 1981 edition.
Algorithms, The λ Calculus and Programming
IAS Admin
developed a model to understand ... Hence the ¸ calculus also served as an alternate model ...... Practical programming using usual languages based on .... and return values as. 'answers'. This style of programming that emerges is therefore.
Sandboxing in a Distributed Pi-Calculus
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....
Area Regge calculus and discontinuous metrics
Wainwright, Chris; Williams, Ruth M
2004-01-01
Taking the triangle areas as independent variables in the theory of Regge calculus can lead to ambiguities in the edge lengths, which can be interpreted as discontinuities in the metric. We construct solutions to area Regge calculus using a triangulated lattice and find that on a spacelike or timelike hypersurface no such discontinuity can arise. On a null hypersurface however, we can have such a situation and the resulting metric can be interpreted as a so-called refractive wave
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.
A Graph Calculus for Predicate Logic
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.
A calculus for attribute-based communication
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....
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 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)
Fractional calculus model of electrical impedance applied to human skin.
Vosika, Zoran B; Lazovic, Goran M; Misevic, Gradimir N; Simic-Krstic, Jovana B
2013-01-01
Fractional calculus is a mathematical approach dealing with derivatives and integrals of arbitrary and complex orders. Therefore, it adds a new dimension to understand and describe basic nature and behavior of complex systems in an improved way. Here we use the fractional calculus for modeling electrical properties of biological systems. We derived a new class of generalized models for electrical impedance and applied them to human skin by experimental data fitting. The primary model introduces new generalizations of: 1) Weyl fractional derivative operator, 2) Cole equation, and 3) Constant Phase Element (CPE). These generalizations were described by the novel equation which presented parameter [Formula: see text] related to remnant memory and corrected four essential parameters [Formula: see text] We further generalized single generalized element by introducing specific partial sum of Maclaurin series determined by parameters [Formula: see text] We defined individual primary model elements and their serial combination models by the appropriate equations and electrical schemes. Cole equation is a special case of our generalized class of models for[Formula: see text] Previous bioimpedance data analyses of living systems using basic Cole and serial Cole models show significant imprecisions. Our new class of models considerably improves the quality of fitting, evaluated by mean square errors, for bioimpedance data obtained from human skin. Our models with new parameters presented in specific partial sum of Maclaurin series also extend representation, understanding and description of complex systems electrical properties in terms of remnant memory effects.
Fractional calculus model of electrical impedance applied to human skin.
Zoran B Vosika
Full Text Available Fractional calculus is a mathematical approach dealing with derivatives and integrals of arbitrary and complex orders. Therefore, it adds a new dimension to understand and describe basic nature and behavior of complex systems in an improved way. Here we use the fractional calculus for modeling electrical properties of biological systems. We derived a new class of generalized models for electrical impedance and applied them to human skin by experimental data fitting. The primary model introduces new generalizations of: 1 Weyl fractional derivative operator, 2 Cole equation, and 3 Constant Phase Element (CPE. These generalizations were described by the novel equation which presented parameter [Formula: see text] related to remnant memory and corrected four essential parameters [Formula: see text] We further generalized single generalized element by introducing specific partial sum of Maclaurin series determined by parameters [Formula: see text] We defined individual primary model elements and their serial combination models by the appropriate equations and electrical schemes. Cole equation is a special case of our generalized class of models for[Formula: see text] Previous bioimpedance data analyses of living systems using basic Cole and serial Cole models show significant imprecisions. Our new class of models considerably improves the quality of fitting, evaluated by mean square errors, for bioimpedance data obtained from human skin. Our models with new parameters presented in specific partial sum of Maclaurin series also extend representation, understanding and description of complex systems electrical properties in terms of remnant memory effects.
A Step-indexed Semantic Model of Types for the Call-by-Name Lambda Calculus
Meurer, Benedikt
2011-01-01
Step-indexed semantic models of types were proposed as an alternative to purely syntactic safety proofs using subject-reduction. Building upon the work by Appel and others, we introduce a generalized step-indexed model for the call-by-name lambda calculus. We also show how to prove type safety of general recursion in our call-by-name model.
Endoscopic vs. tactile evaluation of subgingival calculus.
Osborn, Joy B; Lenton, Patricia A; Lunos, Scott A; Blue, Christine M
2014-08-01
Endoscopic technology has been developed to facilitate imagery for use during diagnostic and therapeutic phases of periodontal care. The purpose of this study was to compare the level of subgingival calculus detection using a periodontal endoscope with that of conventional tactile explorer in periodontitis subjects. A convenience sample of 26 subjects with moderate periodontitis in at least 2 quadrants was recruited from the University of Minnesota School of Dentistry to undergo quadrant scaling and root planing. One quadrant from each subject was randomized for tactile calculus detection alone and the other quadrant for tactile detection plus the Perioscope ™ (Perioscopy Inc., Oakland, Cali). A calculus index on a 0 to 3 score was performed at baseline and at 2 post-scaling and root planing visits. Sites where calculus was detected at visit 1 were retreated. T-tests were used to determine within-subject differences between Perioscope™ and tactile measures, and changes in measures between visits. Significantly more calculus was detected using the Perioscope™ vs. tactile explorer for all 3 subject visits (pcalculus detection from baseline to visit 1 were statistically significant for both the Perioscope™ and tactile quadrants (pcalculus detection from visit 1 to visit 2 was only significant for the Perioscope™ quadrant (pcalculus at this visit. It was concluded that the addition of a visual component to calculus detection via the Perioscope™ was most helpful in the re-evaluation phase of periodontal therapy. Copyright © 2014 The American Dental Hygienists’ Association.
Bitsakis, E.I.; Nicolaides, C.A.
1989-01-01
The concept of probability is now, and always has been, central to the debate on the interpretation of quantum mechanics. Furthermore, probability permeates all of science, as well as our every day life. The papers included in this volume, written by leading proponents of the ideas expressed, embrace a broad spectrum of thought and results: mathematical, physical epistemological, and experimental, both specific and general. The contributions are arranged in parts under the following headings: Following Schroedinger's thoughts; Probability and quantum mechanics; Aspects of the arguments on nonlocality; Bell's theorem and EPR correlations; Real or Gedanken experiments and their interpretation; Questions about irreversibility and stochasticity; and Epistemology, interpretation and culture. (author). refs.; figs.; tabs
Schoenly, Joshua E.; Seka, Wolf; Romanos, Georgios; Rechmann, Peter
A desired outcome of scaling and root planing is the complete removal of calculus and infected root tissue and preservation of healthy cementum for rapid healing of periodontal tissues. Conventional periodontal treatments for calculus removal, such as hand instrument scaling and ultrasonic scaling, often deeply scrape the surface of the underlying hard tissue and may leave behind a smear layer. Pulsed lasers emitting at violet wavelengths (specifically, 380 to 400 nm) are a potential alternative treatment since they can selectively ablate dental calculus without ablating pristine hard tissue (i.e., enamel, cementum, and dentin). In this study, light and scanning electron microscopy are used to compare and contrast the efficacy of in vitro calculus removal for several conventional periodontal treatments (hand instruments, ultrasonic scaler, and Er:YAG laser) to calculus removal with a frequency-doubled Ti:sapphire (λ = 400 nm). After calculus removal, enamel and cementum surfaces are investigated for calculus debris and damage to the underlying hard tissue surface. Compared to the smear layer, grooves, and unintentional hard tissue removal typically found using these conventional treatments, calculus removal using the 400-nm laser is complete and selective without any removal of pristine dental hard tissue. Based on these results, selective ablation from the 400-nm laser appears to produce a root surface that would be more suitable for successful healing of periodontal tissues.
On the Presentation of Pre-Calculus and Calculus Topics: An Alternate View
Davydov, Aleksandr; Sturm-Beiss, Rachel
2008-01-01
The orders of presentation of pre-calculus and calculus topics, and the notation used, deserve careful study as they affect clarity and ultimately students' level of understanding. We introduce an alternate approach to some of the topics included in this sequence. The suggested alternative is based on years of teaching in colleges within and…
Papastavridis, John G
1999-01-01
Tensor Calculus and Analytical Dynamics provides a concise, comprehensive, and readable introduction to classical tensor calculus - in both holonomic and nonholonomic coordinates - as well as to its principal applications to the Lagrangean dynamics of discrete systems under positional or velocity constraints. The thrust of the book focuses on formal structure and basic geometrical/physical ideas underlying most general equations of motion of mechanical systems under linear velocity constraints.
Burkill, Sarah; Couper, Mick P; Conrad, Frederick; Clifton, Soazig; Tanton, Clare; Phelps, Andrew; Datta, Jessica; Mercer, Catherine H; Sonnenberg, Pam; Prah, Philip; Mitchell, Kirstin R; Wellings, Kaye; Johnson, Anne M; Copas, Andrew J
2014-01-01
Background Nonprobability Web surveys using volunteer panels can provide a relatively cheap and quick alternative to traditional health and epidemiological surveys. However, concerns have been raised about their representativeness. Objective The aim was to compare results from different Web panels with a population-based probability sample survey (n=8969 aged 18-44 years) that used computer-assisted self-interview (CASI) for sensitive behaviors, the third British National Survey of Sexual Attitudes and Lifestyles (Natsal-3). Methods Natsal-3 questions were included on 4 nonprobability Web panel surveys (n=2000 to 2099), 2 using basic quotas based on age and sex, and 2 using modified quotas based on additional variables related to key estimates. Results for sociodemographic characteristics were compared with external benchmarks and for sexual behaviors and opinions with Natsal-3. Odds ratios (ORs) were used to express differences between the benchmark data and each survey for each variable of interest. A summary measure of survey performance was the average absolute OR across variables. Another summary measure was the number of key estimates for which the survey differed significantly (at the 5% level) from the benchmarks. Results For sociodemographic variables, the Web surveys were less representative of the general population than Natsal-3. For example, for men, the average absolute OR for Natsal-3 was 1.14, whereas for the Web surveys the average absolute ORs ranged from 1.86 to 2.30. For all Web surveys, approximately two-thirds of the key estimates of sexual behaviors were different from Natsal-3 and the average absolute ORs ranged from 1.32 to 1.98. Differences were appreciable even for questions asked by CASI in Natsal-3. No single Web survey performed consistently better than any other did. Modified quotas slightly improved results for men, but not for women. Conclusions Consistent with studies from other countries on less sensitive topics, volunteer Web
Fractal physiology and the fractional calculus: a perspective
Bruce J West
2010-10-01
Full Text Available This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. We review the allometric aggregation approach to the processing of physiologic time series as a way of determining the fractal character of the underlying phenomena. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. Fractional operators acting on fractal functions yield fractal functions, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine. Allometric control incorporates long-time memory, inverse power-law (IPL correlations, and long-range interactions in complex phenomena as manifest by IPL distributions. We hypothesize that allometric control, rather than homeostatic control, maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can be described using the fractional calculus to capture the dynamics of complex physiologic networks. This hypothesis is supported by a number of physiologic time series data.
Juan, Hsu-Cheng; Lin, Hung-Yu; Chou, Yii-Her; Yang, Yi-Hsin; Shih, Paul Ming-Chen; Chuang, Shu-Mien; Shen, Jung-Tsung; Juan, Yung-Shun
2012-08-01
To assess the effects of abdominal fat on shock wave lithotripsy (SWL). We used pre-SWL unenhanced computed tomography (CT) to evaluate the impact of abdominal fat distribution and calculus characteristics on the outcome of SWL. One hundred and eighty-five patients with a solitary ureteric calculus treated with SWL were retrospectively reviewed. Each patient underwent unenhanced CT within 1 month before SWL treatment. Treatment outcomes were evaluated 1 month later. Unenhanced CT parameters, including calculus surface area, Hounsfield unit (HU) density, abdominal fat area and skin to calculus distance (SSD) were analysed. One hundred and twenty-eight of the 185 patients were found to be calculus-free following treatment. HU density, total fat area, visceral fat area and SSD were identified as significant variables on multivariate logistic regression analysis. The receiver-operating characteristic analyses showed that total fat area, para/perirenal fat area and visceral fat area were sensitive predictors of SWL outcomes. This study revealed that higher quantities of abdominal fat, especially visceral fat, are associated with a lower calculus-free rate following SWL treatment. Unenhanced CT is a convenient technique for diagnosing the presence of a calculus, assessing the intra-abdominal fat distribution and thereby helping to predict the outcome of SWL. • Unenhanced CT is now widely used to assess ureteric calculi. • The same CT protocol can provide measurements of abdominal fat distribution. • Ureteric calculi are usually treated by shock wave lithotripsy (SWL). • Greater intra-abdominal fat stores are generally associated with poorer SWL results.
Electronic Algebra and Calculus Tutor
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.
Yaprak, E; Yolcubal, I; Sinanoğlu, A; Doğrul-Demiray, A; Guzeldemir-Akcakanat, E; Marakoğlu, I
2017-02-01
Various trace elements, including toxic heavy metals, may exist in dental calculus. However, the effect of environmental factors on heavy metal composition of dental calculus is unknown. Smoking is a major environmental source for chronic toxic heavy metal exposition. The aim of this study is to compare toxic heavy metal accumulation levels in supragingival dental calculus of smokers and non-smokers. A total of 29 supragingival dental calculus samples were obtained from non-smoker (n = 14) and smoker (n = 15) individuals. Subjects with a probability of occupational exposure were excluded from the study. Samples were analyzed by inductively coupled plasma mass spectrometry in terms of 26 metals and metalloids, including toxic heavy metals. Toxic heavy metals, arsenic (p 0.05). Within the limitations of this study, it can be concluded that the elementary composition of dental calculus may be affected by environmental factors such as tobacco smoke. Therefore, dental calculus may be utilized as a non-invasive diagnostic biological material for monitoring chronic oral heavy metal exposition. However, further studies are required to evaluate its diagnostic potential. © 2016 John Wiley & Sons A/S. Published by John Wiley & Sons Ltd.
Probability and statistics with integrated software routines
Deep, Ronald
2005-01-01
Probability & Statistics with Integrated Software Routines is a calculus-based treatment of probability concurrent with and integrated with statistics through interactive, tailored software applications designed to enhance the phenomena of probability and statistics. The software programs make the book unique.The book comes with a CD containing the interactive software leading to the Statistical Genie. The student can issue commands repeatedly while making parameter changes to observe the effects. Computer programming is an excellent skill for problem solvers, involving design, prototyping, data gathering, testing, redesign, validating, etc, all wrapped up in the scientific method.See also: CD to accompany Probability and Stats with Integrated Software Routines (0123694698)* Incorporates more than 1,000 engaging problems with answers* Includes more than 300 solved examples* Uses varied problem solving methods
Time-division multiplexing vs network calculus: A comparison
Puffitsch, Wolfgang; Sørensen, Rasmus Bo; Schoeberl, Martin
2015-01-01
that time-division multiplexing leads to better worst-case latencies, while network calculus supports higher bandwidths. Furthermore, time-division multiplexing leads to a simpler hardware implementation, while dynamically scheduled networks-on-chip allow the integration of best-effort traffic in the on......Networks-on-chip are increasingly common in modern multicore architectures. However, general-purpose networks-on-chip are not always well suited for real-time applications that require bandwidth and latency guarantees. Two approaches to provide real-time guarantees have emerged: time......-chip network in a more natural way....
Tensor calculus, relativity, and cosmology a first course
Dalarsson, M
2005-01-01
This book combines relativity, astrophysics, and cosmology in a single volume, providing an introduction to each subject that enables students to understand more detailed treatises as well as the current literature. The section on general relativity gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes, Penrose processes, and similar topics), and considers the energy-momentum tensor for various solutions. The next section on relativistic astrophysics discusses
Superconformal tensor calculus and matter couplings in six dimensions
Bergshoeff, E.; Sezgin, E.; van Proeyen, A.
1989-01-01
Using superconformal tensor calculus the authors construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. They start from the superconformal algebra which they realize on a 40 + 40 Weyl multiplet and on several matter multiplets. A special role is played by the tensor multiplet, which cannot be treated as an ordinary matter multiplet, but leads to a second 40 + 40 version of the Weyl multiplet. The authors also obtain a 48 + 48 off-shell formulation of Poincare supergravity coupled to a tensor multiplet
Superconformal tensor calculus and matter couplings in six dimensions
Bergshoeff, E.; Sezgin, E.; Proeyen, A. van
1986-01-01
Using superconformal tensor calculus we construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. We start from the superconformal algebra which we realize on a 40 + 40 Weyl multiplet and on several matter multiplets. A special role is played by the tensor multiplet, which cannot be treated as an ordinary matter multiplet, but leads to a second 40 + 40 version of the Weyl multiplet. We also obtain a 48 + 48 off-shell formulation of Poincare supergravity coupled to a tensor multiplet. (orig.)
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. Copyright © 2014 Elsevier Ltd. All rights reserved.
Calculus domains modelled using an original bool algebra based on polygons
Oanta, E.; Panait, C.; Raicu, A.; Barhalescu, M.; Axinte, T.
2016-08-01
Analytical and numerical computer based models require analytical definitions of the calculus domains. The paper presents a method to model a calculus domain based on a bool algebra which uses solid and hollow polygons. The general calculus relations of the geometrical characteristics that are widely used in mechanical engineering are tested using several shapes of the calculus domain in order to draw conclusions regarding the most effective methods to discretize the domain. The paper also tests the results of several CAD commercial software applications which are able to compute the geometrical characteristics, being drawn interesting conclusions. The tests were also targeting the accuracy of the results vs. the number of nodes on the curved boundary of the cross section. The study required the development of an original software consisting of more than 1700 computer code lines. In comparison with other calculus methods, the discretization using convex polygons is a simpler approach. Moreover, this method doesn't lead to large numbers as the spline approximation did, in that case being required special software packages in order to offer multiple, arbitrary precision. The knowledge resulted from this study may be used to develop complex computer based models in engineering.
Regularization of positive definite matrix fields based on multiplicative calculus
Florack, L.M.J.; Bruckstein, A.M.; Haar Romeny, ter B.M.; Bronstein, A.M.; Bronstein, M.M.
2012-01-01
Multiplicative calculus provides a natural framework in problems involving positive images and positivity preserving operators. In increasingly important, complex imaging frameworks, such as diffusion tensor imaging, it complements standard calculus in a nontrivial way. The purpose of this article
Plotnitsky, Arkady
2010-01-01
Offers an exploration of the relationships between epistemology and probability in the work of Niels Bohr, Werner Heisenberg, and Erwin Schrodinger; in quantum mechanics; and in modern physics. This book considers the implications of these relationships and of quantum theory for our understanding of the nature of thinking and knowledge in general
Introduction to probability with R
Baclawski, Kenneth
2008-01-01
FOREWORD PREFACE Sets, Events, and Probability The Algebra of Sets The Bernoulli Sample Space The Algebra of Multisets The Concept of Probability Properties of Probability Measures Independent Events The Bernoulli Process The R Language Finite Processes The Basic Models Counting Rules Computing Factorials The Second Rule of Counting Computing Probabilities Discrete Random Variables The Bernoulli Process: Tossing a Coin The Bernoulli Process: Random Walk Independence and Joint Distributions Expectations The Inclusion-Exclusion Principle General Random Variable
Fuzzy relational calculus theory, applications and software
Peeva, Ketty
2004-01-01
This book examines fuzzy relational calculus theory with applications in various engineering subjects. The scope of the text covers unified and exact methods with algorithms for direct and inverse problem resolution in fuzzy relational calculus. Extensive engineering applications of fuzzy relation compositions and fuzzy linear systems (linear, relational and intuitionistic) are discussed. Some examples of such applications include solutions of equivalence, reduction and minimization problems in fuzzy machines, pattern recognition in fuzzy languages, optimization and inference engines in textile and chemical engineering, etc. A comprehensive overview of the authors' original work in fuzzy relational calculus is also provided in each chapter. The attached CD-Rom contains a toolbox with many functions for fuzzy calculations, together with an original algorithm for inverse problem resolution in MATLAB. This book is also suitable for use as a textbook in related courses at advanced undergraduate and graduate level...
A 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...
Quantum geometry in dynamical Regge calculus
Hagura, Hiroyuki
2002-01-01
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
The giant calculus within the prostatic urethra.
Demir, Omer; Kefi, Aykut; Cahangirov, Asif; Cihan, Ahmet; Obuz, Funda; Esen, Adil Ahmet; Celebi, Ilhan
2011-08-01
The giant calculus within the prostatic urethra is a rare clinical entity in the young population. Most of the calculi within the urethra migrate from the urinary bladder and obliterate the urethra. These stones are often composed of calcium phosphate or calcium oxalate. The decision of treatment strategy is affected by the size, shape and position of the calculus and by the status of the urethra. If the stone is large and immovable, it may be extracted via the perineal or the suprapubic approach. In most cases, the giant calculi were extracted via the transvesical approach and external urethrotomy. Our case is the biggest prostatic calculus, known in the literature so far, which was treated endoscopically by the combination of laser and the pneumatic lithotriptor.
[Does carbonate originate from carbonate-calcium crystal component of the human urinary calculus?].
Yuzawa, Masayuki; Nakano, Kazuhiko; Kumamaru, Takatoshi; Nukui, Akinori; Ikeda, Hitoshi; Suzuki, Kazumi; Kobayashi, Minoru; Sugaya, Yasuhiro; Morita, Tatsuo
2008-09-01
It gives important information in selecting the appropriate treatment for urolithiasis to confirm the component of urinary calculus. Presently component analysis of the urinary calculus is generally performed by infrared spectroscopy which is employed by companies providing laboratory testing services in Japan. The infrared spectroscopy determines the molecular components from the absorption spectra in consequence of atomic vibrations. It has the drawback that an accurate crystal structure cannot be analyzed compared with the X-ray diffraction method which analyzes the crystal constituent based on the diffraction of X-rays on crystal lattice. The components of the urinary calculus including carbonate are carbonate apatite and calcium carbonate such as calcite. Although the latter is reported to be very rare component in human urinary calculus, the results by infrared spectroscopy often show that calcium carbonate is included in calculus. The infrared spectroscopy can confirm the existence of carbonate but cannot determine whether carbonate is originated from carbonate apatite or calcium carbonate. Thus, it is not clear whether calcium carbonate is included in human urinary calculus component in Japan. In this study, we examined human urinary calculus including carbonate by use of X-ray structural analysis in order to elucidate the origin of carbonate in human urinary calculus. We examined 17 human calculi which were reported to contain calcium carbonate by infrared spectroscopy performed in the clinical laboratory. Fifteen calculi were obtained from urinary tract, and two were from gall bladder. The stones were analyzed by X-ray powder method after crushed finely. The reports from the clinical laboratory showed that all urinary culculi consisted of calcium carbonate and calcium phosphate, while the gallstones consisted of calcium carbonate. But the components of all urinary calculi were revealed to be carbonate apatite by X-ray diffraction. The components of
TWO-PHASE EJECTOR of CARBON DIOXIDE HEAT PUMP CALCULUS
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.
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…
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…
Improving Calculus II and III through the Redistribution of Topics
George, C. Yousuf; Koetz, Matt; Lewis, Heather A.
2016-01-01
Three years ago our mathematics department rearranged the topics in second and third semester calculus, moving multivariable calculus to the second semester and series to the third semester. This paper describes the new arrangement of topics, and how it could be adapted to calculus curricula at different schools. It also explains the benefits we…
Giant Calculus In The Mouth Of Partially Edentulous Woman, (Case ...
Objective: This case report is to create awareness of the presence of giant calculus in the mouth, the possible causes and its prevention. Report: This describes the oral condition of a partially edentulous woman with a giant calculus in the mouth. It highlights the effect of such an enormous calculus in the oral cavity.
Computer Managed Instruction Homework Modules for Calculus I.
Goodman-Petrushka, Sharon; Roitberg, Yael
This booklet contains 11 modules (290 multiple-choice items) designed for use in the first course of a three-course calculus sequence using the textbook "Calculus with Analytic Geometry" (Dennis G. Zill). In each module, relevant sections of the textbook are identified for users. It can, however, be used in conjunction with any calculus textbook.…
ASSESSING STUDENTS' UNDERSTANDING OF PRE-CALCULUS CONCEPTS
Dr. Jyoti Sharma
2017-01-01
Calculus is one of the most momentous achievements of the human intellect (Boyer, 1949). It has given a new direction to the work of mathematicians and scientists. Calculus has exponentially expanded the scope and use of mathematics in other fields. Learning calculus is important to pursue career in applied mathematics.
Ito calculus for σ-models and Yang-Mills theories
Patrascioiu, A.; Richard, J.L.
1984-07-01
It is pointed out that the effective continuum action for σ-models and Yang-Mills theories may differ from the naive continuum action by terms of order g 2 or higher, which are non-symmetric. The modifications are produced by a generalization of the Ito calculus to dimensions higher than one
Covariant differential calculus on quantum Minkowski space and the q-analogue of Dirac equation
Song Xingchang; Academia Sinica, Beijing
1992-01-01
The covariant differential calculus on the quantum Minkowski space is presented with the help of the generalized Wess-Zumino method and the quantum Pauli matrices and quantum Dirac matrices are constructed parallel to those in the classical case. Combining these two aspects a q-analogue of Dirac equation follows directly. (orig.)
Kramosil, Ivan
2001-01-01
Roč. 5, č. 1 (2001), s. 45-57 ISSN 1432-7643 R&D Projects: GA AV ČR IAA1030803 Institutional research plan: AV0Z1030915 Keywords : first-order predicate calculus * standard semantics * Boolean-like semantics * frequentistic semantics * completness theorems Subject RIV: BA - General Mathematics
Irreversibility and conditional probability
Stuart, C.I.J.M.
1989-01-01
The mathematical entropy - unlike physical entropy - is simply a measure of uniformity for probability distributions in general. So understood, conditional entropies have the same logical structure as conditional probabilities. If, as is sometimes supposed, conditional probabilities are time-reversible, then so are conditional entropies and, paradoxically, both then share this symmetry with physical equations of motion. The paradox is, of course that probabilities yield a direction to time both in statistical mechanics and quantum mechanics, while the equations of motion do not. The supposed time-reversibility of both conditionals seems also to involve a form of retrocausality that is related to, but possibly not the same as, that described by Costa de Beaurgard. The retrocausality is paradoxically at odds with the generally presumed irreversibility of the quantum mechanical measurement process. Further paradox emerges if the supposed time-reversibility of the conditionals is linked with the idea that the thermodynamic entropy is the same thing as 'missing information' since this confounds the thermodynamic and mathematical entropies. However, it is shown that irreversibility is a formal consequence of conditional entropies and, hence, of conditional probabilities also. 8 refs. (Author)
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
Multivariable dynamic calculus on time scales
Bohner, Martin
2016-01-01
This book offers the reader an overview of recent developments of multivariable dynamic calculus on time scales, taking readers beyond the traditional calculus texts. Covering topics from parameter-dependent integrals to partial differentiation on time scales, the book’s nine pedagogically oriented chapters provide a pathway to this active area of research that will appeal to students and researchers in mathematics and the physical sciences. The authors present a clear and well-organized treatment of the concept behind the mathematics and solution techniques, including many practical examples and exercises.
Dynamical Regge calculus as lattice gravity
Hagura, Hiroyuki
2001-01-01
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 i dl i . On the other hand, using the scale-invariant measure Π i dl i /l i , we can calculate observables and observe a large hysteresis between two phases that indicates the first-order nature of the phase transition
Hybrid Logical Analyses of the Ambient Calculus
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...
Modular invariance and covariant loop calculus
Petersen, J.L.; Roland, K.O.; Sidenius, J.R.
1988-01-01
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.)
Modular invariance and covariant loop calculus
Petersen, J.L.; Roland, K.O.; Sidenius, J.R.
1988-01-01
The covariant loop calculus provides an 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 two- and three-loop results derived using analytic geometry (one loop is known to be okay). We establish identity to 'high' order in some moduli and exactly in others. Agreement is found as a result of various nontrivial 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.)
A sequent calculus for signed interval logic
Rasmussen, Thomas Marthedal
2001-01-01
We propose and discuss a complete sequent calculus formulation for Signed Interval Logic (SIL) with the chief purpose of improving proof support for SIL in practice. The main theoretical result is a simple characterization of the limit between decidability and undecidability of quantifier-free SIL....... We present a mechanization of SIL in the generic proof assistant Isabelle and consider techniques for automated reasoning. Many of the results and ideas of this report are also applicable to traditional (non-signed) interval logic and, hence, to Duration Calculus....
Differential calculus for Dirichlet forms: The measure-valued gradient preserved by image
Bouleau, Nicolas
2005-01-01
In order to develop a differential calculus for error propagation we study local Dirichlet forms on probability spaces with square field operator $\\Gamma$ -- i.e. error structures -- and we are looking for an object related to $\\Gamma$ which is linear and with a good behaviour by images. For this we introduce a new notion called the measure valued gradient which is a randomized square root of $\\Gamma$. The exposition begins with inspecting some natural notions candidate to solve the problem b...
Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus
Aguirre, Alejandro; Barthe, Gilles; Birkedal, Lars
2018-01-01
We extend the simply-typed guarded $\\lambda$-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like Mar...... literature to justify better proof rules for relational reasoning about probabilistic expressions. We illustrate these benefits with a broad range of examples that were beyond the scope of previous systems, including shift couplings and lump couplings between random walks....
Quantum probability for probabilists
Meyer, Paul-André
1993-01-01
In recent years, the classical theory of stochastic integration and stochastic differential equations has been extended to a non-commutative set-up to develop models for quantum noises. The author, a specialist of classical stochastic calculus and martingale theory, tries to provide anintroduction to this rapidly expanding field in a way which should be accessible to probabilists familiar with the Ito integral. It can also, on the other hand, provide a means of access to the methods of stochastic calculus for physicists familiar with Fock space analysis.
Continuous Slice Functional Calculus in Quaternionic Hilbert Spaces
Ghiloni, Riccardo; Moretti, Valter; Perotti, Alessandro
2013-04-01
The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.
A large primary vaginal calculus in a woman with paraplegia.
Avsar, Ayse Filiz; Keskin, Huseyin Levent; Catma, Tuba; Kaya, Basak; Sivaslioglu, Ahmet Akın
2013-01-01
The study aimed to report a primary vaginal stone, an extremely rare entity, without vesicovaginal fistula in a woman with disability. We describe the case of a large primary vaginal calculus in a 22-year-old woman with paraplegia, which, surprisingly, was not diagnosed until she was examined under general anesthesia during a preparation for laparoscopy for an adnexal mass. The stone had not been identified by physical examination with the patient in a recumbent position or by transabdominal ultrasonography and pelvic tomography during the preoperative preparation. Vaginoscopy was not performed because the vagina was completely filled with the mass. As a result of its size and hard consistency, a right-sided episiotomy was performed and a 136-g stone was removed using ring forceps. A vesicovaginal fistula was excluded. There was no evidence of a foreign body or other nidus on the cut section of the stone, and it was determined to be composed of 100% struvite (ammonium magnesium phosphate). Culture of urine obtained via catheter showed Escherichia coli. After the surgical removal of the calculus without complications, a program of intermittent catheterization was started. The follow-up period was uneventful, and the patient was symptom free at 6 months after the operation. We postulate that the calculus formed as a consequence of urinary contamination of the vagina in association with incontinence and prolonged maintenance in a recumbent posture. This report is important because it highlights that, although vaginal stones are very rare, their possibility should be considered in the differential diagnosis of individuals with long-term paraplegia.